The Disembodied Mind: An Exploration of Consciousness in the Physical Universe [1 ed.] 1527541282, 9781527541283

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The Disembodied Mind

The Disembodied Mind: An Exploration of Consciousness in the Physical Universe By

James C. Austin

The Disembodied Mind: An Exploration of Consciousness in the Physical Universe By James C. Austin This book first published 2020 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2020 by James C. Austin All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-4128-2 ISBN (13): 978-1-5275-4128-3

To Alma and Mary Marcia

CONTENTS

The story in a nutshell ................................................................................. x Preface ........................................................................................................ xi Selected notation ...................................................................................... xvi Glossary.................................................................................................... xix 1. Introduction ............................................................................................. 1 1.1 Structure and content............................................................... 7 1.2 Rational overview ................................................................. 26 Part I: The science 2. Logic and mathematics .......................................................................... 30 2.1 Classical logic ....................................................................... 32 2.2 Sets, infinity and the continuum............................................ 57 2.3 Alternatives to the classical Platonist’s view ........................ 67 2.4 Conclusions ........................................................................... 73 3. The logic of science............................................................................... 75 3.1 Classical origins .................................................................... 79 3.2 The Arab Scholars ................................................................. 83 3.3 Medieval Europe ................................................................... 87 3.4 Renaissance science .............................................................. 90 3.5 The Renaissance to 1900....................................................... 94 3.6 Modern viewpoints................................................................ 99 3.7 Summary: the nature of evidence ........................................ 106 4. Space-time ........................................................................................... 109 4.1 Galilean relativity ................................................................ 111 4.2 Special relativity.................................................................. 114 4.3 Minkowski space-time ........................................................ 121 4.4 Curved space-time............................................................... 126 4.5 Gravitational collapse and black holes ................................ 131 4.6 Defining a present in curved space-time ............................. 146 4.7 Summary ............................................................................. 153

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5. Classical mechanics............................................................................. 155 5.1 Generalised coordinates ...................................................... 157 5.2 The Lagrangian formalism .................................................. 164 5.3 The Hamiltonian formalism ................................................ 170 5.4 The Hamilton-Jacobi equation ............................................ 175 5.5 Summary ............................................................................. 179 6. Quantum mechanics ............................................................................ 181 6.1 Early quantum theory .......................................................... 185 6.2 The full quantum theory ...................................................... 192 6.3 Axioms of quantum theory.................................................. 209 6.4 The interpretations of quantum mechanics.......................... 221 6.5 Quantum gravity.................................................................. 243 6.6 Macroscopic-Bell states ...................................................... 253 6.7 Summary ............................................................................. 255 Part II: The mind 7. Physicalism and its motivations........................................................... 260 7.1 The argument from fundamental forces .............................. 263 7.2 The argument from physiology ........................................... 267 7.3 The argument from methodological naturalism .................. 268 7.4 The persistence of epistemic physicalism ........................... 272 7.5 Summary ............................................................................. 277 8. Mind and the principle of localisation ................................................. 280 8.1 Consciousness causes collapse ............................................ 282 8.2 The principle of localisation................................................ 285 8.3 Further support .................................................................... 293 8.4 Summary ............................................................................. 300 9. Competing ideas .................................................................................. 302 9.1 Many minds: a physicalist’s perspective ............................. 302 9.2 Further support for the physicalist’s perspective................. 311 9.3 Traditional support for physicalism..................................... 323 9.4 Summary ............................................................................. 332

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10. Phenomenal change and timeless physics ......................................... 334 10.1 Presentism versus eternalism............................................. 336 10.2 The moving spotlight theory ............................................. 349 10.3 Are current theories of time substantive? .......................... 350 10.4 Objective theories.............................................................. 353 10.5 Mind-dependent theories ................................................... 360 10.6 Discussion and summary................................................... 366 11. Bio-neural systems and artificial consciousness................................ 369 11.1 Consciousness through cognitive neuroscience................. 371 11.2 Quantum aspects ............................................................... 377 11.3 Artificial consciousness..................................................... 381 11.4 Summary ........................................................................... 389 12. Selected consequences of localisation ............................................... 391 12.1 Normal sequences ............................................................. 392 12.2 Isolation of nonmaterial minds? ........................................ 402 12.3 Controversial topics: a final word ..................................... 405 12.4 Conclusions and future direction....................................... 409 12.5 A final thought: where is all the data?............................... 411 References ............................................................................................... 412 Index........................................................................................................ 425

THE STORY IN A NUTSHELL

The idea of encapsulating a book into a single passage is here borrowed from Julian Barbour’s The End of Time (Barbour, 1999). In his story in a nutshell Barbour uses Turner’s 1842 painting Snow storm as a metaphor to illustrate how the dynamics of the universe may be regarded as part of an all-encompassing static reality. The related debate, which has endured since the time of Parmenides and Heraclitus is here extended to include conscious experience. Those scientists who subscribe to the static reality concept refer to our dynamic experience as the grand illusion. Barbour is one such advocate and uses the word illusion in the same context in his summary. This book offers an alternative explanation that treats our consistent dynamic experience seriously while maintaining an overall static physical reality – there is no illusion. The apparent contradiction is resolved by accepting that conscious agents are separate nonmaterial entities that dynamically evolve within the framework of a static physical reality. The earliest reference from modern times that I know of may be summarised by the following famous quote by Hermann Weyl (1949). The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life-line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time.

Hermann Weyl, Philosophy of Mathematics and Natural Science [Princeton: Princeton University Press, 1949], p. 116.

PREFACE

At its core this work is about the relationship between conscious minds and the physical universe that they occupy. My interest in theoretical physics began just after I left school in the early 1970s. At that time, like the formative years of most scientists and engineers I assume, I was curious about the underlying principles that underpinned physical reality. However, for a long time I considered consciousness to be something of a side issue, forever irreducible, a mysterious aspect of reality beyond the probe of science. Like most practical engineers my concerns lay with physical principles at a very superficial level. These limited requirements led to frustration, which prompted me to explore further. However, my limited academic training meant that I would, for many years, follow one blind alley after another. That did not mean that the experience was wasted, far from it, I gained a lot of insight into the workings of general relativity – one of the great pillars on which twentieth century physics is based. As I recall I did acquire a rather geekish reputation amongst my work colleagues regarding my newly found knowledge of the theories formed by Maxwell, Lorentz, Einstein, and Minkowski. On rare occasions conversations would drift towards the deeper aspects of consciousness. Some would agree with me that conscious minds represented a separate aspect of reality while others would, quite aggressively sometimes, insist on the opposite. Even at the level of the layman the polarisation between physicalists and dualists was palpable. The third option of idealism rarely arose, that was a little too deep and sophisticated for us. One interesting observation was a contrast between the attitudes of dualists and physicalists at that layman’s level. Dualists displayed an obvious uncertainty in their views realising the mysterious nature of what they contemplated. Physicalists, on the other hand, were less open. They were absolutely certain of their beliefs, anything else was considered irrational and they seemed completely impervious to persuasion. This may be because physicalism is perceived as a rather simpler viewpoint than dualism, requiring less of an ontological commitment and consequently easier to hold on to. This is where I must alert the reader to my own prejudices. I do not call myself a dualist because I cannot rule out a possible truth with some form of idealism at its core. Instead I,

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shamelessly refer to myself as an anti-physicalist whose certainty in this direction has grown in recent years, not despite my scientific background but because of it. This book is a summary of that personal exploration. Due in part to the increasing de-industrialisation of the UK during the Thatcher years, I realised that opportunities for non-graduate engineers were diminishing. So in 1990 I left the Michelin Tyre Company at Stoke on Trent in the UK, to attend the University of Keele as an undergraduate at the rather mature age of 32. I still viewed electrical and electronic engineering as a potential future career just as I had pursued it in the previous 16 years. Also my interest in mathematical physics was still very much alive, hence my choice of degree in Mathematics and Electronics. On graduation I immediately undertook a PhD in diagnostic ultrasonics within the Electronic Engineering Group at the same institution under the tutelage of Prof. Richard E Challis, which was successfully examined in 1997. My career since then has consisted of two post-doctoral positions in non-destructive testing interspersed with periods of part-time teaching for the university mainly in mathematics and physics. Despite this rather practical/scientific route my interest in some of the more philosophical topics concerning the foundations of mathematics and physics had not waned. Notwithstanding my scientific background in the late 1990s, I realised that it had one glaring weakness, a lack of any formal training in quantum theory. Previously, conversations with fellow contemporary undergraduates who were reading physics as a principal course alerted me to the prevailing state of affairs that conflicting interpretations of quantum theory were still hotly debated. In my naivety I had thought that these issues had been put-to-bed decades earlier. As a consequence I took an interest in the emerging field of quantum computing, which was an obvious way in for an electronics engineer. I had previously read many texts on quantum theory even up to the point of being able to solve basic quantum mechanical problems. But it was not until I had purchased a copy of the well-known text by Nielsen and Chuang (2000) that a deeper understanding of quantum theory really developed. Other texts that greatly influenced my thinking were Barbour’s the End of Time, (Barbour, 1999) and latterly The Physical Basis of the Direction of Time (Zeh, 2007). Coupling this with general relativity my attention was drawn to Hawking’s black hole evaporation mechanism and its consequences for preserving unitary evolution and a possible solution to the information loss paradox. In this I had formed the view that black hole event horizons do not form in a finite coordinate time, a view confirmed by other works (Suggett, 1979; Barcelo et al, 2006; Vachaspati et al, 2007; Mersini-

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Houghton, 2014). With a belief in unitary evolution, I began to form ideas relating interpretations of quantum mechanics with those of consciousness around 2003. These included the realisation that physicalism is completely incompatible with the Everett (many worlds) interpretation (an interpretation most popular with researchers in quantum computation and quantum cosmology), and consequently one may speculate that supporters of other interpretations did so because of their innate physicalist leanings. It was when I considered publishing these ideas that my searches lead me to the works of Albert and Loewer (1988) and EJ Squires (1993). Upon reading these articles I admit to feeling a little deflated that I could not be the first to publish such an idea. Albert and Loewer had beaten me to it by some fifteen years. My career would continue to tick along with part-time teaching, my own researches and the occasional project in non-destructive testing. At the same time I was encouraged that my ideas were supported in published works. During the period, 2004-08, I was privileged to have worked with one of the great scientists of the twentieth century, Prof. Peter H. Plesch. Together we worked on his last article (Plesch and Austin, 2008) before his passing in 2013. My partner, Alma and I had visited Peter and his wife, Traudi on a number of occasions while they lived at their last address in Northampton. As I recall during one visit, I noticed a number of issues of the journal Paranormal Review on a side table in Peter’s study. It was obvious to me that rigorous scientific enquiry into paranormal events was just another of his many interests. We had had many discussions about Peter’s other interests including classics, particularly Roman glass of which Peter and Traudi possessed an impressive collection. Topics in classics were quite often discussed given that Alma is a graduate of classics herself. Indeed, it was on an earlier classics trip to Sorrento, organised by the Head of Classics, Mr Richard Wallace at Keele in 1991, that Alma had met Peter, and it was through Alma that I came to know Peter myself. Although we had many stimulating discussions on various topics it is to my everlasting regret that I never broached the subject of his interest in the paranormal. Peter’s interest in the paranormal was not mentioned again until his funeral in 2013. It was during a eulogy spoken by one of his former colleagues that the logic of such an interest was questioned. Alma and I had attended Peter’s funeral with another good friend of ours, Mr Mark Wiggin. Alma and Mark were Foundation Year students at Keele during the academic year 1989-90, and Mark had taken a short chemistry module taught by Peter during that year. As far as I know that was the only contact Mark had had with Peter. Despite this Peter had left a deep impression on

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Mark. It was during our journey from the funeral that I mentioned Peter’s interest in the paranormal. This is a topic that Alma and I were perfectly open with, but not Mark, who it turns out, is an ardent physicalist. What had struck me about the eulogy was that the word logic, a word so often misused, was quoted with reference to Peter’s paranormal interest. I pointed this out to Mark who was not persuaded. My parting shot, as I recall, was that he (Mark) should ‘google’ Wigner’s Friend, a reference to an extended form of the Schrodinger’s Cat thought experiment, the idea being to pinpoint conscious agents within such a scenario. The discussion was not resurrected again until a few months later when we discussed one of Terry Pratchett’s books The Science of Discworld IV: Judgement Day (Pratchett, Stewart and Cohen, 2013), in which the authors strongly asserted their physicalist views on page 39. Mark was still not persuaded by arguments that certain interpretations of quantum mechanics not only admit but also demand some form of dualism. I forget Mark’s exact response but the words new age were in there somewhere. On reflection I believe it was unfair of me to use quantum mechanical arguments to persuade a friend who is not a scientist and confesses to being somewhat maths phobic. It was at this point that I made the decision to research this problem with a view to writing this book. The way I read it Mark and I agree to differ on this point. He has since read a working paper of mine, but just says, I still don’t agree. Fair enough! We still remain good friends and he is fun to be with at our regular reunions. Later, on further reflection, I realised that justification in dualism did not require the Everett interpretation, indeed classical relativity was enough when we consider that non-material minds are localised in time at a particular instant. In effect my past and future exist yet I only exist in my present. In classical general relativity of course there is only one unknown but predetermined future, which is incompatible with free will. Free will is only regained when we invoke the Everett interpretation. This work is not intended for the lay reader but assumes a modest mathematical knowledge. It can be very exasperating when one opens an interesting text expecting to find an explanation to something in mathematical form, only to find that it is not there. An example is Barbour’s End of Time, when he refers to that damned equation, referring to the Wheeler-DeWitt equation of quantum gravity that is not explicitly shown. This is because it may not have been appropriate for Barbour’s intended readership. At the other extreme the last four (quantum gravity) chapters of General Relativity: An Einstein Centenary Survey, Eds Hawking and Israel (1979), are heavily mathematics laden, yet there is not

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one mention of the Wheeler-DeWitt equation, which is central to quantum gravity. That is because knowledge of it is implicitly assumed. In this work I hope to bridge that gap by explicitly stating equations accompanied by appropriately detailed explanations. That way it will appeal to readers with a mathematical inclination while, at the same time, providing a text readable by a slightly wider audience. In my view Zeh (2007) and Penrose (2004) do this rather well. This is not intended as a criticism, authors consider their intended readership. It has been said that every equation in a book halves its potential sales. Although I have some sympathy with this view I do not think it is really all that bad. Mathematical statements, where appropriate, convey very precise ideas not easily expressed in words. With sufficient explanation this book will hopefully convey its ideas clearly enough for the moderately technical reader. I am minded that I have set myself a difficult challenge. In the end only you, the reader, can be my judge. I am indebted to many for the support I have received in the writing of this book. It is often the case that one may proof read one’s work many times and still leave deficiencies due to over-familiarity or carelessness. It is therefore useful to have someone detached to independently read through the text thereby detecting errors that would otherwise be missed. For this I am grateful to my long-term partner Mrs Alma Wood, Dr Steven J Payne, and Prof. Roger M Whittaker. I am especially grateful to Mr Mark Wiggin whose stimulating discussions spurred me on in the research that led to this text. I am equally indebted to Mr David Wood for alerting me to the work of Michael S Gazzaniga that is featured in the penultimate chapter and I also thank Mr Dennis Wilton and Dr Philip Emery for valuable advice concerning the publication process. Lastly I should not forget Jasper, my ginger and white domestic shorthair cat who features in figure 6.3, and I should make it clear that no animals were harmed during the preparation of this work. Specialist software packages used for the production of illustrations were DazStudio 2.3.3.146 for the front cover, figures 4.9 and 5.2, QCAD 2.00 for figures 6.4-5, and GeoGebra 5.0.413.0-d for figure 11.1.

JCA University of Keele November, 2019.

SELECTED NOTATION

Logical connectives ∧ , And ∨ , Or ¬ , Not Ÿ , Implies (as in A Ÿ B says A implies B or B follows from A) ⇔ , Two-way implication. Also “iff” (if and only if) A , “Turnstile” symbol same as Ÿ but where the left hand side is a conjunction of many premises with a single label, or possibly an overall context. Set theory ∀ , Universal operator (for all) ∃ , Existential operator (there exists) ∈ , Membership (as in a ∈ A says a is a member of A) {}, Elements of a set enclosed by ⊆ , Subset ⊂ , Strict subset ∪ , Union α

*

, Union of indexed terms from 1 to α

i =1

∩ , Intersection α



, Intersection of indexed terms from 1 to

i =1

α

[ a, b ] , ( a, b ) , [ a, b ) , ( a, b ] ,

Intervals: including end points, excluding end points, including a excluding b, excluding a including b respectively Numbers ` , The set of all natural numbers {0,1, 2,"}

] , The set of all integers {" , −2, −1,0,1, 2,"}

_ , The set of all rational numbers { p q : p, q ∈ ], q ≠ 0} \ , The set of all real numbers

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^ , The set of all complex numbers, topologically \ 2 Infinity ℵ0 , Countable infinity

ℵn , Suspected higher orders of infinity following from the general continuum hypothesis ( n > 0 ) Vectors (column-vectors): r ,

, xa

Covectors (one-forms or row-vectors):

, xa , also e.g.

∂ ∂q

Derivatives dy , Ordinary derivatives (e.g. of y with respect to x) dx ∂z ∂z ∂z , , Partial derivatives, where z = z ( x, y ) , also z,a ≡ a etc. ∂x ∂y ∂x

δ , Partial differential operator with respect to configuration space δΦ

variable, Φ ∂ ∂ ∇, , ," Vector derivative operator (actually a covector); e.g. ∂x ∂q

· ∂ § ∂ ∂ =¨ , ," ¸ ∂x © ∂x1 ∂x2 ¹ and the Laplacian operator ∂2 ∂2 ∇ 2 ≡ ∇ ⋅∇ = 2 + 2 + " ∂x1 ∂x2

Aa ;b ≡ Aa ,b + Γ a bc Ac , Covariant derivative of the vector with components Aa on a curved manifold Γ a bc , Connection coefficients defined in section 4.4

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Selected notation

Integral etc.

³ ( ) dx, ³ dx ( ) , Integral operators N

¦

, Sum of indexed terms from 1 to N

i =1 N

∏

, Product of indexed factors from 1 to N

i =1

Brackets E , Expectation value of E

v A v , Scalar from matrix, A, pre-multiplied by a row-vector and postmultiplied by its Hermitian conjugate, see section 6.2.2. [ x, y ] ≡ xy − yx , Commutator

{ x, y} ≡ xy + yx , Anti-commutator

φ , Modulus (magnitude) of vector quantity φ Binary operations ⋅ , Scalar (dot product) between vectors × , Vector (cross product) between vectors ⊗ , Tensor product between matrices of any shape d ( a, b ) , Topological metric distance between a and b Miscellaneous tr , Trace: sum of all entries in the leading diagonal of a square matrix Δx , Change in x , , End of proof

GLOSSARY

Base space

The space in which a system of separate parts resides. In most cases this is the three-dimensional space in which we live.

Born rule

This says that probability of a configuration is the square-modulus of the wave function for that configuration.

Boson

A particle with integer spin. Here the basic spin unit is the Dirac-Planck constant, = = h ( 2π ) .

Category mistake

A misunderstanding of a concept that we are dealing with. For example we are making a category mistake if we treat a pair of gloves as a separate entity from either the left hand, or the right hand glove.

Chaos theory

The study of gross and consistent features exhibited by chaotic systems. Such systems are deterministic so given exactly the same initial conditions a system will always evolve the same way. Thus chaos is not randomness.

Configuration space (C-space)

Space of relative positions for independently adjustable parts/particles of a system. Absolute configuration spaces, as opposed to relative configuration spaces include translational (linear) and rotational degrees of freedom.

Decoherence

The appearance of wave function collapse due to the appearance of a correlation between a system and the outside world. The word derives from the departure of a quantum state from coherence at a measurement event.

xx

Glossary

Differential operator

An operator based on a derivative which is the slope of the graph of a wave function in a given direction, say time or any direction in space.

Dualism

The doctrine that mind has a separate existence from matter, from the dual concepts of mind and matter or mind and body.

Eigenstate

A classical physical state, one that might be expected as a result of a measurement or observation.

Eigenvalue

A numerical value associated with an operator. A single operator may have many eigenvalues.

Empiricism

The view that science begins with gathering data and progresses to the detection of patterns which eventually form new hypotheses and theories.

Energy

The capacity to do work (force times distance). A conserved quantity stored as kinetic (moving) or potential (stationary) forms.

Entanglement

A combined quantum state of two systems in which individual states for each system cannot be defined.

Epistemology

Pertaining to knowledge. For example the squaremodulus of the wave function represents the probability of a specific configuration (Born rule). In epistemic interpretations of quantum theory the wave function/probability is merely knowledge, it has no objective existence.

Eternalism

The view that physical reality is made up of events and that all events, whether past, present, or future exist. Events effectively exist outside of time.

Fermion

A particle with odd-half integer spin in the same units as bosons. Fermions include baryons (making up atomic nuclei) and leptons (electrons and neutrinos).

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Hamiltonian

An expression for the total energy of a closed physical system, generally expressed as kinetic energy plus potential energy.

Hausdorff manifold

A geometric manifold, e.g. a surface, which can curve smoothly but does not branch.

Idealism

A monistic view that reality ultimately consists of mind. Matter emerges from the mind as opposed to the other monistic view, physicalism, where mind emerges from matter.

Lagrangian

A non-conserved energy parameter defined as kinetic energy minus potential energy.

Matrix

An operator written as a rectangular array of numbers. Heisenberg’s original quantum mechanics used matrices of potentially infinite size making the formulation impractical for most applications.

Microtubules

Macromolecular tubular strucures forming the cytoskeleton of neurons. These are formed from α/βtubilin dimers that may be described as “kidney” shaped structures each with two lobes. The lobes may be close or far apart and these represent distinct states which can be in a quantum superposition.

Minisuperspace model

Amodel in quantum gravity in which the configuration space is idealised to a small and therefore managable number of dimensions.

Modal theories

Interpretations of the measurment problem as proposed by van Fraassen (1981) in which sytems possess a dynamic state that may be, and a value state that actually is. Non-local hidden variable theories fall into this category.

Non-locality

The ability of a wave function to exist across large expanses of space for multi-particle systems, and to exhibit instantaneous decoherence across large distances.

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Glossary

Ontology

Objective: objectively real, having an objective relationship with other physical entities.

Operator

A non-numerical mathematical object that generates a numerical quantity when acting on a wave function or state vector.

Orchestrated decoherence (Orch D)

The realisation of one configuration of microtubules against other possibilities. The associated wave function does not collapse, instead a nonmaterial mind transitions from being uncertain as to which state it will finally experience, to selecting a particular state.

Orchestrated objective reduction (Orch OR)

A collapse of the wave function, orchestrated by a material mind, associated with configurations of microtubules forming the cytoskeleton of neurons in the brain. The wave function collapses to a state associated with a choice being made.

Phenomenal time

Time as we experience it–experienced duration. This is an entirely distinct concept from physical time as viewed by those who study relativity theory. It is proposed that phenomenal time is a function of the mind.

Physicalism

Sometimes called monism or materialism, this is the doctrine that mind emerges from, and is therefore part of matter – the physical world is all that exists.

Presentism

The view that only the present exists. The past is merely remembered in the present and the future is yet to come.

Probability

A variable describing the likelihood of a given event whose value lies between zero and one. Zero is the impossibility limit i.e. it can never happen, whereas one is “dead certainty” that it will happen.

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Quantization

A procedure for converting a classical model for a system into a quantum model. Energy, momentum, and angular momentum variables are replaced by factors proportional to differential operators in time and distance respectively.

Quantum gravity

A quantum theory (see quantum mechanics) where configurations or states of a physical system include the geometry of the base space, where gravitational fields are distortions in the geometry of space-time.

Quantum mechanics

A conceptual framework (meta-theory) in which definite classical states of a physical system are replaced by a wave function that can be spread over a range of different states or configurations. Mathematically it is a sub-branch of linear algebra.

Qubit

Short for quantum binary digit. Any quantum system having two classical eigenstates.

Rationalism

A view of science that prioritises the formation of postulates from which theories are deduced. Theories are subsequently tested against observational or experimental data.

Space-C

A timeless generalisation of space-time. This is the product of C-space and base space. Space-time is considered a subset of space-C where time is an ordered sequence of appropriately foliated base space configurations.

State vector

Equivalent to the wave function. Each configuration is represented as a basis vector. Therefore all basis vectors with non-zero probability contribute to the state vector.

Sui generis

Of its own type. Referring to hypothetical special forces in biological systems that have physical effects but originate from a non-physical source.

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Glossary

Supervenience

Dependence: to say that “the mind supervenes on matter” is to say that “the mind depends on matter”. If we qualify that by saying that “the mind depends on matter for its existence”, this is ontological supervenience.

Token minds

Minds thought of a separate independent entities, as opposed to type minds considered to be reflections of the same entity.

Turing test

An artificial intelligence system will be deemed to have past the Turing test when a human in conversation with it is unable to distinguish its responses from that of any other human.

Uncertainty principle

A law from quantum mechanics that says that pairs of associated variables cannot be measured simultaneously with arbitrary accuracy. Such pairs are e.g. position and momentum, time and energy, angular position and angular momentum. Such pairs are known as canonically conjugate pairs.

Unitary evolution

The quantum determinism.

Wave function

An object with one or many numerical components dependent on generalised coordinates of the configuration space. When the number of numerical components is many this may also be called a state vector.

mechanical

version

of

classical

CHAPTER 1 INTRODUCTION

It is said that consciousness is the most mysterious aspect of reality, and that all of the other great questions that science strives to answer pale into insignificance compared with this problem. Does the nonmaterial mind exist or are we just mere matter? All religions both ancient and modern believe in a soul or a nonmaterial mind separate from physicality, with conscious capacity and the power of volition. But can the conscious part of our minds really survive death? If it exists the soul has the faculty to experience many things including that other great mystery of reality–time. But is subjective time the same as physical time? If on the other hand, nonmaterial minds do not exist and we are just parts of the wider material universe, then we are relieved of the burden of answering these questions. If they do however, then do all humans and other animals have associated nonmaterial minds, or are some of us just mere automata? I could say to you that I am conscious, but you only have my word for it. How would you know otherwise? This work is focused on this most mysterious aspect of reality– the mind and its consciousness of the physical world. In ancient times the mind was regarded as something separate from the reality that we immediately experience. More recently in the age of science the idea of a nonmaterial mind remained that, it resisted any kind of investigation that may lead to reliable results. At present, notwithstanding our evident monumental progress in other areas, we are little wiser today. Eminent philosophers in history have attempted in-depth studies of consciousness, yet its nature remains stubbornly illusive. It is what David Chalmers (1995) has dubbed the hard problem. A small first step towards the wisdom we seek is the realisation that consciousness is not an object. It is a property of an object, the mind. More specifically consciousness of another object may be thought of as the mind’s contact with it. Treating the mind as an object indicates that we accept the independent existence of other’s minds–you are not alone. This is not intended to conflict with the traditional term, subject. The two terms are not mutually exclusive they are just used in different contexts–the

2

Chapter 1

mind is objective but its experiences are subjective. One may ask the question what the mind actually is. No one has been able to show that it is reducible to anything more fundamental. So it is better to enquire how it relates to the rest of reality, which is amenable to scientific investigation. We may make inroads towards the resolution of the mind-body problem by considering three course-grained viewpoints that have come to the fore over recent centuries. Taking religious beliefs through the ages one imagines a view of the mind as a completely separate entity from the rest of objective reality. This is dualism, the idea that reality consists of two substances–matter and mind. Personally I know of no religion, ancient or modern, that does not support dualism in some form. The antithesis of dualism is monism, where the two components, mind and matter, are merely different aspects of the same thing. However, monism comes in two forms. The first and most obvious is the view that matter is primary and that mind is emergent from it. This view is sometimes known as materialism although this can be confused with dialectical materialism in situations where the corresponding contexts overlap. In our context it is more illuminating to use the term physicalism as a synonym for the primacy of matter viewpoint. Most debates about the nature of mind, today and in the past, centre on the dualism-physicalism dichotomy. And in this work that is, for the most part, where the focus lies. Physicalism, or scientism as it is sometimes called (Tart, 1998), is a more recent viewpoint and essentially evolved from observation and discovery of patterns of behaviour within the objective world. This pursuit is today what we call science, and because science is closely associated with rationality, then so is physicalism. The result is that nowadays physicalism has a very large if not a majority following amongst scientists and philosophers of science. A term often used by philosophers who subscribe to physicalism is to say that the mind supervenes on matter. This is like saying that the mind depends on matter for its existence. Evidence against this viewpoint has emerged in recent decades. That said it would be a mistake to completely ignore the other form of monism, even though it is not really the focus of this book. Turning physicalism on its head, and effectively saying that matter supervenes on mind, we arrive at this alternative viewpoint. This says that mind has primacy over matter, a viewpoint that has become known as idealism, because it is the world of ideas, objects in the mind, which is primary. If we eventually discover that physicalism is not part of the truth then it may be that, although reality appears dualist, it is at root ideal. In other words dualism may just be a first step on the way to idealism. This is the reason I prefer not to focus too much on idealism, let this book

Introduction

3

represent one step of many from mere appearances towards a deeper reality. As modern science evolved from the late seventeenth century the domain of matter where the soul could exist, narrowed markedly. During the eighteenth century it became accepted that the source of human and animal behaviour within the objective world, the brain, was the only site from which a nonmaterial mind could exert any kind of influence. Therefore it was considered that biological systems were somehow special. From the middle of the nineteenth century the laws of physics were considered completely deterministic, so a nonmaterial mind could exert no influence there. Throughout the late nineteenth and early twentieth centuries the sciences leading to the reinforcement of physicalism–biology and neuroscience, and the emergence of relativity and quantum theory were contemporary. The latter sciences were marking the beginning of a new paradigm, whereas the former were still developing rapidly as new technology became available. For physicalists there did not seem to be a problem. New discoveries in biology only served to reinforce their view. Physicalism was associated with rationality, therefore for ardent physicalists, dualism or any other alternative view was considered irrational. Biologists, neuroscientists, and physicists did not work in isolation, and physicalism was prevalent amongst the physics community also. However, in order to arrive at the truth we need to dissect our own experience of reality with due diligence. A significant aspect of experienced reality is time. A major feature of the early twentieth century paradigm shift was the decline of presentism – the view that only the present exists. This was because it was seriously challenged by relativity. Prior to this the debate between presentism and eternalism – the view that past, present, and future exist on an equal footing, had rumbled on since the pre-Socratics. Now with the emergence of relativity, science was cutting through the presentism/eternalism debate like a scalpel, yet still no alarm bells rang in the physicalist’s camp. Yet why should they? Scientists in the early twentieth century were mainly concerned with their specialist fields. Their intense focus meant that science progressed rapidly, while some of the deeper meanings of their discoveries were temporarily overlooked. Things got even more complicated with the arrival of quantum theory. Results from the double-slit diffraction experiment, simply did not make any sense when set against the backdrop of classical physics. There was the formal part of quantum theory, which accurately predicted the double-slit results with repeatable experimental outcomes. What more

4

Chapter 1

could you ask? But the theory had no coherent interpretation. A little over a decade later the Nazis were on the rise in Europe and before long the world was plunged into war. Scientists were too busy with their part in the war effort on both sides to be concerned with the deeper meanings of quantum theory–it just worked. The sentiment was: shut up and calculate, even though this literal statement may not have been quoted until much later. The first indication to emerge that there might be something wrong with the physicalist’s viewpoint appeared in a book by Hermann Weyl (Weyl, 1949), although it is possible that Sir Arthur Eddington had something to say on the topic also. This is the now famous quote by Weyl that appears in the story in a nutshell at the beginning of this book. For convenience it is repeated here, The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life-line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time. (Weyl, 1949, p116).

This immediately leads to a very simple argument opposing physicalism. If you are not persuaded by physicalism but are in conversation with someone who is, then simply ask them to think of an event in their past, they do not need to tell you what it is. When they answer in the affirmative then merely suggest that that event still exists, but that they are not there anymore–that is it. They have identified part of their physical aspect where their nonmaterial mind is now absent. The main feature of relativity is the four-dimensional block space-time continuum. It is eternal with past, present, and future existing in a contemporary sense. Note the deliberate use of the present tense. The problem however is that it does not explain the very strong feeling of free will, and that we make our own future. In classical relativity the future is completely predetermined, and this does not sit well with many, particularly non-physicalists. That began to change in 1957 when a new article (Everett, 1957) appeared. In the previous years since the emergence of quantum theory, the interpretation that was generally accepted became known as the Copenhagen interpretation. This is not regarded nowadays as particularly rigorous. It just said that when a quantum state was measured, the associated wave function collapsed to a classical eigenstate. There was no specification of the wave function ontology, nor was there a defined mechanism describing the collapse. It was just a convenient way to connect the weirdness of the quantum state to the classical reality of our immediate experience. In 1957, the proposal was that the wave function

Introduction

5

encapsulated the whole of physical reality and remains unaffected by the measurement process. This is all that Schrodinger’s formulation of quantum theory entails–quantum theory does not predict any collapse mechanism. The 1957 article was titled: “Relative State” Formulation of Quantum Mechanics. Its author, Hugh Everett III, was bold enough to follow through and apply quantum theory as it was, to the macroscopic world without any extra assumptions. This ultimately entailed that all of the possible histories to the future of a measurement event would continue to exist but we would only experience one. The eigenstate that you do experience is merely a matter of perspective–it is where you are in a space of configurations. The relative state formulation does exactly what it says, it enables one physical state to be expressed in terms of another, just as relativity enables space-time coordinates of one inertial frame to be expressed in terms of another. There is no mention of the popular designations, manyworlds interpretation, or parallel universes. These terms were coined later in order to make clear to a lay public the idea of similar realities running unseen but concurrently with ours. The problem with these descriptions is they convey the false notion that these many-worlds are discrete, they are not. They differ only by particle and field configuration at any instant. The same can be said of distinct points in time throughout a given history. Therefore the appropriate manifold on which to map all configurations is a configuration space, or C-space. In this work, C-space is treated as a continuum and for all practical purposes it has infinitely many dimensions. C-space is what John Archibald Wheeler referred to as superspace. It encapsulates the entire history of our universe as well as the complete histories of all other possible universes. By envisioning the history of our universe as a continuous ordered set of instant configurations, we begin to see all other universes as similar sets of configurations that are not on our particular timeline. But they are related by the fact that they all occupy the same C-space continuum–one world, which is infinitely larger than anything we are even cosmologically accustomed to. Another way of looking at it is to say that quantum mechanics does for C-space what relativity does for space-time. Further, subsuming general relativity into quantum theory provides a unified framework, namely quantum gravity, which determines how the wave function depends on any configuration. We do not have a complete theory quantum of gravity yet because we do not have sufficient data describing the physics near the Planck scale ( ( =G c3 )

12

 10−35 m).

However, the important point is that C-space is static and eternal, and moreover physical time is subsumed within it. This leads us to two

6

Chapter 1

postulates on which this work is based. The first, which is generally agreed by mainstream physics community reads: PI

Eternalism: Physical reality consists solely of a timeless wave function over a universal configuration space.

This asserts the static nature of reality and that time is emergent. The second postulate is equally important and addresses the subjective nature of our experience, this reads: P II Experience: Our conscious dynamic experience of the world is real. It seems likely that many physicists would regard postulate P II as being most contentious. This would be because they take seriously postulate P I, which asserts that the whole of reality, including us, is a root static. In this context the “us” in the previous sentence refers only to our physical aspect, and may lead one to speculate that those making such an assertion are persuaded by physicalism. Physicists so persuaded often describe our dynamic experience as the grand illusion. However, those taking such a view, many of whom are eminent scholars in their own fields, would do well to remember that everything they have ever learned, including their own historic contributions, have come to them through the filter of their own dynamic experience. To call this an illusion strikes me as the intellectual equivalent of cutting one’s own throat. The real illusion is the appearance that the physical world is dynamic–it is not. And yet our dynamic experience persists, the Sun rises and sets every day as the Earth turns on its axis. We experience the weather in all its forms, our movement and that of others. And we dynamically experience the gaining of new knowledge. If this is an illusion then the reliability of any newly gained knowledge must surely be called into question. The consistency of our dynamic experience makes it the most powerful piece of empirical evidence that we possess, especially given that all other such evidence is dependent on it. This work is presented in two parts. Part I concerns science and scientific methodology, which is dedicated to the justification of the eternalism postulate, P I. The experience postulate P II we take as selfevident. Part II is more concerned with the mind and how it relates to the physical world. What follows is a brief description of each chapter and how to get the most from this book.

Introduction

7

The primary purpose of this work is to alert the reader to new sources of evidence for a mental realm separate from the physical. This is done in a relatively rigorous style and is the reason for its abundant technical content. Looking back from the early twentieth century, traditional sources regarded as evidence consisted of reports of what we think of today as paranormal experiences. Also authority figures purported to be endowed with shamanistic or messianic attributes are another source of this type of evidence. Unfortunately evidence from word-of-mouth alone cannot be regarded as scientific because there is no other readily accessible corroborating source. From the eighteenth century onwards, serious scientific efforts to derive evidence of nonmaterial minds from biological systems eventually came to naught. This is because the laws of physics on which biological systems depend remain stubbornly closed. From the mid twentieth century onwards however a new source of robust scientific evidence for a nonmaterial realm began to emerge. This evidence comes not from biology or neuroscience as one might expect, but from the development of general relativity and quantum mechanics. It is as though biologists and neuroscientists are just too close to the problem. We need to step back and look at physical reality as a whole. The new evidence emerging from such an approach is precisely what this book is intended to highlight. Coupled with our own dynamic experience of the classical world we are beginning to identify properties of our own minds that bear no resemblance to anything in the physical world. One specific property that is most obvious is the localisation of nonmaterial minds in a space of configurations. It is this localisation property that generates our perception of reality. Moreover, our C-space location changes with our perceived phenomenal time, and this is the source of our dynamic experience. In this way spatialized physical time becomes synonymous with an ordered sequence of configurations, and so our physical time coordinate emerges. For the purpose of this work we refer to this as the principle of localisation of consciousness within C-space. Throughout part II we will explore ways in which this model fits in with more traditional viewpoints, and in many cases we see how these points of view are subsumed into the principle of localisation model.

1.1 Structure and content This work is aimed primarily at graduates in the physical sciences or at philosophy graduates with some familiarity with physics or mathematics. Its main purpose is to disseminate to a wider academic audience, ideas that

8

Chapter 1

currently circulate within a narrow division of the philosophical community. Some of the chapters, particularly those in part I are quite technical. This is because of the rigorous nature of the new evidence for nonmaterial minds, demanded for such a bold claim. This is in keeping with the old adage: Extraordinary claims require extraordinary evidence. As a result there may be those who prefer to skip or at best skim read some chapters, 2, 4-6 say, and rely initially on their summaries provided in this section. A determined lay reader may subsequently attempt a reading of the more abstruse chapters once they are armed with the course-grained message provided here. Those more familiar with the content of those chapters will likely be graduates in mathematics or physics and will be in a good position to accept or otherwise challenge any aspect that they find contentious. The book consists of twelve chapters including this introduction. Chapters 2-6 constitute part I, and part II is made up of chapters 7-12. As previously mentioned part I is mainly concerned with the justification of the first postulate and this is where the bulk of the evidence emerges. Notwithstanding the above claim that P I is generally agreed by the mainstream, there are still those for whom it does not sit well. There are two possible reasons for this, (i) the claim of P I needs to be rigorously tested further, and (ii) if P II in conjunction with physicalism is taken seriously then there is an imperative to deny P I. The latter would entail a return to presentism or some hybrid theory of time with elements of presentism and eternalism, an issue that is explored to some extent in chapters 4, 6 and 10. The opening chapter of part II considers motivations for mindbody physicalism (chapter 7). It is demonstrated how physicalism cannot survive in the face of postulates P I and P II (chapter 8). Beyond that, ideas are presented that compete with the new dualism, and it is shown that these fail by implicitly denying P II (chapter 9). In chapter 10 we highlight some of the more contentious issues regarding time. Due to the decline of presentism, attempts to tacitly preserve mind-body physicalism lead to a plethora of hybrid theories of time with a timeless backdrop. These theories seek to explain the passage of time as an irreducible element of the physical world, while the mind is demoted to nothing more than an entity emergent from matter. In chapter 10 we show that these theories can be identified with aspects of our localisation model. Chapter 11 discusses the consequences of the latest ideas in neuroscience, and it is shown that they do not threaten the new dualist model. Moreover it is seen that this augments the prospects for artificial consciousness. Finally, in chapter 12, this is followed by a discussion of what the principle of

Introduction

9

localisation allows. A recap of the more contentious topics helps to place the principle of localisation into the context of a unified scheme based on current physical theories.

1.1.1 Logic and mathematics (chapter 2) All rational thought begins with initial assumptions referred to either as axioms or postulates. In general we need to be aware that axioms and postulates are not proven statements, they are merely assumed. Therefore the postulates above may be called into question irrespective of any motivation for doing so. Moreover, even if the postulates are accepted, classical logic, which has been the basis of rational thought since the ancient Greeks, may also be questioned, see for example Brouwer (1908) or McCall (1976). This and other issues regarding the machinery of rational thought are discussed. The main purpose of this chapter is to clarify the process of rational thought. The application of logic to mathematics is clearer than it is in the other sciences. Mathematicians deal with ideal problems and do not have to contend with noisy data for example. This is why we take a brief look at the foundations of mathematics to illustrate the mechanism of rationality. In this work we are claiming that the mind is a distinct entity from the physical domain. Challenges to this claim are likely to come in many forms. Traditional ways to dispute the conclusion of an argument are to examine either its premise(s) or the logical process leading to the conclusion. However, with extraordinary claims such as ours we may also encounter extraordinary challenges, by which we mean challenges to the logical process itself. Classical logic, which originated with Plato, Aristotle, and Chrysippus has been the basis of rational thought throughout the intervening ages, and in particular it is the mainstay of all of the sciences. A particularly important function of this chapter therefore, is to justify the continued use of classical logic by the mainstream as a basis for scientific and philosophical enquiry. The assertion that meaningful statements are either true or false is at the root of classical logic, that is to say the true-false dichotomy, also known as the bivalence principle, is exhaustive. We can either prove a statement true or false directly, or for example we may prove a statement true by directly showing that it is not false. This latter non-constructive approach is considered not reliable by intuitionists for example, because in some circumstances they claim, it is not legitimate to say that an unproven statement is either true or false. The unproven status is regarded as a third logic state, which is neither true nor false. Unfortunately the proof of a statement not being false is at the root

10

Chapter 1

of scientific methodology, and near the end of chapter 2 we briefly explore why the intuitionist approach is questionable. This chapter is couched to a moderate degree in terms of set theory and symbolic logic. The methodology of certain example proofs, particularly the use of truth tables, does rather betray my background in electronics and may not suit those accustomed to more traditional approaches. However, these methods are no less valid. To the uninitiated some of the proofs are not easy to follow, even when the principle of proof is well understood. However, a reader having such difficulty should be able to skip certain proofs without losing the essential message of the chapter.

1.1.2 The logic of science (chapter 3) The principal concern here is with the logic and methodology of science. This chapter is written mainly in a historical context, and begins the discussion starting from classical times. It continues through to the Arab scholars, medieval Europe, the renaissance, and the twentieth century. Since the classical age there has existed friction between rationalism and empiricism. Here the former is based on the deductive process from postulates to a general theory or hypothesis and then to more specific consequents. The latter is concerned with the practicality of gathering data and seeing the patterns within. From these patterns initial hypotheses are formed. Even though a starting point may exist, the problem with rationalism is that the starting point can itself be challenged. On the other hand a particular problem with empiricism is that we may be able to gather large bodies of consistent data without any inkling of how it may be used to predict further consequences. Planting your flag on one side or the other is a bit like trying to clap with one hand–we need both. Views on this are varied. Personally I feel that it is too simplistic to refer to a single general scientific method. Details of procedures often depend on the research field and the circumstances in which a particular study takes place. However, towards the end of chapter 3 three forms of inference are considered: deduction, induction, and abduction. With deduction the truth of a general theory guarantees the truth of its consequents (rationalism), while induction starts with the consequents and infers a general theory by recognising patterns in the data (empiricism). Abduction is a variation of induction in which a general theory is already established, but we wish to infer a lower level hypothesis from patterns in the data. In both induction and abduction we infer in the opposite direction of deduction, therefore the truth of the consequents does not guarantee the

Introduction

11

truth of general theories or hypotheses arrived at. In the end it is the continuous interplay between rational and empirical approaches through which science progresses. The truth of any general theory, like axioms and postulates, is never guaranteed, and this is why theories are continuously tested and sometimes modified.

1.1.3 Space-time (chapter 4) This chapter deals with classical space-time in a physical context. Metaphysical considerations are deferred until chapter 10. The central thread throughout this chapter is the physical nature of time. We begin by discussing Galilean relativity, which describes space and time as they appear in the more familiar everyday world. This was the prevailing view of reality prior to the twentieth century. This is followed by a discussion of relativity theory beginning with Minkowski space-time and progressing to curved space-time where, not only do we argue for an existing past and future, but also for a space-time that is singly connected. In other words the space-time manifold has no holes, boundaries, or discontinuities in it. This has implications for gravitational collapse, the information loss paradox, causality, and unitary evolution in quantum theory. The chapter concludes with a discussion of a modern form of presentism popularly known as the growing block (Sorkin, 2007a). In this model the past exists but the future does not, therefore the present is represented by the future-most boundary of space-time. Sorkin’s model is based on assumed space-time discreteness. Moreover, the discrete spacetime models based on causal set theory and loop quantum gravity do predict the same effects as a positive cosmological constant with regard to an expanding universe. This includes an accelerating universe, which has already been empirically confirmed (Riess et al, 1998), see also chapter 6. However, it is demonstrated that discrete space-time models do not necessarily preclude an existent future. Moreover, an existent future does not preclude an indeterminate future. This nicely leads to the main discussion in chapter 6 on quantum mechanics.

1.1.4 Classical mechanics (chapter 5) Although classical mechanics predates either relativity or quantum theory, some of its subtleties can be difficult to follow, and it was felt that an early discussion on the nature of space-time was more pressing. So a short discourse on classical mechanics is left until chapter 5. The term classical mechanics conjures up Newton’s work on the laws of motion and

12

Chapter 1

universal gravitation. For most practical purposes this is sufficient. However, to the mathematical physicist this is only the beginning. What is really meant by classical mechanics begins more that a century after Newton with the work of Joseph-Louis Lagrange in 1788. Classical mechanics in this context is more general than the Newtonian approach. In Newtonian mechanics particles are considered to move, with or without constraints, in the familiar three-dimensional space according to certain laws. In the more general mechanics of Lagrange however, the root space is the configuration space with generalised coordinates in which the evolution of a closed system is represented by a single path with time mapped onto it as a parameter. The first section after the introduction provides a short discussion of generalised coordinates that form configuration spaces. Throughout most of this book these are referred to as C-spaces, which is a vitally important concept when describing the general backdrop of physical reality and the minds that occupy it. Classical mechanics can be presented in one of three equivalent formalisms including that of Lagrange, the other two being by William Rowan Hamiltonian published in 1833 and the Hamilton-Jacobi equation. All of these are based on what is known as the principle of stationary action. Each are discussed in turn and at the end it is briefly shown that the Hamilton-Jacobi equation, when appropriately manipulated, gives rise to Schrodinger’s wave equation of quantum mechanics, thus hopefully preparing the reader for the chapter to follow.

1.1.5 Quantum mechanics (chapter 6) This is the longest and most detailed chapter of the book. A reason for this is that quantum mechanics and its literal interpretation are central to the arguments presented here, and in particular, a certain amount of detail is needed to justify postulate P I. Although it would be impossible to address every aspect of quantum theory in one chapter, I have attempted to condense the story of the theory by confining the discourse to a small number of key developments. In this way the chapter reflects my personal view of the theory, and it is only right that I lay my cards on the table in this regard. This is particularly true when it comes to interpretations of quantum mechanics. My use of italics here is deliberate because my views, being similar to those of David Deutsch, are that there is only one literal interpretation of quantum theory. Other interpretations are variations and therefore deviate from quantum theory itself. There is an appropriate historical analogy to this situation, which concerns Galileo’s interpretation of the heliocentric model of the solar system. His use of the model to

Introduction

13

generate accurate predictions of future celestial events was considered acceptable. Galileo only fell foul of the ecclesiastical authorities when he interpreted the heliocentric model as the literal truth. Nowadays we have a similar situation with quantum mechanics. What we call the Copenhagen interpretation is a view held by workers in the field who are content to use the formalism of the theory to produce required results. The theory has never failed in this regard. However, the only literal interpretation of the theory is that of a pure wave function over a universal relative configuration space. Other interpretations require contrived modifications or additions to the theory to reconcile it with our immediate experience of reality. I believe those taking such views are looking for physical explanations for that experience, thus betraying their physicalist bias. But if the mind is nonmaterial as this work suggests, then no such explanation can be found. After a brief introduction this chapter begins by describing the early quantum theory from 1900 onwards. This concludes with the wave particle duality of Einstein and DeBroglie, which leads to a description of the full quantum theory, which appeared in 1925 in the forms of Heisenberg’s matrix theory and the entirely equivalent wave theory presented by Schrodinger in the following year. After describing certain key aspects of the full theory the discussion turns to a set of axioms (or postulates) for quantum theory provided by Nielsen and Chuang (2000). These axioms are not unique, but are sufficient to entail the full theory. In this section, some of what are considered to be the more bizarre consequences of the theory are explored including the Schrodinger’s cat and Wigner’s friend thought experiments. This is immediately followed by a section, which categorizes and describes interpretations of quantum theory. For many readers no doubt, this section will be the most contentious. In the context of this book, a discussion of quantum mechanics could not be considered complete without some consideration of gravity. Quantum gravity is variously regarded as incomplete, or as a future theory. The main obstacle to a full theory of quantum gravity is generally considered to revolve around ultraviolet divergences that cannot be removed by the usual mathematical tricks employed for the same purpose in, for example quantum electrodynamics. The best we have done so far is canonical quantum gravity (CQG), which is formulated in terms of the Wheeler-DeWitt equation, +Ψ = 0 , (DeWitt, 1967). This theory has been criticized in the past for not being able to say anything about the physics near the Planck scale. However, like any theory we can only get out what we put in, and it works nicely as a unified version of existing

14

Chapter 1

theories. Its appeal is that the universal Hamiltonian, + , is open ended and it can be modified with additional terms as data from new experiments is acquired. Ultraviolet divergences occur because theorists assume that space-time is a continuum. However, a likely candidate for an updated form of the theory is loop quantum gravity (LQG), which assumes spacetime to be discrete and thereby circumvents problems associated with ultraviolet divergences. Another criticism is that the Wheeler-DeWitt equation cannot be used for practical calculations without some form of idealization, for example the construction of minisuperspace models. Such models consider wave functions of limited systems over C-spaces with an appropriately small number of dimensions. However, similar idealizations are often applied in general relativity and quantum field theory, so it seems that this criticism does not carry a lot of weight. Finally in quantum gravity, given that space-time is deconstructed into space and time, and that physical time is merely a parameterised path through C-space, we address the question of how we might define a unified manifold to replace the space-time of general relativity. It is shown that time may be treated as a differential metric in C-space and this is represented with opposite sign and algebraically added to the metric for the base space. The opposing signs of the two metric components reflect the hyperbolic nature of the space-time that emerges. The new manifold is called space-C where the C represents the C-space that replaces time. This allows us to think of C-space configurations in relation to specific inertial frames in space-C, just as hypersurfaces of simultaneity relate to spacetime in general relativity. The penultimate section of the chapter deals with an important aspect of quantum theory, the existence of macroscopic-Bell states. Until very recently macro-Bell states had never been produced in the laboratory. Because of the potential discreteness of space-time it is possible to approximate the quantum volume, the number of discrete space-time elements, of the known universe. This figure can be compared with configuration volumes (C-volumes), which are the number of individual eigenstates of macro-Bell states. In this work we define macro-Bell states to be quantum states with C-volumes exceeding that of the known classical universe. In terms of the number of qubits this is about 800. In the past it was speculated by researchers who are persuaded by collapse theories that a wave function collapse would occur long before any quantum state reached such a volume. However, a research group led by Prof. Gerd Leuchs at the University of Erlangen-Nurenberg (Iskhakov et al, 2012) was the first to exceed this limit, and indeed did so by a large

Introduction

15

margin ( 105 qubits). In the following year the same group generated a significantly larger quantum state ( 106 qubits), thus proving their repeatability and reinforcing the existence of macro-Bell states in nature (Kanseri et al, 2013). At the time of writing no other research group had generated such quantum states. However, in my view these results represent a significant boost for the literal interpretation of quantum mechanics. More importantly for us this is a strong pointer to the objective nature of the universal configuration space (C-space).

1.1.6 Physicalism and its motivations (chapter 7) This is the first chapter of part II, which is concerned with the relationship between minds and the physical domain they inhabit. It explores motivations for physicalism specifically through the accounts of Papineau (2001), Stoljar (2016), and Seager (2014). David Papineau discusses two motivations for physicalism, the argument from fundamental forces and the argument from physiology. I could be accused of bias here because of my leanings towards the physical sciences as opposed to the biological. That said I still believe that the argument from fundamental forces is the strongest of all motivations for physicalism. This is also referred to as the causal closure of physics, which says that every physical effect has a physical cause and is entailed by the conservation of energy, momentum, and angular momentum, as well as electric and nuclear charges. Causal closure implies that there is no room for sui generis vital forces posited by Descartes, and thought to exist within biological systems. This ties in nicely with Papineau’s discussion of the argument from physiology. The nineteenth century physiologist and physicist Hermann von Helmholtz was motivated to propose a general conservation of energy in order to analyse the physical behaviour of biological systems, which were considered too complicated in structure to investigate by any other means. The second author considered is Daniel Stoljar. More specifically it is his claim that the argument from methodological naturalism is an even stronger motivator for physicalsim. The first of two premises for this argument is that the methodology of science should also be applied to metaphysics. The second says that scientific methods applied to metaphysics naturally leads to physicalism. It is this second premise that seems rather weak as a motivator for physicalism. Stoljar claims the argument to be stronger than that of causal closure and also that it has not received much attention in the literature. I believe the weakness of the second premise is at the heart of the reason for the lack of interest. In

16

Chapter 1

chapter 7 this is considered and analysed symbolically via the knowledge argument, which is also discussed in some detail. It is far from clear how the strength of this argument should be comparable with that of causal closure. The penultimate section deals with the persistence of epistemic physicalism, which asserts the explicability of everything in physical terms. William Seager discusses this at length and highlights four motivations linked to reductive epistemological hierarchy of the empirical sciences. In the end we arrive at a relationship in which ontological physicalism, the doctrine that everything in existence is physical, entails reductive epistemological physicalism. Seager believes that epistemological physicalism is supremely successful provided we do not apply it to the mind-body problem. This means that ontological physicalism will be falsified if we can find an example that denies epistemological physicalism. According to Seager such an example is consciousness itself. Seager also applies a Bayesian analysis suggesting the relationship, p ( E P ) ≈ p ( P E ) ≈ 1 , between conditional probabilities, where E and P assert the truth of epistemological physicalism and ontological physicalism respectively. However, what is not known are the values of the absolute probabilities, p ( E ) and p ( P ) . Physicalists assume that these probabilities are high. However, they could just as easily be arbitrarily close to zero and still satisfy the relationship between conditional probabilities. In this case physicalism would be falsified.

1.1.7 Mind and the principle of localisation (chapter 8) As the title suggests this chapter is dedicated to minds and the localisation of consciousness principle. It is the core of the book, which brings together the two main postulates, P I and P II, and synthesises the principle of localisation. This principle asserts the localisation of consciousness in the context of the universal C-space. By itself this is not a new idea and the earliest reference I have been able to locate was due to Zeh (1970). Although Weyl (1949) did not use the term localisation, his famous quote certainly implied it in a space-time context. However, relativity alone does not explain the very powerful feeling of volition that we all possess because Weyl’s idea was couched in terms of a classical theory with a predetermined future. This is likely to be the meaning when it is asserted that free will is an illusion. The illusion is not necessarily an assertion of physicalism, it merely recognises that in relativity there is only one pre-

Introduction

17

existing future. This rather unacceptable state of affairs is resolved by the introduction of quantum theory, particularly its literal interpretation. The first overt recognition of a relationship between quantum theory and consciousness appears to be due to Wigner (1961). It seems likely that other distinguished colleagues of Wigner’s did consider such ideas possibly as early as the late 1920s, but without committing anything to writing. The main problem is that Wigner’s ideas were understood in terms of collapse theories. This is suggested by the popular designation of this interpretation as consciousness causes collapse. But without any understanding of a collapse mechanism this hypothesis seemed to linger without any prospect of a successful breakthrough confirmation. What is interesting is that Wigner noticed the persistence of physicalism to be more prominent in the fields of biology and biochemistry than in mathematical physics. Nowadays we see this as no surprise given the central role that physics now plays with regard to the mind-body problem. With pure wave theories representing the literal interpretation of quantum mechanics we have to accept that the physical domain is unaltered by the presence of minds. Pure wave theories entail a distributed static wave function. This completely contrasts the properties of minds that include dynamic experience and their localisation in C-space. This is how a conjunction of the main postulates, P I and P II, entails mind-body dualism and the principle of localisation of consciousness in C-space. It was Albert and Loewer (1988) who were the first to express this idea. However, they could not justify a complete metaphysics. The metaphysics they did propose was one of branched histories (Everett branches) in which the population of nonmaterial minds is dense. This became known as the many minds view (MMV). Further support for a dualist metaphysics come from other authors who propose alternative configurations of minds. Michael Bitbol (1990) did not offer any particular suggestions in this regard, but he did arrive at the principle of localisation through what he calls perspectival realism. This relates to the experience of reality from one’s own perspective, in other words the configuration experienced at any instant reflects one’s position in C-space. In 1991 and 1993 Euan J Squires proposed what we refer to here as a universal mind view (UMV), in which each individual mind is merely a component of a single universal mind that navigates the universal Cspace. Each individual still has control over its position in a C-subspace corresponding to all physical variables with which it has immediate access, for example one’s own body. Moreover, all minds are free to independently navigate the base space in which they reside. However,

18

Chapter 1

nonmaterial minds are constrained to occupy only one of the Everett branches, in other words they move together in C-space. The physics of this model is in complete agreement with that of Albert and Loewer, and Bitbol. However, the metaphysics of minds in the UMV completely contrasts that of Albert and Loewer’s MMV. Further it could be argued that the UMV represents a form of nonlocal hidden variables–see section 6.4.3 for a discussion of hidden variables theories. It would appear that there is a spectrum of proposed models where the extremes are represented by the MMV and UMV. Hemmo and Pitowsky (2003) arrive at what might be termed weak nonlocality by examining the reports of two participants in a thought experiment who each measure the spins of an EPR pair of electrons at different spatial angles. On the basis of correlations between the minds of one observer with the reports of the other, Hemmo and Pitowsky claim a nonlocality that is weaker than in the UMV model while in complete agreement on the physics with both Squires (1993) and Albert and Loewer (1988). The problem is that this is based solely on the physics of the situation, and while we cannot rule out the possibility of a weak but direct connection between nonmaterial minds, Hemmo and Pitowsky’s model asserts a distribution of minds satisfying the Born probabilities on each Everett branch, perfectly in line with Albert and Loewer.

1.1.8 Competing ideas (chapter 9) Given the profundity of the claims made by Albert and Loewer and subsequent authors, it is only to be expected that equally learned colleagues would publish robust challenges to the rational processes leading to metaphysical mind-body dualism. The very fact that such challenges are considered necessary perversely demonstrates the efficacy of those neo-dualist models. Moreover, should these models survive the kind of challenges described here then they would emerge even stronger. The strongest opposing arguments are considered and it is the purpose of this chapter to show precisely where these fail. It was the work by Michael J Lockwood (1996) that represents the most serious assault on the dualistic conclusions of Albert and Loewer. This work entitled ‘Many Minds’ Interpretations of Quantum Mechanics, models individual minds as subsystems of their brains by considering a mind’s corresponding configuration space. Like all physical systems they persist through time and are subject to quantum mechanical branching within its particular C-space. Token minds, a term used by philosophers to indicate that each mind possesses a sense of individual identity, are

Introduction

19

represented by paths through this C-space that are representations of time locally. Other orthogonal variables are condensed into what Lockwood calls the superpositional dimension. Together with time this forms the experiential manifold. However, in order to emphasise its multidimensionality we rename Lockwood’s superpositional dimension as a superpositional manifold. However we represent the physical subsystems of the brain, we can say that they are all extended in C-space. Lockwood is perfectly in tune with postulate P I by providing an impeccable description of the physics. However, the mistake is to implicitly deny P II. In other words, the localisation of minds in C-space is not acknowledged. Under any circumstances considered normal, minds do not experience superpositions of states. Like any other physicalist Lockwood is burdened with the question of the physical difference between a specific event when you experience it, and the same event when referred to at some future time. To deny that a past event exists at some future time is to succumb to presentism, which denies postulate P I. Other authors who follow similar lines of thinking to Lockwood are David Deutsch (1996), Simon Saunders (1998), Jenann Ismael (2003), Hilary Greaves (2004), and Peter J Lewis (2007). These authors are considered in chronological order as presented here. Deutsch and Saunders are dealt with in separate sections followed by the latter three who are considered under a single heading: Probability in a globally deterministic universe. All of these authors consider individual inhabitants of the universe to be entirely physical. And given the modern physical paradigm that we subscribe to nowadays, our physical selves are considered to be static branched structures extended in C-space. Without exception these authors provide faultless descriptions of the physics entirely in line with postulate P I. However, like in Lockwood’s account, there is no focal point of consciousness in C-space. The most threatening objection to our enterprise is arguably from Saunders (1998) who accounts for our experience via a relational theory of time. The reason is that in so doing he seems to come closer than any other to the views expressed in this book without actually endorsing them. In the relational theory of time the claim is that a mind can be wholly present at one event and then at some time later be wholly present at a future event. The mistake is that this is done without acknowledging the existence of the earlier event while the mind is present at the later one. In other words his model slips into presentism, while he exalts eternalism in a later publication (Saunders, 2002). The reasons for this inconsistency are never made clear.

20

Chapter 1

More traditional support for physicalism is significantly easier to deal with. By and large this is because traditional physicalists are antithetic to Cartesian dualism, an idea long since discredited. Another point is that some may espouse views that only appear to support physicalism, while on closer inspection these ideas may be subsumed into our localisation model. While it would be impossible to account for every angle in the physicalist’s programme in one book, I have taken a small sample of four that I believe to be a fair representation. These are Gilbert Ryle (1949), Derek Parfit (1984), Victor Stenger (1993; 2002), and Richard Dawkins (2006). Of these, Ryle is the only one to which we can firmly attach the physicalism label. He dismisses Cartesian dualism by showing that its claim is a category mistake or type error. However, such type errors are linguistic mistakes. They are not mistakes about the ontology of the objects that the language describes. Further it can be shown that Ryle presupposes physicalism, hence making his whole argument circular. Parfit, on the other hand, is a personal identity reductionist whose views can easily be subsumed into our localisation model. This is readily seen when we realise that personal identity is attached solely to our physical aspect and not to our minds. In our short description we borrow Peter J Lewis’ road analogy to examine Parfit’s thought experiments. In his article Stenger vociferously attacks the notion of quantum consciousness. However, we cannot easily lay a charge of blind allegiance to physicalism at his door. This is because his objection appears to be against the idea that quantum systems can be conscious. And in this work we take it that only nonmaterial minds are conscious, we are therefore in tune with Stenger on this point. Stenger only appears to support physicalism if we assume that he is denying the existence of nonmaterial conscious entities. However, he never says this outright. He does appear to set up a straw man in Robert Lanza (1992) who makes some valid points but demonstrates a limited understanding of quantum theory. Lanza effectively implies that the existence of elementary particles is conditional on conscious observation. In our localisation model such an observation actually betrays the C-space location of the mind doing the observing while physical objects remain entirely unaffected. Perhaps the best known of these authors is Richard Dawkins whose famous work The God Delusion (2006) proclaims his atheism. However, he has very little to say about physicalism, and as far as I can see he makes no claim either way in this regard. Atheism does not imply physicalism, but the converse is true. Because their model constitutes infinitely many independent minds all on a par, Albert and Loewer could

Introduction

21

just as easily claim to be atheists despite their claimed dualist stance at the time of their 1988 publication.

1.1.9 Phenomenal change and timeless physics (chapter 10) This chapter addresses some of the more philosophical issues concerning the nature of time. Without it we could be accused of relying too heavily on the physical nature of time addressed in chapter 4, which does little to consider how and why we seem to feel duration. This chapter echoes the work of Henri L Bergson (1910) who makes a clear distinction between the extended external time recognised by physicists and the internal phenomenal time that we experience as duration. Essentially there are only two primary views of time, eternalism and presentism. The timeless nature of eternalism means that it does not address the fact that we experience a dynamic present. This has led to a number of theories that are referred to as hybrids consisting of a timeless backdrop combined with a dynamic present. If such a present is considered to be universal then by definition it is also absolute, as opposed to relative, and immediately we see a problem reconciling it with relativity. This chapter discusses four broad types of temporal theory: presentism, the moving spotlight theory, the irreducible fact theory, and mind dependent theories. The section following the introduction argues against presentism. More precisely it maintains that relativistic arguments, for example by Eichman (2007), are stronger than the purely metaphysical arguments of McTaggart (1908) and Ewing (2013) because the former are based, at least indirectly, on empirical input. What follows is a brief discussion of the moving spotlight theory introducing the idea of a moving now against the backdrop of a spatialized objective time coordinate. When we identify experienced time with its coordinate counterpart we arrive at Ewing’s claimed incoherence due to the notion of moving through time. However, by following Bergson’s example of treating coordinate time and experienced duration as distinct concepts, then this incoherency is overcome (Bergson, 1910). In section 10.3 we consider whether theories of time are substantive (Figg, 2017). By this we mean, are they debatable? If they are obviously true or false then they are not substantive. This is used to analyse the debate between eternalism and presentism using spatial analogies, anywhereism and dynamic hereism (Figg, 2017, pp180…), where in dynamic hereism only those places visible to you exist. It is concluded that although dynamic hereism may be seen as consistent, no

22

Chapter 1

one would take it seriously because it is always possible to revisit the same place, whereas in time we ostensibly cannot revisit a particular event. In the section following that we consider objective theories of time. Essentially these imply that change takes place independently of any mind. Given the growing body of evidence since around the turn of the twentieth century, that there is a timeless substrate to the physical world then this implies a second substance that moves in phenomenal time, relative to it. Moreover, phenomenal time remains an irreducible element of the theory. Section 10.4 provides a brief discussion of two leading objective theories: McCall’s (1976) objective dynamic theory and Ewing’s (2013) irreducible fact theory. McCall describes four theories of time that are consistent with interpretations of quantum theory covered in section 6.4. Theory A is purely classical, theory B corresponds to nonlocal hidden variables, theory C is representative of pure wave theories that we favour, and theory D, the objective dynamic theory, corresponds with wave function objective collapse theories favoured by Pearle (1976), Penrose (1979), and Ghiradi, Rimini, and Weber (1986). In section 6.4 we argue strongly against such theories on the basis that they are generally incompatible with a relativistic regime, and that the collapse phenomenon must be necessarily nonlocal. McCall overcomes this problem by suspending the principle of bivalence. As argued in chapter 2 this is one principle that we rigidly adhere to. Section 10.4.2 discusses Ewing’s (2013) favoured irreducible fact theory. Ewing defines this theory as a combination of a timeless backdrop with the objective passage of time. The question addressed in this section centres on the meaning of objectivity in this context. Unless otherwise stated we should take the word objectivity to refer to anything independent of a given individual mind, including other minds. Within that definition we may define material objectivity to refer only to the physical world. When she refers to the objective passage of time Ewing does not make this clear. However, if she is following a physicalist agenda then material objectivity is meant. On the other hand if she is making no commitment to physicalism then the source of the objective passage of time could just as easily be Squires’ universal mind. The one thing we can be certain of is that phenomenal time is certainly objective in the general sense. As the old saying goes time marches on. After a short discussion that emphasises the distinction between parametric time and phenomenal change in section 10.4.3, section 10.5 considers mind dependent theories. This section addresses objections to mind-dependent theories by Kroes (1984) and Ewing (2013) by showing that they cannot apply to the principle of localisation. Both of these

Introduction

23

authors assume that a universal present is necessary in mind-dependent theories. It is shown that in our theory a universal present is entirely redundant and may only appear in the special case of a universal mind localised in C-space. A further objection by Ewing is that mind-dependent theories divorce change from the passage of time. Our response is to show that a displacement in C-space by a token mind manifests itself as phenomenal change, and this is identical to the passage of time.

1.1.10 Bio-neural systems and artificial consciousness (chapter 11) This chapter begins by reminding us that consciousness is merely a property of the mind and in this way reaffirms the difference between the two. From this it is seen that many of the claims by neuroscientists are erroneously interpreted as physicalist. The explanatory gap (Nagel, 1974; Levine, 1983) is discussed along with the associated hard problem (Chalmers, 1995) of closing that gap. A much used example of the explanatory gap is the impossibility of describing colour in purely objective terms, in other words without referring to a particular colour. This gap still remains and is symptomatic of the distinction between mind and matter. The question of how a nonmaterial mind may interface with the physical is addressed, and it is speculated that it is in regions of the brain with the highest level of connectivity. In addition it is also speculated that modules exist consisting of feedback loops, which provide arbitrarily high sensitivity to their inputs (Tanoni and Balduzzi, 2009; Ward, 2011). It may also be conjectured that a nonmaterial mind may contact only the left hemisphere of the brain via a module called the left-brain interpreter (Gazzaniga, 2012). Physicalists may be tempted to argue that this is where the mind emerges. However, the role of an interpreter is to interpret for something or someone else, the question that remains is what? If it is for some other part of the brain then physicalists still carry the responsibility to explain how the mind emerges from that other part. Another piece of evidence cited by physicalists is the readiness potential. Here it is argued that a potential build up of intent initiates well before a subject is aware of the action to take place. This is regarded as strong evidence supporting the illusion of free will. However, it has also been pointed out that the capacity to inhibit this potential before the action is realised, also exists (Gazzaniga, 2012 and associated references therein). A targeted search for consciousness in bio-neural systems has been initiated and is still ongoing (Hameroff, 1998; Hameroff and Penrose,

24

Chapter 1

2014). This work involves the quantum aspects of α/β-tubulin dimers, which form microtubules, a constituent of the cytoskeleton within neural cells. Each dimer is crudely modelled as having two states dependent on its morphology and these are regarded as being classical states, which can exist in quantum superposition. The problem is that the model is couched in terms of objective collapse theories via what Penrose calls orchestrated objective reduction (Orch OR). In this section it is shown that it can just as easily be reinterpreted in terms of pure wave theories, where we now have orchestrated decoherence (Orch D). Apart from this minor objection it is believed that Hameroff and Penrose are looking in the right place to clarify our understanding of the mechanism by which nonmaterial minds make choices in the physical world. Before concluding, the prospects of artificial consciousness are considered. These involve manufactured systems possessing the same properties of the brain that provide an appropriate interface to host a nonmaterial mind. The problem both from top-down and bottom-up perspectives is discussed, where in the former we consider the work of Hofstadter (1999) who focuses on strange loops, loops of reference spanning hierarchies within any system. It is speculated that these structures are conducive to forming interfaces with nonmaterial minds. In the bottom-up approach we show that artificial neural networks are universal binary logic systems that can be programmed with a set of analogue weights. The behaviour of these systems is entirely deterministic but still has the potential to pass the Turing test. However, such systems cannot be conscious because there is no quantum substrate to provide a mechanism from which minds can make choices. In bio-neural systems such weights are continuously adjusted via a mechanism that is poorly understood. It seems likely that further work of the type conducted by Hameroff and Penrose, for example, will in due course provide the link between these weights and the quantum substrate. Once this mechanism is understood in sufficient detail then there will be few barriers to realising such systems artificially.

1.1.11 Selected consequences of localisation (chapter 12) In this the last chapter consequences of the localisation of consciousness model are explored. We begin by showing that experiences of life considered normal are a consequence of localisation. Indeed that is the theme throughout this book, and particularly in part II. However, it is also shown that there exist classifications of events outside what we think of as

Introduction

25

normal, and in many cases these coincide with familiar reports of incidents regarded as paranormal. We begin by defining a normal sequence of events. Our lives are no more than a time ordered sequences of matter configurations. Sequences are considered normal when adjacent configurations are as close as the resolution of C-space will allow (continuous paths) and that the next configuration has a nonzero probability conditional on the present one. Two further constraints are that, nonmaterial minds can only interact with physical avatars and not with other minds directly, and that they are ostensibly unable to carry their own memories. The single minds view (SMV) and model variations due to Albert and Loewer (1988), Hemmo and Pitowsky (2003), and Squires (1993) are revisited, and it is shown that in all of these, nonmaterial minds experience normal events. This is why none of these models can yet be eliminated from the list of possibilities. In section 12.2 we address the question of whether nonmaterial minds are isolated as demanded by normal constraints, or possibly linked at some deeper unconscious level. The latter is certainly the case for weak nonlocality and the universal mind theories. Moreover, if some kind of unconscious communication is possible then this would imply that a configuration of minds exists at any instant. Otherwise it becomes impossible to define a configuration of minds in space-C. Moreover, if the principle of localisation is accepted then it is shown that the truth could be, not just one of the theories of mind just mentioned, but a combination of all four. Section 12.3 summarises the controversial aspects mentioned in this book. Essentially our conclusions rest on three issues that are not universally agreed. The first two of these are the so called interpretations of quantum mechanics and, allied to this, the information loss paradox in gravitational collapse processes. Both of these can be interpreted to consistently arrive at unitary evolution being the basis for all change in the universe. The third and final subject of controversy is the nature of experienced duration and its relation to the physical temporal coordinate. An overall consistent model is arrived at when we treat experienced time as a function solely of the nonmaterial mind. In section 12.4 the principle of localisation and the theories of mind that are consistent with it, are summarised. It seems that future investigations of the mind will only be in terms of itself. That is the nonmaterial mind can never be probed using material scientific instruments. Other techniques will be needed. Specifically, the Mind and Life Institute in their biennial proceedings hosted by HH the Dali Lama, report scientific investigations in relation to experiences of Buddhist

26

Chapter 1

practitioners during deep meditation. These are discussed and analysed at meetings of delegates from a broad range of disciplines. The book ends with a final thought on the quantity of data in the universe required to realise each of the four theories of mind. It seems that the extremes, MMV and UMV, require the least data, and that some would argue therefore that one of these are the most likely. Here I would urge caution, and to keep our options very firmly open.

1.2 Rational overview In section 1.1.2 above chapter 3 on the logic of science describing scientific methodology was summarised. There are those who subscribe to the view that science can only be applied to the materially objective (physical) world. Consequently the subject of nonmaterial minds can never be part of science and will always be beyond rationality. However, in this work we see that the physical domain is merely a subset of an objective world consisting also of non-physical elements–minds. For this reason we may apply scientific methods to the nonmaterial aspects of reality, thereby allowing the mind to be treated as part of science. From an empirical standpoint we may consider following relationship between a general theory, T, and the results, R, of an experiment, E, given by

T ∧EŸ R.

(1.1)

In a traditional scientific setting this says that ‘under the general theory, T, performing experiment, E, generates the result, R’. However we may also translate this to ‘under the general theory, T, if you perform experiment, E, then you will experience the result, R’. This would allow for subjective elements within scientific endeavours. Provided such an experiment is carefully designed and its results are repeatable with different subjects, then the latter interpretation of (1.1) is no less scientific. Alternatively by focussing on the deductive approach we may consider the two main postulates, on which this work is based. The main conclusion is the principle of localisation of consciousness (L) in the universal C-space. Therefore we must be very clear as to the relationship between L and physicalism (Phys). The localisation of consciousness does not deny physicalism if we relax P I, and it is the purpose of this book to persuade the reader towards such a denial. Postulate P II directly implies the principle of localisation, and this is considered self-evident. This relationship is succinctly presented as

Introduction

P II Ÿ L .

27

(1.2)

In addition the relationship of the postulates with physicalism is

P I ∧ P II Ÿ ¬Phys .

(1.3a)

That is the conjunction of P I and P II imply that mind-body physicalism is false. Therefore if we deny P I, as is the strategy of, for example Sorkin (2007a), then there will be a physical feature localised in C-space to which we can attach a material mind. This feature is the future most boundary of space-time. The same argument can be applied to any theory of time featuring a universal present. Wave function collapse models may also generate localised features within the physical domain. The work of Hameroff and Penrose (2014) is a case in point. This may direct us to an underlying motivation for interpretations of quantum mechanics other than pure wave theories. In part I P I is justified and assuming its acceptance, this postulate is assumed throughout part II. Therefore a denial of P I would not allow us to deny physicalism on the basis of the relationship (1.2). In this case physicalism and L could both be true. If we assume physicalism from the start then the relation (1.3a) may be equivalently written as

Phys Ÿ ¬P I ∨ ¬P II .

(1.3b)

In other words physicalism entails that we must deny either P I or P II. We have seen that Sorkin (2007a) and Hameroff and Penrose (2014) follow the former strategy, whereas Michael J Lockwood (1996) spearheads a reactionary movement following the latter. As mentioned in the précis of chapter 2 there are those that may question the mechanics of classical logic itself, for example Brouwer (1908). However, as far as I am aware there is no evidence that this was Brouwer’s motivation, he just questioned the reliability of classical logic generally. On the other hand this does alert us to the possibility that those desperate enough to preserve physicalism in a mind-body context may follow such a strategy. One such example, briefly mentioned in section 1.1.9, is McCall (1976) who suspends the principle of bivalence in order to render his objective dynamic theory of time, compatible with relativity. Such a move we consider equally incompatible with the current methods of science. The discussion of these alternative logics, amongst other things will be briefly explored in the next chapter.

PART I: THE SCIENCE

CHAPTER 2 LOGIC AND MATHEMATICS

In order to understand the relationship between the universe and ourselves, we must systematically dissect every aspect of that endeavour we call science. At its root that includes the very apparatus of our reasoning–logic and mathematics. The purpose of this chapter is to understand how we know that something is true, and if we do know, how much of that knowledge is belief and how much is certainty? Is the meaning of certainty that we are absolutely certain that something is true, or is it that that something really is absolutely true? We can start by answering this question immediately. The answer is the former, the certainty that something is true is merely a state of mind–we are certain. Certainty is no guarantee against being dead wrong, and we can never assert that anything is absolutely true. A common thread running throughout this book is relativisation. That is, many things we often assume as absolute are actually relative, i.e. dependent on something else. As we shall see later, relativisation as applied to aspects of space-time resulted in the now famous theories of relativity. Relativisation of the quantum state leads to the many worlds interpretation of quantum mechanics and its more recent variants. It also appears that the truth of a proposition has a similar dependence on other more fundamental propositions, or we may say that something is true with respect to some overarching model or context. The way we arrive at the truth of a proposition, called a proof, is dependent on other true premises. A simple proof, “If A then B”, is symbolically represented by

AŸ B

(2.1)

where B is the proven statement and A is the premise from which B holds true. The simplest proofs take this form. This is because given A, B is considered self-evident. However, in many cases mathematical theorems, stated in the form (2.1), are far from obvious. In such cases their proofs must be expanded to something like

Logic and mathematics

A Ÿ P1 Ÿ P2 Ÿ " Ÿ Pn −1 Ÿ Pn Ÿ B

31

(2.2)

where the individual steps between intermediate statements Pk ( k = 1," , n ) are considered self-evident. Whatever the circumstances the truth of the statement B is contingent on the truth of A. In logic or mathematics you will never see a proof of the form ‘ Ÿ B ’ in isolation, because this is an assertion that B has been proven absolutely true. Such an assertion is meaningless, if such a form as this is written in a mathematical text then the context will always suggest what should be written to the left of the ‘ Ÿ ’ symbol. This is often the case when the double turnstile ‘B’ symbol is employed. In these cases the assertion and justification of truth is one of semantics, however there will always be an overarching model or context. Asserting the truth of a statement always prompts the question …based on what? When we ask this question repeatedly, assuming that we always get an answer, then we will be backtracking right-to-left along the chain of implications shown in (2.2), towards more general or more fundamental statements. In practice we always end up at a statement where we can go back no further. Such a statement is referred to as an axiom or a postulate. It is such a statement that we intuitively accept as true without proof. This does not say that axioms are absolutely true they are merely accepted. This leads us to the foundational structure of mathematics and other axiomatic bodies of knowledge. So when we say that something is true, it is only true in the context of a particular formal model or theory. One objective of this chapter is to explore the various methods of deduction that have evolved since classical times. Without doubt a single chapter of a book can hardly do justice to centuries of intellectual endeavour since the appearance of the laws of thought originated by Plato. What we can do however is to look at the standard logic, which has evolved from the works of Chrysippus (c 280-207 BCE), through Boole (1847) and Frege (1879) to the present day. Modern propositional and predicate logics, from Boole onwards, represents a formal and rigorous method of deduction in all branches of mathematics and many other axiomatic subjects falling under the heading of analytical philosophy. From a rational standpoint this is how science in general has been built and how, hopefully it will ultimately lead to a resolution of the mind-body problem. However, there are other deductive models with their own rules of inference. Although most mathematicians ally themselves to the doctrine of Platonism where mathematical objects are considered to have an independent existence outside of the human mind, there are those who adhere to the alternative philosophies. Examples include intuitionism and

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Chapter 2

formalism. The underlying deductive processes, associated with intuitionism especially, deviate significantly from standard propositional calculus. This is expressed in the algebra of Heyting, considered later in this chapter, particularly with regard to the denial, by intuitionists, of the law of the excluded middle.

2.1 Classical logic In this section we consider only unambiguous propositions that have definite truth-values (either true or false). As we will see ambiguous statements or collections of logically connected propositions can be constructed that have the property of self-reference. In these circumstances there will be statements that cannot be assigned any definitive truth-value. Around the beginning of the twentieth century various strategies to exorcise self-referencing were explored with varying degrees of success. These included the works of Zermelo (1908), Russell and Whitehead (1910, 1912, 1913), Fraenkel (1922), Skolem (1922), von Neumann (1925), and Gödel (1931), which will be considered in the next section. The important point here is that once we are sure that we have a consistent model generating unambiguous propositions from accepted axioms then we can further explore the logic governing the relationships between those propositions. The lowest order of symbolic logic, propositional logic, has its origins in the laws of thought originating with Plato (c 427-347 BCE), and the syllogisms of Aristotle (c 384-322 BCE). Syllogisms have a very specific structure consisting of a major premise, a minor premise, and a conclusion. An often-quoted example by Aristotle is: All men are mortal (the major premise), Socrates is a man (the minor premise) therefore Socrates is mortal (the conclusion). Sometimes the major and minor premises are universal and particular respectively–universality being encapsulated in the word All in All men are mortal. The minor premise here is certainly particular given that Socrates is a particular man. However, it turns out that not all forms of reasoning can be reduced to the form of a syllogism, to see why we need only to consider the classical laws of thought. The three classical (Platonic) laws of thought, including their symbolic representations, are:

Logic and mathematics

1. 2. 3.

33

Identity: A ⇔ A The principle of contradiction: ¬ ( A ∧ ¬A ) The law of the excluded middle: A ∨ ¬A

The first law simply states that the proposition A is logically equivalent to itself. At first sight it might be argued that this principle is so obvious that stating it has little value. But consider the symbol ⇔ , which is equivalent to the simultaneous use of Ÿ and ⇐ . In other words A ⇔ A is equivalent to ( A Ÿ A ) ∧ ( A ⇐ A ) , where ⇔ can also be replaced with “iff”, which is shorthand for “if and only if”. Examples of English meanings for these symbols are given in Table 2-1 below. Table 2-1: English equivalents of Ÿ , ⇐ , and ⇔ .

AŸ B A implies B If A then B A only if B

A⇐B A is implied by B If B then A A if B

A⇔ B A implies and is implied by B A is logically equivalent to B A if and only if B (A iff B)

As we can see the symbol ⇔ has hidden meaning that becomes clear when we consider its constituents, Ÿ and ⇐ , separately, and then express their English equivalents. We will encounter these connectives again in this section when we consider Chrysippus’ indemonstrable syllogisms, and a fourth law of thought. The principle of contradiction is a rule that asserts that a proposition cannot be both true and false simultaneously. The rule contains two connectives, ¬ and ∧ . The English equivalent of ¬A is “not A”, this is the negation of the proposition A. The English equivalent of A ∧ B is “A and B” or more technically “the conjunction of A and B”. So the expression in the brackets asserts that both A and not A is true–an absurdity, so the principle of contradiction asserts the exact opposite by way of the negation symbol outside of the brackets. The principle of the excluded middle asserts that the combined proposition of a statement “or” its negation is always true. The English equivalent of A ∨ B is “A or B” or more technically expressed as “the disjunction of A and B”. So if ¬A is false then A is true. Likewise if A is false then ¬A is true. The principles of contradiction and the excluded middle are known to have their origins with Plato. As far as I am aware these are the earliest references to logic and the laws of thought, as we would recognise them today.

34

Chapter 2

The first realisation of a coherent propositional logic is often credited to Stoic school and in particular to the revival of Stoic logic by Chrysippus of Soli (c 280-207 BCE). Chrysippus made a distinction between simple propositions such as “it is day”, and non-simple or compound propositions consisting of simple propositions joined together using logical connectives. Chrysippus named five types of compound proposition according to the connective used (Gould, 1970) that are listed in Table 2-2. Table 2-2: Chrysippus’ logical connectives. Connective if and either…or because more/less likely…than

Example if it is day, it is light it is day and light either it is day or night because it is day, it is light more likely it is day than night

Here we see several types of connective familiar to modern logic including the conjunction (and), disjunction (or), and conditional (if). However, there appears to be no negation here. Also the fifth connective using the word “likely” suggests a stochastic property more familiar to users of fuzzy logic. More importantly at the heart of Chrysippus’ propositional calculus are the indemonstrable syllogistic forms of deduction. These are listed in Table 2-3 along with their symbolic forms.

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35

Table 2-3: The five indemonstrables. Name1 Modus ponens

Description2 ( ( A Ÿ B ) ∧ A) Ÿ B If A then B. A. Therefore B

Modus tollens

( ( A Ÿ B ) ∧ ¬B ) Ÿ ¬A If A then B. Not B. Therefore not A

Modus ponendo tollens

( ¬ ( A ∧ B ) ∧ A ) Ÿ ¬B Not both A and B. A. Therefore not B.

( ( A ∨ B ) ∧ A ) Ÿ ¬B Either A or B. A. Therefore not B. Modus tollendo ponens

( ( A ∨ B ) ∧ ¬A ) Ÿ B Either A or B. Not A. Therefore B.

Example If it is day, it is light. It is day. Therefore it is light. If it is day, it is light. It is not light. Therefore it is not day. It is not both day and night. It is day. Therefore it is not night. It is either day or night, It is day therefore it is not night. It is either day of night. It is not day. Therefore it is night.

Some remarks regarding the content of Table 2-3 are very pertinent here. Examining the symbolic description we see the syllogistic structure of these arguments

( ( Major premise ) ∧ Minor premise ) Ÿ Conclusion . In each case the major premise is itself a compound proposition. This is connected by a conjunction to a simple proposition, which is the minor premise. Together they imply the conclusion. Also all of the common logical connectives are at least implicit in these syllogisms. This prompts the question: is this set of indemonstrable syllogisms universal? By 1 These latin names were unknown to Chrysippus. They only came into common into use during the middle ages. 2 It is unlikely that the symbols used here were known to Chrysippus. A fully mature symbolic logic did not appear until the middle of the nineteenth century with the works of Boole (1847) and Frege (1879).

36

Chapter 2

universal we mean that any conceivable truth function of the propositions A and B can be constructed from the connectives used here. Although Chrysippus is unlikely to have known this, we can use that fact that the connectives ¬ and ∧ are themselves universal. So all we need to show is that these connectives can be constructed from combinations of the connectives used in the indemonstrables. The first thing that we note is that there are two forms of modus ponendo tollens, and their major premises are not equivalent. That is the first verbal description of the major premise “not both A and B” is not equivalent to the modern connective “exclusive or” ( ∨ ) that forms the major premise in the second form (Either…or…). This is demonstrated in the truth Table below (table 2-4) where a true state is denoted by “1” and a false state by “0”. Table 2-4: Inequivalence of “Either A or B” and “Not both A and B”. A

B

A∧ B

A∨ B

A∨ B

0 0 1 1

0 1 0 1

0 0 0 1

0 1 1 1

0 1 1 0

Either A or B 0 1 1 0

Not both A and B 1 1 1 0

In the sixth column of Table 2-4, the meaning of “either…or…” has an implicit extension, namely “either…or…but not both”. This is reflected in the bottom row where the “not both” extension changes the true (1) state in the fourth column to the false (0) state in the sixth column. Note the equivalence of “exclusive-or” (column five) and “either…or” (column six). In the seventh column the explicit “not both” prefix refers to A and B not both being true. This is the same as “not A and B” or ¬ ( A ∧ B ) where the word “both” acts as a bracket around “A and B”. The inequivalence of both forms of modus ponendo tollens in terms of their major premises is reflected in the difference between truth states in the first row of columns six and seven. This also reveals the difficulty of working with natural language when constructing arguments of this kind. We are now in a position show the validity and universality of the indemonstrables.

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37

Theorem 2-1: Chrysippus’ five indemonstrables are both valid and universal. Proof: The set of connectives, which constitute a function of the form

f : {0,1} → {0,1} , is defined as being universal when any combination of 2

outputs from f can be generated from all combinations of inputs. This is done with the aid of Table 2-5. Table 2-5 A

B

f ( A, B )

0 0 a 0 1 b 1 0 c 1 1 d Examples of f ( A, B ) include:

( a, b, c, d ) = (1, 0, 0, 0 ) ( a, b, c, d ) = ( 0,1, 0, 0 )

iff f ( A, B ) = ( ¬A ∧ ¬B ) , iff f ( A, B ) = ( ¬A ∧ B ) .

Taking the disjunction of these two functions say, gives ( a, b, c, d ) = (1,1, 0, 0 ) iff f ( A, B ) = ( ¬A ∧ ¬B ) ∨ ( ¬A ∧ B ) .

Therefore in principle, any combination of outputs can be generated from the connectives {¬, ∧, ∨} , and so are universal. The disjunction connective ∨ can be represented in terms of {¬, ∧} . This is

shown

by

¬A = ¬ ( A ∧ A ) .

A ∨ B = ¬ ( ¬A ∧ ¬B ) .

In

addition

we

can

say

Therefore, not only is the set {¬, ∧} universal, but also so is any function,

f ( A, B ) , generating any combination of outputs can be constructed from

binary functions of the form ¬ ( A ∧ B ) . This form is the major premise of modus ponendo tollens. So this demonstrates the universality of Chrysippus’ five indemonstrables.

38

Chapter 2

To show their validity we need to show that all of these syllogisms yield true states for every truth combination of A and B. In other words we say that all of the syllogisms are necessarily true, known as tautologies. We will show that this is the case for modus ponens using Table 2-6. The rest is left to the reader. Table 2-6: The validity of modus ponens. A

B

0 0 1 1

0 1 0 1 1. 2. 3. 4. 5.

AŸ B

( A Ÿ B) ∧ A

( ( A Ÿ B ) ∧ A) Ÿ B

1 0 1 1 0 1 0 0 1 1 1 1 All truth combinations are shown in the columns 1 and 2 for A and B. The truth for implication from A to B is given in column 3. Column 4 shows the truth states for the conjunction of columns 1 and 3. Applying the rule for implication from column 4 to B is shown in column 5. All truth states in column 5 are in the “1”-state demonstrating the necessary truth of modus ponens.

This demonstrates that modus ponens is a tautology.

, The five indemonstrables form the core of an early propositional logic developed by Chrysippus to better understand the nature of reality and our place within it. By showing the universality of the connectives used in the indemonstrables we see the completeness of Chrysippus’ system of deduction. However, although significant further developments took place up to the nineteenth century, it was not until then that a symbolic logic revolution took place with the main contributors being Boole and Frege. Before this logicians had to manage mainly with the use of natural language, which is unwieldy and not always conducive to rigorous, formal deductions. The use of symbols to represent distinct propositions and connectives is less likely to lack clarity and fall foul of ambiguities. The use of brackets removes any uncertainty as to the order in which operations in a compound proposition should be performed. Where

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39

mistakes do appear these are the result of human error and are promptly corrected with little or no protracted debate. We have already mentioned the three classical laws of thought: identity, the principle of contradiction, and the law of the excluded middle. A fourth law, implicit in Chrysippus’ logic, was not stated until after the introduction of symbolic methods (Hamilton, 1860). This law states: Infer nothing without ground or reason. This is reminiscent of the expression (2.1)

AŸ B

(2.1)

where B is that which is inferred and A is the ground or reason, or in other words, B because A. In essence this is nothing more than the major premise of modus ponens and modus tollens presented in Table 2-3. Indeed, the fourth law is, in effect, modus ponens. It is interesting that all of the laws of thought and the five indemostrable syllogisms can themselves be deduced from the definitions of the connectives {¬, ∧, ∨, Ÿ} by the use of a Boolean algebra over the set {0,1} , where a “1” indicates truth, affirmation or tautology and a “0” indicates an absence of truth, a falsehood or contradiction. In this way we can quantify truth in the context of the propositional calculus, which is simply a Boolean algebra over a bivalent set. The definitions of the connectives in classical logic are shown in Figure 2-1.

A ¬A

∧

0

1

0

1

0

1

0 0 0

1 0 1

∨ 0 1

0 0 1

1 1 1

Ÿ 0 1

0 1 0

1 1 1

Fig 2-1: Definitions of connectives

The general definition of a Boolean algebra can be succinctly given by the following (Hamilton, 1982):

40

Chapter 2

Definition 2-1: A Boolean algebra is a set A with two binary operations {∧, ∨} satisfying the following axioms. 1. 2. 3. 4.

∧ and ∨ are both commutative and associative. There exists elements 0 and 1 in A such that for each a ∈ A , a ∨ 0 = a , and a ∧ 1 = a . The distributive laws hold. For each a ∈ A there is an element ¬a such that a ∨ ¬a = 1 and a ∧ ¬a = 0 .

We need to make a few cautionary remarks here. A Boolean algebra is a much more general object than propositional calculus. Here the operations {∧, ∨, ¬} are not and, or, and not. They are merely a set of, two binary and one unary, operations satisfying the rules given in the definition. The set A is not generally bivalent either. When A is a larger multivalent set the elements 0 and 1 are extremes suggesting that A is an ordered set. An ordering relation, R, on A may be defined by

aRb if and only if a ∧ b = a . It is not my intension to explore Boolean algebras beyond what has already been said. This is an extremely detailed and rigorous topic in itself. For further details on Boolean algebras and ordered sets see Hamilton (1982). What we can do is to look at an important consequence of the fourth law, the rejection of circular reasoning. It can be seen that with the use of the conditional symbol, Ÿ , we can construct networks of logical dependence of the type seen in Figure 2-2. The proposition A is seen to have a whole network of conditional dependences that could extend indefinitely throughout the logical space. If A is accepted as an axiom then all of the consequent propositions, Pabc" , would be accepted as true theorems. But the possibility exists that propositions can form

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41

A Ÿ P1 Ÿ P2 Ÿ P3 Ÿ " P21⇔ P211Ÿ "

" ⇐ P221⇐ P22

BŸC

I

D

H

E G⇐F

Fig 2-2: Networks of truth dependence formed using the conditional connective.

isolated islands within the logical space. For an example consider the propositions B to I that form an isolated circle. Unless at least one of these propositions is accepted as an axiom, then the truth value of these propositions is unknown and any theory based on them would be highly suspect. The critisism that a particular theory, G say, is based on circular reasoning, is certainly justified. However, it is a mistake to assume that such a circle generates untruths. To demonstrate this, the circle would need to be conditionally connected to a proposition known to be false. Circularity of itself does not guarantee a contradiction. This mistake is made by McTaggart (1908) in his essay concerning the theories of time. However, as we shall see later, I do accept McTaggart’s conclusions for entirely different reasons. Now that we have briefly considered the origins of propositional calculus, we are in a position to get a feel for how most working mathematicians apply it. Essensially the two proof methods are, direct proof (modus ponens) and proof by contradiction (modus tollens), which make considerable use of the conditional connective, Ÿ . From Chrysippus’ proof methods shown in Table 2-3 it is possible to show that

42

Chapter 2

Theorem 2-2: Given any two propositions A and B,

( A Ÿ B ) ⇔ ( ¬B Ÿ ¬A ) .

(2.3)

Proof: We can show this in Table 2-7 based on the assumption that truth-values are bivalent. Table 2-7: Equivalence of direct proof and proof by contradiction. A

B

0 0 1 1

0 1 0 1 1. 2. 3. 4. 5.

¬B 1 0 1 0

¬A 1 1 0 0

AŸ B

¬B Ÿ ¬A

1 1 0 1

1 1 0 1

All truth combinations are shown in the columns 1 and 2 for A and B. Columns 3 and 4 show the corresponding states for ¬B and ¬A . Applying the rule for implication from A to B generates the states in column 5. Applying the same rule to ¬B and ¬A respectively produces the states in column 6. The similarity of the states in columns 5 and 6 demonstrates the equivalence of A Ÿ B and ¬B Ÿ ¬A .

This proves the validity of

( A Ÿ B ) ⇔ ( ¬B Ÿ ¬A )

and hence

demonstrates the equivalence of direct argument and proof by contradiction. , Let us now consider simple examples of each type of proof. As an example of a direct proof, I shall show a proof of Pythagoras’ theorem and, a little later, I present a well-known proof by contradiction showing that 2 is irrational.

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43

Theorem 2-3: Pythagoras’ theorem For any right angled triangle with sides of lengths a and b at right angles, the length of the third side (hypotenuse), c, is given by c2 = a 2 + b2 .

(2.4)

To prove this we begin by assuming that we are working in twodimensional Euclidean geometry (EG), which would be a short hand representation of Euclid’s axioms. Once we have the correct context then one way is to consider Figure 2-3 below. The procedure in an informal proof would be to note that the area outside the inner square consists of four triangles each with area

1 2

ab . So c 2 + 2ab = ( a + b ) ≡ a 2 + b 2 + 2ab . 2

Therefore c 2 = a 2 + b 2 for each and therefore any right angled triangle.

A a H

b

E a B

c

b D

F G

Fig 2-3: Geometric part of a proof of Pythagoras’ theorem.

C

44

Chapter 2

Proof of Pythagoras’ theorem: The axioms of plane Euclidean geometry are as follows: A1. A straight line can be drawn between any two points A2. Any straight line may be extended A3. A circle may be described with any centre and radius. A4. All right angles are equal to one another. A5. If two lines cross a third line subtending interior angles whose sum is less than two right angles, then the former two lines converge to a point on the same side as the interior angles. Let b and a be lengths of lines placed end-to-end forming a single line of length b + a horizontally from point A to point B. (A1, A2) Drop a second identical vertical line from B to point C. (A1) Project a third horizontal identical line from C in the opposite direction to the first line, to point D. (A1) Project a fourth identical line vertically up from D to A. (A1, A4) Define point E, on the line between A and B, at distance b from A. (A1, A2) Project a line from E at such an angle that it meets the line BC at F, a distance b from B. (A1) Project a line from F at such an angle that it meets the line CD at G, a distance b from C. (A1) Project a line from G at such an angle that it meets the line DA at H, a distance b from D. (A1) Project a line from H at such an angle that it meets the line AB at E, a distance b from A. (A1) C1. This completes the construction of the squares ABCD and EFGH. Ÿ Included in this construction are the four triangles: AEH, BFE, CGF, and DHG, each with area 12 ab .

Ÿ the total area of the four triangles is 2ab. (C1) Ÿ area ( ABCD ) ≡ ( a + b ) = area ( EFGH ) + 2ab ≡ c 2 + 2ab . (C1) 2

Ÿ c 2 + 2ab = ( a + b ) = a 2 + 2ab + b 2 2

Ÿ c2 = a2 + b2 So therefore for each and any right angled triangle c2 = a 2 + b2 .

,

Logic and mathematics

45

In the above proof the axioms, A1…A5, are quoted first. The construction statements, each of which is followed by their axioms labels in brackets, follow these. The statement concluding the construction is labelled C1. All subsequent statements are preceded by the implication “ Ÿ ” sign. Such a statement followed by “(C1)” indicates that it follows directly from the construction. Otherwise it follows from the preceding statement. The first thing to note is that this is not Euclid’s original proof. Indeed there are many proofs of Pythagoras’ theorem. This form of proof is based on one found in the Chinese text, Zhou Bi Suan Jing, (Cullen, 2007, 139) but it shows how Figure 2-3 can be constructed from Euclid’s axioms. The only criticism I can see in this proof is that there is no explicit notion of area within the axioms. The concept of area is implicitly assumed, although a sixth axiom could have been added stating: “A6. The area of a rectangle is the product of the lengths of any two of its adjacent sides”. The statement immediately following C1 would have been followed by “(A6)”. Putting this criticism aside this is a nice example of a, quite literally, constructive proof. A shorthand way of expressing the fact that Pythagoras’ theorem is syntactically provable from Euclid’s axioms is by

A1 ∧ A2 ∧ " ∧ A5 Ÿ Pythagoras' theorem

(2.5a)

However, even this can be rather unwieldy especially if a particular theorem is dependent on many axioms. In such cases the turnstile ‘A’ symbol is invoked. If Euclid’s axioms are represented by EG, then (2.5a) can be succinctly expressed as

EG A Pythagoras' theorem

(2.5b)

However, we need to be very cautious here when we express a particular theorem. It is very much a theme of this book that when stating any proven proposition, that some kind of context is evident. It most certainly is not sufficient to quote Pythagoras’ theorem in isolation, even when we explicitly state the meanings of every symbol. This is because there are non-Euclidean geometries where Pythagoras’ theorem does not hold, hence the statement quoted in (2.5b). So later in this book when I justify, my admittedly very bold claim that conscious minds are entities separate from the physical, it will be a proposition justified within a very

46

Chapter 2

specific context. As soon as that context changes then everything will be wide open once more. The proof that Chrysippus’ five indemonstrable syllogisms are both valid and universal is one proving a number of propositions by applying truth tables. Despite this, it is a direct proof similar to the proof of Pythagoras’ theorem and could be written in the form (2.2). If we take part of that proof that could be written formally, for example the proof that modus ponens is a valid method of deductive inference, then the proof of that could be written as

CL A ¬ª( ( A Ÿ B ) ∧ A ) Ÿ B ¼º (from table 2.6) ⇔ Modus ponens is a valid form of deduction

, where ‘CL’ is a shorthand symbol for classical logic. Given the bivalent truth-values of classical logic it is possible to prove a range of inference rules just from the definitions of the connectives {¬, ∧, ∨, Ÿ} already provided in Figure 2-1. As an example, a collection of rules listed below are condensed from Hofstadter (1999, 187) 1. 2. 3. 4. 5. 6.

¬¬A ⇔ A ( B ⇐ A) Ÿ ( A Ÿ B )

( ( A Ÿ B ) ∧ A) Ÿ B

(modus ponens)

( A Ÿ B ) ⇔ ( ¬B Ÿ ¬A) (equivalent to modus tollens) ( ¬A ∧ ¬B ) ⇔ ¬ ( A ∨ B ) (DeMorgan’s rule) ( A Ÿ B ) ⇔ ( ¬A ∨ B )

So essentially the axioms of classical logic are the definitions of the connectives {¬, ∧, ∨, Ÿ} . Let us now consider a proof using modus tollens. This is proof by contradiction otherwise known as reductio ad absurdum. The equivalence of this to direct argument is seen in rule 4 above. As mentioned previously, I will prove the following theorem. Theorem 2-4:

2 is irrational.

Logic and mathematics

47

Proof: There is no known way to prove this directly so we begin by assuming in contradiction that 2 is rational.

¬

(

) (

2 is irrational ⇔

2 is rational

⇔ 2=

)

p , p, q ∈ ` q

Ÿ 2q 2 = p 2 Ÿ p 2 is even ⇔ p is even Ÿ p = 2 s, s ∈ ` Ÿ p 2 = 4 s 2 = 2q 2 Ÿ 2 s 2 = q 2 Ÿ q is even by the same argument that p is even By repeating this procedure we continue to divide by 2 getting a smaller even number each time. This gives us our desired contradiction because by repeatedly dividing any even number by 2 we eventually arrive at an odd number. So therefore

2 is irrational.

, With reductio ad absurdum, so long as we arrive at a false statement we can conclude that the proof is successful. In some cases we do directly contradict the negated proposition that we started out with. This is fine but it is not necessary, arriving at any absurdity, as in this case, is sufficient. Certainly one of the most profound influences in the history of mathematics and mathematical logic is a procedure developed by George Cantor between 1874 and 1895, which has become known as Cantor’s diagonal argument. Originally it was used to prove that there are distinct orders of infinity with increasing magnitude starting with the lowest, countable infinity denoted by ℵ0 . More will be said about this when we come to discuss set theory. The procedure has also been used to prove other theorems, most notably it was an essential element in the proof of Gödel’s incompleteness theorems and has found other applications in for example artificial intelligence.

48

Chapter 2

In its original form the argument was used to show that no directory of real numbers could ever be regarded as complete. Essentially, the notion of a complete set of real numbers, even over an interval of finite measure, was proven to be a contradiction in terms. This is a clue that its structure is in the form of reductio ad absurdum, like the previous proof of the irrationality of 2 . So let us consider the following theorem: Theorem 2-5: There is no complete list of real numbers. Proof: We begin by assuming, in contradiction, that we can catalogue every real number in the range [ 0,1) . As we can see 1 is not included in this range so every real number in this range has a representation of the form N = 0.n1n2 n3 " , where ni ∈ [ 0,9] ∩ ` . We then create a table of candidates in one column with an index number r [ N ] in another. The

order of these numbers is not relevant, i.e., they do not have to be in numerical order. As long as we can, in principle, list an infinite number of them, we can show that this list is always incomplete by constructing another real number not in the original list. Table 2-8: List of real numbers

r [N ]

N

1 2 3 4 5 6 7 8 9

0.711276639 " 0.344791179 " 0.539348535 " 0.842381217 " 0.287233627 " 0.487813792 " 0.948479163 " 0.992623731 " 0.417552516 "

#

#

The list in Table 2-8 is randomly ordered and each entry consists of an infinite number of decimal places. We now construct another number from the first decimal place in the first entry followed by the second decimal place in the second entry, and so on… This is formed from all of the bold numbers in Table 2-8

Logic and mathematics

49

0.749333136 " The next step is to construct a corresponding real number by systematically changing each digit using some simple rule. The effect is to make the new number different from the first by changing the first decimal digit, followed by changing the second decimal digit to make it distinct from the second, and so on… In principle we have constructed a new number distinct from any in the list. In other words the original list must have been incomplete. A simple rule might, for example, be ni → ( ni + 1) mod 9 . The new number then becomes 0.850444247 " The next step is to add this new number to the list then to repeat the procedure. This way we can go on generating new numbers indefinitely. This contradicts our original assumption that we can construct a complete catalogue of real numbers in the range [ 0,1) . The final stage of the proof is to show that this applies to any interval in \ . To do this we merely choose any interval and show that there is a bijection between that interval and [ 0,1) . For example choose

[ a, b ) , then for

f : [ 0,1) → [ a, b ) we could choose f ( x ) = (b − a ) x + a .

In addition we could use this in conjunction with the bijection, f ( x ) = tan x , to map to or from, the whole of \ or, for example, f ( x ) = log x , to map to or from semi-infinite intervals

[ a, ∞ )

or ( −∞, a ] . This concludes the proof that any catalogue of real numbers will always be incomplete.

, What does this actually tell us? The largest set of real numbers that can be catalogued is a countable infinity. This is illustrated by the index r [ N ] . However, what Cantor showed was that the set of real numbers cannot be counted in this way, in other words the set \ is larger than any countable set including infinite ones with order ℵ0 . In short \ is not countable, it is

50

Chapter 2

a continuum, and it is customary to designate the infinity of the continuum by the symbol C. Researchers in artificial intelligence use another interesting application of the diagonal procedure. Here we consider how a mathematician decides whether a particular algorithm, C p ( n ) , acting on the number n terminates or not. Is there an algorithm running in the mathematician’s head which convinces him/her that the algorithm C p ( n ) will or will not stop, or is there some other process taking place that cannot be categorized as an algorithm? Sir Roger Penrose (1994; 1997, 109-113) has discussed this issue at length and it is interesting enough to revisit the problem here. A few examples that Penrose mentions are 1. 2. 3. 4.

Find an odd number that is the sum of two even numbers Find a natural number that is not the sum of three square numbers Find a natural number that is not the sum of four square numbers Find an even number greater than 2 that is not the sum of two primes

The arguments n for these algorithms would be 2, 3, 4, and 2 respectively: two even numbers, three square numbers etc. Their designations or index p might be 1, 2, 3, and 4 respectively as in the above list. However, provided the designations are unique it does not matter how we label them. So let us consider each in turn, algorithm 1 is easy, this will never stop. It will go on searching through pairs of even numbers adding them together to check whether the sum is odd. It is obvious to anyone with a minimal mathematical training that the sum of two even numbers is always even. In program 2 we take sums of three square numbers a 2 + b 2 + c 2 where a, b, c ∈ ` and try each possible outcome i.e.

0 = 0 2 + 02 + 0 2 1 = 02 + 02 + 12 2 = 02 + 12 + 12 3 = 12 + 12 + 12 4 = 02 + 02 + 22 5 = 12 + 02 + 22 6 = 12 + 12 + 22 It turns out that it is not possible to go any further. Try it. Since 0 2 + 2 × 2 2 > 7 then all we need to do is check every combination of 0, 1,

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51

and 2, square and add them, and there will be no combination that will return 7. This program will stop at this point. What about program 3 where we sum four square numbers, will this stop? It turns out that it will not. This is due to a famous theorem that Lagrange proved in the eighteenth century, that every natural number could be expressed as a sum of four square numbers. Will program 4 ever stop? It is believed that it will not but the truth is nobody knows. The belief that it will not stop i.e., that all even numbers greater than 2 can be expressed as a sum of two primes, is the famous Goldbach conjecture. Of the four programs listed above, we can decide whether the first three will or will not terminate. Ultimately we may eventually find a proof or a disproof of the Goldbach conjecture thus answering the question for program 4. But the central question here is: is there an algorithm that will decide whether any algorithm, with a single natural number argument, will or will not terminate? The surprising answer is no. Theorem 2-6: Given any algorithm C p ( n ) with a unique designation p

operating on n ∈ ` , there is no algorithm that can decide whether C p ( n ) will terminate.

Proof: As in the previous case we begin by assuming, in contradiction, that such an algorithm does exist. Each algorithm C p ( n ) is distinct therefore the set of all such algorithms is countable. So we can label all possible designations using p ∈ ` . This provides us with a countable infinity of such algorithms: C0 ( n ) , C1 ( n ) , C2 ( n ) ," . Let the assumed algorithm, operating on program p that, in turn, has argument n, be represented by A ( p, n ) , and let us further require that if A ( p, n ) stops then C p ( n ) does not stop. Let the proposition sP say that “program P stops”. Then in symbols the preceding statement is sA ( p, n ) Ÿ ¬sC p ( n ) . The proof continues:

¬ª sA ( p, n ) Ÿ ¬sC p ( n ) ¼º If A ( p, n ) stops then C p ( n ) does not stop. Ÿ ª¬ sA ( n, n ) Ÿ ¬sCn ( n ) º¼ Applying the diagonal procedure*. Here we imagine a matrix with the first argument of A ( p, n ) representing one coordinate and the second argument representing the other. A similar

52

Chapter 2

matrix was formed in Table 2-8 where we could form a diagonal. The diagonal in this case is represented by A ( n, n ) . But this is an algorithm with a single argument n. Therefore it must be included in the set, {C p ( n )} . Let its designation be Ck ( n ) for some fixed k. So

A ( n, n ) = Ck ( n ) .

Therefore A ( k , k ) = Ck ( k ) . But the diagonal statement * Ÿ ª¬ sA ( k , k ) Ÿ ¬sCk ( k ) º¼

Ÿ ¬ª sA ( k , k ) Ÿ ¬sA ( k , k ) ¼º This gives us our contradiction since the last proposition here says that if A ( k , k ) stops then A ( k , k ) does not stop. So there is no escape from the conclusion that no such algorithm A ( p, n ) exists.

, When we intuitively and correctly see that a particular algorithm does or does not terminate, it has been argued that this is evidence that the mind is operating in a non-algorithmic way. Given that the behaviour of matter at all scales is known to be algorithmic, the apparent non-algorithmic behaviour of the mind, relating to Penrose’s examples above, is intriguing. It is certainly true that no general algorithm exists to decide whether any routine will stop. Whether such an algorithm runs in the mind of a mathematician for specific cases, I think is another question. Where they are known, it is true that algorithms can be run that prove the answer to the questions posed by Penrose above. However, the construction of those proofs in the first place may not be algorithmic. This would involve a search of the truth space the nature of which may not be known. The diagonal argument in the above example has led us to assert that a proposition implies its own negation. A similar argument, employing Gödel numbers, is part of the proof of Gödel’s first incompleteness theorem. This theorem asserts the incompleteness of any formal axiomatic theory of the natural numbers. I will state it here but it is not necessary for my case to repeat the proof, only to give a very brief outline. Gödel’s first incompleteness theorem is as follows (Rucker, 1982).

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Theorem 2-7: If T is a formal system such that 1. 2. 3. 4.

T is finitely given, T extends P, T is consistent, and T is ω −consistent,

then T is incomplete. The first thing to note is that T is merely a collection of axioms. These axioms fall into four distinct groups, let us denote these groups by L, Q, E, and P. Every formal system consists of the first three of these axiom groups. The fourth, P, is specific to the formal system that is often invoked when discussing Gödel’s theorems: Peano Arithmetic. But any set of axioms can be substituted for P, probably the best-known example is ZFC (Zermelo-Fraenkel axioms with axiom of Choice), but more about this later. The group L consists of those axioms relating to first order logic. They could be the definitions of the logical connectives {¬, ∧, ∨, Ÿ} previously mentioned (Figure 2-1). We have already covered these at length so there is no need to say any more here. The group Q relates to the quantifiers ∃ and ∀ . For variable x, constant term k and any meaningful formula F, the axiom schemas relating to these are Q1. ∀x : F ( x ) Ÿ F ( k ) . For all x, F ( x ) implies F ( k ) .

Q2. F ( k ) Ÿ ∃x : F ( x ) . F ( k ) implies there is some x such that F ( x ) .

Q3. F ( x ) Ÿ ∀x : F ( x ) . F ( x ) implies that for all x, F ( x ) .

The group E relates to axioms governing the use of the equals “=” sign. The schemas are E1. For any term k, k = k . E2. For any terms k and l, k = l Ÿ l = k . E3. For any meaningful formula F ( x ) with free variable x and constant terms k and l,

( k = l ) Ÿ ª¬ F ( k ) ⇔ F ( l )º¼

54

Chapter 2

In all algebras as implemented today, an equivalence relation, ~, satisfies the three rules: reflexivity ( k ~ k ), symmetry ( k ~ l Ÿ l ~ k ), and transitivity ( ( k ~ l ) ∧ ( l ~ m ) Ÿ ( k ~ m ) ). In the axioms above, instead of transitivity we have E3 from which transitivity can be proven. A theorem for transitivity is given and proven as follows Theorem 2-8: Transitivity of “=”: given any three terms, k, l, and m, ( k = l ) ∧ (l = m) Ÿ ( k = m) . Proof: Define the following simple propositions as, A = ( k = l ) , B = ( k = m ) , and C = ( l = m ) .

Table 2-9: Truth table for the proof of transitivity. A 0 0 0 1 1 1 1 1

B 0 0 1 1 0 0 1 1

C 0 1 0 1 0 1 0 1

D 1 0 0 1 1 0 0 1

E 1 1 1 1 1 0 0 1

A∧C 0 0 0 0 0 1 0 1

A∧C Ÿ B 1 1 1 1 1 0 1 1

F 1 1 1 1 1 1 1 1

Three further compound propositions are defined as follows: D = ( B ⇔ C ) = (( k = m ) ⇔ (l = m ))

E = ( A Ÿ D ) = ª¬( k = l ) Ÿ ( ( k = m ) ⇔ ( l = m ) ) º¼

This is E3 for “ F ( x ) = ( x = m ) ”.

F = ª¬ E Ÿ ( A ∧ C Ÿ B ) º¼ = ª ( k = l ) Ÿ ( ( k = m ) ⇔ ( l = m ) ) Ÿ ( ( k = l ) ∧ ( l = m ) Ÿ ( k = m ) )º ¬ ¼

(

)

To prove transitivity from E3 we need to show that F is a tautology. This is done with the aid of Table 2-9. Applying rules for the connectives used we need to show that every state in column five for E entails the corresponding state in column seven for transitivity ( A ∧ C Ÿ B ).

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Nowhere do we see “ 1 Ÿ 0 ” between columns five and seven. Therefore column eight yields a true state for every combination of truth state. Consequentially the compound proposition, F, is a tautology. This proves transitivity of “=” given the axiom schema E3. , This shows E3 to be a more powerful proposition than transitivity. The collections of axioms (L, Q, and E), being common to all formal systems, are often not explicitly stated, but they are nearly always assumed. Exceptions may arise if someone is following an intuitionist’s path for example. In cases like this, the group L will vary and be stated accordingly (i.e., the law of the excluded middle is not assumed). Now we can consider the conditions 1-4 in theorem 2.7, and just say a few words about each. Condition 1, T is finitely given, means that there is some number which codes for each axiom in T and this number can be acquired by some algorithm, in a finite period, running on an ideal Turing machine. Condition 2, T extends P, states that the language of T includes all of the symbols and characters in the language of P (Rucker, 1982, 289). Condition 3, T is consistent, enforces the proposition that only true statements can be derived from T. That is if T A G then ¬ (T A ¬G ) i.e., no false statement can be derived from T. Condition 4 is stronger than condition 3 in that, if T is ω −consistent then T is consistent. It turns out however, that condition 4 can be dispensed with (Rosser, 1936). When we consider theorem 2.7 informally it is basically saying that for any formal system T, which is consistent then T must be incomplete. By incomplete I mean that there exist true propositions that are true in the language of T, but cannot be derived from it. Let us consider such a formal system and to prove its incompleteness by stating a true but not provable proposition in its own language. This way we gain, at least a course-grained idea as to how Gödel’s proof works. Consider the following theorem

56

Chapter 2

Theorem 2-9: Given a formal system T, there exists a true proposition G that is not derivable from T. In symbols: Given a formal system T, ∃G : ¬ (T A G ) . Proof: Let G = ¬ (T A G ) and suppose, in contradiction, that T A G .

(T A G ) = (T∧ Ÿ G ) where So G ⇔ ¬ (T∧ Ÿ G ) .

T∧ is the conjunction of all the axioms of T.

(T∧ Ÿ G ) ⇔ (T∧ Ÿ ¬ (T∧ Ÿ G ) ) by substitution ⇔ ( (T∧ Ÿ G ) Ÿ ¬T∧ ) contrapositive ⇔ ( (T A G ) Ÿ ¬T∧ ) Therefore T A G implies that at least one axiom in T is false. This is our contradiction. , In showing that T A G is false we have shown that G is both true and not provable from T. This is essentially how Gödel’s proof is structured. Note that the proposition G is self-referencing. This is reminiscent of the proposition sA ( k , k ) Ÿ ¬sCk ( k ) seen in the proof of theorem 2.6. Indeed it is a feature of Gödel’s proof that every symbol or character in the language of T is encoded by a unique natural number, and the diagonal argument applied. This has become known as Gödel numbering, for details see (Rucker, 1982; Hofstadter, 1999). At this point we should say that Gödel’s theorem applies to formal systems above a certain threshold of complexity. As already mentioned the formal system often invoked is Peano arithmetic, whose language includes the natural numbers. The set of natural numbers is a transfinite system. As far as I know any transfinite formal system will exhibit this kind of incompleteness. Classical logic, (L) on the other hand, which only operates on the set, {0,1} , is an example of a complete formal system. In other words there are no true propositions in L that are not provable in L. Another point is that classical logic is a very limited formal system consisting only of the “L part”. There are no Q or E axioms. Importantly, there are no axioms beyond the L, Q, and E groups,

Logic and mathematics

57

called P, for T to extend. Therefore Gödels’ incompleteness theorems would not apply to L alone anyway. Another point is that completeness is, in some sense, the compliment of consistency. That is consistency says that every proposition proven by the system T, is true in T, whereas completeness says it the other way round, every true statement in T is provable in T (Hofstadter, 1999). And Gödel’s incompleteness theorems assert the incompleteness of formal systems assumed consistent. This section has provided a brief overview of the development of classical logic from Plato to Gödel. Logic provides us with a means of arriving at new truths from established ones, each truth being akin to a stepping-stone in the truth-space, where the conditional implies “ Ÿ ” symbol points the way from one truth to the next. However, as already mentioned, we need to be very cautious when we talk of established truths (hence the italics). Even established truths are dependent on propositions (axioms or postulates), which are accepted as true without proof. We can search the truth-space ad nauseum but we will never find a bedrock of absolute truth–absolute truth simply does not exist. Notwithstanding these limitations, classical logic has become one of the central pillars of the current scientific paradigm and this is the reason why I have included some discussion of it here. The next question is: what can we apply logic to? In general it finds applications throughout the sciences, but most especially amongst mathematicians and philosophers of science. As it turns out in general all mathematical objects can be represented in terms of sets. This includes the natural numbers that represents the substance on which mathematics is seen to operate. Set theory itself is concerned with the question: what defines a set? It is this question, among others, that we now turn our attention to.

2.2 Sets, infinity, and the continuum The foundations of mathematics as presently understood, consist of a finite collection of axioms from which it is possible to prove every known theorem. These axioms are used to define sets. However, as we have already seen, an extended set of axioms also consists of the rules of inference–logic, the axioms of the quantifiers, and the axioms defining equivalence. From such a collection it is possible to prove all of the known theorems of mathematics. The problem with this is that there is no unique set of axioms. This is illustrated in (2.6)

58

Chapter 2

X 1 ∧ X 2 ∧ " ∧ X m Ÿ ª All of the known º « theorems of » « » Y1 ∧ Y2 ∧ " ∧ Yn Ÿ «¬ Mathematics »¼ Here we see that two distinct sets of axioms,

{Yk : k = 1," , n} ,

{X

j

(2.6)

: j = 1," , m} and

that independently imply all the known theorems of mathematics. A question that arises here is: what are these axioms and what do they refer to? The axioms in question are those of set theory, this is because the most general way of thinking about mathematical objects is in terms of sets or collections of them. Mathematics is couched in terms of sets, e.g. the set of all natural numbers, ` = {0,1, 2,"} 3, the set of all integers,

] = {" , −1, 0,1, 2,"} , the set of all rational numbers,

_ = { p q : p, q ∈ ]} etc. and, probably not so well known, is the set

{φ , {φ} , {φ ,{φ}} ,"}

used to represent the natural numbers in non-

numerical terms, where φ is the empty set φ = { } . We do not, however, need to delve into the details of set theory here. We just need to know that the axioms exist and there are at least three common collections of axioms in use today. Zermelo first suggested the standard set in 1908, with modifications by Skolem and Fraenkel in 1922 (Hamilton, 1982). These are the Zermelo Fraenkel axioms, or simply ZF, of which there are nine. They are usually labelled as ZF1,…,ZF9. The axioms in any set of axioms should be independent of each other, that is, there should not be any logical links of the type shown in (2.1), between them or their negations. However, it turns out that axioms ZF6 and ZF7 are so related. ZF6 is one of Zermelo’s original, which is known as the separation axiom. Zermelo’s original set does not allow for the construction of sets whose elements are defined by a function or rule. Axiom ZF7, the replacement axiom, is designed for just such a purpose. To cut a long story short it turns out that ZF6 can be derived from ZF7, i.e.,

ZF7 Ÿ ZF6 3

(2.7a)

Notice that the definition of natural numbers includes 0, this is a common but by no means a universal convention. When constructing proofs it is important to adhear to the particular convention demanded or the wider context in which the proof is constructed. Otherwise misleading or unreliable results may be generated.

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So essentially ZF6 is redundant, or alternatively can be expressed as a theorem in ZF. Most researchers in the foundations of mathematics accept ZF6. From (2.7a) we can see that any denial of ZF6 immediately implies a denial of ZF7 since (2.7a) can be equivalently written as

¬ZF6 Ÿ ¬ZF7 .

(2.7b)

A distinct set of axioms can be constructed by simply adding another axiom. One axiom not generally included in ZF, known as the axiom of choice (AC), was included in Zermelo’s original list, but was regarded with some suspicion. When it is stated in words its intuitive truth is hard to argue against: (AC)

Given any non-empty set x whose elements are pairwise disjoint non-empty sets, there is a set which contains precisely one element from each set belonging to x.

The rather contentious nature of AC in the past has meant that the axiom system known today as ZF does not include it. The addition of AC to ZF provides a new canonical axiom set, the Zermelo Fraenkel axioms with axiom of Choice (ZFC). In recent years AC has been more widely accepted with the result that ZFC is, at the time of writing, regarded as the most widely accepted axiom set amongst researchers of the foundations of mathematics. The description of theoretical knowledge built up by branching chains of proof, as seen in the previous section, looks all well and good but there are pitfalls that have not yet been considered. These are what ZF and latterly ZFC were designed to avoid. So what was the problem? Up to this point we have only considered statements as meaningful propositions, namely statements that are either true or false irrespective of whether we know the truth of a particular proposition. However, when we ignore the axioms and allow an ad hoc method of constructing sets, then such sets will have the form, { x : x has property P} . This is known as naïve set theory (NST). The construction of a set, x, with property, P, is embodied in what is known as the comprehension principle. This is stated as Given any property, there is a set consisting of all objects with that property. (Hamilton, 1982, 111).

60

Chapter 2

On the face of it this seems very reasonable. So let us consider the set, { x : x ∉ x} . This states that we have a set x that has the property of not being a member of itself. If we let

A = { x : x ∉ x} ,

(2.8)

then in line with what we expect from meaningful propositions, we can say that either A ∈ A or A ∉ A . If A ∈ A then according to (2.8) A cannot be a member of those sets that are not members of themselves, therefore A ∉ A - a contradiction. Alternatively if A ∉ A , then from (2.8) A is not a member of a set that is not a member of itself–a double negative, therefore A ∈ A - another contradiction. This is known as Russell’s paradox, and is a consequence of considering the comprehension principle. Naïve set theory is a theory of predicate logic consisting of the existential quantifier ∃ , the universal quantifier ∀ , and the binary membership predicate ∈ . More formally we can state the axiom schema of unrestricted comprehension (the comprehension principle) as

∀x∃y ( x ∈ y ⇔ P ( x ) )

(2.9)

This says that for all x there exists a set y such that x is a member of y if and only if x has the property P. If we substitute P ( x ) = x ∉ x into (2.9) then we have

∀x∃y ( x ∈ y ⇔ x ∉ x ) Ÿ ( y ∈ y ⇔ y ∉ y ) . We immediately see that this leads to a contradiction. Therefore NST is not consistent, that is given A = { x : x ∉ x} then A ∈ A ⇔ A ∉ A . Russell’s paradox generates statements that are simultaneously both true and false–an absurdity, and is also an example of a statement with the property of being self-referential. This was touched on when I discussed Cantor’s diagonal procedure and the proof of Gödel’s first incompleteness theorem. The classic example of this is Epimenides’ paradox or the ‘liar’ paradox, which is encapsulated in the following sentence: This sentence is false.

(2.10)

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61

If we assert the truth of this sentence then it denies us, saying it is false. Alternatively if we assert that it is false then declaring a false statement false then it must be true. Either way we arrive at a contradiction. Indeed Russell’s paradox is just Epimenides’ paradox in a set theoretic context. As previously seen canonical Zermelo-Fraenkel set theory (ZFC) is just one way of dealing with this problem. Other proposals for dealing with unruly collections such as (2.8) include Russell’s Type Theory, which in turn motivated the foundation axiom (ZF9) in ZFC, and the adoption of a more general collection known as a class. Classes are collections that include all sets plus those that are not sets such as (2.8). Classes that are not sets are known as proper classes. This formulation of set theory allows objects like (2.8) to be treated within the system. Collections of axioms developed by Bernays and Gödel from von Neumann’s original version represent the foundations of the set theory that includes classes. The most widely accepted version of this theory, with nine axioms, known as von Neumann-Bernays theory (VNB) includes an almost unrestricted comprehension principle, which states (Hamilton, 1982): (VNB9) (The comprehension axiom) Let A ( X ) be a well-formed formula in which the only quantifiers used are set quantifiers. Then there is a class consisting of all sets x for which A ( x ) holds. Essentially sets are elements of other classes and proper classes are not. Now let us consider the class

A = { x : x is a set and x ∉ x} .

(2.11)

Now suppose A ∈ A :

( A ∈ A) Ÿ ( A is a set and A ∉ A) . So we have a contradiction just like the previous case. However, suppose A∉ A :

( A ∉ A) Ÿ ( A ∉ { x : x is a set and x ∉ x} ) .

62

Chapter 2

That is, it is not the case that A is a set and A ∉ A . Since A ∉ A then A is not a set. VNB admits non-sets (proper classes) and there is no contradiction. A is a proper class and A ∉ A . So what has happened here? Russell’s Type Theory and ZF9 effectively consider sets and their elements to be of different types of object while VNB theory introduces a different type of collection called a proper class, which has the property of not being a set. Both strategies have the effect of breaking the self-referential loop that is so apparent in the ‘liar’ paradox (2.10). Throughout the remainder of this book it is imperative that we guard against inadvertently creating self-referencing loops in the construction of any arguments supporting our case. Sentences of the type This sentence is true.

(2.12)

do not create a contradiction but are inherently circular. A brief analysis of this sentence is as follows

( A = This sentence is true ) Ÿ A AŸ A So A ⇔ A . This tight circle consists of one sentence asserting its own truth through a logical reference to itself. There is no contradiction here. But it is not anchored in any other known truth nor does it lead to any useful proposition. As previously discussed and highlighted in Figure 2-2 circularity of itself does not lead to a contradiction, but unless at least one proposition on the circle is logically anchored, at least indirectly, to an accepted axiom, then nothing useful can be gained from such arguments. Such circular arguments however, do not exhibit such pathological behaviour as propositions of the form

A Ÿ ¬A

(2.13)

which is just a symbolic representation of (2.10). A great intellectual effort was made in exorcising such self-referencing loops from set theory in the early twentieth century. I therefore hope that such loops are not found anywhere in this work. While considering set theory I now address the question of the existence of infinite sets. This will be touched on again in the section on intuitionism. What is apparent is that there seems to be an ongoing debate

Logic and mathematics

63

between Platonism and finitism where the latter considers ` , for example, to be only a potentially infinite set rather than actually infinite. A finitist would argue that you or I could only write down a finite number of symbols representing an infinite set. No one can actually point to such an object and say, “Look! There’s an infinite set”, and behold it in all its glory. We can only write down finite representations. Similar arguments could be aimed at claims for the existence of irrational numbers. Do we really need irrational numbers? Any irrational number, for example e, π or 2 , can be approximated to arbitrary accuracy, by a rational number truncated to a, sufficiently high, finite number of decimal places. There is no computer anywhere in the universe with a register large enough to accommodate an irrational number. So why bother? Can we assume the non-existence of infinite sets and irrational numbers and obtain a contradiction? After all we can always pick a point on the interval [ 0,1) say, and we can always find a rational number arbitrarily close to that point. Mathematicians say that the set _ of rational numbers is dense on \ . So from a practical standpoint surely, _ is all we need. However, most mathematicians are Platonists. Gödel certainly was and I, at least incline towards it. The fact is that infinite sets either exist or they do not. So the answer to the question has two possible outcomes–existence or non-existence. When we assume non-existence we do arrive at contradictions. For example consider the following theorem. Theorem 2-10: The set of natural numbers, ` , is infinite. Proof: Suppose in contradiction that there is an upper bound N to ` . Ÿ ` = {0,1, 2," , k ," , N }

Ÿ N + 1 ∉ ` contradiction Therefore it is always possible to add an extra number, N + 1 , irrespective of the size of N, and consequently, ` is an infinite set. , Here, again we have used proof by contradiction to show that there are infinitely many natural numbers. Therefore infinite sets do exist, as do irrational numbers. The proof of theorem 2.4, which says that 2 is irrational, also employed proof by contradiction. This is because both infinite sets and irrational numbers are infinite systems (they are “transfinite”). The existence of such systems resists proof by direct

64

Chapter 2

construction. Maybe this is why intuitionists do not accept them. However, as a method of proof, modus tollens (proof by contradiction), is accepted by the majority of the mathematical sciences community. As we have seen, modus tollens, is a logical compliment of modus ponens (proof by direct argument). It allows the proof of a rich variety of accepted truths far beyond the layman’s view of infinity. One such accepted truth is that there is not just one level of infinity, but instead a rich hierarchy of infinities beginning with the lowest (countable) order, ℵ0 (Rucker, 1982). Before moving on to consider more conservative philosophies of mathematics, I take a little sojourn to discuss the next known level in the hierarchy of infinities, that of the continuum. I begin by stating and proving the following theorem Theorem 2-11: The cardinality of the continuum, n ( \ ) = 2ℵ0 . Proof: We begin by proving that 2ℵ0 ≤ n ( \ ) . Let the elements of 2` be represented by the decimal expansion

0.n1 2n2 2n3 2" 2nk 2" , ni ∈ {0,1}

(2.14)

where the binary digits, ni , are interleaved by the decimal digit, 2, in order to remove any ambiguity in representation. Let

the

set,

 2 = { x : x = 0.n1 2n2 2"}

then

 2 ⊆ ª¬ 992 , 334 º¼

since

inf  2 = 2 99 and sup  2 = 4 33 . But ª¬ 992 , 334 º¼ ⊆ \ therefore  2 ⊆ \ . So 2ℵ0 = n ( 2 ) ≤ n ( \ ) . We now need to prove that n ( \ ) ≤ 2ℵ0 .

Every real number in ( 0,1) has the binary representation

0.n1n2 n3 " nk " , ni ∈ {0,1} Therefore n ( 0,1) ≤ 2ℵ0 .

A bijection between ( 0,1) and \ is f : x → tan (π x − π 2 ) .

(2.15)

Logic and mathematics

65

Therefore n ( \ ) ≤ 2ℵ0 . Given that 2ℵ0 ≤ n ( \ ) also, then 2ℵ0 = n ( \ ) as required.

, What we have shown here is that the cardinality of the set of real numbers is C = 2ℵ0 . What we have not shown is that ℵ0 and C are distinct cardinalities. Sure, we have proven that any catalogue of real numbers will always be incomplete, but that is not quite the same thing. What we need to do is to prove the following theorem. Theorem 2-12: Given any transfinite cardinal α , then α < 2α . Proof. The proof has two stages: first we begin by showing that α ≠ 2α , and in the second stage we show that α ≤ 2α . First stage: Suppose, in contradiction, that α = 2α . Let α be the cardinality of some set A, i.e., α = n ( A) .

Ÿ There is a 1-1 correspondence between a ∈ A and S ( a ) ⊆ A , where S ( a ) is uniquely associated with a.

Let Q = {a : a ∉ S ( a )} = S ( q ) for some q ∈ A . Suppose that q ∉ Q . Ÿ q ∈ Q because contradiction.

it

satisfies

the

property

that

q ∉ S (q) = Q ,

Now suppose that q ∈ Q . Ÿ q ∉ Q because it does not satisfy the property that q ∉ S ( q ) = Q , another contradiction.

Because we obtain a contradiction in both cases the above 1-1 correspondence does not exist. We have therefore satisfied the first requirement by proving that α ≠ 2α . For the second requirement we prove α ≤ 2α by direct argument.

66

Chapter 2

Suppose for every a ∈ A there is a subset S ( a ) = {a} ⊆ A .

Ÿ This provides a 1-1 correspondence between S ( a ) and a, giving us

the following:

α = n ( A) §α · = n ¨ * {a} ¸ © i =1 ¹ §α · = n ¨ * S (a) ¸ © i =1 ¹ ≤ n ( P ( A ) ) = 2α , where P ( A ) is the set of subsets of A. Because this does not exhaust every subset of A, then α ≤ 2α . We have proved both requirements that sufficient to show that

α

α Physics > Chemistry > Biology

(3.8)

is something that we would recognise today as being an order of dependence. As we go right to left along this chain we are heading towards more fundamental sciences. Other sciences can be included in the scheme as well. Geology, for example, is dependent on physics, but there is also some dependence on chemistry as well. So including all of the sciences known today would not lead to a straightforward linear relationship of the type shown in (3.8). Indeed there are biological processes with physical dependence but where no chemical reaction takes place. Therefore lists like (3.8) should be treated very cautiously. Grosseteste and his contemporaries would not have been burdened with such complications because science in those days was not as mature and significantly less demarcated. For our purposes it is important to note that Grosseteste recognised the significance of induction from empirical data to universal laws and the deduction or prediction of other measured results from those laws. This is a thread that runs right back to Aristotle. Although Grosseteste may be regarded as the originator of western scientific methods, he did not seem to be as generally well known as Roger Bacon (1214-1292), even though Bacon drew much of his inspiration from the writings of Grosseteste. The exact circumstances of

The logic of science

89

Bacon’s birth are unclear. He was born in Ilchester, Somerset, either in 1213 or 1214, possibly at Ilchester Friary. It seems very little is known about his life outside of his cleric and scholastic activities. He was an English philosopher and Franciscan friar who placed considerable emphasis on the study of nature through empirical methods. Certainly since the end of the nineteenth century he was often credited as one of the earliest advocates of the modern scientific method inspired by Aristotle and the later Islamic scholars. Bacon studied at Oxford and may have been a student of Grosseteste. He was a well-known critic of the so-called argument from authority, championing experimental study over reliance on authority, arguing that thence cometh quiet to the mind. He rejected the blind following of past authorities in all fields of study, contrary to the widely accepted method of study at the time. In the field of optics, Bacon draws heavily on the works of Claudius Ptolemy, Alkindus, and Alhazen. He includes a discussion of the physiology and anatomy of the eye, the brain and considers light, distance, angular position, size, direct vision, reflection, and refraction. His studies were directly influenced by the works of Alhazen, drawing on the Latin translations of Kitlab al-Manazir (The Book of Optics). Alkindus also influenced Bacon’s work in optics indirectly through the writings of Grosseteste. Bacon’s investigation of a magnifying glass at least partly rested on the legacy of Alhazen who was in turn influenced by Ibn Sahl’s tenth century study in dioptrics (El-Bizri, 2005). With regard to the search for truth Bacon listed, in his Opus Majus, four sources of error that any investigator should bear in mind: • • • •

Reliance on authority Reliance on popular opinion Reliance on personal bias or vanity Reliance on rational argument

Reliance on authority has already been mentioned. Continued reliance on authority is one way of becoming stuck in the past. In the late twelfth and early thirteenth centuries this was considered by Bacon to be a very real danger. Reliance on popular opinion is dangerous for similar reasons. Both of these sources of error rely on the knowledge and opinions of others. Equally suspect is reliance on personal bias or vanity. You may be relying on your own knowledge and experience, but there is always more to be learned. Falling into this trap can be described as relying on one’s own pet theories. An excellent piece of advice given to me by my senior colleague, the late Prof. Peter H Plesch, was, don’t be afraid to murder your darlings.

90

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This is essentially the advice handed down from Bacon more than seven centuries previously. The fourth source of error in this list was considered by Bacon to be the most dangerous. At first sight it may seem to be perfectly in order to rely on rational argument. However, we need to be aware of the context in which Bacon issues this warning. One famous quote by Bacon reads Argument is conclusive, but it does not remove doubt, so that the mind may rest in the sure knowledge of the truth, unless it finds it by the method of experiment. (Dingle, 2000, 21)

I think this clarifies Bacon’s position. There is nothing wrong with arguments of the form given in (3.7). If there were then we would have to rethink all of our science, mathematics, and philosophy going back at least as far as Plato, this is the reason for the statement, argument is conclusive. However, it does not remove doubt, because the theory, T, in (3.7) might be false. As was discussed in the previous chapter, starting points for arguments (initial premises) are never proven. This is why Bacon insisted on experiment being the final arbiter in the search for truth.

3.4 Renaissance science Arguably the renaissance period, as far as the development of science is concerned, is delimited by the publication of Copernicus’ De Revolutionibus (1543) and Newton’s Philosophiæ Naturalis (1687). This period is often referred to as the scientific revolution, where the post Newton period may be considered as the age of modern science. Other contributors during this transition period included such names as: Tycho Brahe, Galileo Galilei, and Johannes Kepler already mentioned, as well as René Descartes (1596-1650), Robert Hook (1635-1703), and Gottfried W Leibnitz (1646-1716). One contributor in particular, who was famous for his attempt to systemize a scientific method, was Francis Bacon (15611626). Francis Bacon, 1st Viscount St. Alban PC KC, was born on the nd 22 January 1561 at York House near the Strand in London, the son of Sir Nicholas Bacon by his second wife Anne (Cook) Bacon. He was an English philosopher, statesman, scientist, and author, whose scientific interests were natural philosophy and philosophical logic. During his early life he was educated at home owing to poor health. At the age of 12 he entered Trinity College Cambridge, with his older brother Anthony, under the personal tutelage of Dr John Whitgift, the future Archbishop of Canterbury. Bacon’s education was conducted mainly in Latin and

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followed a medieval curriculum. He is also known to have spent time at the University of Poitiers. Apart from being elected to parliament in 1584, Bacon was not well recognised until the ascension of James I in 1603, after which he was knighted. He was made Attorney General of England and Wales in 1613, and became Lord High Chancellor of England in 1617. He was made Baron Verulam in 1618 and created Viscount St Alban in 1621. With no heirs, both of his titles became extinct upon his death on the 9th April 1626. Francis Bacon lived in a time when investigators were just starting to look at the stars through telescopes and peer down microscopes at samples, both biological and inanimate, but too small to be seen with the unaided eye. It was also a time of the protestant reformation where many across Europe were starting to question the legitimacy of the Catholic Church. Within this volatile mix was the appearance of another technological development–the printing press. Effectively the internet of its day, the printing press allowed dissemination of new ideas employing standardised fonts, both alpha and numeric, as well as diagrams. This technology enabled an unprecedented level of cooperation among philosophers who could now build upon each other’s ideas over long periods of time. It would be difficult to overstate the effects of the printing revolution. Astronomers such as Copernicus, Galileo, Brahe, and Kepler were able to share and build on their experimental outcomes. Religious reformers began to publicize new, and increasingly radical, protestant ideas. Indeed the protestant reformation and the scientific revolution encouraged philosophers to discover all they could about nature as a way to learn more about the Divine, an undertaking that promoted a break with the past. As already discussed the process of scientific enquiry consisted of an inductive step from a set of empirically derived facts to generalised principles that entailed those facts. This is followed by the application of the general principles to predict the outcomes of new experiments. It is the inductive step on which Bacon focused. Bacon’s method of induction was more complicated than just making generalizations from observations. In the first instance Bacon advocates making careful and systematic observations to produce quality data. The next step is to generalize using induction, to one or more postulates. At this point Bacon cautions against generalizing too far beyond what the facts actually demonstrate. In further steps a researcher may gather more data, or maybe use the existing data and the new postulates to generate further hypotheses. Certain experimental facts Bacon regarded as useful were exceptional cases and negative instances, where results do not conform to expectations. The

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whole process may be repeated in an iterative fashion to build an increasingly complex body of knowledge supported entirely by empirical data. In Novum Organum Bacon argues that the only hope of building true knowledge is through his careful method, based on facts from observation and controlled experiment. He rejected the old knowledgebuilding methods, which were often not based on facts but on vague illproven deductions and metaphysical conjecture. Here Bacon’s target appears to be Aristotelian methods biased towards deduction using classical logic. While there may be nothing wrong with classical logic, Bacon would have made the point that the axioms, from which Aristotle and his contemporaries deduced, were ill-founded. Using Bacon’s methods man could start afresh, setting aside old superstitions and overgeneralizations. Bacon claimed that researchers could accurately build a base of knowledge from the ground up in a step-by-step way. Describing existing knowledge at the time, Bacon claims: There is the same degree of licentiousness and error in forming axioms as in abstracting notions, and that in the first principles, which depend on common induction; still more is this the case in axioms and inferior propositions derived from syllogisms. (Bacon, 1620, XVII, AphorismsBook I).

Bacon was above all an empiricist and the purpose of his method was to do-away with metaphysical conjecture. The method consisted of procedures for isolating the form or cause of a particular phenomenon. This included methods of agreement, difference, and concomitant variation (Hess, 1964). The method essentially consists of drawing up a list of contributing factors in which the phenomenon you are trying to explain occurs, and a list of factors where it does not occur. The next step is to rank your lists according to the degree in which the phenomenon occurs in each factor. This way it should be possible to detect a pattern and thereby infer what factors contribute to the phenomenon under investigation. From this it should be possible to suggest an underlying cause, referred to as the form, for the phenomenon. This form, in its first approximation, Bacon called first vintage. This is not the final conclusion about the cause of the phenomenon, but is a provisional hypothesis. Further data may be added and the hypothesis modified or changed altogether. In this manner the truth of natural philosophy is approached by gradual degrees. What is presented here is little more than a very course grained description of the Baconian method. An important point is that the

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Baconian method consists of elements that were not new at the time. For example the methods of agreement, difference, and concomitant variation were, as mentioned earlier, known to the Persian scholar Avicenna (Goodman, 1992). Based on facts like these, it could be argued that the Baconian method is no more than a refinement of the methods undertaken by earlier scholars going as far back as Geber in the eighth century. Moreover, before the idea of the controlled experiment, bias toward the empirical side of science goes back as far as Epicurus in the fourth century BCE. So Bacon’s contributions could be regarded as the culmination of an evolving enterprise dating right back to classical times. The principal legacy that Bacon left to posterity appears as a rigorous approach to the inductive side of the scientific process. He criticized his forebears for relying on deduction from what he believed to be ill-founded metaphysical assumptions. This criticism was likely aimed at the schools of Plato, Aristotle, and the Stoics. I know of no remarks he may have made about the school of the Epicurians who argued for a bias towards empiricism. It would certainly seem unfair of Bacon to level similar criticism at the Arab schools who originated the controlled experiment, or at his name sake (or possibly a distant relative) Roger who is known to have counselled against reliance purely on rational argument. Bacon seems to forget, in his criticisms, that the classical scholars especially, were people of their time. They did not have the scientific insight or the means of generating large bodies of experimental data, which was becoming available in the seventeenth century. Throughout the centuries leading to the renaissance period there seems to have been conflicting viewpoints as to the merits of the inductive and deductive parts of the scientific cycle. The fact is that in isolation neither is adequate. Deduction, by its self, does not lead to new knowledge because propositions deduced from general principles are already implicit in those principles. It does however, increase understanding, which is one of the primary goals of science. But there would be a limit to this process. Without new empirical data science would stagnate. Data from experiments and observations provides the means to feed science so that it can grow. However, without the deductive path we would be left with ever increasing bodies of data coupled with little or no understanding as to its interpretation. I do not think Sir Francis envisaged such a scientific future. In his political role Francis Bacon called for the establishment of an institution that would promote and regulate knowledge acquired through experimental means. Alas Sir Francis would never see it happen in his lifetime. After considerable delay due to the civil war, the execution of Charles I and the restoration of the Monarchy; the Royal Society for

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Improving Natural Knowledge was created in 1660. At the time of its founding it was considered to be little more than a Gentleman’s Club for tinkering aristocrats. The Royal Society promoted Bacon’s principles of rigorous observation and experimentation in its periodical Philosophical Transactions of the Royal Society, generally credited as the first scientific journal. In a fairly short time Bacon’s philosophies became widely accepted, which led to the harnessing of nature for profit. A development that ushered in the industrial revolution that began with the first mass smelting of iron, by Abraham Derby in 1709. Depending on how we define the scientific revolution, it culminated with the publication of Isaac Newton’s Philosophiæ Naturalis Principia Mathematica in 1687. What Bacon did for experimentation, Newton did for theoretical and mathematical physics. The fusion of these two approaches into a hypothetico-deductive process was to drive science forward right up until the present day.

3.5 The Renaissance to 1900 The declaration that the process of scientific enquiry consists of repeated cycles of induction followed by deduction, has survived in one form or another since the time of Aristotle. This prompts the question as to where the debate in the philosophy of science should be focused. The deductive part of the cycle operates on theories designed to explain data acquired empirically, often through mathematics or other logical apparatus, as discussed at length in the previous chapter. Throughout history there does not seem to be any contradiction of the notion that a theory or hypothesis entails the empirical data those theories are designed to explain. Indeed the term explanation is just another way of expressing implication, entailment, conditional-if, or whatever with an additional narrative. So when we see expressions like T Ÿ R , the focus should be on the inductive process itself through which we arrive at T given R. As we have seen in section 3.2 Avicenna was critical of Aristotle for not doing enough to arrive at his claimed hypotheses. Francis Bacon was similarly critical of Aristotle for relying on deduction from assumed metaphysical premises. However, the underlying building process of knowledge was exactly the same in classical times as it is today. The only thing that changes is our increased knowledge and understanding, and this is bound to have an effect on the way we view scientific enquiry. Avicenna had far more to go on than Aristotle, and Francis Bacon had even more to work with than either Avicenna or Aristotle. Let us see how the history and philosophy of science appears from the perspectives of two

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influential nineteenth century personalities William Whewell (1794-1866) and John Stuart Mill (1806-1873). William Whewell was one of the most important figures in nineteenth century Britain. He was a polymath who wrote extensively on mechanics, geology, astronomy, theology, educational reform, international law, and architecture. This is aside from the work for which he was most famous, moral philosophy and the philosophy of science. Whewell was born in Lancaster the eldest child of a master carpenter. His father was persuaded by the Headmaster of his local school to allow him to attend Heversham Grammar School in Westmorland, where he could qualify for a scholarship to Trinity College Cambridge. He came up to Trinity in 1812, and in 1816 he was placed second Wrangler and won one of the two Smith’s Prizes. In the following year he won a college fellowship, and in 1820 he was elected as a Fellow of the Royal Society. In 1841 he was named Master of Trinity College, a post that he held until his passing. In his philosophy of science, Whewell acknowledged that all knowledge was ideal or subjective, as well as objective. He was critical of Kant and the German Idealists for their almost exclusive focus on the subjective element, and equally critical of Lock and the Sensationalist School for their strong bias towards the empirical element. Whewell believed that the acquisition of knowledge required attention to both ideal and empirical elements. In this way Whewell claimed to be seeking the middle way, known as antithetical epistemology. It appeared that what Whewell saw as fundamental ideas, he believed to be supplied by the mind itself–they are not, as JS Mill protested, merely abstractions from observations of the material world (Snyder, 2012). Of course no one can be absolutely sure of his reasons for these views but I would speculate that he was influenced by his rigorous mathematical education at Trinity. It seems Whewell tended towards a Platonist’s view, at least as far as fundamental ideas are concerned. In his work Philosophy of the Inductive Sciences, founded upon their History, Whewell refers to discoverer’s induction used to acquire both phenomenal and causal laws. According to Snyder (2012), Whewell considered himself a follower of Francis Bacon, and claimed to be renovating his method. This is well attested by the fact that one of the volumes of his third edition was titled Novum Organon Renovatum. Certainly as far as practical experimental approaches are concerned it is fair to say that Whewell would have been a follower of the Baconian method. For Whewell the process of induction was not just enumeration of a set of measured results say, but included an act of thought referred to as colligation. Colligation, according to Whewell, is a mental process of

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superinducing upon a set of empirical facts that enables them to be explained by a general law. An example used by Whewell and highlighted by Snyder (2012) is the analysis of the orbit of Mars by Kepler. The empirical data provided by Brahe was sufficiently dense and high quality for Kepler to colligate toward the orbit of Mars forming an ellipse. The mind of Kepler was an active participant in the colligation, and it can be argued that the concept of an ellipse was already present in the mind of Kepler. Today we may call this pattern recognition, where the a priori existence of a conceptual ellipse is acknowledged by the use of the word, recognition. In a way I think we have been here before. The ellipse could be regarded as a metaphysical object belonging to the Platonic forms. This is precisely what Francis Bacon criticized Aristotle for doing, even though Whewell claims he was following and updating the Baconian method. It seems that Whewell’s grounding in mathematics at Trinity made him more sympathetic to the rationalist viewpoint than Bacon. To me it does seem that Whewell was trying to tread the middle ground between empiricism and rationalism, as he claimed. According to Hall (1960), Whewell recognised induction and deduction as the reverse of each other, both being encapsulated in the relation T Ÿ R , whereas JS Mill described induction and deduction as separate forms of reasoning. Here and elsewhere he is in effect describing the hypothetico-deductive method…It led directly to controversy with JS Mill because Whewell maintained that the formation of an explanatory hypothesis was the induction, and since the facts explained could then be deduced from the hypothesis, induction and deduction were not different kinds of reasoning as Mill thought, but each the other in reverse…(Hall, 1960).

This view contrasts with that of Snyder (2012) who claims Whewell did not support a hypothetico-deductive approach. Moreover, Whewell explicitly rejects the hypothetico-deductive claim that hypotheses discovered by non-rational guesswork can be confirmed by consequentialist testing. (Snyder, 2012, 4).

It seems that what Snyder is getting at here is that Whewell did not consider induction to be a haphazard process. Snyder goes on to say… In other mature works he noted that discoveries are made “not by any capricious conjecture of arbitrary selection” (1858a, I, 29) and explained

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that new hypotheses are properly “collected from the facts” (1849, 17). (Snyder, 2012, 4).

Irrespective of how we define a hypothetico-deductive process, it seems to me that given a set of empirical facts, R, we can just as easily arrive at a hypothesis, T, by a lucky guess as by Whewell’s systematic colligation. If we can perform the calculation, T Ÿ R , then that constitutes a verification of T. According to Wilson (2016) it seems that this was all that was necessary for Whewell. That is, a hypothesis, T, arrived at in some systematic way that not only explains the data at hand, but also provides explanations in other areas (consilience) was sufficient to accept T as being part of a divinely inspired truth. While Mill might accept this, by itself it is not sufficient. Such an approach Mill describes as an enumerative induction… …“from all observed As are Bs infer that all As are Bs” —is unreliable, often leading us to accept as true generalizations that later turn out to be false. (Wilson, 2016).

The fact is that all induction is unreliable. Mill introduces an additional eliminative procedure where we may have several competing hypotheses explaining the same set of data. By measuring and observing new data in an extended range, we can in principle eliminate all but one of the competing hypotheses. While Mill argues that this is more reliable, which seems very reasonable to me, it still does not guarantee the truth of T. Ultimately all axioms, theories, and hypotheses are provisional. Notwithstanding Whewell’s published views on scientific methodology, it is inconceivable that Whewell was not aware of something like an eliminative procedure. It is just that Whewell preferred to emphasise an enumerative approach under the heading colligation. Another influential scholar of the late nineteenth century, often described as the farther of pragmatism, was Charles Sanders Peirce (18391914). Peirce was born at 3, Phillips Place, Cambridge, Massachusetts on the 10th September 1839. He was the son of Sarah Hunt Mills and Benjamin Peirce, himself a professor of astronomy and mathematics at Harvard University. Charles’ fascination with logic and reasoning began when, at the age of 12, he borrowed his older brother’s copy of Richard Whately’s Elements of Logic. He graduated from Harvard in 1862 and obtained a BSc degree in chemistry from the Lawrence Scientific School the following year. His later academic record was regarded as undistinguished. During the period 1859-91 he was intermittently

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employed by the United States Coast Survey and its later incarnation, the United States Coast and Geodetic Survey. The unfavourable influence of his senior academic colleagues Charles William Eliot and Simon Newcomb may have been responsible for his failure to obtain a tenured academic position. Peirce was difficult to work with, sometimes aloof and irascible. This was likely to be due to the medical condition trigeminal neuralgia from which he suffered throughout his adult life. It is also speculated that his, mostly unpublished, work was far ahead of its time making it controversial within the academic environment of the day. One of the fields of study for which he is best known is the logic and philosophy of scientific enquiry. For our purposes it is his analysis of the categorical syllogism Barbara (AAA) that is of interest. In particular he identifies a mode of inference variously known as retroduction, presumption, or hypothesis, as opposed to deduction and induction. This is more widely known as abduction. The structure of the Barbara syllogism is one we have encountered before in the previous chapter. It has both major and minor premises that imply a conclusion. In symbols if we suppose that T, H, and R are the major premise (a general rule), minor premise (a particular case), and conclusion respectively, then the structure is

T ∧H ŸR

(3.9)

Here the form that the syllogism takes is one of deduction. In words we say “Given T and H infer R” In this mode the truth of T and H guarantees the truth of R. The other two modes of inference are induction: “Given H and R infer T” and abduction: “Given T and R infer H” We cannot, for example, write “ H ∧ R Ÿ T ” for induction because that would change the relationship between T, H, and R. That relationship (3.9) is fixed. So for induction we infer a general rule (T) from a particular case or hypothesis (H) and a conclusion (R). For abduction we infer a

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hypothesis (H) from a general rule (T) and a conclusion (R). So, in both abduction and induction we are backtracking in the opposite direction of the arrow shown in (3.9). So what we have termed induction for most of this chapter we could reclassify as induction or abduction depending on what part of a particular syllogism is being inferred. A relationship similar to (3.9) is seen at the beginning of section 1.2 where H is replaced by the fact that experiment, E, had taken place. We can also envisage more general structures that might appear when analysing scientific enquiry. For example consider

T ∧ ( H1 ∨ H 2 ∨ " ∨ H k ) Ÿ R

(3.10)

Here we see a general theory, T, in conjunction with a disjunction of alternative hypotheses, H i ( i = 1" k ) , that imply a set of observed results, R. In this model the general assumption is that just one hypothesis out of the set, { H i } , under T, is actually true. The criteria we usually apply to close in on the correct hypothesis may be: simplicity–i.e., not containing statements not relevant to the entailment of R (Ockham’s Razor), prior probability or explanatory power. The statement (3.10) may be generalised further in that T may itself be a conjunction of many propositions, while not forgetting that R is also a conjunction of results from a large body of experimental data. Apart from his numerous other contributions, this is one of Peirce’s most notable legacies.

3.6 Modern viewpoints In this section I make some brief remarks regarding the views of philosophers of science during the twentieth century. For me the four names that stand out are Karl Popper, Imre Lakatos, Thomas Kuhn, and Paul Feyerabend. Karl Raimund Popper was born on the 28th July 1902 in Vienna to upper-middle class parents. Popper’s farther was Simon Sigmund Carl Popper, a lawyer from Bohemia and doctor of law at the Vienna University. His mother Jenny Schiff was of Silesian and Hungarian descent. Popper left school at the age of 16 after which he attended lectures in mathematics, physics, philosophy, psychology, and the history of music as a guest student at the University of Vienna. During this period he also completed an apprenticeship as a cabinet maker. In 1922 he attended the University as an ordinary student and graduated as an elementary teacher in 1924. While working at an after-school care club for

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socially disadvantaged children, he continued to study philosophy and psychology and was awarded a doctorate under the supervision of Karl Bühler in 1928. His dissertation was entitled Die Methodenfrage der Denkpsychologie (The question of method in cognitive psychology). Within scientific circles he is best known for his conjecture and refutation views of scientific methodology, and is regarded by many as the originator of empirical falsificationism. In his work Logik der Forschung (The Logic of Scientific Discovery, 1934) he rejected inductionism and logical positivism, and supported a rationalist approach to science, much in the same mould as William Whewell. In Popper’s philosophy of science, theories are constructions of the mind that must be continuously tested within a rigorous experimental regime. A good analogy is Darwinism. Theories are never proved true, they may be assumed true as long as they are useful. But once a theory is falsified by an experimental or observational result then it should be discarded. This became the criterion that determined whether a theory could be regarded a scientific. Scientific theories are always falsifiable. For Popper this solved his problem of demarcation, and became a useful tool in assessing whether or not a theory belonged to science. He was particularly critical of certain wellused theories in psychoanalysis as not being falsifiable. Although such an approach cannot be refuted on logical grounds, it is far too rigid for some, of which I must include myself. For me a classic example concerns the falsified theories of Newtonian Mechanics and Universal Gravitation. These were completely eclipsed by Einstein’s General Relativity, which, for example, fully accounted for the precession of the perihelion of Mercury, an effect that could not be accounted for by Newtonian Gravity. In this case Newton’s theories were not discarded, and probably never will be, because they are still useful in the regime where relativistic effects are small, and much easier to apply. Other examples are presented by Sir Roger Penrose in his The Road to Reality (2004, 1020). The first of these concerns supersymmetry, an idea that makes life much easier for theorists constructing renormalizable quantum field theories. In this theory every elementary particle species has a super-partner that differs in intrinsic spin by 12 = . So every boson has a fermion partner and vice versa. It is very well established amongst the mathematical physics community, and is an essential element in superstring theory. However, at the time of writing it has absolutely no observational support whatsoever. As Penrose explains, due to symmetry breaking, the super-partners possess rest energies well above the reach of modern accelerator technology. And it may be that their rest energies are comparable to the Plank scale, putting supersymmetry

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beyond Popperian falsification at least for the foreseeable future. As Penrose suggests it would be inconceivable for supersymmetry to be considered unscientific, yet by applying Popper’s criterion, it certainly is. The same arguments could be levelled at Penrose’s other examples, one of which concerns the existence (or not) of magnetic monopoles, and another that the universe beyond the cosmic horizon possesses a similar structure to that part we can observe. Both of these hypotheses are considered unrefutable and therefore un-Popperian, but like Penrose, I believe it would be too harsh a judgement to deem them unscientific. For me, while bearing Popper’s ideas in mind, theories and hypotheses may be considered scientific as long as they have a close connection to, or represent an attempt to explain certain observational facts. Also a theory may be consigned to history without being falsified if a better one is found. While Popper’s ideas fully expose the logic of scientific enquiry, it is my contention that we should apply falsificationism with a little less rigidity. Another well-known twentieth century philosopher of science was Thomas Samuel Kuhn. Kuhn was born in Cincinnati, Ohio on the 18th July 1922 to engineer Samuel Kuhn and Minette Stroock Kuhn. He was educated at The Taft School in Watertown, Connecticut from which he graduated in 1940. At this point Kuhn realized his strong inclination towards physics and mathematics. He attended Harvard University where he obtained his BS degree in 1943. His MS and PhD degrees in physics soon followed in 1946 and 1949 respectively under the supervision of John Van Vleck. Kuhn’s best known and most influential work was The Structure of Scientific Revolutions (1962), which was originally published as an article in the International Encyclopedia of Unified Science by the logical positivists of the Vienna Circle. Here Kuhn argues that science does not progress with gradually increasing knowledge, but instead consists of long periods of normal science delimited by shorter periods of revolution in which the theoretical landscape is radically transformed. These short periods of revolution became known as paradigm shifts. Periods of normal science are, for most of the time, very productive. A theory at the centre of a particular paradigm is continuously verified and strengthened during such a period. At some point, however, an anomalous result is encountered. Accordingly, Popper’s criterion applied rigidly would consign the central theory to the dustbin of history. However, according to Kuhn the first anomalous result may be regarded as a mistake by a researcher in the field. This interpretation can be applied to a sufficiently small number of such anomalies. Whereupon there reaches a point of crisis where no theory exists to explain the accruing anomalies. This is where a

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paradigm shift occurs, and is a process in which a scientific community endeavours to produce a new theory that entails all previous results and also explains all of the new anomalies. A well-known example of this is the transition period between the end of the nineteenth century and 1925, which led to the emergence of quantum mechanics. In a Popperian sense this falsified classical mechanics. However, classical mechanics is very useful in regimes where quantum effects are small. This is where Kuhn was accused of relativism in which it seems that the language of the new paradigm is incompatible with that of the old. However, Kuhn denied this and I for one believe he was right. It is perfectly possible to compare quantum theories with their classical counterparts, and it is equally possible to quantify and analyse the differences between corresponding calculations from Newtonian and relativistic theories. Moreover, because classical and Newtonian mechanics are still in use today due to their utility, such comparisons are being made all the time. For Kuhn’s defence of his work against its relativist interpretation, see (Kuhn, 1977). We have seen that Popper’s and Kuhn’s ideas conflict in one particular respect. That is that Popper proposed outright rejection of a theory falsified by even a single experimental result, whereas Kuhn proposed a sort of temporary dogmatism in which a theory was allowed to stand in the face of a small number of anomalies. As the number of conflicting anomalies grows the old paradigm reaches a crisis point, whereupon a new theory is required. This is where a contribution by one Imre Lakatos enters the debate, through which a compromise was sought. Lakatos was born Imre Lipschitz to a Jewish family in Debrecen, Hungary on the 9th November 1922. He graduated from Debrecen University with a first degree in mathematics, physics, and philosophy in 1944. The Nazis invaded Hungary that same year, so to avoid Nazi persecution of Jews he changed his surname to Molnár. Later his mother and grandmother died in Auschwitz-Birkenau, and he changed his surname again to Lakatos. After the war he worked as a senior official in the Hungarian ministry of education. He also continued his own education and was subsequently awarded a PhD from Debrecen University in 1948. In 1956 he fled to Vienna and eventually onto England in response to the Soviet invasion of Hungary. He received a second doctorate from Cambridge in 1961 under the supervision of RB Braithwaite. The book Proofs and Refutations: The Logic of Mathematical Discovery, based on his thesis was published posthumously. Lakatos was appointed to a position at the London School of Economics (LSE) in 1960, a post he held until his death due to heart failure in 1974.

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Apart from his Proofs and Refutations, other work for which he is famous is on scientific enquiry in which he defines Research Programmes. A Lakatosian research programme centres on a set of theories, some of which cannot be modified without abandoning the programme altogether. This no-go area within the theoretical landscape was termed the hard core. Other theoretical assumptions that threaten the hard core were known as auxiliary hypotheses. Adherents of the programme consider these hypotheses expendable and may be modified or abandoned as new empirical data is acquired. A well known example used by Lakatos was Newtonian mechanics where the hard core consisted of the three laws of motion. The framework of a research programme as provided by Lakatos is similar to a paradigm in the Kuhnian sense. For adherents of the programme, research may be conducted on the basis of first principles (the hard core) that are shared by the researchers and accepted as true without any further debate. The idea was to replace Kuhn’s idea of a paradigm by a research programme guided by the objectively valid logic of Popper’s falsificationism. Lakatos was inspired by Pierre Duhem’s idea that it is possible to protect a cherished theory, or part of one, from hostile data by redirecting criticism to other theories or other not so valued parts of a theory. Popper’s falsificationism is very black-and-white but is logically valid. We have seen this already in the falsification of Plato’s emission theory of vision expressed in equations (3.5-6). We see the idea of a Lakatosian research programme encapsulated in equation (3.10), where T represents the hard core and H i are the auxiliary hypotheses. And within one of Kuhn’s paradigms it is always possible for a researcher to make a mistake. To me it would have been inconceivable to ditch general relativity in response to the anomalous appearance of superluminal neutrinos observed during the 2011 OPERA experiment at CERN. Had this not been a mistake, it is more likely that general relativity, and other related theories, would have been extended in some way. Maybe it would have supported a superstring theory or some other variant of quantum gravity. Alas though, it was just another mistake. Regarding the works of Popper, Kuhn, and Lakatos in the round, it seems that they all have something to offer. In addition the apparent minor conflict between Popper and Kuhn suggests the very human nature of scientific research, as does Lakatos’ apparent failure to convincingly reconcile this difference. It would seem that Popper, Kuhn, and Lakatos are like the proverbial three blind men touching different parts of the elephant (science). I hasten to add that I do not mean this in any

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disrespectful way. They just perceive different aspects of science because of their distinct perspectives–nothing could be more human. We do not necessarily need new data to form knew theories either. The process of grand unification in the present has generated a plethora of new ideas, all of which are consistent with existing data, but which have no experimental support of their own. Such theories include: supersymmetry, superstring or M-theory, loop quantum gravity, and twistor theory, to name a few. More provisional hypotheses referred to as crazy ideas are not uncommon amongst particle physicists. Provided we do not lose sight of the logical structure of science within expressions such as (3.10) then as far as I can see, there is no problem. There is just one more significant contributor I would like to discuss, Paul Feyerbend. In the field of the philosophy of science Paul Karl Feyerbend is best known for his rather anarchistic views of scientific enquiry. This comes from his best-known work Against Method (1975). Feyerabend was born in Vienna on the 13th January 1924. He attended primary school and high school in Vienna, and graduated from high school in April 1942. Upon leaving school he was drafted into the German Arbeitsdienst. After basic training he was assigned to a unit based in Quelern en Bas, near Brest, France. After a short leave he volunteered for officer training in the regular army (Wehrmacht–Heer). He had hoped that the war would be over by the time his training was complete, however this was not to be. In December 1943 he was stationed in the northern part of the Eastern Front, was decorated with the Iron Cross and achieved the rank of lieutenant. When the Heer began its retreat in response to the advance of the Red Army, Feyerabend was severely wounded by three rounds while directing traffic. One of these had struck him near his spinal column and caused him severe pain, the result of which he walked with a stick for the rest of his life. He spent the rest of the war recovering from his wounds. After the war Feyerabend was temporarily employed in Apolda where he wrote pieces for the theatre. He was influenced by the Marxist playwright Bertolt Brecht, and was invited by Brecht to be his assistant but turned down the offer. He attended classes at the Weimar Academy and eventually returned to Vienna to study History and Sociology. Later having transferred to physics and eventually to philosophy he submitted his final dissertation on observation sentences. In 1948 he first met Karl Popper at the International Summer Seminar of the Austrian College Society in Alpbach. In 1951 Feyerabend was granted a British Council scholarship to study under Wittgenstein. However, Wittgenstein passed away before he arrived in England. So Feyerabend chose Popper as his supervisor instead, and went to study at LSE in 1952. In 1955 Feyerabend

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received his first academic appointment at the University of Bristol. Later in his career he had professorial appointments at Berkeley, Auckland, Sussex, Yale, London, Berlin, and ETH Zurich. It was during this time that he developed his anarchistic views of science. In his book Against Method, Feyerabend argues that any prescribed method in science would place unnecessary constraints on scientists, and would severely curtail progress. He gave several counterexamples to the necessity of method including: the Copernican revolution and the application of renormalization in quantum field theory. Renormalization, for example, is a particular ad hoc procedure where infinite terms appearing in perturbation series for running coupling constants are replaced by finite ones consistent with observation. Within the philosophical community this was seen as radical because it suggests that philosophy can neither succeed in providing a general description of science, nor can there be any criterion for differentiating scientific statements from non-scientific ones. Here I think I would support the former while remaining rather dubious about the latter. More specifically the former statement may refer to the non-existence of a general method in science, where in the process of induction we are attempting to arrive at a theory, T, to explain a body of empirical results, R, as in the relationship (3.7). In other words we can use any ad hoc method to arrive at T. It is the deductive part of the cycle, where we show that T entails R, that grounds science in objective reality. For the latter, implying that there is no criterion differentiating scientific statements from non-scientific ones, if my interpretation is correct then this is where I would part company with Feyerabend. For me, one of the guiding principles for T being scientific, is that (3.7) holds, and that the results, R, be repeatable under appropriate experimental procedures. One of the criteria for appropriate experimental procedures would be that it is reasonably free from subjective influences and to be repeatable by other researchers. A particular principle for evaluating scientific theories that Feyerabend attacks is the consistency criterion. This essentially says that a new theory should be consistent with an old one. Feyerabend’s position is that this gives unreasonable advantage to the older theory. But what does the consistency criterion actually say? Is it not reasonable to define a consistency as one that says that the new theory predicts the same results as the old theory over an empirical data set for which the old theory is valid? If this is the case then it is reasonable that this is all the consistency one requires. Again everything is grounded in empirical findings, and that is as it should be.

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So it is sensible to suppose that the inductive part of the scientific process should not be unnecessarily constrained by any prescribed method. But I would contend that to go beyond this, as Feyerabend does, is a step too far. Feyerabend even questions the utility of truth, reality, and objectivity. For these reasons Feyerabend is described as an epistemological anarchist, and due to his book Against method has been branded an irrationalist (Preston, 2016). In his assessment of Against method, Preston goes on to say The history of science is so complex that if we insist on a general methodology which will not inhibit progress the only “rule” it will contain will be the useless suggestion: “anything goes” (Preston, 2016, 18).

Provided we apply this only to the inductive step of inferring a general rule from a body of data, then this is fine. Each of the sciences has developed its own methodologies. So there can be no one prescribed method. However, to challenge the necessity of pinning any theory or hypothesis to empirical data is a step too far. In attempting to solve the hard problem of consciousness, it is essential that our theories are grounded in reliable empirical observations.

3.7 Summary: the nature of evidence Throughout this chapter I have provided a brief overview of the nature of science from a historical perspective. Since Socrates and Aristotle the debate appears to have switched between empiricist and rationalist viewpoints. In my view these positions are not mutually exclusive. On the contrary they are both necessary for optimum progress in science. This seems to resonate with William Whewell’s position in seeking a middle ground. In the twentieth century Karl Popper’s views were closely allied to those of Whewell. Popper proposed that an essential criterion for any scientific theory is that it is falsifiable. In symbols this is expressed by writing (3.7) in its contrapositive form

¬R Ÿ ¬T

(3.11)

indicating that any empirical result not consistent with the theory, T, will falsify it. For many, this is considered too rigid. Some may even say naïve, even though it is logically indisputable. For me it is just idealized. It could certainly be applied too rigidly–you would not use special relativity to estimate your journey time when you next get into your car, you would use

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the falsified Galilean kinematics instead. So falsified theories are not just cast aside, some are retained especially if they are easy to use and provide a good approximation to reality in common circumstances. Other instances where falsified theories are retained may be where the first measured result appears that contravenes an established theory. Such a result may at first be considered a mistake. But many more such results may follow leading to a crisis or paradigm shift. This was the position of Thomas Kuhn. For me this is not really a denial of falsificationism, it is just taking a softer approach. Imre Lakatos proposed the idea of a research programme, which is similar to a Kuhnian paradigm while being guided by Popper’s falsificationism. Unfortunately Paul Feyerabend denied not only the existence of a method but also anything that could define science. According to a member of the audience in one of Feyerabend’s seminars in the Autumn term, 1974 at the University of Sussex, he even asks: How does science differ from witchcraft? (Preston, 2016). The nonexistence of a universal method for inferring a theory from data, I can accept. However, I would contend that for a theory to be scientific, the data from which it is inferred must be readily available. By this I mean that experiments generating that data must be repeatable, or observational data in the field can be acquired periodically at certain places and times, for example certain astronomical phenomena during an eclipse. The nature of science is in the nature of the evidence (data) from which it feeds. Repeatability or more generally the easy availability of consistent evidence is what makes science science. The study of UFOs, for example, is not considered science–why? There are numerous theories as to what UFOs might be but the problem is that the evidence is not readily available. Their appearance is rare and irregular. Isolated individuals claiming to have seen or experienced this phenomena up close, may very well be honest and reliable in themselves. But the problem is not with the witnesses but with the evidence itself, it is anecdotal and spontaneous– many instances of personal experience but not universally demonstrated. No one can predict when these events will occur, so no one can take a battery of scientific instruments and observe the phenomena in detail as it happens. It may be that UFOs are natural phenomena. One theory gaining acceptability is the Earthlights hypothesis (Devereux, 1982) in which UFOs are a species of plasmoid generated by a high voltage piezoelectric effect due to high pressures in the period leading up to an earth tremor or quake. Here the best evidence is largely of a statistical nature involving correlations between locations of fault lines, periods when disturbances in

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the Earth’s crust are present, and the appearance of lights in the sky. But until we have sufficient understanding about the conditions of their appearance that we can recreate them in the laboratory, then this topic will continue to remain on the fringes of science. But one might argue that we cannot yet predict earthquakes yet they are apart of science. The study of earthquakes is grounded in geology–the study of mechanics of the Earth which is well understood even if unpredictable and chaotic. The evidence for earthquakes is manifest: they destroy property and kill people in numbers that cannot be ignored. The understanding of earthquakes is such that scaled down versions can be recreated under laboratory conditions, making evidence, mostly in the form of analogue models, easily available for further study. I see science in general as an iterative process with a continuing cycle of induction and deduction. Theory can predict new outcomes, which can be used to guide new experiments. Anyone with familiarity of mathematical iteration will know that stable algorithms converge. Even if one makes a minor mistake, an iterative algorithm robust against such mistakes is guaranteed to converge to the same value. So it is with science, the general process will converge to a truth consistent with empirical data because that is where science is based. No theory can ever be proven but evidence for a given theory can be added to, thus increasing its status. Occasionally we arrive at a crisis point from which a more comprehensive theory emerges. And so the process continues… For the purpose of studying the nonmaterial mind and its relationship to the physical world, it is the ready availability of consistent evidence that I will consider as important for the rest of this work.

CHAPTER 4 SPACE-TIME

When discussing the concept of consciousness it will become clear that a treatment of space-time geometry is essential. In modern physics, spacetime is considered to be an extended object with an independent existence. This is key feature of eternalism and it represents a significant problem for the large number of scientists and philosophers subscribing to physicalism in a mind-body context. For example a simple argument against physicalism might go something like: Choose a moment in your past, that event, where you were in that moment, including your body, still exists. But you are not there anymore.

A supporter of physicalism who also believes in eternalism will simply say that your dynamic experience of reality, in particular of time, is an illusion. This is because eternalism and mind-body physicalism are logically incompatible, and no one can offer any mechanism for the illusion. I will expand upon this argument particularly in chapter 10. Here I will concentrate on space-time and its ontological status, right up to modern classical general relativity. This chapter is not intended to be a comprehensive discourse on general relativity. For that, excellent introductions by Rindler (1969) and Schutz (1985) are good starting points. An equally enlightening introduction to relativity theory may be found in chapters 17-19, 27, and 28 of the volume by Penrose (2004). More serious readers, possibly with ambitions to do research in this area professionally, would be advised to consider Hawking and Ellis (1973), Misner, Thorne, and Wheeler (1973), Hawking and Israel (1979) and, of course, references therein. In the following sections I will begin by considering Galilean relativity, which has, as one of its postulates, absolute time. The almost trivial coordinate transformations between distinct inertial observers are discussed. It is shown that Galilean relativity can be couched in terms of a four-dimensional geometric manifold, even though in the past the

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ontological status of the manifold was considered to fall someway short of that for material bodies. By replacing the postulate of absolute time with the constancy of the speed of light, we arrive at special relativity. This switching of postulates resulted from a contradiction between Newtonian mechanics and Maxwell’s electromagnetism. The constancy of the speed of light was supported by the null result of the Michelson-Morley experiment in 1887. Special relativity is encapsulated into the Minkowski metric (Minkowski, 1908) that nicely dovetails into the general theory. This provides us with the modern view of space-time as an extended four-dimensional block. Vector quantities such as force and momentum, can be represented as four-component objects that obey rules analogous to their threecomponent counterparts in Newtonian mechanics. This provides an extremely persuasive narrative supporting eternalism that effectively spatializes time into just another coordinate. Here it is shown that simultaneity is dependent on an observer’s perspective, thus demonstrating that an absolute present cannot be defined. However, presentists postulate the existence of a hidden inertial frame that extends to curved space-time. As part of this chapter I will explore the introduction of a third postulate, the equivalence between gravity and acceleration, the consequence of which was the general theory of relativity published in 1915. When massive objects are placed into an otherwise flat (Minkowski) space-time, the result is a locally curved manifold that manifests itself as a gravitational field. In extreme cases of stellar collapse this leads to black holes that have surfaces (event horizons) on which the escape velocity is the speed of light. In classical general relativity this leads to a singularity that exists for an observer falling through an event horizon. However, I would contend that the singularity that in-falling test particles encounter in the classical theory is not meaningfully present for observers remaining outside the horizon. Moreover, it can be shown that the event horizon is similarly not present for observers remaining at a distance. In essence, space-time remains singly-connected. Although controversial this interpretation does have growing support (Suggett, 1979; Barcelo et al, 2006; Hawking, 1976, 2005, and 2016; Vachaspati et al, 2007; MersiniHoughton, 2014). The persistence of a singly-connected space-time is important because it has consequences for the information loss paradox and unitary evolution in quantum theory. Simply put, objects encoding information cannot just disappear from the universe. Although still considered to be unresolved I contend that unitary evolution is maintained. Such an interpretation also precludes the physical manifestation of closed timelike trajectories (Deutsch, 1991). This topic will also be discussed

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further in chapter 6, and the corresponding section ends with a short discourse on chronology protection in classical general relativity. The apparent physical nature of the space-time manifold provides further support for an eternalist’s interpretation of the universe. However, a presentist’s interpretation known as the growing block (Sorkin, 2007a) is where the past exists but the future does not. Here the future-most boundary of space-time dynamically moves in the future direction, so the space-time manifold is a constantly growing object. I will argue against this by showing that causal set theory, on which the growing block is based, is still compatible with eternalism. In general it is shown that evidence continues to grow, for a non-dynamic eternalist’s space-time manifold being the stage on which classical physics plays out.

4.1 Galilean relativity Physicists often find it convenient to describe physical events in terms of inertial frames of reference. An important question concerns how different observers, occupying distinct inertial frames, describe the same physical events. Essentially the theory of relativity is designed to answer this question. An event, such as two small particles colliding, a photon being absorbed by another elementary particle or an explosion, may be specified by its time and location of occurrence, ( t , x, y, z ) . These coordinates specify an inertial frame, and distinct inertial frames are differentiated by a defined separation, r ′ = r − r0 , or by one moving at a constant velocity relative to another, r ′ = r − vt , where r = ( x, y, z ) . Both of these situations are satisfied by Newton’s first law in the absence of an external force. There also exist frames in which the first law does not hold, these are not inertial frames but are characterised by non-zero acceleration. An example of this would be the coordinates at rest relative to a rotating turn table. The appearance of non-zero acceleration, a, suggests the existence of a force, F, through Newton’s second law of motion

F = ma

(4.1)

for an object of mass, m. However, this force has nothing to do with anything external, it merely indicates non-inertial effects. Equation (4.1) is fundamental to the laws of mechanics and for a large number of particles, may be written as

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d 2 ri . dt 2

Fi = mi

(4.2)

A force may depend on the distance to other particles, for example

Fi = Fi ( ri − r j , ri − rk ,")

(4.3)

see Figure 4-1.

z

Fi rk − ri r j − ri

ri

rj rk y

x Fig 4-1: Vectors describing positions and separations between interacting particles. The force Fi acting in the ith particle is the sum of interactions for each separation between the ith particle and its neighbours. In the figure the interactions are depicted as repulsive. In general they are a mixture of repulsive and attractive interactions.

It should be emphasised that all known forces in nature are dependent only on the relative position of the body acted on with respect to the location of

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113

the source, and not on the distance to or position of some notional centre of the universe. An observer moving relative to an inertial frame at some velocity, v, will assign coordinates r ′ = r − vt to a particle whose coordinates are r in the original inertial frame. The velocity of that particle in the new inertial frame is obtained by differentiating both sides

dr ′ dr = −v. dt dt

(4.4)

Here we see the principle of addition of velocities, which is a consequence of Galilean relativity. Differentiating both sides again we arrive at

d 2r′ d 2r = 2 dt 2 dt

(4.5)

because v is constant the accelerations are the same. Also since

ri′ − r ′j = ri − r j

(4.6)

then the forces that only depend on particle separations remain unchanged under Galilean transformations. This gives us the first postulate, the principle of relativity, which says that The laws of mechanics are invariant under Galilean transformations

r → r ′ = r − vt . The assertion that the laws of mechanics are identical in both frames does not mean that they will have identical solutions. To determine solutions we need initial conditions (both position and velocity) in both frames. When the solutions are examined in their respective frames it can be shown that they are obeying the same laws. Indeed for distinct inertial frames the initial conditions will be different, so it is no surprise that the solutions will differ. Before moving on it might be informative to consider matrix representations of Galilean transformations. Without any loss of generality we can choose the direction of motion for an inertial frame moving with velocity, v = ( v, 0, 0 ) , in the positive x-direction as indicated. In this case the transformation becomes

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Chapter 4

ª t′ º ª 1 « x ′ » « −v « »=« « y ′» « 0 « » « ¬ z′¼ ¬ 0

0 0 0º ª t º 1 0 0»» «« x »» . 0 1 0» « y » »« » 0 0 1¼ ¬ z ¼

(4.7)

From this it is easily seen that x ′ = x − vt with all other coordinates remaining unchanged. In particular there is no change to the time component, and this remains true for motion in any direction. Indeed this is the second postulate, absolute time, which says that Time is invariant under Galilean transformations

t → t′ = t . Here there is no reason to suspect that the time coordinate, t, is part of a more general geometric structure that includes the space coordinates as well. This is derived from the assumption that time is absolute. In other words external time is not affected by changes in the state of motion, it remains totally independent. This independence suggests that it can be treated as an entirely separate entity from space, with a distinct ontological status. It is possible that until the emergence of special relativity, the postulate of absolute time has kept the presentism/eternalism debate on an even keel. However, relativity theory has tipped the debate firmly in favour of eternalism. This and related issues will be discussed more thoroughly in chapter 10.

4.2 Special relativity Considering the Galilean postulates mentioned in the previous section Newtonian mechanics assumes absolute time. However, this conflicts with JC Maxwell’s electromagnetism. To see this we need to examine Maxwell’s equations, which in their traditional form are written

∇×E =

∂B ∂t

a Faraday's law

∂D b Ampére's circuital law ∂t ∇⋅D = ρ c Gauss' law

∇×H = J + ∇⋅B = 0

d No isolated magnetic charge

(4.8)

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115

The speed of light, c, is implicit in the relationships between the vector field variables: E, D, H, and B in a vacuum, by

D = ε 0E B = μ0 H

c=

1

ε 0 μ0

The vacuum permittivity ( ε 0 ) and permeability ( μ0 ) are universal constants of space itself in any inertial frame. As a consequence so is the speed of light, c. This led to a crisis in physics after the publication of the complete set of Maxwell’s equations (Maxwell, 1873)4. The laws of electrodynamics were not invariant under Galilean transformations. One could ask: the speed of light relative to what? It would not be correct to say relative to individual sources. That would imply we would see light travelling at different speeds from sources moving in various ways relative to us, for example stars. Moreover such an assertion contravenes Maxwell’s equations. The popular fix at the time was to suggest the existence of an aether, a notional substance that filled the universe and made wave propagation in a vacuum possible. There was the assumption that the aether only supported transverse oscillations, presumably so that its density remained constant. Whatever its properties it solved a problem by providing a privileged inertial frame relative to which all electromagnetic waves travelled. This is where it was realised that such a claim could be tested. The velocity of the Earth in its orbit around the Sun is approximately 30 km s-1, so it should be possible to measure the differences in velocity from specific sources at right angles to each other at appropriate times of the year. In 1887 a series of brilliant experiments was started by AA Michelson and EW Morley. Details of their apparatus are omitted here since they can be obtained with little difficulty, through for example an internet search. However, the basic idea is given below in Figure 4-2. The description of the experiment was taken from Gasiorowicz (1979).

4

These were not the four condensed vector equations taught in Universities today and presented here, but rather a larger set of 20 equations. The four we know today as Maxwell’s equations were due to Heavyside and independently by Gibbs and Hertz, and first appeared in 1884.

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Chapter 4

vt2 2

Mirror 2

Mirror 2

Aether wind

l2

O

l2

Mirror 1

l1

O

Fig 4-2: Optical paths in the Michelson-Morley interferometer. Left: location of the mirrors and source, O, relative the aether wind. Right: the actual direction that a light ray on the transverse branch travels relative to the aether wind from O to Mirror 2 and back.

In Figure 4-2 the time of flight from O to Mirror 1 and back is given by

l1 l 2l c 2l § v 2 · + 1 = 2 1 2  1 ¨1 + 2 ¸ c+v c−v c −v c © c ¹ for v 0 , then the wavelength at which this peak is found has the form

λmax =

b T

for a universal constant, b. Based on detailed experiments by Lummer and Pringsheim in 1897, Wien obtained a formula for a blackbody distribution given by

u (ν , T ) = Cν 3 e − βν

T

(6.4)

with two adjustable parameters, C and β . However, a problem remained. While this relation could be made to fit perfectly with Lummer and Pringsheim’s data at high frequency ( ν → ∞ ), it did not work in the low frequency regime. In that domain the data agreed with the Rayleigh-Jeans relation given by equation (6.2). In 1900 Max Planck found the correct formula by interpolation between the high and low frequency domains. Planck’s interpolation was backed by established thermodynamics to arrive at his formula given by

u (ν , T ) =

8π h ν 3 . c3 e hν kT − 1

(6.5)

In the limit where ν → 0 it is easy to see that this reduces to the RayleighJeans formula while at the other extreme, ν → ∞ , it satisfies Wien’s formula shown in (6.4), where C = 8π h c 3 and β = h k . With k known, Planck reduced the number of adjustable parameters from two to one, h. The correct value for h became the Planck’s constant we know today, and is used to determine the scale at which quantum effects become significant. This turns out to be true for the full theory as well as for the case of blackbody radiation. Given this formula’s level of success Planck pursued an intensive search for an explanation of its form. Planck’s reasoning was based on the Boltzmann distribution across all possible energies

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e − E kT . ¦ e− E kT

P(E) =

(6.6)

E

Planck also assumed that the energies formed a discrete spectrum given by E = nhν

n = 0,1, 2," .

(6.7)

Using these assumptions as a starting point it is expected that the average energy is determined by

E = ¦ EP ( E ) .

(6.8)

E

Rewriting the distribution (6.6) in terms of a new variable x = hν kT then

P(E) =

e− nx ∞

¦e

=

− nx

n=0

e − nx

(1 − e )

− x −1

= (1 − e − x ) e − nx .

This enables us to rewrite the average energy (6.8) as follows ∞

E = hν (1 − e − x ) ¦ ne − nx n=0

§ d · ∞ = hν (1 − e − x ) ¨ − ¸ ¦ e − nx © dx ¹ n = 0 = −hν (1 − e =

hν e hν

kT

−1

−x

)

d (1 − e − x )

−1

dx

.

This is precisely the form of the Planck formula. The interpretation of the postulate (6.7), given by Einstein, was that radiation consists of discrete quanta, with individual energies given by

E = hν .

(6.9)

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189

This was a radical departure from the picture of continuous electromagnetic waves. At any particular frequency there existed, not a single continuous wave, but a multitude of discrete wave packets, each with its own independent phase. These later became known as photons. So even at a specific frequency coherence was lost. This marked the beginning of early quantum theory.

6.1.2 The photoelectric effect The implied particle nature of radiation gains strong empirical support from the photoelectric effect first explained by Einstein in 1905. Hertz first discovered the effect in 1887. Subsequent experiments by many successors but mainly by Millikan established the following • • • •

When smooth metal plates are irradiated they emit electrons but not positive ions. Emitted electrons are only detected when the radiation exceeds a certain threshold frequency. The emitted electron current is proportional to the intensity of light but not on its frequency, provided that frequency is above the aforementioned threshold. The energy of the photoelectrons is independent of the radiation intensity, but dependent on the radiation frequency.

Classical electromagnetic theory cannot explain these observations. It was known that metals contained electrons. More specifically the dependence on frequency of photoelectron energy is not expected from classical theory. It was expected from classical theory that for low intensity radiation, there would be a time delay between the incidence of the radiation and the emergence of photoelectrons. Even though looked for no such time delay was ever detected. Einstein explained the effect as individual photons liberating single electrons, where the energy of each photon is hν according to equation (6.9). The energy of the liberated electrons is hν − W where W is the work function dependent upon the metal sample. So the work function is the minimum energy required to remove an electron from the metal body. The photoelectric effect can therefore be summed up by expressing the photoelectron energy, E, in terms of radiation frequency, ν , and work function, W, for the mass, m, and velocity of an electron. This is given by

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Chapter 6

E=

mv 2 = hν − W . 2

(6.10)

6.1.3 The Compton effect This effect is seen in another experiment that provides compelling evidence for the particle nature of radiation. The effect, discovered by AH Compton in 1922, is seen in X-rays that are scattered by thin metallic foils. The observed scattering effects are not consistent with classical theory. Given the angle, θ , from the direction of incidence, classical theory predicts that the scattered radiation should have an intensity proportional to 1 + cos 2 θ , independent of wavelength of incident radiation. Compton found that the scattered radiation consists of two components, one at the wavelength of the incident radiation and another whose wavelength is dependent on the scattering angle. Compton treated the radiation as photons elastically interacting with electrons in the metal, in the same manner as colliding billiard balls. In this way it is demanded that total energy and momentum is conserved for each interaction. The derivation is straightforward. Consider an incident photon with energy, E, and momentum, p, colliding with an electron at rest. Conservation of both energy and momentum demands

p = p′ + q

(6.11)

and

(

E + mc 2 = E ′ + q c 2 + m 2 c 4 2

)

12

(6.12)

where the dashed symbols refer to scattered photons and q is the post interaction momentum of the electron. For p = E c and p ′ = E ′ c , because photons have zero rest mass, then for a given scattering angle, θ , we have

q c 2 = p c 2 + p′ c 2 − 2 p p′ c 2 cos θ 2

2

2

= E 2 + E ′2 − 2 EE ′ cos θ . Also equation (6.12) implies that

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191

q c 2 = ( E − E ′ + mc 2 ) − m 2 c 4 2

2

= E 2 + E ′2 − 2 EE ′ + 2mc 2 ( E − E ′ )

Ÿ EE ′ (1 − cos θ ) = mc 2 ( E − E ′ ) . Using E = hc λ and E ′ = hc λ ′ after a little algebra we acquire the dependence of wavelength on the scattering angle in the form

λ′ − λ =

h (1 − cos θ ) mc

(6.13)

in excellent agreement with experiment. The Compton effect shows that photons are individual entities in their own right, and cannot be grouped together to form an electromagnetic wave. Compton also showed that photons could not be split into entities with energy less than hν . This raises conceptual difficulties around interference phenomena in a diffraction grating say, in which each photon would have to know about all of the others. This is a problem to be addressed when we discuss the full theory. Addressing this interference problem demands a radical change in the way we think about the physical world. Embracing the full quantum theory requires that we let go any notion of reality consisting of particles in a three-dimensional space dynamically evolving. As we will see this has been the source of much controversy throughout much of the twentieth century.

6.1.4 The DeBroglie hypothesis In 1923 Louis DeBroglie completed a body of work in which he was guided by analogies between geometric optics and dynamics. This led to his proposal that if light could have both wave and particle properties, then so could matter, which from our perspective has known particle properties. From the above discussion of the Compton effect we see that photons have momentum whose magnitude is

p≡ p =E c. Using the Planck-Einstein relationship in equation (6.9) we find

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p=

hν . c

But from the properties of wave mechanics, wavelength calculated from frequency is

λ=

c

ν

.

Applying this to the equation for momentum and rearranging gives

λ=

h . p

(6.14)

This was accepted for photons whose energy is purely kinetic. However, matter particles, electrons, protons etc., have non-zero rest mass. DeBroglie therefore proposed that the relation in equation (6.14) should also be applied to massive particles. This allows all elementary particles to be endowed with a wavelength when their momentum is known. As an example this provides an explanation for the discrete energies in electron orbitals, requiring integral numbers of wavelengths in their circumferences, that is

2π r = nλ ,

n = 1, 2,3," .

This in turn allows us to write

pr =

h

λ

r=

nh = n= , 2π

showing that angular momentum is measured in discrete multiples of the Dirac-Planck constant, = .

6.2 The full quantum theory In the last section we saw that DeBroglie had proposed matter and radiation with both particle and wavelike properties. We can extend this idea by considering the following general wave ansatz for a particle moving through three-dimensional space

Quantum mechanics

ψ = ψ 0 exp ª¬i ( k ⋅ x − ω t ) º¼ .

193

(6.15)

This is a specific case of the ansatz considered in equation (5.31). Looking at the spatial and temporal derivatives we have

∂ψ = −iωψ ∂t

(6.16)

∇ψ = ikψ ,

(6.17)

and

where k is the vector wavenumber and ω is the angular frequency. Now rewriting the Planck-Einstein and DeBroglie relations in those terms we have

E = =ω

(6.18)

and

p=

= 

which is consistent with the vector momentum

p = =k .

(6.19)

Substituting equations (6.18-19) appropriately into (6.16-17) gives the following operator relations

i=

∂ψ = Eψ ∂t

−i=∇ψ = pψ .

(6.20) (6.21)

Removing the wave function operand from equations (6.20-21) allows us to define the energy and momentum operators used throughout quantum theory

194

Chapter 6

∂ Eˆ ≡ i= ∂t pˆ ≡ −i =∇ .

(6.22) (6.23)

These operators together with their canonically conjugate variables, time, t, and spatial position, x, respectively, define the necessary procedure required to quantize classical mechanics described in chapter 5. From the previous chapter we see that we can express the energy of a closed system from the Hamiltonian

H=

p2 +V (x) . 2m

Quantizing this using the relations (6.22-23), the Hamiltonian operator becomes

=2 2 Hˆ = − ∇ + V (x) . 2m

(6.24)

Equating this with the energy operator (6.22) and inserting the wave function operand allows us to write Schrodinger’s wave equation

i=

=2 2 ∂ψ =− ∇ ψ + V ( x) . 2m ∂t

(6.25)

Aside from the fact that this wave equation is derived for a single particle in three dimensions and equation (5.32) represents an arbitrarily large multi-particle system, these equations are identical. The next stage is to show that the operators (6.22-23) satisfy the commutation relations indicated by equation (6.1).

6.2.1 Commutation relations Because it is easier to appreciate, we begin by considering the time related operator in (6.22). What we will derive is the relation

ªt , Eˆ º = −i = ¬ ¼

(6.26)

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195

where the minus sign has been referred through from the original ansatz in (6.15). Expanding the left hand side and proceeding with the analysis as appropriate, gives

i =t

∂ (ψ t ) ∂ψ . − i= ∂t ∂t

By applying the product rule to the second term we arrive at

i =t

∂ψ ∂ψ ∂t − i= t − i =ψ = −i =ψ . ∂t ∂t ∂t

So by noticing that the first two terms cancel and by removing the wave function operand we obtain the right hand side and arrive at the commutation relation shown in equation (6.26). Accordingly we should be able to similarly justify the commutation relation for position and momentum

[ x, pˆ ] = 3i=

(6.27)

where the factor of 3 comes from the fact that we are working in ordinary three-dimensional space, as follows

[ x, pˆ ] = x ⋅ pˆ − pˆ ⋅ x

= ( xpˆ x + ypˆ y + zpˆ z ) − ( pˆ x x + pˆ y y + pˆ z z ) = xpˆ x − pˆ x x + ypˆ y − pˆ y y + zpˆ z − pˆ z z = 3i=

Again by expanding the left hand side we have

−i=x ⋅∇ψ + i=∇ ⋅ (ψ x ) . Similarly we see that applying the product rule for a scalar-vector product on the second term we get

−i =x ⋅∇ψ + i =x ⋅ ∇ψ + i =ψ ∇ ⋅ x = 3i =ψ

196

Chapter 6

Again notice that the first two terms cancel and we arrive at the commutation relation expected in (6.27) when the operand is removed. As already shown, the factor of 3 implies that the commutation relation shown in (6.1) applies to each Cartesian axis. With the exception of the identity matrix on the right hand side of (6.1) therefore, these relations are exactly to be expected. Related to the above discussion we now consider a wellknown consequence–uncertainty relations.

6.2.2 Uncertainty relations Given a pair of canonically conjugate variables there will always exist an uncertainty in the measurement outcome of each. Heisenberg’s uncertainty principle states that the product of those uncertainties cannot be less than a certain non-zero value. Another way of saying this is that the product of the uncertainties cannot be arbitrarily small. Firstly it should be stated that this has nothing to do with any imperfections in the instruments used to make the measurements. The uncertainty principle is a deep property of nature completely independent of any flaws in the instrumentation. Another common misconception is that accurate measurement of one variable is more likely to disturb, and therefore increase the uncertainty of its canonically conjugate partner. While such disturbances are more likely at atomic resolutions and smaller, this is not the source of the uncertainty principle. Such a statement is likely to be made by someone who has difficulty in letting go of the classical particle model of reality. When analysing a system using the Schrodinger wave formulation, the uncertainty principle is an inherent property of the wave nature of the particles constituting that system. The uncertainty principle has its roots in the Cauchy-Schwarz inequality, which is a theorem in vector algebra, and in ordinary vector notation is given by

v2 w 2 ≥ v ⋅ w

2

(6.28)

for any two complex valued vectors, v and w, in a Hilbert space. With a little thought it is soon realised that we only have equality when these vectors are either parallel or anti-parallel. The Hilbert space referred to here is a space of quantum states where a single point represents a wave function across the C-space for the system being analysed. It is important not to confuse the Hilbert space with the C-space. A wave function spanning large tracts of the C-space represents only a single point or vector in the Hilbert space. This is why the wave function is sometimes

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referred to as the state vector. For this reason, when circumstances dictate, vector notation may be used for the analysis of wavelike constituents of a quantum system. Examining uncertainty relations is one of those circumstances. At this point it is useful to make some remarks about notation. An entirely equivalent notation, often used by physicists, is Dirac’s bra-ket notation. By expanding the dot-product notation in the Cauchy-Schwarz inequality, equation (6.28) becomes

( v ⋅ v )( w ⋅ w ) ≥

2

v⋅w .

In the bra-ket notation this reads

v v w w ≥ v w

2

where, in a matrix context, row vectors (one-forms) are represented by bras written as v and column vectors, by kets written as v . This notation is very flexible because the objects in the bra-kets may be vectors, square matrices or continuous variables. For example given a Hermitian operator A and the wave function, ψ , the expectation value of A is may be written

A ≡ ψ A ψ ≡ ³ ψ † ( S ) A ( S )ψ ( S ) dS . R

The term on the left hand side may be thought of as an average of some function over a region of the space in which it exists. The middle term may be considered to be a matrix product where a square matrix, A, is post-multiplied by a column vector, ψ , and pre-multiplied by its Hermitian conjugate, ψ . In the right hand term these objects are treated as smooth functions over a patch, R, of a multi-dimensional C-space, S. Also given a Hermitian operator, A, we may, using statistical language, referring to its mean as A . In that case its variance is given by

( ΔA)

2

2

= A2 − A .

(6.29)

198

Chapter 6

By defining its uncertainty to be ΔA ≡ A − A

and noting that the

operator, ⋅ , is linear, it is straightforward to prove equation (6.29). Before proceeding further it may be useful to state and prove the Cauchy-Schwarz inequality formally. Most textbooks on linear algebra will present some proof of it. The one I prefer is inspired by that given by Anton (1991, p515), and is extended to the complex case. Here it is important to note that v is the Hermitian conjugate of v not just the transpose. Theorem 6.1: The Cauchy-Schwarz inequality. If v and w are vectors in a complex inner product space, then

v w

2

≤ vv ww .

(6.30)

Proof: If v = 0 then trivially v v = v w = 0 , so that the two sides of (6.30) are equal. Now if we assume that v ≠ 0

and let k be any complex

number, then by the positivity condition for any vector s , then s s ≥ 0 therefore

v − kw v − kw = v v − kw v − v kw + kw kw = v v − k * w v − k v w + kk * w w ≥ 0. Now if we let k = w v

vv −

w w then we have

v w wv w w

−

wv v w ww

+

v w wv ww

≥0.

Multiplying throughout by w w and rearranging we have

vv ww ≥ v w wv + wv v w − v w wv = v w wv

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199

which reduces to

vv ww ≥ v w

2

concluding the proof of the Cauchy-Schwarz inequality.

, Now given two Hermitian operators, A and B, from the definitions of commutators and anti-commutators ( { A, B} ≡ AB + BA ) we can say

2 AB = [ A, B ] + { A, B} .

(6.31)

Taking the expectation values both sides, and noting the linearity of the expectation value functional we have

2 AB =

[ A, B ]

{ A, B}

+

But because A and B are Hermitian then

{ A, B}

.

(6.32)

[ A, B ]

is imaginary and

is real. Therefore by taking the square modulus both sides we get

4 AB

2

=

[ A, B ]

2

+

{ A, B}

2

.

(6.33)

From this we are able to prove the uncertainty principle with the aid of the Cauchy-Schwarz inequality for these operators

AB

2

≤ A2

B2 .

(6.34)

We are now in a position to state and prove the Heisenberg’s uncertainty principle. Theorem 6.2: Heisenberg’s uncertainty principle. For any two canonically conjugate Hermitian operators, x and p, satisfying the commutation relation shown in equation (6.1), the uncertainties of the two operators satisfy

200

Chapter 6

ΔxΔp ≥ 12 = .

(6.35)

Proof: Combining equations (6.33-34) then

4 A2

[ A, B ]

B2 ≥

2

+

{ A, B}

2

.

Since both terms on the right hand side are non-negative we can drop the second term, therefore

4 A2

[ A, B ]

B2 ≥

2

= AB − BA

2

.

(6.36)

Now if we let A = Δx = x − x and B = Δp = p − p , then the expression in the brackets on the right hand side is

AB − BA = ( x − x

)( p −

p )−( p − p

= xp − x p − x p + x

)( x −

)

x

p − px + p x + x p − p x

= xp − px = [ x, p ] . Now considering the left hand side we have

A2 = ( x − x

)

2

2

= x2 − 2 x x + x .

Taking the expectation value both sides gives 2

A2 = x 2 − 2 x + x

2

= x2 − x

2

= ( Δx )

and similarly we have

B 2 = ( Δp ) . 2

Substituting these results back into equation (6.36) gives

2

Quantum mechanics

4 ( Δx ) ( Δp ) ≥ 2

2

[ x, p ]

201 2

.

Because both sides of the inequality are positive we can square root both sides to arrive at

2ΔxΔp ≥

[ x, p ]

==.

This concludes the proof of the uncertainty principle shown in equation (6.35). , Throughout the above arguments leading to the proof of the uncertainty principle, there is no reference to any physical situation. This is the reason that many physicists, particularly Nielsen and Chuang (2000), state that the source of the uncertainty principle is in the mathematics of quantum mechanics and not in the physics. The fact that measuring one variable of a canonically conjugate pair will affect the uncertainty of the other is not relevant to the uncertainty principle. Because the wave function is spread over a whole set of measurement outcomes and each measurement can only have one outcome, then we need to ask how particular measurement outcomes are realised. Knowing that this is a stochastic process we now consider how measurement probabilities are related to the wave function.

6.2.3 The Born rule This rule directly relates the wave function to the probability density function for particles across the C-space of a particular system. This interpretation, known as the Born rule, is now regarded by the general physics community as the correct one. Because of the close relationship between the wave function and probability density functions, it is possible to understand phenomena such as the superposition principle and interference in relation to probabilities of particle locations. These topics will be discussed below in relation to the double-slit experiment. Here we just provide details of the relationship and look at how it relates in a relativistic context. In general the wave function is a complex function over the Cspace. However, for most laboratory situations it is useful to think in terms of single particles in one, two or three dimensions along with a separate

202

Chapter 6

time variable, t. In one spatial dimension the most general wave function has the form

ψ ( x, t ) = R ( x, t ) eiS ( x ,t ) =

(6.37)

where R and S are real valued functions of x and t. In this setting the Born rule states that the corresponding probability density function is given by

ψ *ψ ≡ ψ ( x, t ) = ( Re−iS = )( ReiS = ) = ª¬ R ( x, t ) º¼ . 2

2

(6.38)

Notice that now we are treating the wave function as a continuous function of its independent variables rather than as a state vector, which would use bra-ket notation. Because the modulus function, R, is real and nonnegative then its square in equation (6.38) must be non-negative. This is the first requirement of a probability density function. The second is that, in the one dimensional single particle case, it must satisfy

³

∞

ψ *ψ dx = 1 .

−∞

(6.39)

That is to say, it is certain that the particle under consideration is somewhere in the system’s C-space. Where we are considering N particles, then we have

³

∞

ψ *ψ dx = N

−∞

where a normalised wave function becomes

1

ψ′=

N

ψ .

Because we may not always be dealing with normalised wave functions, equation (6.39) is replaced by the more general and weaker condition

³

∞

ψ *ψ dx < ∞ .

−∞

(6.40)

That is the square integral of the wave function must converge, and is known as the square integrability condition. The number, N, for the

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203

system must be a constant otherwise ψ would not be a solution of the Schrodinger wave equation. As an example we will show that N must be truly constant in time, i.e., conserved, for free particles in a one-dimensional Schrodinger equation. This has the form

i=

∂ψ = 2 ∂ 2ψ ∂ψ i = ∂ 2ψ =− Ÿ = . 2m ∂x 2 2m ∂x 2 ∂t ∂t

So the complex conjugate of this reads

∂ψ * = ∂ 2ψ * = . ∂t 2im ∂x 2 Now defining a probability density by P ( x, t ) = ψ *ψ , then * ∂P ∂ (ψ ψ ) = ∂t ∂t ∂ψ * ∂ψ ψ +ψ * = ∂t ∂t = ∂ 2ψ * = * ∂ 2ψ ψ− ψ = 2 2im ∂x 2im ∂x 2 ∂ ª = § * ∂ψ ∂ψ * · º ψ ψ =− « − ¨ ¸» . ∂x ¬ 2im © ∂x ∂x ¹ ¼

(6.41)

Now we may rewrite (6.41) in the form of a conservation law

∂P ( x, t ) ∂t

+

∂j ( x, t ) ∂x

=0

(6.42)

where

j ( x, t ) =

= § * ∂ψ ∂ψ * · −ψ ¨ψ ¸ ∂x ∂x ¹ 2im ©

(6.43)

204

Chapter 6

is a probability flux density. We may now consider the change of probability in finding a particle in the interval ( a, b ) , which is given by b ∂ d b Pdx = − ³ j ( x, t ) dx = j ( a, t ) − j ( b, t ) . a ∂x dt ³a

This represents an inflow at a and an outflow at b. If we let − a, b → ∞ , where ∞

N = ³ Pdx −∞

then dN dt must vanish for any realistic physical system, due to the square integrability condition. This treatment may also be found in Gasiorowicz (1979, 172). Let us now turn to a relativistic context. The form of Schrodinger equation given in (6.25) is not Lorentz invariant. If we apply the quantization rules to the relativistic energy-momentum relation, E 2 = p 2 c 2 + m 2 c 4 , then we obtain the Klein-Gordon equation

−=2

∂ 2ψ = −= 2 ∇ 2ψ + m 2 c 4ψ . ∂t 2

(6.44)

Considering the form of the energy-momentum relation we can define two energy states, E = ± p 2 c 2 + m 2 c 4 . Using the same procedure as above to determine the probability density and vector probability flux, it is found that these two quantities are proportional to E and p respectively. This leads to difficulties in interpreting the negative energy states since they lead to negative probability densities. A solution for fermions was found by Dirac in 1928 where the Klein-Gordon equation is factorised to produce expressions linear in ∂ ∂t and ∇ . This led to the now famous Dirac equation where, in the general Schrodinger equation

i= we substitute the Hamiltonian

∂ψ = Hψ ∂t

(6.45)

Quantum mechanics

H = Į ⋅p + β m .

205

(6.46)

In this Hamiltonian the components of the vector, Į , and the scalar, β , are 4 × 4 matrices accounting for the two spin states of an electron repeated twice, with positive and negative energy states. The negative energy states turn out to represent the antiparticle of the particle seen in the positive energy state. For example positrons are the antiparticles of electrons with all the charges, but not mass, negated. Dirac interpreted this as all of the negative energy states being occupied by the fermions in question, according to the Pauli exclusion principle. When an electron, for example, is excited from a negative energy state it leaves a hole, which is interpreted as a positron. To extract an electron from such a state requires energy of at least 2mc 2 , resulting in an electron and positron each with a rest mass of m. Note the fact that this reaction requires at least twice the rest energy of an electron to produce an electron-positron pair. The positron does not have negative energy it just represents a negative energy state as far as electrons are concerned. However, when considering bosons, there is still an interpretational problem because the exclusion principle does not apply. This was solved firstly in 1941 by Stückelberg and then again by Feynman in 1948. What has become known as the Feynman-Stückelberg interpretation models antiparticles as negative energy particle solutions propagating backwards in time. To me this looks like an attempt to interpret antiparticles as ordinary particles in a negative energy state. But we could equivalently say that antiparticles are distinct types of particle with all of their charges negated. In this view antiparticles are considered to be positive energy solutions in their own right, propagating forward in time. By examining the temporal exponential factor of the wave function we see the equivalence of these viewpoints

exp [ − iEt = ] = exp ª¬ − i ( − E )( −t ) = º¼ . It is also useful to consider these interpretations diagrammatically. In Figure 6-1 we see two scenarios for double electron scattering. The second, involving pair creation and annihilation, illustrates the two viewpoints just discussed.

206

Chapter 6

e-

Pair annihilation

e-

ee-

e+ ee-

ea

Pair creation

b

ec

Fig 6-1: Double electron scattering: a A single electron is deflected twice. b The positron from a pair creation event annihilates with another incoming electron. c The same events in b interpreted as a single electron being deflected backwards in time.

At this point it may be useful to remark that particles do not really propagate in either direction in time. They are just there as world lines fixed in space-time, or in C-space depending on your viewpoint. In C-space they would not be linear structures either, but branched structures where distinct branches would have their own probability of occurrence and where for all the possibilities, the probabilities sum to unity. However, this is a discussion for section 6.4. When we apply the same procedure that resulted in equation (6.43) in the relativistic case, and then consider the temporal component, we arrive at an expression for charge density not probability density. Therefore the potential for negative values is no longer objectionable, and for a four component spinor, ψ , the probability density can still be defined by ψ ψ

as required. For details see Halzen

and Martin (1984, 71,74,103).

6.2.4 The double slit diffraction experiment The DeBroglie hypothesis, which represents the root of wave-particle duality, attracted much attention around the time that quantum mechanics emerged as a developed theory. Experiments in 1927, by CJ Davisson and LH Germer (Davisson and Germer, 1928) in the United States and GP Thompson in Great Britain, established preferential directions for electrons scattered by crystals. Using wave mechanics it is possible to determine the Bragg conditions that involve the phase difference between waves reflected from adjacent crystal planes. Therefore wave mechanics

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207

offered an explanation for the preferential directions observed in the scattered electrons. The double slit experiment is similar to diffraction of electrons in crystals, because they both produce scattered waves emerging from distinct points in space. In this way, wherever those waves are detected there will be a different path length to the points of reflection resulting in a phase difference. Where we have a phase difference of 2π n for n ∈ ] , there will be constructive interference. Therefore in turn we see destructive interference when the phase difference is ( 2n + 1) π . The double slit experiment for electrons, in the way it is illustrated in Figure 62, shows both electron and wavelike features. The way those features relate to each other is consistent with the DeBroglie hypothesis and quantum theory in general.

Electron wave interference pattern

Electron source

Electron wave crests (wave function)

Electron stream (particles) Metallic screen with slits Diffracted electron waves

Fig 6-2: The double slit diffraction experiment. Everything in red represents all features related to particles (electrons), whereas in blue we see features pertaining to the wave function for electrons, which is diffracted through a metallic screen with two slits. The electron wave interference pattern is the probability distribution across the observation screen relating to the likelihood of the end of flight position for a particular electron.

Because of the linearity of the Schrodinger equation we can simply add waves from sources separated in space and local phase to obtain an interference pattern on any plane upon which they may impinge. If we consider the wave function to be of the general form given in

208

Chapter 6

equation (6.37), then diffracted waves emerging from the two slits could be labelled, ψ 1 and ψ 2 . So the wave function seen at a point on an observation screen has the form

ψ = ψ 1 +ψ 2 .

(6.47)

With ψ k = Rk exp ( iSk ) , k = 1, 2 , the square modulus of equation (6.47) becomes

ψ = R1 exp ( iS1 ) + R2 exp ( iS2 ) 2

2

= ª¬ R1 exp ( −iS1 ) + R2 exp ( −iS2 ) º¼ ª¬ R1 exp ( iS1 ) + R2 exp ( iS 2 ) º¼ = R12 + R22 + R1 R2 ª¬exp ( iS1 − iS2 ) + exp ( −iS1 + iS 2 ) º¼ = R12 + R22 + 2 R1 R2 cos ( S1 − S2 ) . The last line represents the probability distribution of electrons striking the observation screen. Each of the first two terms represents a contribution when only the corresponding slit is open. Only when both slits are open do we see the appearance of the third term due to interference. This result is probably the simplest example the superposition principle and, as such, is the reason why it is so well known. The understanding of this principle is straightforward provided we think only in terms of waves. But attempting to understand it in terms of particles, which is a classical concept, is fraught with difficulty and provokes questions like: how can an electron passing through one slit know whether the other slit is open or closed? The result of this experiment has contributed to decades of misunderstanding and controversy regarding its interpretation. Thinking about the universe in classical terms, that is imagining that the world is made up of particles that, at root, behave dynamically according to the rules of classical mechanics, is where the confusion originates. The experiment has generated a contradiction, which says that the world is really not like that. What we observe as the classical world, where quantum effects are too small to be appreciated, is emergent rather than being fundamental. This problem will be revisited after we have considered the axioms of quantum mechanics.

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209

6.3 Axioms of quantum theory As discussed in chapter 2, pure mathematics is rooted in sets of axioms such as those of classical logic, quantifier axioms, axioms of the equals sign and the axioms of set theory. Also, in chapter 4, we saw that general relativity is similarly based on what physicists call its postulates. These are: the principle of relativity, the constancy of the speed of light, and the principle of equivalence. Quantum theory is no different, although many workers in the field do not generally think of quantum mechanics as having an axiomatic structure per se, they seem to draw instinctively on a set of recognised principles. Another point to be aware of is that the axioms for any body of knowledge are not unique. For example set theory can be understood in terms of VNB or ZFC. Similarly the axioms for quantum theory that we discuss here may not be universally recognised, but are sufficient to deduce the results that are so far known. Much of what is presented here is taken from Nielsen and Chuang (2000). Their discussion of the postulates of quantum mechanics begins on page 80 of their text. Further along, they recast these postulates in terms of the density operator (page 102). Here we briefly do the same, and then examine some of the consequences concerning superposition and entanglement. This treatment reinforces the fact that quantum systems exist in a configuration space. There are four axioms presented by Nielsen and Chuang, which are: Axiom 1. The state space: Associated with any isolated physical system is a complex vector space with an inner product (a Hilbert space) known as the state space of the system. The system is completely described by a state vector, which is a unit vector in the system’s state space. Axiom 2. Evolution: The evolution of a closed system is described by a unitary transformation. That is the state, ψ , of a system at time, t1 , is related to its state, ψ ′ at a later time, t2 , by a unitary operator, U, which depends only on t1 and t2 . This relation is

ψ′ =U ψ .

(6.48)

Axiom 3. Quantum measurement: Quantum measurements are described by a collection, {M m } , of measurement operators. These are operators

210

Chapter 6

acting on the state space of the system being measured. The index, m, refers to the measurement outcomes that may occur in the experiment. If the state of the system is ψ immediately before the measurement then the probability that result, m, occurs is

p ( m ) = ψ M m† M m ψ ,

(6.49)

and that state, immediately after the measurement is

Mm ψ

ψ M m† M m ψ

.

(6.50)

The measurement operators satisfy the completeness relation

¦M

† m

Mm = I

(6.51)

m

reflecting the fact that probabilities sum to unity

¦ p ( m) = 1 .

(6.52)

m

Axiom 4. Composite systems: The state space of a composite physical system is the tensor product of the state spaces of the constituent physical systems. Moreover, if we have systems numbered 1 through to n, and the system number i is prepared in the state ψ i , then the joint state of the total system is ψ 1 ⊗ ψ 2 ⊗ " ⊗ ψ n . (Nielsen and Chuang, 2000, 80-94) At this point I reiterate a word of caution regarding the state space in axiom 1. Without careful consideration it is certainly possible to confuse the meaning of the state space with the C-space mentioned here and in previous chapters. They are entirely different as was explained when discussing the Cauchy-Schwarz inequality. A state vector in the state space, as described in axiom 1, represents a single quantum state, whereas the same quantum state may also be represented by wave function distributed over C-space. This is a superposition of classical states.

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211

Individual points in the C-space represent classical states, sometimes called eigenstates, which are well approximated for macroscopic systems. However, the uncertainty principle forbids arbitrarily accurate specification of such states. Zooming in on what we consider to be a classical state would reveal a wave function spike with non-zero width across a small neighbourhood of the C-space. Each point in this neighbourhood represents a basis vector in the state space. So for any quantum state, it can be seen how a wave function is represented as a single unit vector in the state space. This is the meaning of the state (Hilbert) space, which is the subject of axiom 1. Axiom 2 states that the evolution of the wave function over time is unitary, where the unitary operator, U ( t0 , t ) , is given by

U ( t0 , t ) = exp ª¬ − i ( t − t0 ) H = º¼ .

(6.53)

So the wave function, ψ , at some later time, t, is related to an initial wave function, ψ 0 , at the earlier time, t0 , by

ψ = exp ª¬ −i ( t − t0 ) H = º¼ ψ 0 .

(6.54)

As we saw in chapter 5 the Hamiltonian is constant in time, which is interpreted as conservation of energy in a closed system, so we can differentiate accordingly. Differentiating both sides with respect to t gives

∂ψ ∂t

=

ª −i ( t − t0 ) H º −iH −iH exp « ψ » ψ0 = = = = ¬ ¼ Ÿ i=

∂ψ ∂t

=Hψ .

This is the Schrodinger equation (6.45) in bra-ket form. From this we can see how the Schrodinger equation specifies unitary evolution for the Hamiltonian, H. Interpretations of what happens during a measurement process will be discussed in detail in the next section. What axiom 3 does is to describe what we see when we observe a quantum system after measurement. It does not say what actually happens, as we will see that is

212

Chapter 6

open to interpretation. What we can say is that whenever we observe a quantum system we will always see a classical state. We can never see a full quantum state prior to measurement because the act of seeing itself constitutes a measurement. One way of testing quantum mechanics in situations like this is to repeat the same measurement many times, while applying the identical procedure of setting up the quantum state each time. In this way we can observe the eigenstate of the measurement operator each time and build up a probability distribution using equations (6.4950). We then compare the results with what we expect from the Born rule in a particular experiment. As far as I am aware no experiment has ever failed this test. As an example let us take a look at measurements specifically concerning single qubits. We can define the eigenstates in column vector form

ª1 º ª0 º 0 = « » and 1 = « » . ¬0¼ ¬1 ¼

(6.55)

For these vectors the ‘0’ position is at the top with the ‘1’ position at the bottom. In general column vectors represent configuration spaces for their respective systems. In this way classical states are characterized by column vectors having one ‘1’ with the rest of the positions occupied by ‘0’s. In general quantum systems, the vector components are complex with the restriction that the vector modulus is unity. A general quantum state for a single qubit is represented by

ªa º

ψ = « », ¬b ¼

a, b ∈ ^

(6.56)

where the probabilities for the two eigenstates are p ( 0 ) = a* a ≡ a

2

and

p (1) = b b ≡ b . The completeness relation is therefore a a + b b = 1 . *

2

*

The post measurement states corresponding to equation (6.50) are

a 0 a

and

b1 b

*

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where the normalised complex coefficients of 0 and 1 are not physically significant. For me axiom 4 is the most exciting aspect of quantum theory, its implications are profound. It is best understood in terms of many generic qubits, which are subsystems, each with two possible eigenstates. As mentioned earlier nature’s examples are elementary fermions and massless nonzero spin bosons. However, I believe things are clearer when we consider idealized qubits based on the two eigenstates, 0 and 1 , while ignoring other complications such as particle location for example. Here we may imagine a massive charged fermion held at the bottom of a potential well and exposed to a uniform magnetic field, the direction of which determines its spin state. The details are unimportant, it is sufficient to know that the system can be seen in one of two spin states. Hence the quantum binary approach. For composite systems we may, for clarity, consider a two-qubit system. As a whole there will be 22 = 4 eigenstates, where a generic quantum state may represented by

ª ac º « ad » ªa º ª c º ψ = « »⊗« » ≡ « ». ¬ b ¼ ¬ d ¼ « bc » « » ¬bd ¼

(6.57)

When this is measured it will reduce to a vector with one ‘1’ and three ‘0’s. For N qubits the corresponding column vector will have 2 N components. In this way we see how large, in terms of dimensionality corresponding configuration spaces are. However, although (6.57) may represent a two-qubit quantum state, it is not the most general representation. Let us examine the following two-qubit state

ª1º « » 1 «0» ψ = 2«0» « » ¬ −1¼

(6.58)

by attempting to factorize it into its qubit components. Equating this with the column vector in (6.57), we have bd ≠ 0 , ad = 0 , and bc = 0 , which

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implies that a = c = 0 , contradicting the first component that says ac ≠ 0 . This state therefore cannot be decomposed into two individual qubits. The qubits from which the state in equation (6.58) was formed are entangled. This is more than just a mere superposition of states. The original qubits have completely lost their identities within the entangled pair. This provides a solid reason why classical particles cannot be considered fundamental. For any entangled pair the C-space is only slightly larger than for two separate qubits, which still has four degrees of freedom. That is, the respective configuration spaces have the same dimensionality, but the Cspace for separate qubits is a smaller set. For three qubits however, we see a substantial difference between three separate qubits having six degrees of freedom and an entangled triplet being a 23 = 8 dimensional system. This generalises to 2N for N separate qubits as compared to 2 N components for a state vector representing a general entangled N-tuple. So we begin to see that the exponential size of the C-spaces involved is the reason that quantum computers have so much more potential capacity than their classical counterparts. Returning to axiom 1, the only restriction placed on the state vector is that it has unit length. Otherwise it can take on any value. In the simple case of a two-state system a consequence of axiom 1 is superposition, which is represented in equation (6.56). For us this means that it can appear in two states at once, contradicting the common sense notion that the states are mutually exclusive. The mutual exclusivity of the states is a classical concept based on the way we experience the world. This is why superpositions seem so weird. The classic thought experiment, Schrodinger’s cat, could be ridiculed as a parody by those wedded to the classical world, thereby exposing the nonsense of quantum theory. However, when he originated this thought experiment, Schrodinger’s purpose was to clarify how best to view superposition and ultimately to show the contrast between the classical and quantum worlds. It is an idealized thought experiment designed to show that quantum systems can simultaneously exist in two or more, otherwise mutually exclusive, classical states, alive and dead in the case of the cat (Figure 6-3).

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Fig 6-3: A cat is locked in an ideal enclosure with apparatus consisting of a timer, a radioactive source and a vial of poison. During a predetermined period governed by the timer a mechanism, designed to smash the vial, is exposed to the radioactive atom, which has a 50% chance of decaying thus activating the mechanism. Hence the probability of 0.5 for each of the states alive or dead .

In this idealized experiment, prior to opening the enclosure the cat is in the quantum state ( alive + dead ) 2 . Upon opening the enclosure (the measurement) this state of superposition decoheres into either of the classical states, alive or dead . In reality of course there would be many classical configurations associated with the cat being alive or dead due to the cat being a macro-system. But for our purposes the cat is representative of a two-state system, a single qubit. Thought of in this way we can represent the Schrodinger cat experiment as a quantum circuit (Figure 6-4). This is the reason why qubits in a superposition are sometimes called Schrodinger cat states.

Fig 6-4: A quantum circuit representation of the Schrodinger cat experiment.

In Figure 6-4 the upper qubit is the atom which when decayed triggers the death of the cat, the lower qubit. Time runs from left to right where each

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gate represents an event in time. Both qubits start in the 1 state, indicating an intact atom and a living cat. The operation in Figure 6-4 is represented mathematically by

ψ = ( I ⊗ X ) C12 ( H ⊗ I ) 11 .

(6.59)

In the tensor products on the right hand side the first and second factors operate on the first and second qubits respectively. Also the subscripts in the CNOT gate, C12 , respectively refer to the control and target qubits. In expanded form equation (6.59) becomes

ª0 1 « 1 «1 0 ψ = 2 «0 0 « ¬0 0 ª1º « » 1 «0» . = 2«0» « » ¬ −1¼

0 0 º ª1 0 0 »» «« 0 0 1 » «0 »« 1 0¼ ¬0

0 0 0 º ª1 1 0 0 »» ««0 0 0 1 » «1 »« 0 1 0 ¼ ¬0

0 º ª0º 1 0 1 »» «« 0 »» 0 −1 0 » « 0 » »« » 1 0 −1¼ ¬1 ¼ 0

1

This is the same entangled state that we used as an example in equation (6.58). The cat and the radioactive atom become entangled through their mutual interaction. A more general version of this experiment known as Wigner’s friend (Wigner, 1961) is where Wigner asks a friend to perform the Schrodinger’s cat experiment. A symbolic representation of Schrodinger’s cat may be qO, where q is the cat and O is the observer who performs the experiment. Using the same symbolic scheme Wigner’s friend would be represented by qFO, where O is Eugene Wigner himself, F is his friend who performs the Schrodinger’s cat experiment, and again q is the poor cat. From Wigner’s point of view we have to imagine that his friend performs the experiment in an isolated chamber, within which a coherent quantum state exists. This state is a three qubit system with a quantum circuit representation shown in Figure 6-5, where the three qubits are the radioactive atom at the top, the cat in the middle, and the friend at the bottom whose state, sad or happy, is determined by the contents of his/her memory upon witnessing the death or otherwise of the cat.

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Fig 6-5: A quantum circuit representation of the Wigner’s friend experiment.

From the circuit in Figure 6-5 we can deduce that the equation for the final state prior to measurement is

ψ = ( I ⊗ I ⊗ X ) C23 ( I ⊗ X ⊗ I ) C12 ( H ⊗ I ⊗ I ) 111 . (6.60) Each of the matrices on the right hand side are 8 × 8 which when used to transform the initial state vector gives the following maximally entangled final state

ª1º «0» « » «0» « » 1 «0» ψ = . 2«0» « » «0» «0» « » ¬« −1¼» I leave it to the reader to satisfy him/herself that this is in fact the case. This final state shows the correlation between the states of the radioactive atom, the cat, and the friend, where the corresponding classical states are intact, alive, happy or decayed, dead, sad . Here we see that the delegation of the experiment to a friend effectively increments the number of qubits by one, thereby doubling the size of the system, a consequence axiom 4. In the next section we will see the full consequences of exponential growth of quantum systems with the addition of extra elements, and we will return to the Wigner’s friend scenario in section 8.2 to elucidate the role that a conscious mind plays both in this scheme and throughout reality as a whole.

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At this point it is useful to remark that the Schrodinger’s cat experiment could never work in practice. The reason has to do with the mass of the cat coupled with the fact that gravity cannot be screened out in the same manner as other field types. Any physical body having a mass greater than the Planck mass will possess a significant gravitational field in its neighbourhood. Significant in the sense that its gravitational field will, in principle, betray its state in the absence of any other effect. As a thought experiment Schrodinger’s cat is idealised in that the box in which it is imprisoned screens out all physical fields thereby isolating the cat from the rest of the universe. This perfect isolation is a requirement for the experiment to work, otherwise any signal leakage to the outside world would result in the establishment of a correlation between the spaces inside and outside the box, constituting decoherence. However, although the interactions considered in most elementary particle physics examples might in principle be screened out, gravity cannot. Any configuration of masses whose total mass is greater than the 22 μg Planck mass is increasingly likely to betray its state by gravitational leakage through the walls of any sealed container. This includes cats most of which have masses in the range 3.5-5 kg significantly higher than the Planck mass. It is this leakage (including the gravitational interaction) that forces the state of the contents of a container to become correlated with the outside world. There would be many near copies of this situation in distinct but neighbouring regions of the C-space, and in each one the contents correlate with the outside world consistently. Gravitational leakage is sometimes cited as a mechanism for objective reduction (Penrose, 2004). However, interpreted as decoherence it becomes a mechanism for ensuring consistency between different regions of a classical universe. So, as will be discussed in more detail in the next section, this is no argument against pure wave theories. A simplified version known as Schrodinger’s lump is where the two states of an isolated system are represented by two distinct positions of a particular mass (Penrose, 2004, 846). This has materialised into a concrete proposal appropriately called FELIX (Free-orbit Experiment with Laser-Interferometry X-rays) (Penrose, 2004, 856). The experiment involves the displacement of a tiny mirror (the lump) by a single photon from an X-ray laser on the same satellite, which is reflected back from another satellite about 108 m distant. Crucially the mass of the mirror is around 5 ×10 −12 kg, well below the Planck mass. So it is likely that this experiment will work by placing the lump into a superposition of the two locations. The resulting decay of the superposition due to gravitational self-energy is likely to be detected, and it is suggested that this would

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provide evidence of gravitational objective reduction. But for me this would not mean that conventional quantum mechanics is wrong. It just means that gravitation is playing a part in the decoherence process, which is expected for quantum system whose isolation with the outside world is compromised. From this I would expect that gravitational leakage from objects as large as cats would mean that Schrodinger’s cat could never become a practical experiment. Therefore as far as this experiment is concerned the universe remains a safe place for cats. Before moving on it will be useful to recast the axioms in terms of the density matrix formalism. There are many cases where we can have full knowledge of the state of a composite system, while at the same time not having full knowledge of the state of its constituents. This is where the expression of quantum states in terms of density matrices is most useful. Much of the details are left out here because they are not germane to our purpose, we merely express the axioms in density matrix form for completeness. For the details of this formalism see Nielsen and Chuang (2000). Given a pure state, ψ , for any particular system then its density matrix is given by ρ = ψ ψ . By itself this tells us very little because all we are doing is to express a pure state of which we have full knowledge and recasting it in a different form. All that we can really see here is that we can express any pure state in either formalism. However, suppose that we have any number of possible pure states, ψ i , with index, i, each with probability, pi . The object,

{ p , ψ } , is referred to as an ensemble of i

i

pure states. If we do not know which of these states a particular system is in, then its density matrix is given by

ρ = ¦ pi ψ i ψ i .

(6.61)

i

This density matrix has the properties that it is positive. That is given any vector, φ , in the state space then φ ρ φ ≥ 0 , and its trace is unity ( trρ = 1 ). This allows us to express the axioms in the density matrix formalism. Axiom 1. The state space: Associated with any isolated physical system is a complex vector space with an inner product (Hilbert space) known as the state space of the system. The system is completely described by its

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density operator, which is a positive operator, ρ , with trace one, acting on the state space of the system. If a quantum system is in the state, ρi , with probability, pi , then the density operator for the system is

¦

i

pi ρi .

Axiom 2. Evolution: The evolution of a closed system is described by a unitary transformation. That is, the state, ρ , of the system at time, t1 , is related to the state, ρ ′ , at time, t2 , by the unitary operator, U, which depends only on times, t1 and t2 ,

ρ ′ = U ρU † .

(6.62)

Axiom 3. Quantum measurement: Quantum measurements are described by a collection, {M m } , of measurement operators. These are operators acting on the state space of the system being measured. The index, m, refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is ρ immediately before the measurement then the probability that the result, m, occurs is given by

p ( m ) = tr ( M m† M m ρ ) ,

(6.63)

and the state of the system after the measurement is

M m ρ M m†

tr ( M m† M m ρ )

.

(6.64)

The measurement operators satisfy the completeness equation,

¦M

† m

Mm = I .

(6.65)

m

Axiom 4. Composite systems: The state space of a composite physical system is the tensor product of the state spaces of the component physical systems. Moreover, if we have systems numbered 1 through n, and system number, i, is prepared in state, ρi , then the joint state of the total system is

⊗

n i =1

ρi . (Nielsen and Chuang, 2000, 102).

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This presentation of the axioms in density matrix form is entirely equivalent to the earlier one in terms of the state vector. The density matrix formalism comes into its own when we are presented with a composite system whose state may not be entirely known, or if knowledge of subsystem states is incomplete in some way. In the next section we discuss the theories and interpretations of axiom 3 (measurement), which has been beset by much controversy in the decades since quantum mechanics first appeared.

6.4 The interpretations of quantum mechanics Since the inception of the full quantum theory researchers in the field have sought ways to reconcile our experience of the world with predictions of the theory. The most obvious consequence of quantum theory is superposition. For example the position of an electron can be in a superposition of several positional states. At an instant in time the electron is effectively smeared across a relatively large region of space. It is only when we try to measure its position that we obtain a definite reading. As mentioned in the previous section it is this process that requires interpretation. It is customary to think of the various points of view as interpretations of quantum mechanics. The reality however, is that they are interpretations not of quantum mechanics, but of the measurement process. As Deutsch (1996) points out, the only interpretation of quantum mechanics is the many worlds view. The implication being that other interpretations require modifications, therefore they are not quantum mechanics, they are merely closely related alternative theories. The many worlds view is, as Deutsch points out, a logical consequence of quantum theory. Falsify many worlds and you falsify quantum mechanics. So although I will continue to use the word interpretation, it is important to bear in mind that they are distinct theories, and that the other interpretations represent rivals to quantum mechanics. There are numerous alternate theories or interpretations but for our purposes it is helpful to group them into four distinct categories. Much of what we can say about one theory, we can also say about others in the same category. These categories are • • • •

Epistemological theories Objective collapse theories Nonlocal hidden variables Pure wave theories.

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Originators of epistemological theories either deny the ontology of the wave function or they are agnostic about it. Essentially the wave function is a mathematical entity from which we derive a probability distribution over a range of classical states representing our state of knowledge about the system in question. In technical parlance such theories are called ψ epistemic. In these interpretations there is no suggestion that the wave function has an independent physical existence. Originators of such theories include Albert Einstein and Niels Bohr. Indeed the well-known Copenhagen interpretation is generally attributed to Bohr In contrast to epistemological theories the remaining three groups of interpretations postulate the wave function as an ontological entity. That is to say it is an entity with an independent existence. The technical term is ψ -ontic. When we measure a quantum state and subsequently see an eigenstate, there is the perception that the wave function has collapsed. In objective collapse theories it is claimed that this collapse is real. Champions of these theories include Pearle (1976), Ghiradi, Rimini and Weber (GRW, 1986), and Penrose (1979, 2004). In order for the wave function collapse to be real, there must be an additional mechanism associated with the collapse. This mechanism is part of a distinct rival theory because it is not predicted by quantum mechanics. Hidden variables theories are distinct from either epistemological or collapse theories in that they assume an ontic wave function without its collapse. Local realist variants have been ruled out by Bell’s theorem (Bell, 1964), which have also been empirically verified (Aspect et al, 1981, 1982). Hidden variables theories postulate that the classical world, which is based on particles, also exists as an extra physical entity alongside the wave function. Therefore we imagine that the actual trajectory (history) of the universe as a special path through the C-space, which is guided by the universal wave function. This gives rise to their alternative designation–pilot wave theories. The existence of a special history is, like the objective collapse mechanism, not predicted by quantum theory. Removing this privileged path gives rise to the pure wave theories governed solely by the Schrodinger wave equation.

6.4.1 Epistemological theories Since the publication of an article (Pusey, Barrett, and Rudolph (PBR), 2012) proving the ontology of the wave function given a rigorous definition of a statistical model (Harrigan and Spekkens, 2010), there has been lively debate throughout the foundations of physics community regarding the scope of its validity. Together these two publications form

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the basis of the ontological model framework. Taken at face value the proof of the PBR-theorem should have been the last nail in the coffin for the epistemological theories. However, not only has the debate continued to revolve around the definition of a statistical model consistent with ψ epistemic theories, suggestions have also included realist ψ -epistemic models, which are epistemological as far as the wave function is concerned, but do not deny the ontology of the C-space (Leifer, 2014). PBR begin by proving their theorem for an idealized toy system which can be prepared in one of two states, ψ 0 or ψ 1 such that

ψ 0 ψ1 = 1

2 . A basis is chosen so that ψ 0 = 0

where ± = ( 0 ± 1

)

and ψ 1 = +

2 . Now consider two identical systems in which

their physical states are uncorrelated. This is achieved by preparing each system independently of the other, resulting in the composite system being in any one of the following four states: 00 , 0 + , +0 , and + + . For a statistical model, subsequent measurements in the basis defined by

ξ1 = ξ2 = ξ3 = ξ4 =

1 2 1 2 1 2 1 2

( 01 + 10 ) ( 0−

( +1 + ( +−

)

+ 1+ −0

(6.66)

)

+ −+

)

would be expected to yield any one of these outcomes with a non-zero probability. However, it can be verified that

ξ1 00 = ξ 2 0 + = ξ3 +0 = ξ 4 + + = 0 . So for any of the original four states, quantum mechanics predicts that one of the four outcomes listed in equation (6.66) will be excluded. This is the contradiction arrived at by PBR. The conclusion being that the wave function is not a mere statistical object and therefore not epistemic. PBR then go on to show that a similar contradiction is acquired when n uncorrelated systems are prepared and appropriately measured. And

224

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finally procedures are suggested which would render a real experiment immune to noise, wherein orthogonality is decided when quantum probability falls within some small bound, ε , where 1  ε > 0 . The implications of an ontic wave function are profound given that the C-space in which it resides must also be ontic. Realist ψ epistemic theories also support an ontic C-space explaining their support for exponential sized Hilbert spaces. As far as I can see, realist ψ epistemic theories represent the only retreat for epistemists, a view largely based on the growing body of empirical evidence (Iskhakov et al, 2012; Britton et al, 2012; Kanseri et al, 2013) for macroscopic-Bell states. We will say more about this in section 6.6. In response to the suspicion that quantum superpositions might be epistemic, Allen (2016) rules out epistemic superpositions for quantum systems with dimensionality d > 3 and further shows that for systems with ψ φ < 1 2 then ψ

and φ

must approach ontological distinctness as d → ∞ . Although it would have been tempting to rule out epistemological theories with the arrival of the PBR-theorem, the debate goes on. In the long run the trend seems to be in the direction of a ψ -ontic interpretation. After all it seems implausible that the wave function is ontic in some circumstances and epistemic in others or that only certain superpositions are epistemic. The wave function is either ontic or it is not, there can be no middle ground. Moreover, most supporters of ψ -ontic theories would point to the effect of physical interference seen in the double slit experiment. I have yet to see a plausible argument to counter this. This is another point where I must alert the reader to my own bias in this regard. I strongly suspect that assumptions in the ontological model framework are too strong. The interference pattern seen in the double slit experiment is a genuine physical effect independent of any observer. While it is true that ψ -epistemic models can be made to generate interference, as Spekkens’ toy model does (Spekkens, 2007), uncertainty relations can also be artificially inserted into such theories. In his toy model Spekkens introduces uncertainty in the form of the knowledge balance principle. This is a balance between knowledge and ignorance effectively saying that the most we can know about any given system is balanced by an equal quantity of unknown data. So to fully specify a system with 22 N states we can know at most N bits. This is the uncertainty principle in quantum theory. By considering two canonically conjugate variables, x and p, a precise specification of x precludes any knowledge of p. Yet both variables require equal quantities of data to specify them. This

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is precisely the knowledge balance principle at work. However, for physical systems we have seen that uncertainty relations are a direct consequence of the mathematics of quantum theory. In Spekkens’ toy model there are only uncertainty relations, there is no quantum mechanics. The wave function in quantum mechanics is constrained by other factors beside the uncertainty principle, strongly suggesting that the wave function is at least influenced by something ontic. To show that quantum theory is more than just uncertainty relations, we prove a theorem saying that functions exist satisfying an uncertainty relation while not being consistent with quantum mechanics. That is, injecting uncertainty relations by hand into an already contrived model does provide a clue to the source of some interference effects, but is by no means the whole story. Theorem 6.3: There exists a function exhibiting an uncertainty relation, which is not further constrained by the Schrodinger equation. Proof: All is required here is to provide an example of a function with an implicit uncertainty principle that does not satisfy the requirements of a wave function in quantum theory. Such a function is

φ ( x ) = A exp ( − a 2 x 2 ) ,

a, x ∈ \ .

(6.67)

By definition we determine the expectation values of x and p respectively to be ∞

x = ³ ψ † xψ dx −∞

∞ dψ · § p = ³ ψ † ¨ −i = ¸ dx. −∞ dx ¹ ©

(6.68)

The wave function we are considering is purely real valued, so equations (6.68) become ∞

x = ³ φ 2 x dx −∞

∞

p = −i = ³ φ −∞

dφ dx. dx

(6.69)

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Chapter 6

In equation (6.69) φ 2 x is an odd function therefore consequently x

2

x = 0 and

= 0 . The expectation value for p can be expressed as p = −i = ³

x =∞

x =−∞

so similarly we have

p

2

φ dφ = − 12 i=φ 2

x =∞ x =−∞

=0

= 0 . We now need expectation values for x 2

and p 2 , or at the very least we need to show that their product is > 0 . ∞

x 2 = ³ φ 2 x 2 dx −∞

= A2 ³ x 2 exp ( −2a 2 x 2 ) dx ∞

−∞

= The situation for p 2

π A2 4 2a 3

>0

is similar, in this case we have ∞

p 2 = −= 2 ³ φ −∞

d 2φ dx. dx 2

d 2φ = A ( 4a 4 x 2 − 2a 2 ) exp ( − a 2 x 2 ) dx 2 so the expectation value of p 2 is given by

p 2 = 2= 2 a 2 A2 ³

∞

−∞

=

π = 2 A2 a 2

(1 − 2a x ) exp ( −2a x ) dx 2

2

2

2

.

>0

Therefore the uncertainty in this case is given by

Quantum mechanics

ΔxΔp = =

(x

2

− x

= π A2 2 2a

2

)(

227

p2 − p

2

)

(6.70)

> 0.

Now we show that this function is not a solution of the Schrodinger equation. Assuming, in contradiction that it is, φ ( x ) is obviously time independent therefore the Schrodinger equation is given by H φ = 0 , which becomes

−=2b

d 2φ =0 dx 2

(6.71)

where b is just a coupling constant to guarantee dimensional consistency. Substituting (6.67) into (6.71) gives

2a 2 ( 2a 2 x 2 − 1) φ ( x ) = 0 This is only valid for two specific points in x, which is not physically meaningful and thereby provides our expected contradiction. So functions do exist which satisfy uncertainty relations but are not solutions of the Schrodinger equation. , In quantum theory when probability is calculated from the Born rule, phase information from the wave function is lost. It is interesting that phase information is manifest in the double slit experiment, and we see a loss of phase information when one of the slits is blocked. It is considered by many that quantum mechanics is ψ –ontic. If we accept the result of the double slit experiment as evidence for an ontic wave function, then we should equally recognise all aspects of the wave function as being ontic. As has been mentioned already it cannot be acceptable to recognise the wave function as being ontic in some circumstances and epistemic in others. It is either one or the other–there can be no ontic/epistemic superposition. The PBR theorem does indeed demonstrate an ontic wave function based on the ontological model framework. However, as Gao (2015) points out, the ontological model framework depends on auxiliary assumptions when projective measurements are considered. For example

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Chapter 6

the PBR theorem relies on a preparation independence assumption. In response to this Gao’s argument considers protective measurements with only a weak but reasonable criterion of reality. Unlike projective measurements, protective measurements do not lead to an immediate decoherence. This is because protective measurements are based on the interaction Hamiltonian, H int = g ( t ) PA , where A is the variable to be measured, P is the canonical conjugate momentum of the pointer variable, and g ( t ) is a very weak time dependent smooth coupling factor, normalised to

τ

³ g (t ) dt = 1 0

with

g ( 0 ) = g (τ ) = 0 . So the longer we make the measurement duration, τ , the weaker the coupling factor needs to be. Protective measurements always yield the expectation value, A , without decoherence. In addition to basing his argument on protective measurements, Gao also drops the ontological model assumptions. Gao’s weaker criterion of reality, replacing these assumptions, effectively asserts that an eigenstate, which is the outcome of a measurement, has a reality of its own, provided that the eigenstate does not change during the period of the measurement (Gao, 2015, 201). As far as I can see the source of the objections to more recent ontology proofs is not the use of protective measurements as such, but the inclusion of the ontological model assumptions originally used to prove the PBR theorem. Gao does not rely on these assumptions but instead uses his weaker reality criterion. This provides a much stronger objection to ψ -epistemic theories generally.

6.4.2 Objective collapse theories Wave function collapse is a perceived consequence of a measurement operation that can be considered in the context of both ψ -epistemic and ψ -ontic theories. In the epistemic case wave function collapse is generally associated with the Copenhagen interpretation. In this context there is no problem apart from objections to ψ -epistemic theories generally. In a ψ -ontic interpretation however, we encounter problems associated with superluminal signalling due to the nonlocal properties of quantum theory associated with an actual history. However, this problem may vanish when the wave function is assumed to be ontic but not physical. Also these theories, mainly due to Pearle (1976) and Ghiradi et al (1986), are amenable to experimental testing.

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If we proceed under the assumption that the wave function is ontic then we need to consider objective collapse. Essentially this can only mean one thing, quantum mechanics is an incomplete theory. The fact that at any instant we experience only one configuration, and not some superposition of many, is a testament to this. With objective collapse models we add a new mechanism alongside unitary evolution governed by the Schrodinger wave equation. This mechanism comes in the form of extra terms in the expression on the right hand side, shown in the following simplified form (Okon and Sudarsky, 2014)

i=

∂ψ = ª H + i= ( Aw ( t ) − λ A2 ) º¼ψ . ∂t ¬

(6.72)

Here λ is a constant, A is an appropriate Hermitian operator, and w ( t ) is a time dependent stochastic process. According to Ghiradi and colleagues this constitutes a unified description of microscopic and macroscopic systems that forbids linear superpositions of states localised at large distances (macroscopic) and induces evolution agreeing with classical mechanics for large systems. When applying an objective collapse interpretation, we model a series of collapse events along a history line in C-space. The wave function for an open system within our universe collapses in response to signal leakage across the boundary of the system with the outside universe. As an example we could consider our own brains as open systems exchanging information, through our senses and actions, with the rest of the universe. The almost continuous collapse of our brain state to classical eigenstates gives us our perception of the world. This is a very attractive idea that provides us with our familiar view of the world. However, apart from that the scheme is totally contrived and there is no other a priori reason for an objective collapse approach. There is no known mechanism for the collapse process, and quantum mechanics certainly does not predict it. The fact that objective collapse theories of GRW and Pearle appear to violate the conservation of energy (Adler and Brun, 2001) would appear to seal their demise. However, a later study, producing a discrete model of wave function collapse (Gao, 2013), appears to solve that problem, although Gao’s energy-conserved collapse could equally be reinterpreted as energy-conserved decoherence. To conclude the discussion of objective collapse there is one inconsistency that does appear insurmountable– nonlocality.

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Chapter 6

Nonlocality is a property of the quantum mechanics of multipartite systems, involving potentially large spacelike separations. Such systems have become known as EPR (Einstein-Podolski-Rosen) systems. If we proceed under the assumption that the wave function is a physical entity, then there are irresolvable problems associated with EPR systems covering large spatial expanses.

O1

O2

Time

m1

m

p1 Entangled pair

m2 p2

Spatial separation Fig 6-6: Incompatibility of quantum nonlocality with objective collapse theories. Thin lines are timelines of particles in entangled superposed states, the thick line is the timeline of a particle in a classical eigenstate post measurement, thin dashed lines link null separated events and the thick red dashed lines link spacelike separated events.

Considering the situation in Figure 6-6, we have a pair of entangled particles, p1 and p2. A measurement, m, takes place on p1, news of which

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reaches observers O1 and O2. O1 immediately sees a change of state from superposed to classical in p2 at m1. This is because the entangled particles are correlated in some way. If we know the eigenstate of p1 then we immediately know the eigenstate of p2. We can demonstrate such a correlation by handing to you, a box containing a glove. The glove is one of a pair of which the other has been placed into an identical box. We have no way of knowing which box contains the right hand glove. If I open my box, after being separated by a large distance, then I immediately know which glove you have. This by itself is not entanglement, just correlation. But it is this kind of correlation that is related to entangled quantum states. Due to their nonlocal nature these states appear to collapse at superluminal speeds. These apparent spacelike signals are shown as thick red dashed lines in Figure 6-6. However, we are assuming that the wave function is a real physical object, which in an objective collapse context, is incompatible with relativity. Note that this would not be a problem in a ψ -epistemic interpretation. But if we insist on objective collapse, assuming a physical wave function, we have to drop relativity. I do not think this is going to happen anytime soon. Therefore to keep relativity we must deny physical collapse. Indeed, for similar reasons, Penrose himself, a supporter of objective collapse, casts some doubt over these theories on page 606 of his work, The Road to Reality (2004), but then goes on to say that the collapse takes place in such a way as to avoid transmitting superluminal information. This seems like one law for one type of physical phenomena and one law for another. Put another way it seems like it is in order for a physical object to propagate at superluminal speeds as long as no one sees it happen. If this is the case for wave function collapse then, by definition, it cannot be objective because that effect is observer dependent. In the case of observer O2 the situation is even more extreme. Upon seeing the results of the measurement m, O2 sees a change in p2 at m2 occurring before the measurement. Moreover, the events, m1 and m2 take place at distinct times along p2’s trajectory–a contradiction. At this point I can see the ψ -epistemists jumping up and down shouting I told you so…. If the collapse is objective then events, m1 and m2, must occupy the same point in space-time because they represent a single event–the collapse of the wave function at p2. Even assuming an objective but nonphysical wave function (Akshata and Srikanth, 2014) does not provide any escape from this. The only apparent way to reconcile an ontic wave function with the effects of nonlocality is to adopt the decoherence approach that is central to nonlocal hidden variables and pure wave

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theories. However, we cannot dismiss objective collapse theories without some discussion of their relativistic variants. In 2006 a relativistic form of objective collapse (Tumulka, 2006) appeared, and later Christian Beck reviewed various theories of relativistic collapse (Beck, 2010). The object of such theories is to describe Lorentz invariant collapse or, at the very least, to acquire consistency by the specification of a time slice for the collapse on which all observers agree. We do not see this in non-relativistic collapse theories. Two solutions emerging from Beck’s work are: Solution I wave function collapse along arbitrary time slices and Solution II distinguished foliation. Tumulka’s model is commensurate with Solution I and is based on a multi-time Dirac equation with additional collapse events referred to as flashes. The model is completely Lorentz invariant. However, as Beck points out, there are still complications with regard to wave function ontology of the type encountered above with non-relativistic models. Admittedly this does rather gloss over a lot of detail regarding attempts to establish such ontology under these circumstances. For details the reader is referred to (Tumulka, 2006) and (Beck, 2010 and references therein). With Solution II the implied preferred frames represent what appears to be a significant obstacle to the aim of achieving a fully relativistic objective collapse theory. However, these preferred frames are not intrinsic to the space-time fabric they are due solely to the configuration of matter. For any globally hyperbolic space-time the metric can be expressed in the form

ds 2 = dt 2 − gij dxi dx j

(i, j = 1, 2,3)

(6.73)

where the universal time variable, t, is locally orthogonal to the spatial coordinates, x i (Schutz, 1985, 323). While this collapse model works in the sense that it produces results consistent with quantum theory, there is still no mechanism for the collapse process. With a preferred frame not intrinsic to space-time I find it difficult to conceive of a collapse mechanism that knows which way to go to generate an objective reduction along the orthogonal time slices of a preferred frame. While I believe this to be a serious problem for objective collapse theories, a preferred frame based on matter configuration is a blessing when it comes to the (1+3) decomposition of space-time in the ADM formalism. But this is a discussion for the next section. Models based on decoherence, lacking the above-mentioned complications, seem to be on a more secure footing.

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6.4.3 Nonlocal hidden variables Previously we have considered theories that call into question the ontology of the wave function, known as ψ -epistemic theories. The famous Copenhagen interpretation is considered by some to fall into this category. However, there are also followers of this view that are somewhat agnostic about it. For this reason the Copenhagen interpretation is difficult to classify, but this does not present us with any particular difficulty. The Copenhagen interpretation is either an objective collapse theory or it is ψ epistemic. On the basis of the PBR theorem and similar arguments it seems that most physicists would likely adopt ψ -ontic theories. Gao (2015) has shown that ψ -ontic theories can be proven using a less stringent reality criterion than the ontological model framework. It seems reasonable that any suggestion of objectivity using this weaker criterion, in one set of circumstances, should rule out ψ -epistemic theories in others. That is the wave function cannot be ontic in some circumstances and epistemic in others. While we need to treat this approach with caution it seems that most of the physics community accepts the wave function as an ontic entity. Objective wave function collapse models seem to suffer from inconsistencies with relativity. Even when the wave function is considered to be ontic but non-physical, inconsistencies can still arise (Figure 6-6). The only way out suggests adopting a preferred frame based on the configuration of matter. The problem here is that no mechanism exists for the collapse process, which needs to be sensitive to the direction of the local energy-momentum flux 4-vector. So rejecting objective collapse theories as well leaves us with two remaining categories: nonlocal hidden variables and pure wave theories. The best-known hidden variable theory is by De-Broglie and Bohm (Bohm, Hiley and Kaloyerou, 1987 and references therein). This falls within a class called modal theories in the sense that the physical (classical) state is treated as distinct from the quantum state (Bub, 1996; van Fraassen, 1981). In this and other hidden variable/modal theories, quantum mechanics is considered incomplete and therefore is supplemented by the evolution of a separate classical particle configuration. When Schrodinger’s formulation of quantum mechanics appeared in 1926 it consisted of an evolution equation (Schrodinger’s) and its solution–the wave function. There is no suggestion of any other ingredients, see for example the derivation leading to equation (6.25). Anything else is additional with nothing to justify it except our own

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preconceived ideas and observation of an apparent classical reality. If we reject ψ -epistemic and objective collapse theories then what do we replace them with? We still have to explain why we see a classical reality in the form of one configuration at a time instead of some confusing superposition of many. Bohm’s approach is to keep Schrodinger’s linear wave equation, as it is, then to consider our history as special in some way. The suggestion being that there are two components to physical reality: the wave function and the actual particle configuration at any instant. These theories will always predict results that are consistent with Schrodinger’s equation because that remains unchanged. But when we make a measurement, instead of a collapse we obtain a decoherence event where we see a smaller part of the universal wave function over a reduced neighbourhood of the actual configuration. During periods when there is no measurement, the wave function is considered to guide the particle configuration. Hence the term pilot wave theories (Figure 6-7). Present configuration

Configuration space Dimensionality ~10123

α -point “Big Bang”

Actual history

Configuration space boundary

Fig 6-7: Evolution of the universe as depicted by nonlocal hidden variable theories. The actual history is similar to that predicted by classical mechanics, except it is being guided locally by the wave function (in blue) which spans the entire configuration space, and evolves according to Schrodinger’s equation.

Bohmian mechanics is often strongly associated with hidden variables theories, but before discussing this we should eliminate models that fall under the heading local hidden variables. Such theories can only ever be nonlocal in spatial terms (Nielsen and Chuang, 2000, 115…). To see why, consider the scenario describing Alice and Bob who make

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independent measurements on each of a pair of entangled particles prepared by a third party, Charlie. We begin by treating the whole system classically, that is we make reasonable assumptions based on our worldview. Alice receives her particle and measures parameters Q and R while Bob makes measurements on his particle of parameters S and T at an event spacelike separated from Alice’s measurement. We begin by assuming that these parameters can only take values ±1 . Then it can be shown, using a simple truth table method, that QS + RS + RT − QT = ±2 .

(6.74)

If the parameters ( Q, R, S , T ) have values ( q, r , s, t ) respectively then we can compute the expectation value of (6.74) using

QS + RS + RT − QT =

¦ p ( q, r, s, t )( qs + rs + rt − qt )

q , r , s ,t

≤

¦ p ( q, r , s, t ) × 2

q , r , s ,t

=2 where p ( q, r , s, t ) is the probability that the parameters take the values

( q, r , s, t ) . From this and the linearity of expectation values we find that QS + RS + RT − QT ≤ 2 .

(6.75)

Equation (6.75) is one of the famous Bell inequalities. Because we have made no assumption regarding the preparation of the particle pair then the inequality (6.75) must be true under all circumstances. Let us now treat the same system using quantum mechanics. Suppose Charlie prepares an entangled pair of particles according to

ψ =

01 − 10 2

.

One of the pair goes to Alice who measures Q = Z ⊗ I and R = X ⊗ I .

(

Upon receiving his particle Bob measures S = − 1

)

2 (I ⊗ Z + I ⊗ X )

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Chapter 6

(

and T = 1

)

2 ( I ⊗ Z − I ⊗ X ) . Expanding the factors in the first term of

(6.75) gives

QS = ψ QS ψ where

ª0º « » 1 «1» ψ = , 2 « −1» « » ¬0¼

ª1 «0 Q=« «0 « ¬0 ª1 1 « 1 « 1 −1 S=− 2 «0 0 « ¬0 0

0 0 0º 1 0 0 »» , 0 −1 0 » » 0 0 −1¼ 0 0º 0 0 »» . 1 1» » 1 −1¼

and

After a short calculation this reduces to

QS =

1 2

.

Similar calculations give

RS = RT =

1 2

and QT = −

1 2

.

This allows us to compute the left hand side of (6.75) which gives

QS + RS + RT − QT = 2 2 > 2 yielding an obvious contradiction with the Bell inequality (6.75). This contradiction has been verified by experiment (Aspect et al, 1981, 1982), showing that nature supports quantum mechanics over our commons sense classical reasoning. It is illuminating to specify our classical assumptions of which there are two:

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•

237

We assume that the parameters, ( Q, R, S , T ) , have definite values independent of our observations. This is the assumption of realism. We also assume that Alice and Bob in no way influence each other’s measurements. This is the assumption of locality.

These two assumptions taken together have become known as local realism. By definition, hidden variables models are realist–they have a classical element, which does indeed have a definite configuration independent of measurement. This is why, when considering hidden variables theories, we can only accept nonlocal variants. The assumptions of local realism certainly do fit the common sense way that we think about the world. Yet Bell inequalities and the empirical results of Aspect and colleagues have shown that, in a quantum context, at least one of these assumptions cannot be correct. It has been said that most physicists would reject classical realism (Nielsen and Chuang, 2000, 117). Indeed this is my own position and moreover, more recent loophole free results have appeared (Hensen et al, 2015; Shalm et al, 2015; Giustina et al, 2015) which should mark the final death knell for local realism. At this stage it is informative to consider Bohmian mechanics and its association with hidden variables. Bohmian mechanics is based on Bohm’s original proposal (Bohm, 1952). While we could provide a full relativistic treatment of Bohmian mechanics it is clearer to explain it in a more familiar nonrelativistic context. We assume a wave function of the form given in equation (6.37), which in the context of a multi-particle system becomes

ψ ( q, t ) = R ( q, t ) eiS (q ,t ) =

(6.76)

where q is a vector in generalized coordinates (C-space). Substituting this into a non-relativistic Schrodinger equation Eˆψ = pˆ 2ψ ( 2m ) + V ( q )ψ we follow similar steps to those beginning with equation (5.32) at the end of the last chapter. Note that for clarity we assume all particles to have the same mass, m. With this in mind the Schrodinger equation becomes

i=

∂ψ =2 2 =− ∇ ψ + Vψ . 2m ∂t

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Substituting the ansatz in (6.76), after a little algebra the Schrodinger equation becomes

∂S ψ ∂R + i= ∂t R ∂t . ψ ª 2 ∇2 R 2 § ∇S ⋅ ∇R 2 ·º = = ψ S i S V = − + ∇ − + ∇ + ( ) « ¨ ¸» 2m ¬ R © R ¹¼ −ψ

(6.77)

Notice that by factoring out the wave function and equating the real parts on both sides we obtain the Hamilton-Jacobi equation with the extra term

−

=2 ∇2 R . 2m R

This term is called the quantum potential, and is the first of two main features of Bohmian mechanics, the second being the particle trajectories. Particle trajectories can be defined in terms of the wave function as follows. Using the ansatz in (6.76), we compute lnψ = ln R + iS = . Differentiating with respect to generalized coordinates then taking the imaginary part, we get

§ ∇ψ · ∇S = = Im ¨ ¸. © ψ ¹

(6.78)

From the discussion in the previous chapter we notice that the left hand side defines generalized momentum, which for a particle of mass, m, is mq , thus adequately specifying particle trajectories. When dealing with many particles with differing masses then equation (6.78) becomes

§ ∇ψ mk q k = = Im ¨ © ψ

· ¸. ¹

(6.79)

So particle trajectories can be completely specified by the wave function. This procedure can be extended to a relativistic context. Without resolving issues of spin we may consider the Klein-Gordon equation

Eˆ 2ψ = c 2 pˆ 2ψ + m 2 c 4ψ

(6.80)

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with the substitutions Eˆ = i=∂ t and pˆ = −i =∇ . Again for clarity we consider a system composed of identical particles. Using the ansatz ψ = ψ 0 exp ( i ( k ⋅ q − ωt ) = ) we obtain the energy-momentum relationship = 2ω 2 = = 2 k 2 c 2 + m 2 c 4 .

From wave mechanics the definition of group velocity is q ≡ ∂ω ∂k . Therefore we can specify the particle trajectories purely in terms of wave properties, by

q =

=ck = k 2 + m2 c 2 2

.

(6.81)

This illustrates the point again that particle trajectories can be extracted from the wave function alone. Along with empirical results this removes the possibility of classical realism, but at the same time allows for locality to remain intact. Bohm’s hidden variable theory is based on two postulates, the first asserting the existence of a particle configuration and the second, that this particle configuration is distributed according to the Born rule. However as we can see here, there is no need for the prior existence of a particle configuration. It emerges naturally from the wave function, and can therefore be regarded as redundant (Vink, 1992; Zeh, 1999; Brown and Wallace, 2005). It has more recently been claimed that Bohm’s theory is not self-consistent (Frauchiga and Renner, 2016), an idea based on a theorem asserting that no single world theory is consistent. However, other authors have countered by showing that the Frauchiga-Renner theorem only shows collapse theories to be inconsistent (Baumann, Hansen, and Wolf, 2016; Sudbery, 2017). Moreover, as we have seen Deutsch (1996) had already remarked that Bohm’s theory is really a multi-world theory, it is just that one of those worlds happens to be special. The postulated hidden variables, as a concept is similar to the hidden frame idea in relativity discussed in chapter 4. Both are superfluous to requirements. However, like the hidden frame proposition, hidden variables are equally impossible to disprove given our current state of knowledge and despite the efforts of Frauchiga and Renner. Their redundancy is presently the only objection we have to their existence. If we accept particle

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configurations as emergent rather than fundamental, then we are left with the minimal ontology implied by quantum theory–pure wave theories.

6.4.4 Pure wave theories These are ψ -ontic theories that were first made explicit by Everett (1957), although it is likely that these would have occurred to Schrodinger, Heisenberg, and Dirac as well. This is because pure wave theories represent quantum mechanics in the raw. As Deutsch (1996) points out, pure wave theories are a logical consequence of quantum mechanics. Other interpretations should be more appropriately regarded as rival theories that either introduces extra ingredients (hidden variables), modifications to the Schrodinger equation (objective collapse) or deny the reality of the wave function (epistemological theories). This is based on the idea that any theory possesses both a formulation, allowing us to calculate, and an interpretation that provides a narrative to facilitate understanding. With the exception of objective collapse theories, all other models use the precise formulation of quantum mechanics. As we have seen objective collapse models introduce extra terms into the wave equation, so in this case quantum mechanics is approximate, see equation (6.72). But all of the competing theories have a different narrative, which is why they are regarded as interpretations. On this point therefore, I agree with Deutsch in the assertion that pure wave theories are quantum mechanics, and the other interpretations are rivals to it. In the early years of quantum theory, particularly during the war years, the standard interpretation with regard to the measurement problem was that an observer caused the wave function of a system to collapse when making a measurement. At that time no one could provide details of the mechanism behind this phenomenon, nor could anyone establish an ontological status for the wave function. Most physicists at that time just accepted the effect without an agreed interpretation. In particular, during the Manhattan Project for example, physicists were not interested in the wave function ontology or literal interpretation of quantum theory. The priority was to apply the established formulation. However, it was not long before scientists considered bringing gravity into the quantum arena. The latest theory of gravity, general relativity, was the prevailing theory then just as it is now. The problem was that when we considered the universe as a whole, general relativity could not be avoided. Cosmology became a sub-field of general relativity. So it was not possible to quantize general relativity without including cosmology in the scheme. As far as we, as inhabitants of the universe, are

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concerned there is no outside laboratory with god like observers that cause the universal wave function to collapse at regular intervals. Another way of looking at this is that gravity, being the weakest of all interactions is almost non-existent when we consider atoms and molecules. Gravity only becomes significant on the scale of planets and stars. These are macroscopic objects, so when we contemplate quantizing gravity we need to consider quantum theory as applied to all macroscopic objects including ourselves. In this way we, the observers, become part of the quantum universe. In the old standard interpretation of quantum mechanics (the Copenhagen interpretation) it is outside observers that affect a wave function collapse. There is no provision for internal observers, and if internal observers caused a wave function to collapse then, from the outside, it would look like a spontaneous collapse without an external cause. In quantum mechanics (even in the Copenhagen interpretation) wave functions do not spontaneously collapse, and there lies an inconsistency. It was Everett who provided a way out of this dilemma without modifications to quantum mechanics. Here the imagined wave function collapse mechanism did not exist. There was only one way for the wave function to evolve, through unitary evolution according to the Schrodinger equation. DeWitt coined the phrase many-worlds because it resulted in a plethora of alternative histories (Bohmian trajectories). Visualisation of coexisting histories like this was difficult for many reasons. At first it could be asked: what separates the histories? Are these histories discrete? These questions are prompted by the appearance that any interaction between distinct histories or timelines is impossible. So, it could be imagined, that gaps existed between these parallel universes. What seemed to happen instead of wave function collapse was that a history or timeline branched or bifurcated. From an observer’s viewpoint the wave function would become narrower at the beginning of each new timeline from a branch point. That is, at the start of each new branch, the wave function would cover a smaller part of C-space. But for all of the new branches from a given observation event it may not narrow at all. Indeed it is more likely to spread out further as unitary evolution progressed. It is as though the wave function is shared between the new branches. But because of the perceived narrowness of the wave function at the beginning of each branch, this gave the appearance of wave function collapse. This is the same decoherence mechanism that we see in Bohm’s theory. A question arising now concerns the best way to visualise this branching phenomena. Effective but naïve ways of imagining how the

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history of the universe may branch are shown in Figure 6-8. In 6-8a, we see schematically how the history of the universe evolves from the past at the bottom of the figure to the future at the top. Note that the number of branches increases with time. This multiple branching has been connected with the perceived arrow of time and the previously mentioned wave function sharing between the branches, for further details see (Zeh, 2007). The problem here is that we may visualise such a structure without giving any thought as to the space in which it is embedded. This can be further illustrated by Figure 6-8b where we visualise a region of space-time branching at a given event, P. This branching is confined to the future of P, which is bounded by a cone generated by a light shell as it propagates in all directions from a momentary flash at P. A space-time diagram for this situation is shown in Figure 6-8c in order to illustrate the future of P and its boundary. The branching shown in 6.8b has been illustrated by Penrose (1979) as a way of making Everett type branching consistent with relativity theory. Having made this description he then criticises the resulting space-time manifold as non-Hausdorff, a naïve definition of a Hausdorff manifold being one that does not branch in any way. For Penrose this was a step too far. Indeed for many relativists, reality is viewed as a single block four-dimensional space-time continuum with a very specific geometric structure. In this scheme it is impossible to completely separate space and time, and in relativity they have merged to become the single entity we know as space-time. For Penrose, and possibly many of his colleagues, to assert the existence of the branched space-time shown in Figure 6-8b was to butcher an already beautiful geometric structure. The resulting non-Hausdorff space-time became an unacceptable feature of the Everett interpretation. However, if we are to quantize gravity we must deconstruct space-time back into its original components–space and time. This procedure (Arnowitt, Deser, and Misner, 1962) is necessary in order to reformulate general relativity in such a way as to allow a Hamiltonian to be expressed. Once this procedure is completed then we can apply the usual quantization rules to obtain a theory of quantum gravity. Bryce S DeWitt first achieved this in 1967, and the theory became known as canonical quantum gravity (CQG). This has, more recently, evolved into loop quantum gravity (LQG), arguably the best strategy leading to a unification of physics as a whole.

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243

Time

Time Future of P P

P Spa t

a

ial

b

axi s

Distance

c

Fig 6-8: Relativistic branching a Branching paths through the configuration space with the time direction up. b A section of space-time continuum branching at the event, P, as envisioned by Penrose (1979). c Space-time diagram showing an event, P, and its future. The boundary of its future propagates away at the speed of light.

6.5 Quantum gravity Before proceeding with the main discussion, there is one glaring misconception that needs to be addressed. This is the notion that whatever the final theory of quantum gravity turns out to be, it will be a unification of general relativity with quantum mechanics. Nothing could be further from the truth. The reason this is not true is because these two theories are on distinct levels. By this, I mean that a particular model or theory, Tk ( k ∈ ` ), can be part of an overarching theoretical framework or scheme, FT . So two theories, T1 and T2 , could in principle be unified under FT . In this way it would make more sense to regard quantum gravity as a unification of general relativity with quantum field theory (QFT). Where these two theories are represented by T1 and T2 say, with quantum mechanics being represented by FT . Thus quantum mechanics is a metatheory that exists on a very distinct level to either general relativity or QFT. This is one possible reason why there is so much debate regarding the ontology of certain elements of quantum mechanics. Quantum mechanics is, sort of, up there in the clouds, whereas lower level theories, like QFT, are much more matter of fact. The current theory of CQG (DeWitt, 1967) may be regarded as a unification of general relativity with the standard model of particle physics, where the latter could eventually be

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expanded into a wider QFT encompassing, for example, dark matter once its composition is known.

6.5.1 Canonical theory and the problem of time Traditionally quantum mechanics was considered a theory of atoms because its effects are stronger at small scales. This is because it requires higher energies to disrupt quantum states at a small scale, whereas the effects of gravity are only manifest at a large scale where quantum states are more fragile. As we increase energy at microscopic scales, corresponding wavelengths shorten. Continuing this extrapolation, a point is reached where the characteristic wavelength is comparable in magnitude to the radius of a black hole of the corresponding mass-energy. This is the regime where the effects of quantum gravity are expected. The characteristic length scale is mass of ( =c G )

12

( =G c )

3 12

 10−35 m , corresponding to a

 1019 GeV/c 2 . This is the Planck scale, which is many orders of magnitude beyond the capacity of any practical accelerator. It is for this reason that we have no data and only plausible hypotheses for the physics at this scale. Being an empirical discipline, physics requires experimental confirmation of a particular hypothesis before it can be accepted as a theory in the true sense of the word. This is also why we have such a plethora of distinct theories that are not in conflict with known physics. So we do not have a complete theory of quantum gravity yet. However, we do have a unification of gravity with the rest of known physics–the previously mentioned CQG, which is a quantization of the ADM formalism sketchily discussed at the end of chapter 4. Here we will proceed with a very brief derivation of the governing equation for quantum gravity–the Wheeler-DeWitt equation. What follows, covers the main steps without becoming embroiled in the detail. For that, the interested reader is referred to DeWitt (1967), Wiltshire (2003), Zeh (2007), and references therein. We begin with the metric given in equation (4.36), which provides us with the necessary 3+1 decomposition of spacetime. The metric of a spacelike hypersurface orthogonal to the local 4momentum flux vector, is denoted by hij = gij ( i, j = 1, 2,3) , and represents a preferred frame as opposed to a privileged (hidden) frame. These are C-space variables at each point in the hypersurface, which provide degrees of freedom additional to those of matter. Given matter

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fields, Φ ( r ) , then a C-space, Riem ( Σ ) , on the spatial hypersurfaces, Σ , may be given by

Riem ( Σ ) = {hij ( r ) , Φ ( r ) : r ∈ Σ} .

(6.82)

From this we may define a physically more meaningful C-space given by Riem ( Σ ) Diff 0 ( Σ ) , where Diff 0 ( Σ ) is the set of all diffeomorphisms (coordinate transformations) connected to the identity. This is what is traditionally referred to as superspace6, a term originally coined by Wheeler. Having established an appropriate C-space we may re-express the Einstein Lagrangian density, / , in terms of the new variables. In Planck units ( = = c = G = 1 ) we have

( − g ) ( R − 2Λ ) 12

/≡

16π 1 ª 12 = Nh ( Kij K ij − K 2 + 3 R − 2Λ ) − 2 ( h1 2 K ) ,0 ¬ 16π +2 ( h1 2 KN i − h1 2 hij N, j ) º + /matter ,i ¼

(6.83)

where

g ≡ det ( g ab ) = N 2 h ( a, b = 0,1, 2,3) , h = det ( hij )

K ij = 12 N −1 ( Ni ; j + N j ;i − hij ,0 ) , K ij ≡ hik h jl K kl , K ≡ hij Kij and Λ is the usual cosmological constant. Here the symbol K ij is the second fundamental form characterising extrinsic curvature of the hypersurface within the embedding space-time, and the semicolons in the corresponding equation represent covariant differentiation. The last two terms in the square brackets of equation (6.83) play no role in the dynamics of the hypersurface (DeWitt, 1967, 1117), so for simplicity they may be dropped. The expressions, K ij K ij − K 2 and 3 R , are extrinsic and 6 The use of the term superspace refers to the configuration space in a quantum gravity context and predates the discovery of supersymmetry.

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intrinsic curvature scalars respectively for the hypersurface, and in general are independent, although in a flat space-time 3 Rij is completely determined by K ij . If we allow the extrinsic curvature scalar to play the role of kinetic energy with negative potential energy corresponding to 3 R − 2Λ , then the Lagrangian is given by

L≡

1 Nh1 2 ( K ij K ij − K 2 + 3 R − 2Λ ) d 3 x . 16π ³

(6.84)

Now that we have a Lagrangian we may appeal to classical mechanics to determine conjugate momenta corresponding to geometric variables already encountered along with their constraints. These constraints are given by

π=

δL =0 δ N,0

πi =

δL =0 δ Ni ,0

π ij =

δL = −h1 2 ( K ij − hij K ) , δ hij ,0

πΦ =

δL h1 2 = ( Φ,0 − N i Φ,i ) δ Φ ,0 N

and are analogous to equation (5.21), p = ∂L ∂q , in the previous chapter. This allows us to express a Hamiltonian of the form

H = ³ (π N ,0 + π i N i ,0 + π ij hij ,0 + π Φ Φ ,0 ) d 3 x − L = ³ (π N ,0 + π i N i ,0 + N + + Ni χ i ) d 3 x

where

+ = 16π *ijkl π ij π kl −

h1 2 16π

(

3

R − 2Λ − 16π T 00 )

h1 2 = ( 2G 00 + 2Λ + 16π T 00 ) = 0 16π

(6.85)

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247

and

χ i ≡ −2π ij ; j ≡ −2π ij , j − hil ( 2h jl , k − h jk ,l ) π jk =

(6.86)

h1 2 ( 2G 0i + 16π T 0i ) = 0. 16π

The expression, *ijkl ≡ 12 h −1 2 ( hik h jl + hil h jk − hij hkl ) , is the superspace metric tensor, sometimes known as the supermetric, at a specific point in Σ . This is a hyperbolic expression with signature − + + + + + irrespective of the signature of the space-time metric. Equations (6.85-6) are referred to as the Hamiltonian and momentum constraints respectively. On inspection we can see how these constraints relate to the (00) and (0i) components of Einstein’s equations. Cutting a very long story short we can apply the standard quantization procedure to equation (6.85) to arrive at the Wheeler-DeWitt equation. In a superspace context the quantization rules are

πˆ = −i

δ δ δ δ . , πˆ i = −i , πˆ ij = −i , πˆΦ = −i δN δ Ni δ hij δΦ

From these it is straightforward to write out the Wheeler-DeWitt equation

+Ψ

ª δ2 h1 2 − « −16π *ijkl δ hij δ hkl 16𠬫

(

3

º R − 2Λ − 16π Tˆ 00 » Ψ = 0 . (6.87) ¼»

)

Here the quantized matter fields, Tˆ 00 , represent the Hamiltonian density of any other fields in the theory. This renders CQG open ended as far as unification with gravity is concerned. That is the Hamiltonian can be modified as new physics is discovered. Operating on the universal wave function, Ψ , equation (6.87) is all that is required to specify the dynamics of the universe. However, in this context, we need to be very careful how we use the word dynamics. The universal wave function may be denoted in full by Ψ ( hij , Φ ) . That is, it is dependent only on the intrinsic geometry, hij , of the hypersurface and the matter fields, Φ . Importantly there is no explicit dependence on time, and this is what many physicists refer to as the problem of time. It is related to a similar problem with classical general relativity, which is a time parameterised theory, that

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treats time as a coordinate in an overall static four-dimensional block as discussed in chapter 4. In quantum gravity we see the same static wave function over a timeless superspace. The question here is: can we recover a time coordinate? The short answer is yes. We can obtain what Zeh (2007, 189) calls a semiclassical description of space-time geometry by substituting the WKB ansatz

Ψ ( hij , Φ ) = χ ( hij , Φ ) exp ª¬iS0 ( hij ) º¼

(6.88)

into equation (6.87). The Wheeler-DeWitt equation is then reduced to a Tomonaga-Schwinger equation (Lapachinsky and Rubakov, 1979; Banks, 1985),

i *ijkl

δ S0 δχ = H matter χ δ hij δ hkl

(6.89)

where we have used Einstein summation convention. The action, S0 , is an approximation obtained as a solution of the Hamilton-Jacobi equation of geometrodynamics (Peres, 1962). Inspection of the left hand side of equation (6.89) reveals that it may be succinctly written in the form i∇S0 ⋅∇χ . This expression can be rewritten by appealing to the definition of momentum in classical mechanics, see equation (5.20). The appropriate substitution is

i∇S0 ⋅∇χ = iq ⋅∇χ ≡ i

dχ dt

where we have used q to denote a superspace vector. From this it is straightforward to write equation (6.89) as

i

dχ = H matter χ . dt

(6.90)

With = = 1 we see that the familiar Schrodinger equation has been recovered, having a dependency on the universal time parameter, t, and where χ is the wave function for matter. If we let 3 R − 2Λ be small compared to the first term in equation (6.87), we see that the Hamiltonian,

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H matter , at a point in ordinary space, becomes the (00) component of the energy density operator, Tˆ 00 , as expected. This goes a long way to justifying the claim that physical reality is static at a fundamental level, an assertion accepted by most mainstream physicists. Moreover, many versions of the four-dimensional block universe described by general relativity are implicit in the Wheeler-DeWitt equation. Each of these versions or timelines, are what Zeh (2007) refers to as WKB trajectories. This is due to the use of the ansatz in equation (6.88) to approximate the universal wave function. Essentially this shows that time is merely a parameter along a path through the C-space whose trajectory is governed by the Wheeler-DeWitt equation. For me this nicely sidesteps Penrose’s original objection to non-Hausdorff space-time. These trajectories are DeWitt’s many worlds, residing in C-space, which is a Hausdorff manifold and a more fundamental backdrop than space-time. Because we still lack data for the structure of physical reality at the Planck scale, we are still left with many questions and many plausible hypotheses describing geometries of space-time at this scale. Such theories fall under various headings such as superstring or the related M-theory, twister theory, loop quantum gravity, and causal set theory to name just a few.

6.5.2 Superstring and M-theories M-theory (Duff, 1999, and references therein) evolved through the application of supersymmetry to higher dimensional space-time manifolds. The deployment of higher dimensional geometry is a throwback to Theodore Kaluza’s unification of general relativity and Maxwell’s electromagnetism in 1919, wherein general relativity is formulated on a (4+1) dimensional space-time. When the familiar (3+1) space-time of general relativity is stripped away we are left with a theory describing a single, spin 0 scalar field and a 4-vector electromagnetic field satisfying Maxwell’s equations. Later, in 1926, Oscar Klein suggested that the extra fourth space dimension be compactified to subatomic scales, similar to the surface direction normal to the length of a drinking straw (Založnik, 2012; Miller, 2013). Such theories became known as Kaluza-Klein theories and, upon learning of their existence, I have to admit I was sold. The emergence of Maxwell’s equations plus the idea that elementary particle masses could be determined through quantum theory by the scale at which extra dimensions were compactified, was just too much of a coincidence. Indeed it seemed as though parameters such as electric charge could be explained

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as a momentum of otherwise neutral particles orbiting around in a compactified dimension, with polarities determined by the direction of travel. However, I came to realise that the number of extra dimensions (Calabi-Yau geometries) needed to account for the whole of physical reality, seven in all, constituted too much potential variation. Picking out the correct Calabi-Yau topology and vacuum out of a possible 10500 (Blumenhagen et al, 2005, p85) makes the proverbial needle-in-a-haystack task seem rather trivial. For this reason I later became more persuaded by the grainy geometries of loop quantum gravity and causal set theory.

6.5.3 Grainy space-time: LQG and causal set theory Given the characteristic scale at which we expect significant deviations from classical behaviour due to quantum gravity, one cannot help wondering if the notion of a space-time continuum breaks down completely. If this is the case then this raises questions regarding details of geometry at the Planck scale. Causal set theory makes no assumption about such a geometry, as mentioned in chapter 4, it only assumes that discrete points are connected if they are timelike or null separated. Otherwise they are spacelike separated. This conjures up an image of a space filled by grains, possibly in the manner of grains of sand, but connected together by tiny threads in roughly one direction. In this way physical reality would be based upon a large collection of space-time atoms packed in at the Planck scale. However, mathematically the continuum is a more primitive object than any discrete structure. So it is legitimate to enquire as to the properties of the continuum in which such a discrete space is embedded. As far as I know this question currently has no answer, an issue that fortunately has little bearing on our conclusions. Causal sets have been adequately discussed already at the end of chapter 4, and we have seen that the growing block interpretation leading to a form of presentism, can be reinterpreted with a future consisting of a superposition of causal sets as opposed to one which is non-existent. Moreover, a superposition of causal sets could be thought to exist to the future of a past null cone with respect to a reference observer. Loop quantum gravity (LQG) on the other hand is based on a structure known as a spin network (Rovelli, 2003). Spin networks describe the spatial component of space-time, and when it evolves in time it is described as a spinfoam. Spin networks consist of nodes with links to other nodes, similar in some ways to causal sets. However, spin networks do not map to or exist in ordinary space, but rather space emerges from the network. This idea results from background independence, a key concept

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in relativity theory. The length of a particular link is proportional to the quantized area, in ordinary space, to which it represents. It turns out that these lengths are discretized in multiples of 1 2 and behave in a way similar to the spins of elementary particles, hence the designation spin network. For example a spin network consisting of a node with four links would represent a tetrahedron in ordinary space, with each link representing a two dimensional side. In general the area, A, of a surface, S, is related to a collection of links, representing S, with lengths, ji ( i = 1, 2," , n ) which, in Planck units ( G = = = c = k = 1 ), is given by n

A = 8πγ ¦

ji ( ji + 1)

i =1

(6.91)

where γ is a dimensionless quantity known as the Barbero-Immirzi parameter. It turns out that the value of γ has implications for black hole thermodynamics, and its value can be fixed by considering equation (4.35) for the Bekenstein-Hawking entropy. In Planck units this is S BH = A 4 . Now taking one link of a spin network with minimal length, equation (6.91) gives us the quantum area, A0 = 4πγ 3 . So we may regard a black hole enclosing n bits of entropy as possessing an event horizon with area A = nA0 . The number of states therefore, is

N = 2n = 2

(

A 4πγ 3

)

.

So the Bekenstein-Hawking entropy is given by

S BH = k ln N

=

A

4πγ 3

ln 2, ( k = 1) .

Applying equation (4.35) allows us to fix the Barbero-Immirzi parameter

γ =

ln 2

π 3

for further details see (Rovelli, 2003, 222).

(6.92)

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There is a remarkable consequence of discrete space-time theories, which is the observationally confirmed, cosmological inflation. Inflation is predicted without any reference to a scalar field acting as a cosmological constant. Application of the discrete form of the WheelerDeWitt equation to the early universe results in a greatly accelerated initial phase as suggested by cosmological observations (Rovelli, 2003, 216). Moreover causal set theory has been seen to predict low magnitude inflation in the present universe purely as a result of the assumed discreteness (Sorkin, 2007b). This is consistent with observations of supernovae at varying distances (Riess et al, 1998). So we already have some empirical evidence of the discreteness of space-time.

6.5.4 Space-C: a replacement for space-time in a timeless reality Given that we are accepting that reality is at root timeless, we need to address the question of the status of space-time. In general relativity time is treated as a coordinate of the geometry in the same manner as the spatial variables. Further considering the literal interpretation of quantum mechanics in conjunction with the discussion of this section, we are left with a C-space and a three-dimensional base space without any designated time variable. Moreover, C-space and base space appear as separate, incompatible objects. The question is: can they somehow be merged? In section 5.1 we introduced the notion of time as a single path through C-space. And prior to that in section 4.7 it was shown that a universal time parameter could be defined when we assume that spacetime is globally hyperbolic. This was illustrated by equation (4.38)

ds 2 = dt 2 − gij dx i dx j

(4.38)

in units where c = 1 and indices take on the values ( i, j ) = (1, 2,3) . We can now ask: can we define a metric for C-space, which will replace the time term in (4.38)? Using generalised C-space coordinates, q A , where A = 1, 2," , then a differential metric for C-space would, in general, look like

dt 2 = ζ AB dq A dq B

(6.93)

where ζ AB is the C-space metric tensor. This allows us to define the following metric for the manifold on which our augmented reality resides,

Quantum mechanics

ds 2 = ζ AB dq A dq B − gij dx i dx j .

253

(6.94)

This we describe as the differential metric for space-C, were the term ‘space’ stands for the familiar base space, but instead of adding time to provide a space-time manifold, we add the C-space metric defined in equation (6.93). This is denoted by the term ‘C’ in space-C. In this way our space-time becomes a sub-manifold of space-C with a combined metric tensor that can be represented in matrix form by

ªȗ 0 º « 0 −g » . ¬ ¼ The top right ( N × 3 ) and bottom left ( 3 × N ) elements, for an Ndimensional C-space, may be chosen to be zero matrices because every space-time sub-manifold is globally hyperbolic. In general however this does not need to be the case and consequently space and C-space may be merged into a general metric for the single manifold, space-C. This approach does not have any immediate bearing on our conclusions, but does provide a useful thinking tool when attempting to visualise the problem of democracy of basis in a quantum gravity context. Moreover, this enables us to view our own position within context of a self-consistent and significantly broader reality. In essence time is nothing more than an ordered sequence of timeless configurations.

6.6 Macroscopic-Bell states With the assumption of discrete space-time it is possible to make comparisons between quantum volumes of geometric structures or spaces that are relevant to a particular study. For example, if we look back at Figure 6-7 we see the assertion that the universal C-space has a dimensionality of approximately 10123 . This is a reflection of the population of elementary particles in the known universe, a population dominated by gravitons. It is reasonable to suppose that this approximates the number of bits in the universe. A question we can ask is: how large would a black hole need to be in order to accommodate this much information. We can approximate this through the use of the BekensteinHawking formula. The critical radius of a black hole, in Planck units is r0 = 2M . So its area is A = 16π M 2 , allowing us to write S BH = 4π M 2 . Equating this to the estimated quantity of information in the known

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universe ( 10123 bits) gives a mass of ~ 2 × 1053 kg, with a critical radius of ~ 31 billion light years. This is marginally less than a 46.3 billion light year figure based on observations from the Wilkinson Microwave Anisotropy Probe (WMAP), which factors in dark matter (Gott III et al, 2005). The assumption of discrete space-time immediately leads to a finite quantum volume of ~ 10 244 for the history and volume of the observable universe. With this in mind we may now consider quantum (configuration) volumes of discrete macro-Bell states that, although finite, can be many orders of magnitude larger than this. In April 2012 an article appeared in Nature (Britton et al, 2012) describing a quantum simulator involving a coherent state consisting of 350 qubits. The corresponding quantum state accommodates all possible classical states of 350 bits, that is all 2350 states of a of a 350 bit register. The quantum volume of an n qubit state is therefore n 2n , a form that reflects a product manifold between our three-dimensional base space and C-space (space-C volume). For 350 qubits this is about 10108 . The configuration volume (C-volume) of a 400-qubit state is approximately equal to 10123 , the classical bit count of the observable universe. This prompted renowned physicist Paul Davies to famously speculate that macro-Bell states may not be possible, in which case we would see an immediate objective reduction of any potential state above this threshold. This has been dubbed the Seth Lloyd limit, named after another equally distinguished physicist. No mechanism for such a collapse has ever been offered, but it would mean that the size of the universe is somehow implicit in smaller, more accessible sized regions, in the same sense that the design of the human body is implicit in every one of its cells. Britton’s 350 qubit simulator was tantalizingly close to this limit. I have italicised the word limit here because it tacitly assumes that it cannot be exceeded. However, in October of the same year another article appeared describing the formation of an ephemeral (a few tens of ps) squeezed vacuum state constituting 105 photons (Iskhakov et al, 2012). Notwithstanding its short life, its C-volume is 2100000  1030103 >> 10 244 . This greatly exceeds the Seth Lloyd threshold, as I now prefer to call it, and exceeds the quantum volume of the known universe by a slightly smaller margin. In the following year the same group of researchers generated an even larger coherent state consisting of 106 photons (Kanseri et al, 2013). In general we may consider configurations without any defined base space. Given that the quantum

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volume of our base space is B + 1  10183 then the dimensionality of a relative C-space for this number of space particles configured generally in an infinite dimensional void is B  10183 , thus increasing the dimensionality of the universal C-space by a factor of 1060 . This implies tight constraints on the evolution of the universe at energies well below the Planck scale. At present it seems that macro-Bell states of magnitudes shown are a long way from being deployed for quantum computational purposes. The main point however, is to simply demonstrate their existence. Large macro-Bell states are now an empirical fact, which in turn proves the existence of C-spaces large enough to accommodate them. Of course this does not prove the objective existence of the universal C-space, but it does make its denial increasingly difficult to justify. Detractors generally, are now burdened with the task of suggesting a yet higher ceiling for the size of a macro-Bell state, a limit even more difficult to justify. As far as I am aware no new ceiling has been suggested. This also provides reasonably strong empirical evidence against the classical growing block. In the eternalist model, physical reality is described by a universal quantum state containing a single path through Cspace, which up to the present we call the history of the universe. This state also envelops all possible futures because all possible eigenstates are included. The macro-Bell states so far discovered are microcosms of the universal state, so in just the same way they will contain C-space trajectories describing something akin to classical evolution. Given one such trajectory there would exist a point regarded as the present in relation to an appropriately positioned observer. Asserting the growing block model in this case would have the absurd consequence that the eigenstates contained within all potential futures of that present are magically selected for nonexistence. This is very significant and, as far as I know, represents the first empirical evidence favouring the eternalist’s model. The consequences of this are even more serious for presentists who claim that neither past nor future exists. Even the thick present proposed by Smolin (2013) comes nowhere near to accommodating the macro-Bell states demonstrated by the Erlangen-Nurenberg group.

6.7 Summary In this, the last chapter of part I, we have considered how the process of science initiated the most significant change in our knowledge to date–the beginning of quantum theory. Unlike relativity, which emerged in two distinct significant steps (the special and general theories), quantum

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mechanics grew into a full theory incrementally, involving many empirical inputs over a period of more than a quarter of a century. For the most part, the formalism of quantum theory most used is based on wave mechanics where the wave function, the object of interest, is distributed across a space of configurations accounting for all types of physical field, ultimately including gravity. The inclusion of gravity led to canonical quantum gravity (CQG), which is an effective unification of gravity and quantum field theory, but suffers from ultraviolet divergences and, due to a lack of empirical data, cannot model physics at the Planck scale. However, the Hamiltonian for matter fields can be updated appropriately as new physics beyond the standard model emerges. The Schrodinger form of the full theory can be derived from wave-particle duality emerging from the early quantum theory. However, as was pointed out in chapter 2, any theory can be a logical consequence of a non-unique set of axioms and quantum theory is no different. The axioms presented here were borrowed from Nielsen and Chuang (2000). Arguably the most significant of the four axioms was that describing composite systems, having consequences for the exponential size of coherent states and the C-spaces accommodating them. It seems that the most controversial aspect of quantum theory is the interpretation of the measurement process. It has been pointed out by Deutsch (1996) that, a pure wave theory logically follows from quantum theory. Other interpretations are merely rival theories to quantum mechanics. This of course does not rule out the possibility that one of these rivals is the truth. But until we acquire new evidence to support that, particularly as far as objective collapse is concerned, then we should continue to follow quantum theory. Aside from objective collapse, the other theories have the same formalism as quantum theory but may include extra ontological entities such as hidden variables, referring to the actual history as far as particle configurations are concerned. But like the potential hidden frame in relativity, we do not yet have evidence to rule out a privileged history. The only objection we have to hidden variables is a philosophical one based on the application of Ockham’s razor to minimize our ontology. The inclusion of gravity in quantum theory, which led to CQG became a springboard for a whole plethora of distinct theories consistent with known physics. The simplest of these being causal set theory and loop quantum gravity (LQG). These theories imply that space and spacetime are discrete at the Planck scale. Moreover, through space-time discreteness, they predict inflation and the accelerating universe we observe today. Moreover LQG is manifestly background independent. The

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discreteness of space-time allows us to straightforwardly assess quantum volume of the classical known universe and to show that the C-volumes of macroscopic-Bell states, now known to exist, can exceed this by many orders of magnitude. This provides tentative empirical evidence pointing to an ontic universal C-space. Although it is premature to make any definitive statements regarding the structure of physical reality, the evidence seems to be pointing to a static wave function inhabiting a universal C-space of almost infinite dimensionality. This being the case then the only way to explain our dynamic experience of reality is to consider our conscious selves as separate nonmaterial entities localised within the universal C-space, and in space-C generally. Moreover, our dynamic experience comes about because of our movement through the otherwise timeless landscape of space-C. In part II of this book we consider the philosophy of mind in relation to this landscape, and to use this to address the conflicting views that have emerged in both the recent and more distant past.

PART II: THE MIND

CHAPTER 7 PHYSICALISM AND ITS MOTIVATIONS

From a historical perspective physicalism has its roots in philosophical materialism; the view that everything is material or is entirely dependent on matter for its existence. This doctrine can be divided into two parts: (i) that there is only one kind of reality and that is fundamentally material and (ii) that humans and other animate life-forms consist not of two substances comprising a body and an immortal soul, but are purely body in nature. These ideas have a long tradition dating back to the speculative atomism of Leucippus (fifth Century BCE), Democritus (c460-370 BCE), and Epicurus (341-270 BCE). Briefly atomism was the ancient doctrine designed to give an account of changes and constituents in terms of the ultimate elements existing as atoms, their smallest units that are indivisible and indestructible. With the advance of the physical sciences atomism became the bedrock upon which physics and chemistry in particular, were built. This in turn gave rise to scientific materialism. Being applied to other sciences such as geology, biology, and physiology, the world along with all of the life-forms that occupy it could be explained in terms of configurations of atomic constituents, which included the central nervous systems of all animate life-forms. The German physiologist Karl Vogt (1817-1895) gained notoriety for his statement to the effect that thought relates to the brain just as bile relates to the liver and urea relates to the kidneys. But neither Vogt nor his better-known contemporary Ludwig Büchner (18241899) gave any clear explanation of the mind. During the twentieth century materialism split into dialectical materialism as espoused by Marx and Engels, and physicalism, which is strongly associated with the physical sciences (Acton, 1960). It is the purpose of this chapter to discuss the motivations and persistence of physicalism among many scientists and analytical philosophers up to the present. In chapter 3 we considered the role of science as a means for converging towards truth. In this objective, its efficacy in the material world is unrivalled as can be testified by its contribution to our current civilization. Notwithstanding its success there has been considerable debate over past centuries regarding the details of scientific methods. In

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loose conversation, reference is often made to the scientific method. But as we saw in chapter 3, there is no one such method. However, there is a common pattern that revolves around the relationship

Theory Ÿ Fact

(7.1)

where the Fact refers to evidence acquired by empirical means and the Theory is either a more general contrived statement or one deduced from other theories, designed to have an explanatory capability. For the sciences constructed to analyse the material world: physics, chemistry, biology, geology etc., the methods connected with (7.1) are and have been supremely successful. The resulting iterative process, in which the number of facts increases through experimentation and the theories are augmented or modified, converges inexorably towards the truth. Taking a reductionist view, physics is generally regarded as the most fundamental of the empirical sciences. So in the context of physics the so-called theory of everything, the aim of most theoretical physicists, is representative of the truth. However, this should be qualified so as to say the theory of everything physical. The belief system, in which the theory of everything physical is the theory of everything, is physicalism. It is noted here that physicalism is most emphatically not science. It is a belief system despite its strong association with the physical sciences. A major consequence of physicalism is that the mind is either physical or it is emergent from physical structures, where philosophers describe this by saying that the mind supervenes on the physical. Therefore the whole of reality, including minds, is at root physical. There are a whole host of reasons why this cannot be true, just as there is ample justification as to why it should be. It is the main purpose of part II, and indeed of this book, to discuss and analyse the existing evidence leading to the competing views for and against physicalism. In this chapter we consider the motivations for physicalism and to ultimately ascertain whether they have any reliable foundation. In this chapter we focus mainly on the work of three authors Papineau (2001), Stoljar (2016), and Seager (2014). For Papineau the reasons for the growing popularity of physicalism in the latter half of the twentieth century are centred on the establishment of the causal closure of physics. This is then maintained with an emphasis on arguments from fundamental forces and from physiology. Papineau suggests that for those antipathetic to physicalism, it was just the fashion of the day. Physicalists acquired their belief based on their high regard of the physical sciences. In this respect I think Papineau was right because, as

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I have hinted at already and will be expanded upon in chapter 8, substantial evidence against physicalism emerged during the latter half of the twentieth century. Stoljar claims that arguments from methodological naturalism are stronger than those from causal closure. Here we briefly explore the reasons for this claim and show that it has an Achilles heel due to one of its premises failing to address opposing dualist arguments. This is briefly examined in terms of the knowledge argument (Jackson, 1982). Seager on the other hand takes a somewhat different tack. For Seager the reason for the success of physicalism is based on the acceptance of epistemic physicalism, a doctrine that asserts the physical explicability of everything. Because such an argument cannot be based on observation of anything other than the physical world, it would seem that physicalists are begging the question. However, as long as we keep consciousness out of the equation the physicalist’s programme has been remarkably successful. But as Seager points out, consciousness throws up a significant roadblock on the road to total physicalism. Coupled with the circularity of their argument, the physicalist’s position does seem rather precarious. Furthermore, Seager, in his words, points to a disparity between epistemic physicalism and ontological physicalism, where the latter describes the physical nature of things. Seager provides an argument in Bayesian form, which highlights the disparity between the ontological and epistemic forms of physicalism. The definitions Seager provides for these distinct forms of physicalism are as follows. Let Y denote the physical substrate of the world around us, then for any phenomena, X, the definition of absolute epistemic physicalism reads, X is absolutely epistemologically dependent on Y iff it is impossible to understand X except via an understanding of Y (Seager, 2014, Preprint p5).

A corresponding definition of ontological physicalism may also read as, If7 X ontologically depends on Y then it is absolutely impossible for X to fail to exist if Y exists (Seager, 2014, Preprint p4).

For Seager the persistence of physicalism stems from the success of the epistemic form outside the context of the mind-body problem. For the wider scientific community the ontological form is the one most closely associated with the generic term physicalism. Seager’s disparity between 7

The word “If” is not part of the original quote.

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these two forms and the success of epistemic physicalism as a motive for physicalism generally, will be discussed later in this chapter.

7.1 The argument from fundamental forces This section explains physicalism from the standpoint of Papineau’s first argument, from fundamental forces. For Papineau the dominant reasons for physicalism’s increased popularity stem from this argument, more specifically the causal closure, or completeness of physics. In his preamble Papineau considers other possible causes for the rise of physicalism, most notably the decline in the early twentieth century of phenomenalism, the doctrine that all human knowledge is confined to or derived exclusively from phenomena (sensory input). Phenominalism, as a theory of mind, defines the mind as a cluster of related experiences (Quinton, 1989, 232). However, I do not see how phenominalism is necessarily antithetical to physicalism, especially since Quinton does not state whether or not the cluster of related experiences is supported by a physical substrate. Furthermore, Papineau points out that it is possible to deny phenominalism without embracing physicalism, and he alludes to philosophers who do exactly that. However, the rise of physicalism and the decline of phenominalism do seem to coincide around the middle of the twentieth century. Rather than physicalism filling a void left behind by phenominalism, Papineau believes that, through the force of arguments to follow, physicalism displaced phenominalism. In the late seventeenth century GW Leibniz postulated the conservation of momentum and kinetic energy. For impacting particles this limited physical scheme was complete provided the total energy in the system remained kinetic. All collisions had to be elastic so there was no energy loss due to any thermal mechanism, and there was no potential energy field that could store the energy of moving particles. As long as this remained the case there was no room for sui generis mental or vital forces. The behaviour of the system was perfectly predictable. Of course Leibniz’s system was incomplete. The debate about whether physics is complete or not, has changed several times over the intervening centuries, but nowadays we can be confident that the physical universe is a causally closed system. In chapter 5 we discussed the development of classical mechanics, which appeared in its original form in 1788 due to the work of Lagrange. Subsequently an essay was published (Laplace, 1814), which became colloquially known as Laplace’s demon. It describes an omniscient super intelligence (the demon) possessing knowledge of the

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positions, momenta, and potentials of every particle, large or small at one time. So this scheme, which included potentials, covered a wider spectrum of behaviour than Leibniz’s mechanics. The demon therefore, would be capable of ascertaining the precise configuration of the universe at any other time by following the laws of classical mechanics. In his introduction Laplace states We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. (Translated from Laplace, 1814).

Of course the application of classical mechanics in this way has been superseded by thermodynamics as a way of dealing with open systems. The point being made here is that with the application of classical mechanics, the state of a closed system at one time determines its state at any other time. Again there is no room for sui generis mental or vital forces that may influence such a closed system. This is determinism, but more importantly it is the foundation for the causal closure of physics–the central tenet of the motivation for physicalism as Papineau sees it. So how does Papineau cast his argument for physicalism? The main argument favouring physicalism according to Papineau relies on three premises, (i) the causal closure of physics, (ii) causal influence, and (iii) no universal overdetermination. The whole argument is structured as follows. Premise 1 (the completeness of physics): All physical effects are fully determined by prior physical causes. Premise 2 (causal influence): All mental occurrences have physical effects. Premise 3 (no universal overdetermination): The physical effects of mental causes are not all overdetermined. Conclusion: Mental occurrences must be identical with physical occurrences.

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This causal argument has a number of variations, as Papineau explains, but the canonical form shown here (Crane, 1995; Sturgeon, 1998) is sufficient to cover the main points without becoming embroiled in numerous minor details. The establishment of the conclusion here would undoubtedly seal the physicalist’s programme. If we accept all three premises then we must accept the physicalist’s conclusion that the mind supervenes on matter, or more specifically the mind supervenes on the brain. However, the conclusion may be denied by rejecting at least one of its premises. We can consider each in turn. Personally I would regard rejection of premise 1 with a degree of incredulity. Considering the observations and experiments of natural philosophers going back at least as far as the first millennium BCE, on close inspection there has been no credible instance of a physical effect without a physical cause. Had such an event occurred in history then it would have been investigated further to establish its repeatability, and an affirmative conclusion would have left us with an entirely different paradigm to the one we have today. So it is my intention to proceed as though the physical domain is a closed system from a causal perspective. Papineau follows this causal argument with a few comments relating to the argument overall and more specific remarks about premises 2 and 3. I have already made my position regarding premise 1 clear. Like Papineau I will consider premise 3 before discussing premise 2. The rejection of premise 3 would require that some mental causes of physical effects are overdetermined. This is to say that a physical effect, retracting your arm suddenly as a pain response for example, has more causes than is required for the effect: (i) a physical cause (because we are accepting premise 1) and (ii) an additional mental cause due to the pain felt. Counterfactual dependence of the distinct physical and mental causes (Segal and Sober, 1991; Mellor, 1995, 103-5) is a way of defending this form of overdetermination. This does raise the question of the exact relationship between these causes. Papineau alludes to possible causal mechanisms, but then goes on to say that there seems no good reason for believing in them. I think I would concur. Counterfactual dependence relies on highly improbable coincidences taking place on a regular basis. In a probabilistic context this would seem highly unlikely. Another way of describing premise 3 is that put forward by Stoljar (2016). Here it is described as the exclusion principle8 of which there are a number of different forms. The form expressed by Stoljar is 8

This is completely unrelated to Pauli’s exclusion principle for quantum states of multiple fermions.

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Chapter 7 If an event e causes event e*, then there is no event e# such that e# is nonsupervenient on e and e# causes e*. (Stoljar, 2016, 23).

This is denied by asserting the presence of both events e and e# that both cause e*, therefore the exclusion principle denies overdetermination, agreeing with premise 3. The classic counterexample to this is the one of the firing squad in which both the firings, by soldier A and soldier B cause the death of the prisoner. The firings, being distinct events, constitute overdetermination. While this represents an exception to the exclusion principle it is unclear whether it has any bearing on the mind-body problem. The firings of both soldiers take place within the physical domain, so as an example it is rather moot. There are likely to be many arguments possible that promote overdetermination. However, for the purpose of this work I will accept premise 3 that mental causes are not overdetermined. So in order to deny physicalism in the context of the causal argument it is essential to deny premise 2. Deniers of premise 2 usually resort to some form of epiphenomenalism, which is the claim that mental states do not have physical effects. However, my own position with regard to epiphenomenalism is one of considerable suspicion. Papineau refers to functionalism as a closet version of epiphenomenalism. But functionalism, also according to Papineau, is a popular version of physicalism. In other words, although epiphenomenalists may claim that minds exhibit behaviour independent of the physical, they also claim that minds are dependent on the physical substrate for their existence. Therefore, in the context of this work epiphenomenalism is just another form of physicalism. Papineau refers to mental states as being second order, by using the phrase referring to a mental state as a …state-of-having-some-state-which-plays-a-certain-role, rather than with the first-order physical state which actually plays that role. (Papineau, 2001).

An example may be that of a computer program running on a physical machine. The program may be in a given state at a particular instance. This state has a dependence on the program, which in turn is dependent on the physical machine for its existence. The state is second order. Papineau then goes on to say that functionalism represents a serious threat to epiphenomenalist’s denial of premise 2, for reasons already described. What is essential to the anti-physicalist is to deny premise 2 by arguing in favour of minds that are independent with respect to both behaviour and existence. Epiphenomenalists succeed with the former but fail with the

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latter. With regard to this type of strategy, Papineau in a previous paragraph says I leave it to readers to decide whether this denial of the efficacy of the mental is a price worth paying to avoid physicalism. (Papineau, 2001).

In what context is the denial of the efficacy of the mental, taken? If it were the efficacy within the physical domain then it would seem that, for Papineau, the price is too high. However, as we shall see in chapter 8, given the current paradigm of modern physics, this kind of approach is an entirely natural one for anti-physicalists.

7.2 The argument from physiology This second argument that Papineau cites, represents another line of reasoning in support of physicalism. At root it has a connection with the argument from fundamental forces via the key originator of the conservation of energy, Hermann von Helmholtz (1847). Helmholtz was in fact medically trained and completed his studies under Johannes Müller at the Berlin Physiological Laboratory in the early 1840s. Along with fellow students Emil Du Bois-Reymond (1818-96) and Ernst Brücke (1819-92), Helmholtz pursued a reductionist programme in physiology aimed at showing that all dynamic processes in the bodies of animals, including humans, could be understood in terms of laws that govern the rest of the physical domain. What Papineau describes as noteworthy is the failure of both theorists and experimentalists in the rational mechanics tradition to postulate the conservation of the single quantity, energy. Because of the diverse and varied processes taking place in human and animal bodies, Helmoltz was motivated to make this connection and to propose the conservation of energy where, barring any thermal losses, potential and kinetic energies varied in such a way as to preserve the sum of the two. While other researchers in the field of classical mechanics were investigating the concomitant variation of potential and kinetic energies, it was Helmholtz who finally made the connection. What Papineau also notes is that conservation of energy per se does not rule out the existence of other forces that may be labelled as vital or mental, provided they are conservative. Indeed, although it was possible to reduce all forces to a small number of fundamental interactions, Helmholtz could not have known about the nuclear forces, the effects of which were initially detected towards the end of the nineteenth century.

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Therefore, any number of new interactions could be introduced provided they were conservative. The notion of sui generis mental or vital forces persisted until after World War I. It was not until the twentieth century that experimental technology caught up with the hypotheses developed during the late nineteenth century. Prior to this, the best that could be achieved was the assessment of chemical inputs and outputs to and from specific processes. One example concerned an experiment by Max Rubner in 1889 in which calorimetric measurements of movement and respiration of a small dog exactly corresponded to the food it consumed (Coleman, 1971, 140-3). During the early twentieth century it became increasingly difficult to posit anomalous vital forces. Much became known about the biochemical and neurophysiological processes, even at a cellular level. The protein constituents of enzymes and basic biochemical cycles were identified, which led eventually to the discovery of DNA (Watson and Crick, 1953). During the century between the 1850s and 1950s researchers progressively probed biological organisms towards molecular and atomic scales. At no point either then or since has any anomalous interaction been discovered beyond the forces familiar within the rest of physics. Of course it is always possible to say that absence of evidence is not evidence of absence. However, in this case all of the well-known and repeatable biological processes can in principle be understood in terms of fundamental forces already known. This nicely ties in Papineau’s second argument with his first.

7.3 The argument from methodological naturalism This is an argument put forward by Daniel Stoljar (2016, 23), which is based on the application of scientific approaches to metaphysics. It has two premises and its structure is as follows. Premise 1: The methodological approaches applied to the natural sciences should also be applied to metaphysics. Premise 2: …as matter of fact, the metaphysical picture of the world that one is led to by the methods of natural science is physicalism (Stoljar, 2016, 23). Conclusion: It is rational to believe in physicalism (Stoljar, 2016, 23).

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Premise 2 and the conclusion are exact quotes, whereas premise 1 here is my own phrasing, which embodies the idea that all metaphysical models should be consistent with publicly accessible and repeatable facts as encapsulated by the relation (7.1). Stoljar goes on to discuss how people might respond to this argument. With regard to premise 1 he states that very few people are likely to contest premise 1, a view with which I would entirely concur. Stoljar claims that this argument is rather more persuasive than the one from causal closure, and I would agree that the first premise is a strong one. However, on the face of it premise 2 does look rather weak, and it is tempting for anti-physicalists to reject the second premise. From my own standpoint I see premise 2 as begging the question. This is because the methods of natural science, by and large only probe the physical domain. So they do not address opposing dualist arguments, and we are in danger of projecting the physical world onto our metaphysical model and thereby concluding physicalism. To be fair Stoljar does provide a get-out clause, by stating that methodological naturalism does not deny that other views are possible. Although some of the sciences possess an explanatory autonomy in that in certain epistemological respects some sciences cannot be reduced to others, these are not necessarily incompatible with physicalism. Methodological naturalism merely states that physicalism is the most likely metaphysics at present, while it is still possible to view the world in non-physicalist terms. In an earlier statement Stoljar indicated that in order to appreciate the force of premise 2 it is important to appreciate what physicalism is, and more importantly what it is not. Prior to this Stoljar engages in a lengthy discussion of subtly distinct forms of physicalism of which I will not go into here. So I had better clarify my own view of physicalism. In the introduction to this section I gave a loose definition of physicalism to be: A theory of everything physical is a theory of everything. This was given in the context of the aim of theoretical physicists to unify all of physics from a theoretical standpoint. The problem with this is that it is rather epistemic. What is required here is an ontological definition, which may be read as: Everything physical is everything.

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Although this lacks the sophistication of the definitions given by Stoljar, it should serve our purposes very well. Stoljar’s final point on this concerns the knowledge argument (Jackson, 1982). This describes a thought experiment in which a subject, Mary who happens to be an omniscient neuroscientist, has been isolated her whole life in a closed monochrome world. Her environment consists solely of surfaces that are merely shades of grey. This includes display screens, pictures, literature etc. as well as natural objects, and she is forced to learn about the world solely through the media she has available. Notwithstanding these difficulties she has learned, and therefore knows all that physical theory can teach her. Yet upon release into the wider world she immediately discovers that she did not know everything, because she is exposed to the new experience of seeing colour. The knowledge argument is structured as follows Premise 1: Before her release Mary knows everything physical there is to know about other people. Premise 2: Before her release Mary does not know everything there is to know about other people, but does learn something new about them upon her release. Conclusion: There are truths about herself and other people that escape the physicalist story. For clarity we will take two separate quotes by Stoljar then symbolically analyse the corresponding arguments in turn. For the analysis let P, KA, and MN denote physicalism, the knowledge argument, and methodological naturalism respectively. The first quote is: Finally, one might be inclined to appeal to arguments such as the knowledge argument to show that physicalism is false, and hence that methodological naturalism could not show that physicalism is false. …if successful the knowledge argument suggests, not simply that physicalism is false…. (Stoljar, 2016, 24).

But moreover that any approach based on methodological naturalism must be false. In symbols this becomes:

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KA Ÿ ¬P MN Ÿ P (Argument from methodological naturalism) ¬P Ÿ ¬MN (Contrapositive)

( KA Ÿ ¬MN ) ⇔ ( MN Ÿ ¬KA) So we may conclude that the knowledge argument falsifies methodological naturalism or vice versa. However, Stoljar goes on to say: But if that is so, it is mistaken to suppose that the knowledge argument gives one any reason to endorse anti-physicalism if that is supposed to be a position compatible with methodological naturalism. (Stoljar, 2016, 24).

So if we insist on compatibility with methodological naturalism, we have:

MN Ÿ ¬ ( KA Ÿ ¬P )

( KA Ÿ ¬P ) Ÿ ¬MN (Contrapositive) MN Ÿ ¬ ( P Ÿ ¬KA ) Therefore Stoljar breaks the relationship between physicalism and the knowledge argument through the use of methodological naturalism. In addition he also shows that methodological naturalism and the knowledge argument deny each other. The question is: which of these two arguments is the stronger? Earlier in his article Stoljar discusses three objections to the knowledge argument. The first is the ability hypothesis, which draws a distinction between propositional knowledge such as Mary knows that snow is white, and practical knowledge such as Mary knows how to ride a bike. It is claimed that because Mary does not gain propositional knowledge upon release then the knowledge argument is invalid. My response is that the ability to perceive colour is no less important than any propositional knowledge. Moreover the ability hypothesis does not specify which aspect of Mary gains the practical knowledge. If the ability to perceive colour is one of Mary’s nonmaterial mind and not her physical aspect, then the ability hypothesis is invalidated. If however, we have to assume that Mary’s mind is physical then followers of methodological naturalism are tacitly begging the question. The second involves the distinction between a priori and a posteriori physicalism where the former says that physicalism can be known through purely rational means and the latter requires empirical

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input; see for example Howell (2015). A symbolic definition of physicalism is

S Ÿ S*

(7.2)

where S specifies the physical nature of the world and S* specifies its total nature. The claim is that because Mary has not had the experience to know (7.2), then she does not know (7.2). However, the fact that Mary has not had the experience to know (7.2) does not mean that it is untrue. The flaw in this response is that it equates the status of (7.2) with the knowledge or ability of our material selves to perceive colour, this is circular and therefore lacks any firm grounding. The third response is based on the distinction between a theoryconception and an object-conception of the physical. The first premise of the knowledge argument may be read with either conception in mind. If Mary’s is a theory-conception, then it is argued that an object conception of the physical will escape the knowledge argument. However, this distinction is nonexistent if we assert that the knowledge argument is idealised. Therefore an object-conception of the physical will not evade the knowledge argument. For the above reasons I disagree entirely with Stoljar on his claim that the argument from methodological naturalism is any stronger than the one from causal closure. Moreover it seems weaker than the knowledge argument particularly because of its premise 2. The causal closure of physics seems to be on a much firmer footing, logically, than methodological naturalism, especially given the weakness methodological naturalism’s premise 2, which is inherently circular. To me this is the reason, as Stoljar points out, that methodological naturalism has had little attention in the literature compared with the argument from causal closure. We now consider an entirely different motivation for physicalism.

7.4 The persistence of epistemic physicalism There is no denying that physicalism generally has gained broad acceptance throughout the philosophical and scientific communities. William Seager (2014) provides clear reasons for this by reminding us of the reductive epistemological relationship between the empirical sciences, with physics firmly at the substrate. This is one of four motivations for physicalism mentioned by Seager. These motivations as listed are

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1. 2. 3. 4.

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Unparalleled scope, scale, and explanatory power of physicalist metaphysics Induction Intrusions from below Methodological integration.

Seager lists examples of these, so for details the reader is referred there. However, a few words about each should provide a cause-grained overview of the motivations for physicalism. As Seager points out, the first motivation requires little explanation. It is based on the fact that during the progress of science to date, there has never been an instance of a phenomenon that cannot be reduced to the physical processes involving a few species of elementary particle and the currently understood interactions between them. The second motivation is referred to as induction because of its connection to the process of proposing theories from empirical facts. That is by going from fact to theory in the relation (7.1). This is a general pattern of science, which in its current form has enjoyed incredible success over more than four centuries, a success based in part on the presupposition of physicalism by investigators. Seager calls this optimistic induction and qualifies this in a footnote characterising pessimistic induction by the fact that every theory that has ever been superseded by an up-to-date version, has been false. Optimistic induction is therefore applied to physicalism because it has never failed to survive a paradigm shift, a powerful motivation indeed. The third motivation for physicalism Seager describes as intrusions from below. This relates to the epistemological dependence hierarchy exhibited by the sciences with physics firmly placed at the lowest substratum. This also relates to hierarchies of scale where macroscopic behaviour has a dependence on microscopic processes at a scale that only physics can access. Although in many cases macroscopic behaviour may appear epistemologically independent of the micro world, Seager offers the example of thermodynamics deduced from statistical mechanics by Boltzmann. Here, as Seager points out, if all of the momenta in a macroscopic system were negated, we would witness impossible phenomena such as a shattered wine glass spontaneously reassembling itself on a table from floor based shards. Such a low probability event would be described as an intrusion from below. While such an event would temporarily violate the second law of thermodynamics, because of its stochastic nature it could still be explained on the basis of the more fundamental theory of statistical mechanics.

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The fourth motivation, Seager suggests, is not really an argument at all but an attitude. This is methodological integration in which philosophers side with the methods of science to advance the metaphysics of physicalism. Just to be clear, this is not methodological naturalism discussed by Stoljar, but it could be argued that it is without premise 2. The motivation here is the promotion of a monistic, pure metaphysics unencumbered by external elements including the supernatural. I would agree with Seager that the desire for an affinity with science is a powerful motivator. Many who adopt this stance would say that they are just being rational. However, this is a dangerous position to take, and some may say irrational, especially if it can be shown that the physical domain is somehow incomplete via, for example, stochastic processes in quantum theory. But this is a discussion for the next chapter. Another approach that Seager takes is to examine the relationship between ontological and epistemic physicalism. In the introduction to this chapter definitions of absolute epistemic and ontological physicalism were quoted. Seager seems to prefer working with reductive epistemic physicalism, which is a slightly weaker version than the absolute form. This definition may be read as X is reductively epistemologically dependent on Y iff9 it is possible to understand X via an understanding of Y (Seager, 2014, Preprint p5).

Moving on to ontological physicalism, this is based on ontological dependence, which may be defined as follows …when X ontologically depends on Y then it is absolutely impossible for X to fail to exist if Y exists (Seager, 2014, Preprint p4).

Based on this we can define ontological physicalism as the claim that everything ontologically depends on the physical. We now turn our attention to the relationship between ontological and reductive epistemic forms of physicalism. Seager presents this relation using the following quote If physicalism is true then any phenomenon will stand in (at least) a reductive epistemological dependence relation to the physical (Seager, 2014, Preprint p11).

A more symbolic and systematic way of stating this is to say 9

Seager uses an “=” sign in the preprint here.

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Ontological physicalism Ÿ Reductive epistemic physicalism . (7.3) It is as though the facts of physicalism are presented as knowledge gained from observations while the ontological form is the theory arrived at by induction from the facts. This is entirely consistent with the methods of science, and it therefore comes as no surprise that physicalism has been so popular. Therefore, all we need in order to deny ontological physicalism is to demonstrate a phenomenon that contradicts the right hand side of (7.3). According to Seager that phenomenon is consciousness. However, in the passage relating to this relationship he goes on to suggest that one way of doing this is to highlight an independent variation between some property of the mind and the physical. Seager gives two reasons why this is difficult: (i) there is ample evidence for concomitant variation between conscious states and brain states, and (ii) hypothetical variation between any consciousness-physical links is invisible. As we will see in the next chapter this is no longer the case. Notwithstanding these difficulties Seager points to the lack of any evidence indicating an epistemological dependence of consciousness on the physical. The success of physicalism in other areas coupled with the reluctance of consciousness to fit in with the physicalist’s programme suggests that such epistemological dependence is nonexistent. If this is the case then the claim of ontological physicalism collapses. This is Seager’s disparity between the epistemological and ontological forms, and he presents this as evidence against ontological physicalism. In other words the success of physicalism in an epistemological sense is ultimately its downfall. This may very well be the case but there remains room for doubt. Seager presents these ideas in a probabilistic form as follows. Let E be the statement consciousness is reductively epistemologically dependent on the physical, and let P denote physicalism is true. Then the long history of physicalist success is denoted by the conditional probability, p ( E P ) , being high, that is

p ( E P) ≈ 1 .

(7.4)

Using the identity

p ( E P) ≡

p ( E ∩ P) p ( P)

(7.5)

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we can argue the following. Equation (7.5) implies

p ( E P) p ( P) ≡ p ( E ∩ P) . But by the symmetry of E and P, we have

p ( E ∩ P) ≡ p ( P E ) p ( E ) .

(7.6)

This allows us to write

p(P E) ≡

p ( P) p ( E P) p(E)

.

(7.7)

The problem here is that we have no indication of the absolute probabilities, p ( E ) and p ( P ) . To get an idea for the value of p ( P E ) Seager argues that the general success of reductive epistemic physicalism coupled with the lack of an established reductive relationship for consciousness, suggests that the probability that physicalism is true given that we have a reductive relationship for consciousness, is high. That is it could be argued that p ( P E ) = 1 . With this entirely reasonable conclusion being the case, we see an almost total overlap of the regions for E and P in the corresponding Venn diagram (Figure 7-1). However, this will only tell us that p ( E )  p ( P ) , it gives us no indication what these absolute probabilities are. As suggested by Seager, it is entirely possible that the regions, E and P, are vanishingly small indicating that ontological physicalism is false. However, the absolute probabilities could just as easily be high. Previously we have mentioned that the existence of a property variation between consciousness and the physical would deny ontological physicalism. In the chapter to follow we will see that there is variation in abundance.

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E

P S Fig 7-1: Venn diagram representing probabilities of reductive epistemological dependence (E) and ontological physicalism (P). Significant overlap of these

(

)

(

)

regions suggest p E P ≈ p P E ≈ 1 , while the absolute size of these regions remains unknown. They could, as Seager suggests, be extremely small. Hence denying physicalism. But this is by no means guaranteed. Note p ( S ) ≡ 1 .

7.5 Summary Throughout this chapter we have considered various key motivations for physicalism and how it relates to the active topic of debate, the mind-body problem. Most of what can be said here is best summarised by David Papineau (2001) and William Seager (2014), along with methodological naturalism discussed by Daniel Stoljar (2016). Papineau considered motivations based on arguments from fundamental forces, which is centred on the causal closure of physics. Papineau also discusses arguments from physiology where it is demonstrated that all biological processes can be reduced to physical ones. Following that we considered Stoljar’s treatment of methodological naturalism, which is best described as metaphysics maintaining solidarity with the natural sciences. Finally we considered Seager’s reasons for the popularity of physicalism. Here we see that physicalism can be discussed outside the context of the mind-body problem, where there is little debate. However, for Seager consciousness erects a significant roadblock on the route to total physicalism. For Seager the main motivation here is the success of reductive epistemological physicalism throughout the sciences. There is no debate here as long as consciousness is excluded. However, the ultimate destination of

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ontological physicalism, bringing the mind into the discussion, presents a significant impasse. My own position on motivations for physicalism, is that Papineau’s arguments from fundamental forces are the most convincing. Arguments from physiology rely on the absence of any vital special interactions within biological systems. While I believe this to be true, the argument itself is flawed. As the old saying goes absence of evidence is not evidence of absence. However, biological processes can in principle be reduced epistemologically to physical interactions. This nicely brings us back to arguments based on the causal closure of physics. Stoljar’s methodological naturalism seems fine as long as we leave out premise 2, without which we appear to arrive at the last of Seager’s four motivations, methodological integration. Not many would argue with this, but methodological naturalism’s premise 2 does look rather circular. It reminds me of the logic of the lamppost. Suppose you arrive at your front door in the dark and a little tipsy. Reaching into your pocket you find you’ve lost your keys. The argument goes that when you retrace your steps in your search, it makes sense to look for them under lampposts where the ground is lit, because if they have been lost in the dark you will not find them anyway. So if you apply premise 2 to the search for consciousness, this suggests searching within the physical domain. But this does not rule out the possibility that something with the property of consciousness is non-physical. And if it were you would never find evidence of consciousness within the physical realm anyway. This is the weakness of methodological naturalism. Seager’s main approach is to apply a probabilistic Bayesian argument to show that the success of physicalism assuming we exclude the mind is based on a reductive epistemological argument. But every attempt thus far, to bring consciousness into the physicalist’s programme has failed, strongly suggesting that consciousness is a property of something non-physical. Descriptively this seems fine. However, Seager’s Bayesian analysis provides no way of evaluating the absolute probabilities of either physicalism being true, or of consciousness being reductively epistemologically dependent on the physical. So physicalism is not invalidated, and the reductive epistemological dependence of other empirical sciences on physics is a powerful motivator for physicalism generally. Again, in my view this is not quite as convincing or as powerful as Papineau’s argument from fundamental forces. As long as we are assuming a dynamic physical universe the arguments centred on the causal closure of physics seem fairly watertight. However, the dynamic universe as we perceive it is based firmly on the

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classical mechanics of the eighteenth and nineteenth centuries. As we will see in the next chapter, within the twentieth century paradigm of relativity and quantum theory, the relationship between the causal closure of physics and physicalism is totally severed.

CHAPTER 8 MIND AND THE PRINCIPLE OF LOCALISATION

Up to the time of the renaissance people’s beliefs were heavily dependent on a wide variety of religious doctrines each underpinned by some form of mind-body dualism. That is, after physical death there was believed to be some other non-physical realm to which the self or soul migrated. While alive the soul inhabited or was associated with the body to which it had some measure of control even though no one had any idea of the mechanisms behind this relationship. René Descartes (1596-1650) attempted to narrow a potentially wide range of imagined possibilities by suggesting a separate nonmaterial soul interacting with the body at the site of the pineal gland, located at the top of the brain stem in the centre of the cerebral cortex. Since that time many variations of this approach have become collectively known as Cartesian dualism, that is, a nonmaterial entity with the faculty of experience and free will animates the bodies of humans and animals by some, yet to be determined, mechanism. As we have seen in the previous chapter however, anything resembling Cartesian dualism gradually fell out of favour as new scientific paradigms took hold. Throughout the eighteenth century there still remained the possibility that vital forces of some kind that played a role within biological systems, and this approach kept alive the hope that our biology could be influenced by some non-physical aspect of ourselves. However, the causal arguments highlighted by Papineau (2001) in particular became by and large responsible for killing off any form of Cartesian dualism altogether. The long and consistent road up until the 1950s that bore witness to the decline of Cartesian dualism provided considerable impetus to the opposing view, and therefore physicalism became particularly entrenched. Within the current paradigm in which biological processes can be understood in terms of fundamental physical interactions, the search for anomalous behaviour of such complex systems that might be regarded as evidence for nonmaterial interactions has lost most of its impetus. However, also during the twentieth century the new paradigm of modern physics became well established. This began with relativity, which was

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inspired by Maxwell’s theory of electromagnetism. In 1915 this evolved further into general relativity by the inclusion of gravity via the principle of equivalence. As discussed in chapter 4 a significant aspect of the general theory is the four-dimensional block space-time originated by Hermann Minkowski in 1908. Coincidently this appeared in the same year that the philosopher JME McTaggart published his famous work The Unreality of Time, which arrived at conclusions consistent with Minkowski’s model. Notwithstanding the emerging problem of time, the doctrine of physicalism in the mind-body context had become so entrenched even though it is based on a dynamically interpreted classical mechanics. Throughout the twentieth century many theoretical physicists embraced the static picture that general relativity offered while at the same time, tacitly assuming physicalism. The resulting impasse became known as the grand illusion, and is still described as such today. If we are to take seriously the eternalist model of general relativity, then there is no mystery here. What has happened, in the context of eternalism, is that scientists have assumed physicalism as an axiom then arrived at a contradiction, the grand illusion of change. Classical logic tells us quite simply that physicalism is false. Otherwise we must unpick general relativity and form a new theory in its place. This is summed up in Weyl’s now famous quote presented in at the beginning of this book. Taking Weyl’s quote (Weyl, 1949, 116) seriously entails the existence of non-physical minds moving along their lifelines within a static space-time, hence providing our experience of change. However, although this classical model denies physicalism, there remains the problem of free will. In classical general relativity there is but one predetermined unknown future thereby denying free will. Free will is finally rescued by taking seriously all of the consequences of pure wave quantum theory to the exclusion of all other interpretations. These other interpretations are designed to reconcile our dynamic experience with the static configuration space of quantum theory. It may be speculated that they represent physicalist’s attempts to rescue their metaphysics. As we saw in chapter 6 every one of these rivals has its own problems. But the most difficult of these to dislodge is nonlocal hidden variables. This is because it does not deny the formalism of quantum theory it only posits a special history, which may deny free will but not neo-dualism entirely. To preserve physicalism we must deny physical time and resort to some form of presentism. Furthermore Bohmian mechanics, often associated with hidden variables theories, does not require a privileged history. All possible histories naturally emerge from Bohmian mechanics, and are

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encapsulated within the pure wave theories. Therefore a conscious mind located at a given point in the universal C-space will have a unique past and the freedom to choose amongst a multitude of available futures. At any instant, a mind exists at one location in C-space because we only experience one configuration at a time, not some superposition of many. The principle of localisation, as I have called it, is where minds have the unique property of being localised in C-space. This is not a new idea, Hans Dieter Zeh uses the term localisation of consciousness when referring to Everett’s (1957) interpretation, where conscious minds are localised not only in space-time but also in Hilbert space components (Zeh, 1970, 74). In the previous chapter it was mentioned that Seager (2014) had challenged us to show independent variation between some property of nonmaterial minds and the physical. Well here is that variation–conscious agents are localised in C-space whereas the wave function, representing the physical domain, is continuously distributed over that same space; the variation is manifest. The first to explicitly publish these ideas in the context of pure wave quantum theory was Albert and Loewer (1988) in their seminal work Interpreting the Many Worlds Interpretation. In my view this work has had the greatest influence to date, not least because of the many attempts to discredit it, but consideration of these will be deferred until the next chapter. Moving forward however, it would be a travesty to overlook the contribution made by Eugene Wigner (1961). In the next section we briefly revisit the Wigner’s friend scenario and the context of quantum observations by external observers, which later became known as consciousness causes collapse. This is followed by a longer discussion of Albert and Loewer’s work, which casts the same ideas in the context of Everett’s relative state. Before summarising this chapter we examine other sources of support for a localised consciousness interpretation in the context of relative state.

8.1 Consciousness causes collapse As far as I am aware Wigner’s (1961) article, Remarks on the Mind-Body Question, is the earliest overt publication connecting the quantum measurement problem with consciousness. This essay was also reprinted in Symmetries and Reflections, Scientific Essays in 1967. It is undoubtedly true that many of Wigner’s predecessors would have had thoughts along these lines, but would likely have only mentioned them in passing. It seems that Wigner’s view at the time echoed the existence of a growing belief amongst his colleagues that the spirit of Descartes’ “Cogito ergo

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sum” was once more on the rise. However, according to Wigner, there were still adherents to the old materialism who were persuaded by arguments along the lines of Laplace’s demon. What seemed ironic to Wigner was that the proportion of these adherents seemed larger amongst biochemists than physicists. Nowadays this is no surprise because we know that physicists, particularly those investigating the foundations of physics, were in a greater position to appreciate at least a vague relationship between quantum theory and consciousness. In the section following his introduction Wigner considers the language of quantum mechanics, in which he describes the wave function of an object as representing all of the knowledge we could possess about it10. In particular with full knowledge of an object’s wave function it is possible to assign probabilities to all its possible future states. That is the wave function is entirely deterministic while the behaviour of the object is not. We have already discussed quantum mechanics at length in chapter 6 therefore we do not need to pursue this any further. All that needs to be said here is that Wigner recognised the incompatibility of physicalism with quantum mechanics, which is testified by the sentence Solipsism may be logically consistent with present quantum mechanics, monism in the sense of materialism is not. (Wigner, 1967, 176).

In the following section Wigner provides reasons for physicalism. But he begins by giving two arguments against it, (i) it is incompatible with quantum theory and (ii) the mind and its processes are primary concepts with knowledge of the world being part of its contents. Here Wigner asserts that (ii) is the principle argument against physicalism, though his reasons appear vague. I would certainly agree that the mind is irreducible, but a counterargument could claim that its contents are mere physical memories. In this book we argue for (i) to represent the stronger argument to counter physicalism since it has a stronger grounding within science. Moving on, Wigner goes on to suggest a reason for physicalism that has to do with the devotion that researchers have to their particular disciplines. This is summarised by the following passage, where the part between quotation marks is, for example, a feasible assertion by a scholar of the eighteenth century specialising in classical mechanics. …"Light may exist but I do not need it in order to explain the phenomena in which I am interested." The present biologist uses the same words about 10

This does not deny that the wave function is ontic, only that it has epistemic content, as one would expect.

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Chapter 8 mind and consciousness; he uses them as an expression of his disbelief in these concepts. (Wigner, 1967, 177).

Although we can never be sure of people’s motives this does have a ring of truth about it. Any eminent researcher will only be interested in those factors that have a bearing on his/her particular field of study. Biologists will only find interest in facts and theories that affect growth, decay, and behaviour of biological organisms. It is interesting that Wigner tacitly assumes that consciousness has no direct bearing on biology. This is because he recognises that consciousness only has a bearing at the foundations of physics. If biological systems are affected then it will only be via the physical substrate on which biology depends. However, I am not sure that disbelief in a non-physical consciousness is mere disinterest in every case. Human motives are extremely varied and as we will see in the next chapter it is not difficult to cite individuals who vehemently champion mind-body physicalism. From chapter 6 we find that the Wigner’s friend gedankenexperiment is essentially an extension of the Schrodinger’s cat scenario even though he makes no reference to cats. In Wigner’s scenario his friend measures a quantum state ψ and observes either ψ 1 if he saw a flash or ψ 2 if he did not. Before Wigner interrogates the system as a whole its state is α ψ 1 ⊗ χ1 + β ψ 2 ⊗ χ 2 quantum theory, where

χ1

and

χ2

as expected from orthodox

are the states of the friend’s

memory according to whether he respectively saw a flash or not. The question is: where does the consciousness of all the participants reside in this experiment? This will be more thoroughly examined in the next section, but we reiterate that Wigner took seriously the collapse of the wave function. When Wigner makes his measurement by asking his friend what he saw, the friend’s answer will correspond to either χ1 or χ 2 with probabilities α

2

or β

2

respectively. So for Wigner either one or

the other of these states becomes real. Similarly either ψ 1

or ψ 2

becomes real from the friend’s viewpoint. However, if the friend is an instrument rather that a sentient being then Wigner asserts legitimacy of the original coherent state even after the friend has made its measurement. In other words it is the consciousness of sentient observers that induce a wave function collapse. Wigner was later to change his mind on this issue to one where the photon wave function collapse took place as an objective process

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somewhere within the mechanism of the eye. Subsequently it is the reduced wave function that is finally presented to the conscious mind. Therefore in this new model consciousness plays no role in wave function reduction, and the causal relationship between the mind and matter is removed. For details see Thaheld (2005) and references therein. This does leave the question as to when the collapse takes place. If the friend is an instrument then does a collapse take place in the eye when the cat’s state is measured? If so for the friend but not for Wigner then this suggests relative state. But what is important to bear in mind is that both Wigner’s original model and the one he later supported were solidly based on objective reduction of the wave function, not relative state. As already seen in chapter 6 this is far from unproblematic, and in this work we follow a model where there is no wave function collapse. By reinstating pure wave theories we are lending credence to Wigner’s original instinct that the mind plays a central role in the measurement problem. After all, ultimately we are trying to explain our dynamic experience of a single world, and it would seem absurd to suggest that the mind plays no role in our experience. What is required is a model that does not deny either our consciousness or the physical processes of the universe, as they are presently understood.

8.2 The principle of localisation The title of this section, as previously mentioned, refers to Zeh’s (1970) term localisation of consciousness in the context of space-time and certain Hilbert space components. We can condense this statement by saying that a mind is localised in space-C. Or it may be more accurate to say that the point at which a mind contacts the physical is localised in space-C. This is consistent with what has been termed instantaneous minds (Loewer, 1996; Hemmo and Pitowsky, 2003), and will be consequential for further discussions in the final chapter. For now it is sufficient to say that localisation is entirely consistent with our common experience of a single configuration of matter at any instant. In chapters 5 and 6 we discussed, amongst other things, the emergence of space-time as a single path through an ontic C-space. Therefore any instant during your life is represented not just as a single location in space-time, but that same location in the much larger universal space-C. It is not customary to think of private personal experiences, particularly specific events, as anything other than anecdotal. However, in the case of our dynamic experience as a time-ordered sequence of matter configurations, we are justified in regarding this common experience as empirical evidence in a scientific

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sense. This is because all other empirical evidence has been presented to us within the framework of dynamic experience. To deny this fact would call into question all of the scientific achievements that we have accumulated throughout the ages. Therefore for our purposes we can consider the localisation of consciousness to be as reliable as any other empirical evidence, given the context of modern physics. The great mystery here is why so many eminent scholars would consider this to be an illusion. I am of course assuming that everyone else’s experience of reality is similar to mine. This section is centred on the work of David Albert and Barry Loewer (1988), which for me represents a broadly reasonable account of the physical world and the mental entities that occupy it. They begin by considering five elementary principles that are sufficient to show why quantum mechanics is philosophically perplexing. These are (i)

Any isolated quantum mechanical system is characterized by a state vector ψ S ( t ) .

(ii)

Provided the system, S, remains isolated, ψ S ( t )

(iii)

deterministically according to the Schrodinger equation (6.45). For any complete compatible set of observables, O, of S, the state vector can always be expressed in the form

evolves

ψ S = ¦ ck Ok , ck ∈ ^ k

where

Ok

are the eigenstates of O with corresponding

eigenvalues ok such that if k ≠ l then ok ≠ ol . (iv)

When a measurement of O is carried out on S in the state ψ S 2

the probability of obtaining O = Ok is equal to ck . (v)

When a measurement of O is carried out with the result that O = Ok then the state of S is reduced instantaneously to the eigenstate, Ok . (Albert and Loewer, 1988, 195)

Albert and Loewer highlight two long running difficulties associated with these principles. Firstly there is the problem of interpreting superpositions generally. This has already been addressed by the acceptance that a wave function (state vector) exists in an ontic C-space corresponding to the

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system, S, and we need say no more on this point. The second difficulty is one of interpreting the measurement problem. Again there has been extensive discussion on this issue in chapter 6. All that needs to be said here is that Albert and Loewer describe this as a problem in reconciling principles (i) and (ii) with principles (iv) and (v). As we have seen there are those who are prepared to sacrifice (i) and (ii). Epistemological and objective collapse theories are cases in point. As Albert and Loewer point out, and also as we have argued extensively in chapter 6, these interpretations face very considerable difficulties. As their title suggests Albert and Loewer’s work is grounded in the relative state formulation as Everett (1957) called it, but they use DeWitt’s more common designation the many worlds interpretation. These authors consider three approaches to what we have termed pure wave theories. These are the splitting worlds view (SWV), the single minds view (SMV) and the many minds view (MMV). The first of these is the most common way to imagine the relative state formulation. Here we think of an initially single sheeted universe, which bifurcates in a manner shown in Figure 6-8b. Albert and Loewer level three criticisms at this, with which I generally concur. The first of these concerns an interpretation of the splitting process where all of the particles in the universe are duplicated at the bifurcation point. The appearance is that the universe is blatantly violating the conservation of mass-energy in contravention of the Schrodinger equation. In my view the best way to deal with this is to consider the splitting as implicit in the wave function, which inhabits a non-splitting (Hausdorff) C-space. Conservation laws then refer to individual branches and not the whole wave function. The second criticism relates to stochastic processes taking place within an entirely deterministic framework. How do we assign probabilities to branches subsequent to a splitting point when all of these branches are realised? I believe this problem is the most difficult, if not impossible to answer as long as we presuppose physicalism. As we will see in the next chapter, this is a reason that many physicists and philosophers of science find this problem extremely difficult to address. Albert and Loewer deal with this later in their text by suggesting a dualist approach. Personally I like to use a pinball machine analogy, where like the balls taking various paths through the machine, minds similarly move through the wave function. Like the pinball machine it is in principle, possible to calculate probabilities for every available path at every junction, without reference to a ball moving through it. This adds weight to DeWitt’s assertion that probabilities are implicit in the wave function.

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However, I do not think, as Albert and Loewer put it, that the probabilities are measures of actuality. All of the branches are equally actual. I admit that here I have rather jumped the gun by suggesting a dualist solution to this difficulty as Albert and Loewer eventually do. But without localised mental entities to act as pointers moving through the wave function, uncertainties associated with the stochastic process are completely meaningless. Even in a dualist context the most difficult of Albert and Loewer’s criticisms of the SWV is the democracy of basis. Personally I do not think this is particularly serious. A similar democracy of basis exists in general relativity. This is easier to picture because of the small number of dimensions involved. The background independence of general relativity means that we can choose our coordinates, and therefore our basis to solve particular problems. In cosmology for example, a natural choice of basis is one where the spatial coordinates align along time slices orthogonal to the direction of local energy flux. As we have discussed before this in no way suggests a hidden frame, it is just a natural choice to define a universal time. In a similar way we expect a natural choice of basis in space-C to be one where we perceive physical reality to have particle constituents. Given the space-C dimensionality approaching a countable infinity it would be difficult to imagine what the universe would look like from the viewpoint of another basis. We can see the democracy of basis in action when we analyse the spin states of an electron pair for example, in either a computational or a Bell basis, the analysis is perfectly valid in either. But in an eigenstate of the Bell basis the notion that the pair consists of two distinct elementary particles is completely lost. This is very difficult to imagine on a macroscopic scale, and as far as I can see there is very little to be gained by applying such analyses to macro-systems. However, as Albert and Loewer rightly point out, it is difficult to make sense of basis independence in the SWV. I broadly agree with Albert and Loewer’s criticisms of the SWV. However, although it may be considered naïve I believe it can be a useful thinking tool when trying to visualise physical reality in the wider context of a wave function occupying a near infinite dimensional space-C. In the final section of their paper Albert and Loewer discuss the non-physical aspects of their interpretation under the heading of the many minds view. In this section the authors address the problem of how we never perceive either macroscopic objects or our own mental states to be in superpositions of eigenstates. They assume that introspection is entirely trustworthy, which is in line with the central claim of this chapter that the denial of the localisation of minds undermines the whole of our

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experience. In addition to quantum theory they consider three further conditions, which they use to clarify their argument. (i)

Let A be an observer who can perfectly measure an electron’s xspin. (That is with instrument, M, the post measurement state of the M-electron system is M ↑ x ⊗ ↑ x iff A believes the electron state to be ↑ x , and similarly for the electron state ↓ x )

(ii)

If A can correctly report her mental states corresponding to definite eigenstates according to whether she measures ↑ x or

↓x

(iii)

(She never reports beliefs that the electron’s state is

uncertain, ill-defined or superposed.) then A does believe that the electron is in a definite eigenstate. The state where A believes ↑ x and the state where A believes

↓ x are identical with certain physical states of A’s brain. (That is A’s mental state exactly corresponds with A’s brain state.) We call those states B ↑ x and B ↓ x . (Albert and Loewer, 1988, 203-4) These conditions are slightly rephrased but they carry the same message intended by the original authors. If we assume quantum mechanics and all three of the above conditions then we will necessarily arrive at a contradiction. More specifically condition (iii) is inconsistent with quantum theory and the other two. This is because condition (iii) identifies mental states with, potentially superposed, brain states. If we are to accept that mental states cannot be superposed then we must drop (iii) from this list. Albert and Loewer continue with This response together with the assumption that all physical states are quantum mechanical commits us to a modest (so far) non-physicalism. (Albert and Loewer, 1988, 205).

I admit I am not sure what is meant by modest non-physicalism. For me theories, in this context, are either physicalist or they are not, I do not think we need to equivocate here. They later arrive at the following proposition, which encapsulates what they have termed the single mind view (SMV),

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(P) The probability that A’s mind ends up believing that spin is up = c12 (sic) and the probability that A’s mind ends up believing that spin is down = c22 (sic). (Albert and Loewer, 1988, 205).

This is based on the combined state of the system, A’s brain-electron, which is given by

ψ = c1 B ↑ x ⊗ ↑ x + c2 B ↓ x ⊗ ↓ x

(8.1)

The proposition, (P), generalises to any perfect measurement by an observer, although in the above quote I think the corresponding probabilities are intended to read c1

2

and c2

2

respectively.

Up to this point I broadly agree with Albert and Loewer’s thesis. Beyond this they invoke what they term as the many minds view (MMV), which I do not necessarily disagree with either, it is just that I have certain misgivings regarding their reasons for such a move. But before discussing this I had better make clear the essentials of the SMV and MMV. The SMV is best illustrated with an example. If we use the Wigner’s friend scenario as described in section 6.3, we can locate single minds of all the participants within the process at any given instant. Figure 8-1 depicts a vertical axis showing the discrete eigenstates as combinations of the states of the radioactive atom, the cat and the friend. The horizontal axis is time with the state vector evaluated at each instant. As we know there are only two possible outcomes to this experiment, which are beyond the right hand ends of the two branches illustrated. On the lower branch the atom does not decay, therefore the cat survives and the friend is happy. At the branch point there is a probability of 0.5 that the atom will decay. The first corner on the upper branch is where the atom decays and the next corner is the death of the cat. The absence of the cat’s mind beyond this point is illustrated by the change in colour from red to black. Minds will generally follow either of these paths including those of the friend and Wigner himself. In the SMV all of the three minds will follow either of these paths but it is important to note that they will do so independently. The minds are under no obligation to remain together, and the probability of 0.5 at the junction applies to each mind individually. As a consequence, from the point of view of the friend’s mind for example, there will be a fact of the matter as to whether the cat is alive or dead before opening the inner enclosure. This in no way contradicts the

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fact that the state vector is in a superposition prior to any measurements. This is because in the SMV minds are not considered physical and are therefore not part of the state vector. Friend opens the cat’s box

000

Friend is sad

001 010 011

Cat dies

100 101 Atom decays

110 Atom is intact

111 111

1 2

( 110

− 111

Cat is alive

)

1 2

( 100

− 111

Friend is happy

)

1 2

( 000

− 111

)

Fig 8-1: Time evolution of the Wigner’s friend thought experiment. The vertical axis shows the states of the three qubits of which the experiment consists. The right hand qubit is the radioactive atom (intact or decayed); the middle qubit is the cat (alive or dead) while the left hand qubit is the friend (happy or sad). The horizontal axis (time) shows the state vector with sharp unitary changes delimiting each stage.

In this way the SMV can be used to illustrate the paths of minds through and within mini C-space models like this. However, Albert and Loewer object to the SMV on the grounds that it lacks any form of supervenience. From the fact that they are proposing a non-physicalist model it is evident that they would reject the strongest form of supervenience, which asserts that the mind supervenes on the brain. However, there is a weaker form of supervenience asserting that mental states supervene on the brain states. But brain states, being physical, can be in a quantum superposition whereas mental states cannot. Therefore this weaker form of supervenience fails also. To rescue this they postulate a form of MMV in which individual minds are as densely packed along paths in C-space as rational numbers are packed along the real line. So on any appropriate path in C-space, there will always be a mind arbitrarily close to any given point. In this way the density of minds can be said to

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respect the Born rule. Mental states therefore supervene on brain states en masse. While the principle of localisation does not deny this I believe we can explore other possibilities within the mental realm. Such possibilities may include, for example, a version of MMV where finite non-zero intervals of a timeline contain only a finitely many minds. But possibilities like this will be discussed more extensively in the final chapter. For now it is worth remarking that the SMV, though limited in its possibilities, represents a reasonable starting point for the exploration of non-physicalist models. Above we have seen that two forms of supervenience failed under the SMV. To rescue the weaker form Albert and Loewer proposed their version of MMV. However, this is where I have serious misgivings, why should there be any kind of supervenience on a coherent quantum state? I for one would challenge anyone to provide evidence of such supervenience–there is none. The necessity for supervenience stems from the need to explain why our mental experience is consistent with the physical world we inhabit. This requirement was not originated in the context of quantum theory. In the mind-body context it started with token identity theory in which token mental events are identical to token physical events (Davidson, 1970). Without reference to quantum theory the world is thought of as a single time ordered sequence of matter configurations. The only experience we have of the wave function is the single branch representing our own history and personal biographies. We do not see branches that are separate from ours. It is in this context that we should apply our supervenience criteria. I therefore propose a form of supervenience that is consistent with our experience while remaining compatible with the current quantum paradigm. This I call local supervenience (LSU), that is local in a C-space sense, and it states that (LSU) where a mind is present, individual mental states supervene on brain eigenstates. This removes the necessity for mental states to exist in superpositions, and places the SMV firmly within the principle of localisation. Therefore (LSU) should go a long way to circumventing objections to dualist theories on supervenience grounds. Before considering further support for Albert and Loewer’s thesis, it is worth remarking that it is entirely consistent with the causal closure of physics. Minds, neither individually nor en masse, have any affect whatsoever on the physical world. Minds occupy certain branches of

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the wave function because they follow the Born rule at bifurcation points, or in some circumstances they are able to make choices. Most important of all, the strongest motivations for physicalism have been eliminated. It is time to move on.

8.3 Further support In this section we consider other support for non-physicalist models. However, such backing is likely to be extensive just as we would expect wide support competing views. We therefore restrict our discussion to supporting documentation within the context of pure wave quantum theory. Here again it would be difficult to do justice, even to this narrower context within such a small section. So for our purposes we consider the contributions of Michel Bitbol (1990), Euan J Squires (1993) and, Meir Hemmo and Itamar Pitowsky (2003).

8.3.1 Perspectival realism Bitbol does not reference Albert and Loewer at all, so it could be argued that he arrives at his conclusions independently. If so this would add weight to the non-physicalist thesis generally. However, I think it is unlikely that Bitbol was unaware of Albert and Loewer’s work. Bitbol shows that the Everett interpretation is a limiting case of a series of interpretations of the measurement problem, which leaves progressively less of the observer out of the quantum process. He goes on to show that in this limit a residue of the observer still remains outside of the measurement description. In his abstract, referring to the measurement description, he states Something is still needed besides this description: pure cognitive capacity, the subject, or, in a very abstract sense: "mind". (Bitbol, 1990).

Bitbol refers to his interpretation as perspectival realism, in which the central idea describes a “real object” to be but the class of all its aspects, seen from different points of view. This interpretation of Everett’s relative state formulation is also known as the possible-points-of-view interpretation. At root Everett’s interpretation was considered to be a transition from many possibilities to one actuality, but all of the elements of any superposition are actual. Understanding of Everett’s position requires us, not to eliminate the idea of possibility, but to remove the idea of a transition from the possible to the actual. Bitbol understands Everett’s

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interpretation to be a distribution of actuality to every element of a superposition, each of which is defined as a possible point of view. In this interpretation the final stage of any experiment is at a cognitive level in which the mind of an experimenter is identified with one of the points of view it can adopt from the physical system in question. So far so good but I do find it strange that Bitbol describes the perspectival view as lacking any meta-level from which points of view can be described. In other words there are no points of view of points of view. Yet as far as I can see Albert and Loewer do precisely that. Of course it is possible that there is a confusion of contexts here and I do not think this is particularly serious. The main point is that Bitbol describes minds as becoming irreversibly identified with specific points of view at the conclusion of every measurement. In addition he considers mind to be lacking any inherent point of view, just as it lacks a spatio-temporal location. Perhaps it is more accurate to state that we are unable to define points of view (C-space locations) or spatio-temporal locations either generally or at any instant prior to a measurement. This is why, at the beginning of the previous section, I suggested that maybe only points of contact of a mind with the physical world are definable. Bitbol describes a mind that is somehow detached from the physical world as being point-ofview-less. Unfortunately at one point he describes minds as being not within the world; neither do they stand outside the world. To me this reads more like a Buddhist koan rather than anything concrete. In this work we treat the mind as an entity that objectively exists even though all of its experiences are purely subjective. So a mind is either within the world or it is outside it. But we will leave any discussion of possible configurations of, or relationships between minds until the final chapter. Keeping to the spirit of Bitbol’s ideas we can describe pure wave theories as versions of a limiting case in which all but the most subjective aspects of the observer are subsumed into the quantum measurement process. As Bitbol points out, when we analyse this limiting case we are left with a residue remaining outside the measurement process. This residue may be described as the limiting case of mind where we have stripped away all of the attributes that can be described physically. This includes many of our mental processes such as accessing memories and logical thought etc. that minds may also share with modern AI systems. However, the residue referred to here is the pure cognitive capacity that witnesses such thoughts and memories as qualia that cannot be translated into anything physical. But I digress, the main feature of a point of view post measurement is the localised point of contact between a mind and the physical–Bitbol is describing the principle of localisation.

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8.3.2. Selection and the universal mind At the beginning of researching this topic the first article I encountered was Quantum theory and the relation between the conscious mind and the physical world by Euan J Squires (1993). A significant main feature of this work, in contrast to Albert and Loewer, is the notion of a single universal mind (Squires, 1991). Squires’ reasoning is as follows. Suppose we both take part in a single experiment where we measure the spin of an electron along a vertical axis, the wave function for the system consisting of us both plus our apparatus including the electron being measured, is

ψ = α ↑ ⊗me ↑ ⊗ you ↑ + β ↓ ⊗me ↓ ⊗ you ↓ 2

where as usual α + β

2

(8.2)

= 1 . According to quantum theory we would

both report that we agree on the post measurement result. However, in Albert and Loewer’s MMV there is the possibility that your consciousness would, for example, read ↓ while my mind would experience seeing ↑ . It is this type of outcome that Squires sets out to avoid. Both Albert and Loewer, and Squires agree totally on the physics of this process. The disagreement has solely to do with mental configurations. However, at our current stage of development we have no way knowing anything about the arrangement of mental entities, single or otherwise, outside of our physical experience. Squires’ motivation for proposing a universal mind view (UMV) is to avoid the possibility that my mind could end up talking to a version of you that is no longer a conscious being. This is an alternative approach, besides the MMV, to avoiding what has become known as the mindless hulk problem, which has been objected to many times throughout the philosophical literature. In the UMV the universal mind occupies one branch of the Everett multiverse, while its individual tokens each occupy a single brain within that branch. The tokens can make individual choices within their branch of the wave function without compromising the causal closure of physics. Another way of describing this is to suppose that you make a particular choice between A and B. At some point there will be a coherent wave function in some part of your brain with the form ψ = α A + β B . Because your mind is fully associated with your brain and this wave function, you will have full control over the choice, most particularly if α  β . All other sentient beings in the universe will bare

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witness to your choice, and that will become part of our history. As Squires puts it, In this model it is clear that the conscious mind, through such deliberate choices, would have a significant effect upon the experienced world, i.e., it would be efficacious. (Squires, 1993, 119).

Further, Squires goes on to say that there would be no mysterious effect by the conscious mind on the physics, which remains a closed causal system. This is because we are dealing with one-off choices, not numerous choices that would be subject to the probability weightings. So, just as we have been saying all along, Squires goes on to state, The conscious mind does not change physics, it selects from it. (Squires, 1993, 119).

Of course the mechanism surrounding choices and coherent quantum states in the brain is far from fully understood, but is a field of intense current research (Hameroff and Penrose, 2014), which will be discussed further in section 11.2. There is an interesting parallel here. Recalling the passage in subsection 6.4.3 with regard to nonlocal hidden variables, this is a noncollapse model that includes a special history. In the UMV all of the minds within the universal mind always occupy the same Everett branch. Therefore it is always possible, in principle, to define a unique history for every individual of the universal mind. The presence of a universal mind reintroduces realism, so by Bell’s theorem such a model must exhibit nonlocality. Looking at it the other way nonlocal hidden variables require a pointer to move through the universal C-space in order to generate a unique history. This pointer cannot be part of the wave function it is a separate ingredient. Therefore nonlocal hidden variables models assert substance dualism, exactly what Squires advocates. Another interesting point is that in this model all the components of the universal mind are contemporary in a universal time. The universal mind selects a preferred frame. So there may be no need to object to such models if you are persuaded by mind-body dualism, hidden variables and hidden frames are implicit in the UMV. The main point however, is that the UMV is an alternative proposal to the MMV and neither of these denies the laws of physics, as they are presently understood. They are however, distinct in their interpretation of minds. Until we know more, there will be many such interpretations of the mental realm. Squires, like many others, go to great

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lengths to avoid the possibility of mindless hulks entering their models. I do not find this a particularly serious problem. You can be sure of your own consciousness but not anyone else’s. In our current state it is difficult to conceive of the methods needed to make any headway. But in order to investigate scientifically, whether minds are one or many, or the way, it or they are distributed across C-space, will require direct access to the mental realm. One thing of which we can be certain is that we cannot answer these questions by investigation of the physics. These and related issues will be explored later in the final chapter.

8.3.3 Probability and nonlocality in the MMV The MMV of Albert and Loewer, and the UMV proposed by Squires represent opposing extremes of a spectrum of possibilities, MMV is completely local and UMV is totally nonlocal. The question is: could the truth be somewhere in between? Could there be a form of weak nonlocality? Hemmo and Pitowsky (2003) fully support a dualistic model that is similar in many ways to Albert and Loewer’s MMV. However, in contrast to Albert and Loewer they do indeed claim a form of weak nonlocality amounting to a direct correlation between the minds to a limited extent. They claim that this is necessary in order to solve an MMV version of the mindless hulk problem. Let us begin with the simplest case of EPR experiment. In the general case an entangled pair of electrons is separated. One is received and its spin measured by Alice along the x-axis, while Bob similarly measures the other along the axis x∠θ . In the simplest case we have θ = 0, π . The entangled pair is thus characterized by the wave function

ψ =

1 2

(+

a

−b− −

a

+

b

).

(8.3)

When Alice and Bob perform their measurements and subsequently compare results, this simple case will have two Everett branches, one where Alice and Bob measure + a − b and the other where they get

− a + b , that is they always get opposite readings. In the SMV it is possible to obtain a mindless hulk scenario. This is when Alice’s and Bob’s single minds respectively take separate branches. In Albert and Loewer’s MMV, Hemmo and Pitowsky claim that an analogous situation is possible. However, as far as I can see Albert and Loewer only claim a correlation between Alice’s minds and Bob’s reports, and vice versa. It is

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claimed that Albert and Loewer do not allow us to say, for example, that Bob’s +reports, as perceived by Alice’s − minds, are associated with Bob’s +minds. But Albert and Loewer only state that exactly half of Alice’s minds and half of Bob’s minds will appear on each branch. Therefore Bob’s +reports will always be received by Alice’s − minds, and in that particular branch those reports will be associated with Bob’s +minds. Although Albert and Loewer do not specifically say this, it is my reading of it that they do not state anything to the contrary. Now let us look at the more general situation where 0 < θ < π . In this case there will be four post-measurement Everett branches. These correspond to Alice and Bob’s respective measurements, + a + b ,

+ a − b , − a + b , and − a − b . This is diagrammatically shown in Figure 8-2. The pre-measurement wave function of the system composing of Alice, Bob, Alice’s electron, a, and Bob’s electron, b, is given by

ψ = Ax ⊗ Bx∠θ 1 2

sin 2 (θ 2 ) +

a

+

b

1 2

cos 2 (θ 2 ) +

a

−

b

1 2

(+ 1 2

−b− −

a

cos 2 (θ 2 ) −

a

+

b

a

+

b

1 2

).

(8.4)

sin 2 (θ 2 ) −

a

−

b

ψ Fig 8-2: Schematic of Everett branching in an EPR experiment where Bob’s measures electron spin on an axis at an arbitrary angle, θ , to Alice’s. The premeasurement wave function, ψ , is as given in equation (8.4). The postmeasurement eigenstates,

+

a

+

b

etc., are shown on each branch, and the

associated probabilities are given at the top of each branch.

The probabilities given at the top of the branches in Figure 8-2 are those predicted by quantum mechanics, they are decided solely by the physics. The fact that these probabilities are reported consistently over many trials by sentient experimenters would suggest purely stochastic dynamics of

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their individual minds, exactly as stated by Albert and Loewer. Therefore weak nonlocality does not follow. Felline and Bacciagaluppi (2013) arrive at the same conclusion via a more detailed analysis. Hemmo and Pitowsky go on to discuss GHZ (Greenberger, Horne, and Zeilinger, 1989) experiments where three experimenters each measure one of three maximally entangled electrons. The quantum state is given by

ψ =

1 2

(↑

z

⊗ ↑z ⊗ ↑z − ↓z ⊗ ↓z ⊗ ↓z

).

(8.5)

Here the z-spins of the three electrons are parallel, and each of the three electrons are distributed among Alice, Bob, and Charlie who receive and measure x-spins and/or y-spins according to four possible procedures, ( x, x, x ) , ( y, y, x ) , ( y, x, y ) , and ( x, y, y ) . However, it is not my intention to become embroiled in the details of GHZ states other than to say that they represent generalisations of EPR states, where for N qubits in the computational basis they always have the form

ψ =

(1 2

1

⊗N

− 0

⊗N

).

Each of the four measurement scenarios associated with equation (8.5) have four Everett branches with equal probability. In general it may be considered that all of the measurement procedures are completed in a particular order. This would result in 256 Everett branches in each of which there is an intersection of minds that have passed through particular combinations of spin states at each of the four stages. As in the EPR case the distribution of minds across all 256 branches of the final state is determined solely by the physics. My reading of Albert and Loewer’s MMV suggests that on sufficient repetition of the four measurement procedures, any single mind will obtain a probability distribution consistent with quantum theory. According to Hemmo and Pitowsky’s reading of the MMV a subset of the minds of one of the observers (Alice, Bob, or Charlie) will register…  …different measurement results (for the same measurement), depending on the settings chosen by the other observers… (Felline and Bacciagaluppi, 2013, Preprint p18).

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Felline and Bacciagaluppi go on to say that this would correspond to weak nonlocality. However, it is unclear whether they are referring to minds or bodies when they quote: chosen by the other observers. If it is minds then indeed we do have weak nonlocality, but if it is bodies then such dependence is to be expected on physical grounds. In the following paragraph they state that this conclusion does not follow, and then go on to give an appropriate example. For details see Felline and Bacciagaluppi (2013). The point here is that weak nonlocality cannot be ruled out but, as Felline and Bacciagaluppi have shown, it does not necessarily invalidate Albert and Loewer’s MMV. This will be explored further in subsection 12.1.4 of the final chapter.

8.4 Summary In previous discussions we have shown that there is substantial evidence that coordinate time is an emergent rather than a fundamental property of reality (Barbour, 1999; Zeh, 2007 and references therein), and therefore this would lead us to seriously question the dynamics of the universe from a purely physical standpoint. Not only has such evidence been acquired theoretically, there is now tentative empirical evidence for such a model via the existence of macroscopic-Bell states (Iskhakov et al, 2012; Kanseri et al, 2013). Through the works of Weyl (1949), Wigner (1961), Albert and Loewer (1988), Bitbol (1990), Squires (1991; 1993) and, Hemmo and Pitowsky (2003) it can be shown that physicalist theories are in a state of tension with the growing body of evidence highlighted here. The statement made by many physicists that our dynamic experience constitutes a grand illusion demonstrates that they tacitly assume physicalism as an axiom then apply rational arguments that conclude in contradiction. The answer that they should be looking for is that your dynamic experience at any instant is a direct pointer to where you are in C-space. Our common experience of a dynamic particulate reality consisting of one matter configuration at a given instant is reconciled with the mainstream model of physics describing a purely static reality, by the acknowledgement of a mind-body dualistic model. As we will see in the next chapter there are many scientists and philosophers who would regard such a claim as a desperate expedient as termed by Lockwood (1996). However, it would appear that it is physicalists who are desperate to prove their point by the proposition of neo-presentist models and/or wave function collapse theories as alternatives to what is becoming the mainstream.

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As we have seen however, this does not mean that the champions of neo-dualist approaches will be free of disagreement. The MMV of Albert and Loewer is in complete contrast to Squires’ UMV, where the latter could be regarded as a nonlocal hidden variables/frame model. In addition Hemmo and Pitowsky adopt a many minds model, somewhere in between these extremes, that exhibits a weaker form of nonlocality. But Hemmo and Pitwowsky base their model on their reading of Albert and Loewer’s work. Notwithstanding their differences all of these authors concur on the deployment of pure wave theories. This shows that until we have an effective means of effectively investigating the mental components of reality, there will always be subscribers of dualist or mental monist models who will differ in their views. There is still much to be learned and this is symptomatic of the fact that science requires the food of reliable data in order to thrive. Until we have effective means of gathering data on mental aspects, both individual and collective, there will always be those who will regard such rational investigations to be in the realms of dragons and unicorns. In the following chapter it is these views to which we turn our attention.

CHAPTER 9 COMPETING IDEAS

There is an old saying that goes something like: the most convincing lies are those that are closest to the truth. We can apply similar thinking to physicalists whose work takes seriously the literal interpretations of modern physics. Such physicalist views are the most difficult to argue against and require thorough examination in order to dislodge them. In the first two sections of this chapter we examine those views and to show where they fail given the postulates that our dynamic experience is real in conjunction with the literal interpretation of modern physical theories. These neo-physicalists, as I refer to them, provide impeccable descriptions of physical situations in a quantum context, while either tacitly or overtly assuming that the mechanism of consciousness is accounted for in those descriptions. In the process they appear to disregard the fact that their theories fail to predict our common conscious experience. On that basis it is the main purpose of this chapter to challenge their claims. Before summarizing, the third section examines other, possibly better known, authors whose views are either distinctly physicalist, or could be construed as being so. It is likely that scientists and philosophers who tacitly or openly argue for physicalism outside the context of modern physics, are greatly more numerous than those alluded to above. In these cases it is found that either, they are in some sense easier to argue against, they are not promoting physicalism, or that physicalism is not necessary for their own arguments to succeed.

9.1 Many minds: a physicalist’s perspective From previous chapters we see a gradual realisation by some that a dualistic metaphysics is needed to reconcile the contradiction between the nature of the physical world and our experience of it. Notwithstanding these developments there remains a tenacious adherence to the physicalist’s cause, this is despite the nullification of its original motivation namely the causal closure of physics. One of the leading lights in the physicalist’s movement post Albert and Loewer is Michael J

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Lockwood (1996 and relevant references therein). In this section we critically examine this work. Approximately the first two thirds of Lockwood’s article are entirely consistent with the description of physical reality so far described in this book. More specifically Lockwood’s thesis is entirely consistent with quantum theory, general relativity, and canonical quantum gravity. The divergence between our respective views begins in Lockwood’s paper on page 176. The first paragraph beginning on this page starts with a sentence indicating his full acknowledgment of reconciliation between unitary evolution of the universal state vector and our dynamic experience provided by Albert and Loewer’s model. This approach plainly succeeds in reconciling the universal occurrence, within the physical universe, of unitary evolution, with the appearance of state vector reduction in accordance with the usual statistical rules. (Lockwood, 1996, 176).

This is then immediately followed by But the appeal to dualism, in order to make sense of quantum mechanics, strikes me as a rather desperate expedient. (Lockwood, 1996, 176).

So is it really possible to reconcile quantum theory, dynamic experience, and physicalism? To answer this question we need to critically examine the content of Lockwood’s paper in the paragraphs following these quotes. The first thing that I should make clear is that I am not using dualism to make sense of quantum theory, rather I am suggesting that the only way to make sense of our classical dynamic experience in the light of quantum theory is to invoke a robustly dualistic metaphysics. This might seem like splitting hairs, but we need to appreciate that quantum theory, being more fundamental than classical physics, provides only the stage on which the play is enacted, the classical world as experienced by the players (minds) is dependent upon their positions (Bitbol’s points of view) on the stage. However, let us see, using Lockwood’s text, if we can counter this position. The first thing that Lockwood does is to emphasize his use of the concept of a mixed state. This is something that we have not done so far because to do so earlier we would have run the risk of compromising the clarity of an already difficult topic. However, for those unfamiliar with mixed states we will consider them now. Mixed states can only be described in terms of density operators (what Lockwood calls projection operators) that were briefly touched on in chapter 6. Our discussion is

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based on Nielsen and Chuang (2000, 100) who provide a brief but clear explanation of mixed states. The density operator of a given state vector, ψ , is written as ρ = ψ ψ . However, any state expressible as a state vector can only be a pure state, a state of which we have full knowledge. There are in addition, states that cannot be so expressed. These are states that are epistemologically incomplete but can still be written in the general form

ρ = ¦ pi i i , i j = δ ij .

(9.1)

i

Of course pure states may also take this form and one reason that Lockwood uses this notation is because the probabilities, pi , associated with each term are explicit. In this notation a mixed state is defined by the condition

tr ( ρ 2 ) < 1

(9.2)

where quality here would define a pure state. We may use this to describe the spin of an electron in any particular direction. For example, in the computational basis positive spin may be denoted by 1 where negative spin is 0 . A corresponding mixed state may be ρ =

1 1 + 12 0 0 where we have a probability of 12 for each of the eigenstates. This is what we get when, for example, we measure the x-spin of a z-spin electron. The two resulting eigenstates are associated with Everett branches, where we see positive (negative) x-spin in one branch only. In matrix form this state is given by ρ = 12 I , from 1 2

which we get tr ( ρ 2 ) = 12 < 1 satisfying the condition for a mixed state. An observer, A, measuring the electron x-spin will be in a corresponding mixed state 12 A1 A1 + 12 A0 A0 . Lockwood then goes on to consider a

pair of entangled electrons each of which is in a similar mixed state. Subsequent measurements of each electron by independent observers in independent directions will obtain corresponding results locally. It is not possible to tell if there has been any measurement of one electron by interrogating in any way the state of the other. This is due to the no signalling theorem as Lockwood correctly illustrates.

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The next stage of Lockwood’s discussion considers his favoured version of many minds view (MMV). To begin with, just as we need to differentiate between the universe and the multiverse clearly, we must also distinguish between our mind and our multimind. In his words this is not very euphonious, so for multimind he adopts the capitalised term, Mind, and for clarity we will do the same. At any instant, a token mind can have a fully conscious maximal experience, of which there is a limiting case of unconsciousness in, for example, dreamless sleep or coma. All instances of unconsciousness are considered to be identical. We are now in a position to state Lockwood’s three main assumptions (axioms) upon which his MMV is based. (i) (ii)

(iii)

My Mind is a subsystem of my brain. There exists a set of mutually orthogonal pure states, comprising a basis for my Mind, which forms what Lockwood calls the conscious basis. Let ϕ be a pure state belonging to the consciousness basis of my Mind, and let E be the corresponding maximal experience. And suppose that the mixed state of my Mind, at time t, is represented by a weighted sum of projection operators corresponding to mutually orthogonal pure states. Then, it is contended that my Mind at t will contain a continuous simultaneous infinity of tokens of E, with a measure that is proportional to s. (Lockwood, 1996, 178-9)

These assumptions as represented here are considerably condensed compared with Lockwood’s originals. However, I believe they convey his intended message adequately. We can now explore their meaning in more detail. Assumption (i) is reasonably self-explanatory. In Lockwood’s model there is no independent conscious observer. The existence of the Mind fully supervenes on the existence of the brain. That is we assert the existence of a neural network within the brain, which is the Mind. Because, from a quantum mechanical perspective any physical system, including brains, is subject to Everett type branching, the subsystem we call the Mind also branches. This is schematically represented in Figure 91.

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t2 Boundary of the body or brain

Boundary of the Mind

t1

Fig 9-1: Schematic showing the time evolution (bottom to top) through a single branching event. The thin black lines may represent the outside surface of either the body or the brain. The region in green represents the Mind as a neural network existing as a subsystem of the brain.

The subsystem we are calling the Mind is a physical system, which possesses a distinct (mutually orthogonal) set of states. This would be expected for any physical system, and in the C-space for the Mind each location represents a basis vector, the full set of which forms the conscious basis. Assumption (ii) above asserts the existence of this conscious basis, where the set of basis vectors form a projective Hilbert space. Because assumption (ii) is descriptive of a physical system as implied by assumption (i), this provides the following logical relationship

Assumption (i) Ÿ Assumption (ii) .

(9.3)

So far so good, but with regard to assumption (iii) Lockwood further clarifies his description with further explanation, so we will do the same. In what followed Lockwood introduced the idea of a twodimensional experiential manifold, where time is plotted vertically from earlier events at the bottom to later events at the top, as is consistent with convention in relativity. In addition he introduces a superpositional dimension plotted horizontally. This is where I would imagine particular points on a horizontal time slice to represent what Lockwood calls

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maximal experiences. On any given vertical timeline within the experiential manifold, your token mind will have a maximal experience, e, at a specific time, t. At the same time your Mind, being distributed continuously over the superpositional dimension, will have a maximal experience, E. This provides us with a two-dimensional array of maximal experiences where the vertical axis representing time is clear. The superpositional axis however, does require more explanation. If we consider Figure 9-1 for a moment, the measure of a horizontal slice through the region marked in green, representing the Mind, is the variable, s, quoted in assumption (iii). Of course Figure 9-1 is greatly simplified. In reality we can imagine a high density of Everett branching events occurring from the earliest time of your existence. A simplified example of the experiential manifold is taken to represent the green region in Figure 9-1. This however, does not really conjure up the image that one gets when reading Lockwood’s article. Lockwood describes the experiential manifold for an individual as a two-dimensional array as seen in Figure 9-2, which does not explicitly show the Everett branching. Death line

t2

t1 Birth line

Fig 9-2: Lockwood’s experiential manifold for Alice’s Mind (in green). Time is plotted vertically and shows earlier ( t1 ) and later ( t2 ) time slices, where the superpositional dimension is on the horizontal. The black curve shows the path of one of Alice’s token minds where she enters existence at the birth line to where she exits existence at the death line.

In Figure 9-2 we see a typical schematic of the experiential manifold for the Mind belonging to Alice say. The birth line shows all of the possible states of Alice’s Mind at birth all coming into existence at approximately

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the same time, hence the straight horizontal nature of the birth line. The various paths possible through the experiential manifold however, will eventually lead to her exit from existence at the death line. This occurs for any of a broad range of ages due to variations of possible circumstances in her life. Hence the rather ragged nature of the death line as Lockwood describes. The black curve in the figure indicating the path of just one of Alice’s token minds will have other branches stemming from it as a result of choices and chance events, but these are not explicitly shown in the figure. If we consider the time, t1 , for a moment, then where that horizontal line,

{t = t1} ,

cuts the experiential manifold, EM (in green),

there will indeed be a simultaneous infinity11 of tokens of the experience, E1 , which is proportional to its measure, s1 . In symbols this is

s1 = m ({t = t1} ∩ EM )

(9.4)

where the operator m ( ⋅) indicates the measure of the set inside the brackets. This is similar to how Lockwood describes his assumption (iii). Now let us see if we can expand on Lockwood’s description of the experiential manifold. I believe Lockwood uses the form of a twodimensional manifold in order to maintain clarity for the benefit of the reader. However, I think we can go a little further. If we replace the name superpositional dimension with superpositional manifold then this allows us to think of the set of orthogonal states that constitute the Mind as the truly multidimensional object that it is. In addition let us see if we can give a reason for discriminating time from the other superpositional variables. After all the only difference between distinct points in EM is that they represent distinct mental states. So where does time fit in? To answer this we need to adopt the rationale used in canonical quantum gravity (CQG). Recall that CQG is a timeless theory, which maximises generality by minimising the number of its ingredients. In CQG all we have is a complex valued wave function over a universal C-space (space of eigenstates). There is no explicit time variable. So if we isolate Alice’s Mind then we are left with a system, which possesses a set of distinct (mutually orthogonal) mental states, over which we can plot a wave 11

I hesitate to use the word continuous here, in the manner of some other authors, because I consider myself as a discrete unit, and an infinite set of such units I would regard as being countable.

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function. However, Alice’s Mind is not an isolated system. It is immersed in an external reality that includes autonomic functions of her larger brain. This will include systems acting as clocks with which we can correlate the variable, t. So if we take the tensor product of the Mind and a subsystem representing a clock, this may very well provide us with our sense of time within physical reality. Admittedly this does seem like a bit of a cheat because we are relying on an external system to provide our time variable. What we are calling the Mind may include a subsystem that we can identify as a clock. Either way we are still discussing physical systems, and whether or not the Mind includes a clock-like system just depends on where we choose to place the boundary of the Mind to be within the larger brain. So what does all of this really tell us? An important point to come out of the above discussion is that the state space for the Mind, EM , we are calling the experiential manifold is an extended subset of the universal C-space. This may be consistent with quantum mechanics, and indeed with other branches of theoretical physics. However, it is not consistent with our experience. Yes it is that old grand illusion chestnut again, except I would implore you, the reader, not to regard your own experience as an illusion. The way you experience reality, as an ostensibly continuous time ordered set of well-defined configurations, is something that is life-long and consistent. Moreover it should be emphasized that this experience is the filter through which we acquire all of our truths. So what has Lockwood actually described? Lockwood’s is a perfect description of a physical system, the Mind, as a subsystem of a brain within a biological organism such as the physical you and me. Like any physical system it can be described as having a large but finite set of mutually orthogonal eigenstates. But because these eigenstates are distinct then the system is extended in C-space. Your mind is not, because if it were your mind’s implied extension in C-space would mean that you experience many configurations at defined instants. That is we would experience being in superpositions–this never happens. What Loewer (1996), in his reply to Lockwood, calls the instantaneous mind is an entity that has no discernable extension in C-space. This includes the path followed by an instantaneous mind through C-space, which it experiences as time. But the mind is not extended in time, if it were we would experience many distinct points in time at single instants. This would constitute a temporal superposition, which again never happens, at least not under normal circumstances. This reminds me somewhat of another quote by Lockwood from this same paper,

310

Chapter 9 Moreover, one thinks of one's mind (in the usual sense of that term) as being wholly present at each of these times. One doesn't think of part of one's mind as existing at one time and part at another. (Lockwood, 1996, 179).

This to me looks like a veiled acknowledgement of the principle of localisation. However, there are some philosophers who would respond by saying something like: yes my mind is wholly present at the earlier time and then, in relation to that it is wholly present at the later time. This comes from the relational theory of time, which is a thinly disguised form of presentism, a doctrine rigorously denied by modern theoretical physics. But we defer that discussion until the next section when amongst other physicalists we discuss the work of Simon Saunders. Ultimately Lockwood is providing a perfect description of the physics closely associated with our experience. I do not think he would deny this, but he is not describing the experience itself. Recall that in chapter 6, we are regarding everything physical as part of a whole, which at root, is fundamentally static. This is due to eternalism, the view supported by most physicists, and which has, admittedly indirect, empirical support (Iskhakov et al, 2012; Kanseri et al, 2013) through the existence of macro-Bell states. There is neither dynamics in the physics of the universe as a whole nor locally within the realms of our immediate experience. Dynamics only emerges when something changes relative to something else. That change takes place in our experiential time, which is internal to our own minds, and is only correlated with physical time (the parameter on the path we trace through C-space) during the periods of our lives. In Lockwood’s description there is no pointer or index. If I experience an event, e1 , at time, t1 , and then subsequently experience an event, e2 , at the later time, t2 , then it is perfectly legitimate to refer to e1 when I am experiencing e2 . Therefore the question that physicalists need to answer is: what is the difference between the physical circumstances locally to e1 when I experience it, and later when I experience e2 ? The metaphysical difference is that when I experience e1 , I am present at e1 , but when I experience e2 I am not present at e1 . For those subscribing to eternalism, there is by definition no physical difference between these distinct set of circumstances at e1 . In the next section we explore the writings of other authors in this field who either support or appear to support Lockwood’s conclusions as summarised in his 1996 article.

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9.2 Further support for the physicalist’s perspective In this section we consider work that supports Lockwood’s views, particularly as presented in his 1996 article. In chronological order these are: David Deutsch (1996), Simon Saunders (1998), Jenann Ismael (2003), Hilary Greaves (2004), and Peter J Lewis (2007). For clarity we confine ourselves to this order, and address each in turn.

9.2.1 Deutsch’s Comment on Lockwood In his introductory paragraph Deutsch (1996) makes reference to a scandal in the philosophical foundations of physics that reflects badly on some of the leading scholars of the twentieth century. This relates to interpretations of quantum theory of which there has indeed been fierce debate. The use of italics in the previous sentence reflects my sympathy for Deutsch’s view that pure wave theories are really the only interpretation of quantum theory. Following this he then proceeds to say that Lockwood’s article is, in his words, untainted by it. Deutsch fully concurs with Lockwood, not only on his interpretation of the physics but also on the metaphysics. By now the reader will be well aware of my position, which is one of complete agreement on the physics, but total disagreement regarding the metaphysics. However, it is not sufficient for me to simply take issue with a particular viewpoint and leave it there. Just as I have done previously I must precisely point to the source of the problem. So here I propose to examine Deutsch’s response to Lockwood particularly with regard to consciousness alone, since I think all three of us would entirely concur on the physics. While writing his reply we know that Deutsch was drinking tea, simply because he says so in his article, a rather mundane fact that we have no reason to doubt. What is a little less mundane is the notion that, just as there would be other copies of Deutsch drinking tea there were similarly numerous copies drinking coffee, or any other beverage that might come to mind, in otherwise very similar circumstances. This is an example that Deutsch uses to illustrate the multiplicity of physical reality in the form of the wave function, which I view as being endowed with a structure reminiscent of a wood grain. This grain consists of branching paths where, on any particular branch, the objects that have mutually interacted with each other become entangled and are obliged to be configured according to an approximation of classical mechanics. Of course the formulation of quantum mechanics is uncontroversial, just as the church authorities allowed the formulation of the heliocentric model of

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the solar system in Galileo’s time. However, it was Galileo’s literal interpretation of the heliocentric model that bought him into conflict with the Vatican authorities. So it is today that the literal interpretation of quantum mechanics, as Deutsch puts it, …is still greeted with cynicism, incomprehension and even anger. Quoting Lockwood, Deutsch then goes on to say in the following paragraph, The particular implication of quantum theory that all the fuss is about is, of course, as Lockwood puts it, 'the simultaneous existence of distinct... experiences' (of a single person). (Deutsch, 1996, 222).

As we have seen it is certainly true that, in pure wave models, there are distinct simultaneous experiences of individuals with a claim to the same identity. However, in order for this quote to be entirely correct we must define the single person as encompassing the whole set of copies distributed throughout C-space. This is what Lockwood does when he uses the term, Mind, for a collection of simultaneous token minds. All of the token minds, constituting the single person branching out from a single birth event in C-space, make up the experiential manifold possessing a fractal structure. This is what Deutsch appears to be defining as a single person, where each branch exists in a distinct universe or timeline. Deutsch goes on to explain that he does not have all of the experiences of his single self because, as he puts it, …the laws of quantum mechanics restrict the operation of our brains so as to confine, as Lockwood puts it, ‘the gaze of consciousness to a kind of "tunnel vision" directed downwards in the experiential manifold. (Deutsch, 1996, 222).

There seems to be a veiled suggestion here that this represents the mechanism of consciousness. However, we can apply a similar description to entirely inanimate objects. Julian Barbour does precisely this in his description of time capsules (Barbour, 1999, 30…). Essentially time capsules are configurations of matter that are products of some past process, the details of which can, in principle, be extracted. Examples include written records, archaeological artefacts, analogue, and digital electronic data etc. These are man made records but the time capsule concept is also applicable to the natural world independent of man. Obvious examples include geological and fossil records as well as cosmological records in the form of radiation signatures from space. These extend into the deep past–in the case of the cosmic microwave background almost all the way to Barbour’s α-point (the big bang)–tunnel vision on a

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grand scale indeed. For example one of the asteroid belt objects would have a number of copies that increase geometrically with time. A set that includes all of the copies over time would form the equivalent of an experiential manifold for that object. However, we cannot designate it as such because it is inanimate. It might be more appropriate to refer to it as an existence manifold. When we think this way then we see similar structures for all physical objects, both animate and inanimate. The body of Alice, for example, has an existence manifold. The existence manifold of Alice’s brain is a subset of that for her body. And Alice’s experiential manifold is similarly a subset of her brain’s existence manifold. This is a reasonable course grained description of Alice’s physicality. We can apply similar thinking to Deutsch as he was drinking tea and drafting his reply to Lockwood mentioned earlier. Through his firm grasp of quantum theory he could visualise neighbouring timelines where he is drinking coffee. He did not experience drinking coffee at that point. That was the experience of a separate individual (token mind) who has the same name, identical DNA, and a near identical biography. In addition there will be branch points in his near past, to the past of which both sets of David Deutschs (tea drinking versions and coffee drinking versions) followed identical paths through C-space. I myself occupy a body, which is entangled with one of Deutsch’s ‘tea drinking’ copies because that is what I recall reading in his article. There will be other copies of me (physically) writing this chapter now but with the words ‘tea’ and ‘coffee’ switched, because those copies are referring to the article written by one of the coffee drinking David Deutschs. But I have no such recollection because in this life I (my token mind) was not in that part of C-space. And when you read this chapter I will no longer be at the point where I am writing these words, I will have moved on because I am localised in Cspace and moving through it along my own timeline. Lockwood’s description and Deutsch’s support of it is entirely correct as far as it goes. However, the structures they describe are all extended in time, or in the language of quantum theory, extended in Cspace. There is nothing in Lockwood’s description that would qualify as Loewer’s instantaneous mind. So what Lockwood and Deutsch perfectly describe is a stage without players, a road with no travellers, or a microscope with no one at the eyepiece; the analogies are numerous. To borrow the words of my late esteemed colleague Peter H Plesch, Lockwood and Deutsch’s account may be described as a fine production of Hamlet without the Prince of Denmark. The analogy with the road in particular is very apt, but this will be dealt with later in subsection 9.2.3. In

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the next part of the discussion we consider the idea of relational time, and the bearing that this has on the consciousness debate.

9.2.2 Relational time Substantivalism says that space and time are like a container that exists independently of the contents, in this case matter and its motion. Relationism, on the other hand, says that if you take away the contents you take away the container. In other words space-time cannot exist without matter and its motion. Dowden (2001) defines these opposing view succinctly by Substantivalism is the thesis that space and time exist independently of physical material and its events. Relationism is the thesis that space is only a set of relationships among existing physical material, and time is a set of relationships among the events of that physical material. (Dowden, 2001, 32).

Notwithstanding the current general theory of relativity, there is still considerable debate as to the correct interpretation. This is because the background independence that general relativity demands has no bearing on the debate. Substantivalists would argue that the space-time manifold possesses geometric properties of its own that are dependent on event location and the local matter configuration. It is as though matter distorts the manifold by its very presence, much like a stretched rubber sheet is distorted when a weight is placed upon it. This conjures up the image of a space-time manifold with material properties of its own. Such an interpretation does not deny background independence, because that refers to the coordinates we choose to map onto the manifold, and not the manifold itself This debate is similarly irrelevant to the arguments being put forward in this book. All that is demanded to satisfy the theoretical framework being endorsed here is background independence (in relation to coordinates), and the literal interpretation of quantum theory (pure wave theories). One of the consequences of this, admittedly not universally accepted framework is the eternalist’s view of the physical world in which a specious present along with past and future events in relation to that present, all exist. Saunders (1998; 2002) is a supporter of this view in conjunction with a relational theory of time. In his 1998 article Saunders discusses time and quantum probability in the context of relational theory. I have no particular issue with relational theory, either of time, space or

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probability. However, in section 4.2 of his article in which he discusses many minds, he attempts a deduction starting with the relational theory of time, concluding that Loewer’s (1996) description of the instantaneous minds view (IMV) must be false. It is this point alone that we need to address here. I should also make it clear that Loewer also criticizes the IMV on grounds that it denies the existence of temporally persisting minds and could therefore lead to mindless hulk scenarios. However, the IMV is not inconsistent with a token mind’s experience of reality in the physical models endorsed here. Let us be clear about what Saunders actually says in the case of an experimenter who measures the x-spin of a z-spin electron. The measurement takes place between an earlier time, t1 , and a later time, t2 . The question is what she1 at t1 will expect to become at t2 . There are three alternatives: either she1 will expect oblivion because she1 does not exist at t2 , she1 will expect to become both she↑2 and she↓2 in superposition or more likely, she1 will expect to become either she↑2 or

she↓2 exclusively. The first option is implausible and the second is not consistent with our dynamic experience, this leaves only the third option. So far therefore both Saunders and I concur. However, Saunders claims that adopting a relational approach is enough to account for the experience that she1 has in becoming either she↑2 or she↓2 at the point of measurement. The implication being that relationism accounts for our dynamic experience generally. I can have some sympathy with the motivation of many philosophers to limit their ontological commitments. However, I would suggest that in some cases ontological minimalism could be taken a little too far. So let us continue with Saunders line of argument to see where it leads. The first thing that Saunders does is to attack our literal experience of time. In my Story in a Nutshell I use the famous quote from Weyl (1949) The objective world simply is, it does not happen. Only to the gaze of my consciousness, crawling upward along the life-line of my body, does a section of this world come to life as a fleeting image in space which continuously changes in time. (Weyl, 1949, 116)

But Saunders describes this as Weyl’s attempt to dislodge the whole idea of our consciousness moving through time. And yet when I examined the surrounding text in his book I could find no evidence that Weyl was trying

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to discredit it. It seems to me that if Saunders is attempting a full account of our dynamic experience then his assault on the above quote is fatal to his enterprise. And if his reply is that the dynamic change we experience is just how things appear then he must answer, appear to whom or to what? Saunders then proceeds to argue as anyone would by making the perfectly reasonable statement, If my consciousness crawls up the life-line of my body, then it departs from one time, t1 and arrives at another t2 ; in which case my body at t1 has no consciousness,… But if it does not, then my consciousness is at both times at once, an absurdity. (Saunders, 1998, section 4.2).

What follows is his attempt to explain our experience of time using relational theory alone. I need to make clear at this point that I have no problem with relational theory, just its use in this context. Immediately after the above quote he follows with, In contrast, on the relationalist account, the movement of consciousness is already described by the life-line, in terms of the relations among its parts; nothing crawls up my life-line, my life-line already depicts change. (Saunders, 1998, section 4.2).

This may seem like splitting hairs but this sentence betrays its own problem. Saunders is correct when he says the lifeline depicts change, but it is not change in itself. It is like saying that I have a photograph of my car that depicts or describes it. But if that is all that I have then I cannot get in it and drive it away. Likewise the lifeline, it only depicts change but nothing actually changes. He then follows this with As for the second horn of the dilemma, consciousness is at both times, but not both times at once. To be at both times is being at the first time, and then, i.e. to stand in a temporal relation, to being at the second time. (Saunders, 1998, section 4.2).

In short this is presentism in disguise. He acknowledges that we cannot be present at two distinct times at the same instant. But in the second sentence he indicates that we are at the first time and then at the second time, and we experience this in the corresponding sequence. There is nothing wrong with this in itself but when we are at the second time, there is no acknowledgement of the first time, and he should not simply ignore it. Either he tacitly assumes presentism, or he needs to explain what the physical difference is at the first time, between when we are at the first

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time and when we are at the second time. In this book we overtly assume eternalism. Saunders seems to do the same (Saunders, 2002). Therefore, because he is also assuming physicalism, he shoulders the burden of having to explain what changes at a specific event from when we are present at it to when we are not. It is no good just saying that we are at t1 when we are at t1 , and we are at t2 when we are at t2 . If we are assuming physicalism and eternalism together then there must be a change at t1 when we pass through t1 because our minds are physical, and when t1 is in the past it still exists even though we are not there anymore. From what has just been said it seems that Saunders falls into the same trap as Lockwood and Deutsch. By tacitly assuming physicalism he provides an impeccable description of the physics and just leaves it there. With Lockwood we have the experiential manifold and Deutsch’s endorsement of it. Equally with Saunders we have the lifeline meandering on a timelike trajectory without any outward acknowledgement of its status as a static geometric structure. Saunders’ lifeline is empty. There are no travellers on the road.

9.2.3 Probability in a globally deterministic universe In her introduction Jenann Ismael (2003) asks the question: how can deterministic laws, together with complete knowledge of the past, leave room for chancy events? Given that we are considering a globally deterministic reality where all future possibilities are realised in superposition, then this may be regarded as a fair question. Indeed she is not on her own. Hilary Greaves (2004) addresses the same question and both of these authors claim to have solutions that do not involve consciousness as an irreducible add-on. My question is: do they? Peter J Lewis (2007) also addresses this issue in the context of branching persons within an Everettian universe. He concludes that subjective uncertainty cannot be grounded in probability within an Everettian landscape. In this subsection we consider whether the probability issue has any bearing on the metaphysics of consciousness within a quantum reality. In section 3 of her paper Ismael provides an approximate description of what we are arguing for in this book. Her particular stance with regard to this model can be summed up in her following statement. The suggestion that we add to the Everett picture, a humunculus,… traveling a unique path through the branches,…and that we interpret the Born probability of an up-result in a spin measurement as the probability

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Here we see a less than veiled assumption that physics is the only game in town. Moreover, by the use of the phrase …isn’t very satisfying as physics, she has forgotten that the humunculus is meant to be non-physical, so in this context she is not talking about physics. Earlier in this paper she stated, almost accusingly, that some Everettians have a tendency to introduce non-physical or non-supervenient elements into the overall scheme of reality in order to explain our classical experience. It should be pointed out that it is not just Everettians that recognise the need for nonmaterial elements additional to the physics. We only need to point to Weyl’s now famous quote, which makes no reference whatsoever to quantum theory. Indeed, looking at modern physics in the round, clues to the absence of our minds from the physics are impossible to ignore. In the section following that she refers to probabilities of potential events being actual in collapse theories, whereas and in an Everettian universe she replaces the word actual with here. Because, in the context of electron spin measurements, here picks out one of the post measurement branches, which looks like a reference to C-space location. Born probabilities tell each observer how typical their situation is at the beginning of each post measurement branch. In the final paragraph of her section 4 Ismael goes on to claim that if her proposal works …it means there is no need for non-supervenient facts about personal identity, it means there is something in the universe (something already there, something there before we add supernatural facts constitutive of personal identity, or non-supervenient relations between person stages) on which to pin the Born probabilities. (Ismael, 2003, 780).

What is not in dispute here is that Born probabilities are determined solely by the physics, and are not influenced by the presence, state or otherwise of any conscious agents. The claim here is that there is no need for nonphysical facts about personal identity. In order to appreciate this claim we need to be fully aware of what it is that we attach a personal identity to. Identities are attached solely to our physical aspects. It is animated human bodies to which identities are assigned and not the humunculi as Ismael calls them. The token minds that progress through the C-space via a multidimensional network of lifelines only have claim to those identities throughout their lives. Using the analogy of a road I could argue that I am driving along the A500 when indeed I am. However, I (the traveller) am not the A500. Of course this analogy is limited, roads are not generally

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one-way and they do not branch and retain the same identity. The identity of a particular road is usually only retained by one of the exits at a particular junction. But in the case of branching lifelines in C-space, all of the branching avatars originating from a particular point are entitled to claim the same identity. This does not cause any problem because the branches are isolated from each other in a communicative context. So when Ismael claims that there are no non-supervenient facts about personal identity, she is right. However, the point she seems to have missed is that personal identity has nothing whatsoever to do with consciousness. A conscious agent is only associated with a particular identity as long as it is on a lifeline possessing that identity. Beyond her section 4 she does make reference to localities in C-space by pointing out problems when referring to post measurement results indexically. This is because distinct post measurement branches can claim the same spatiotemporal location (but not the same space-C location), and are equally continuous with their pre-measurement history. Our language has evolved only to cope with linear time and not branching histories. Therefore we need to add-on extra terminology in order to identify and discriminate between branches or lifelines post measurement. In her final section 11, Ismael discusses the preferred basis problem; the symmetry of which she claims is another source of the tendency to introduce consciousness into an Everettian universe. The way to defend democracy of basis, she claims, is to start from the position that there is no objectively privileged basis and then go on to make any preferred basis perspectival. There is nothing wrong with this either. Moreover this is exactly what Bitbol (1990) does. In other words both Ismael and Bitbol argue for a democracy of basis and that any preferred basis is relative to some perspective, albeit a location or some general orientation in C-space. Unfortunately Ismael does not acknowledge that something must occupy that location or orientation in order to witness reality from the corresponding perspective. Like Lockwood, Deutsch, and Saunders, Ismael has placed herself in the position of describing the physics adequately, and has even acknowledged classical perspective. However, she has not recognised that something like you must occupy a location in space-C at t1 when you experience being at t1 , but when you are at a later time, t2 , there is no physical change at t1 . But there is a metaphysical change because, as I continue to emphasise, you (your token mind) are not there anymore. Hilary Greaves (2004) takes a somewhat different approach to the problem of probability in a branching universe. Greaves is more concerned with rational courses of action by an observer possessing knowledge that

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she is living in a branching reality. For this Greaves uses the apparatus of decision theory (Savage, 1972) as developed in a quantum setting by Deutsch (1999) and Wallace (2002; 2003). This amounts to saying that an observer prior to a branching event should care about all of her postbranching successors in proportion to their respective Born probabilities. By itself there does not appear to be anything in her aims that would conflict with what we are saying in this book. Apart from that, I do not propose to go into the details of quantum decision theory here since it would take us too far from our present course. The interested reader is therefore referred to (Greaves, 2004 and references therein). However, Greaves does seem to lay too many cards on the table, in that as far as I can see she is making statements that are unnecessary for her enterprise. To begin with there is a statement in her introduction exposing all of her physicalist’s credentials, which reads We are to adopt a reductionist approach to personal identity over time and a …strong supervenience of the mental on the physical: once the physics has been specified, there is no further matter of which post-measurement observer will be me, and we are simply to choose a best description. (Greaves, 2004, 425).

This is in complete contrast to the approaches of Albert and Loewer, Squires etc. There does not appear to be anything wrong with a reductionist approach to personal identity over time, since we assume the position that identity applies to physical lifelines and not the token minds that trace their paths. However, we do take issue with a strong supervenience of the mental on the physical. This is because it has already been shown in section 8.2, to be without foundation, a course that has led us to a minimal supervenience (LSU). The stance adopted by Greaves is therefore one where she questions the relevance of subjective uncertainty. The example that she uses is one where the subject, Alice, performs a spin measurement on an electron to obtain either spin-up or spin-down. Alice’s person stage prior to the measurement is labelled Alice1. At the point that she obtains her measurement result, there is a bifurcation that is followed by two person stages, Aliceup2 and Alicedown2, depending respectively on whether she sees spin-up or spin-down. As we have already discussed the three options as to what Alice1 will become post measurement, mentioned by Saunders (1998), are that she will experience: i) ii)

neither (oblivion) spin-up or spin-down

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spin-up and spin-down.

She takes issue with Saunders whose preferred option was spin-up or spindown. Saunders’ option is consistent with common experience, whereas Greaves opts for spin-up and spin-down on the basis that both person stages exist in an Everettian context. For Greaves there is no subjective uncertainty because both sets of classically incompatible events are realized. It can therefore be argued that Greaves’ option is the correct one in the wider context of physics over a sufficiently large patch of C-space. However, this is at odds with her tacit claim that she deals effectively with what Alice1 will experience. In her section 4.1 (Greaves, 2004, p438) where she discusses subjective uncertainty, there is a further statement illustrating perfectly the issue at hand, relating to Alice measuring the vertical spin of an electron. …we get the following: the personal-identity-over-time relations among the person-stages are such that, according to our counterpart-theoretic account of talk of the future, Alice1 will become Aliceup2; and Alice1 will become Alicedown2. Similarly, Alice1 will see spin-up, and Alice1 will see spin-down. (Greaves, 2004, 440).

Careful consideration of the two sentences that constitute this statement should reveal that they are referring to different aspects of the situation. Focussing on the first sentence, the operative phrase is: Alice1 will become Aliceup2; and Alice1 will become Alicedown2. As we have already remarked there is no contradiction here provided that it is made in a quantum physics context. However, the second sentence refers unequivocally to subjective elements: Alice1 will see…. This is about what a mind of Alice1 will experience seeing after the measurement. Any token mind (not just those belonging to Alice) occupying a position in C-space corresponding to Alice1 will, without doubt, be uncertain as to whether she will see spinup or spin-down. Given full prior knowledge of the measurement set up it would be possible to calculate the Born probabilities for each of the two outcomes. A token mind present at the Alice1 stage will be uncertain as to whether she will experience seeing spin-up or spin-down. The uncertainty lies with the nonmaterial mind not with the physics. It should be made clear that common experience dictates that we never witness superpositions of mutually incompatible events. In this work we take this as strong empirical evidence, given that there has never been cited any demonstrable example to the contrary throughout the whole of human history. All of our knowledge including our science is presented to us through the channel of common experience, and any attempt to

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undermine that also challenges all of our prior achievements. We therefore need to put this apparent absurdity into a different context. Greaves is referring solely to the physics of the situation because she adopts the position that minds are totally physical. As we have already discussed there is no contradiction in the wider quantum context. As far as I can ascertain Greaves could have demonstrated an understanding of Born probabilities in the context of quantum decision theory without resorting to maximal supervenience of the mental on the physical. That is she could have left her physicalist’s leanings to one side. However, in a way Greaves has done us a great service. By tempting us to treat the second sentence in the above quotation subjectively she has demonstrated perfectly why physicalism can never be made to work in an Everettian context. Peter J Lewis (2007) holds a similar position to Greaves. In response to Saunders’ three options, neither, both or one or the other, he states that just as the post-junction segments of a forked road are physically continuous with the pre-junction segment, then a premeasurement person stage is continuous with its two post-measurement stages. In Lewis’ analogy the post-fork segments head for Upton and Downham to appropriately reflect the spin eigenstates of an electron after measurement. Here road1 is the road segment at x 1 before the fork whereas road2 ↑ and road2 ↓ are road segments at x2 heading for Upton and Downham respectively. Then he makes the case for which is the appropriate question to ask. The analogous question in the road case is this: Which road (if any) does road1 become at x2? (Note that the question is not ‘‘Where will I get to if I drive along the road to x2?’’; the analogy is between the road itself and the Everettian person.) (Lewis, 2007, 3).

Here Lewis uses the analogy of a forked road to describe Everettian person stages through time. In the note at the bottom of this quote it is clear that Lewis associates the road with the physical aspect of an Everettian person taking a quantum measurement. Again there is a clear definition of the Born probabilities at the measurement junction both before and after it. But the uncertainty lies with token minds following the lifeline, minds do not split therefore they must take only one of the two alternatives. The reason there is difficulty in connecting probability with uncertainty is because they are associated with entirely distinct aspects, probability is associated with the physics only, and uncertainty is connected solely to subjective elements. Moreover, we see that Greaves does not countenance subjective uncertainty because that is not part of the physics.

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Both Lewis and Greaves concur because they are concerned only with the physics encapsulating both quantum states post measurement. Lewis however seems more guarded when it comes to minds and the issue of their consciousness. Indeed that such things are not even mentioned suggests that an understanding of probability in a branching universe is entirely possible without any reference to supervenience, or otherwise, of the mental on the physical. Again I refer to the analogy of the pinball machine described in the previous chapter. A pertinent example would be that of a ball that drops through a junction with two gates each leading to separate paths. In the absence of a ball it is still possible in principle to calculate the probability associated with each gate. Uncertainty as to what will happen is only meaningful when a ball is set in motion within the machine. Similarly uncertainty in an Everettian universe is only meaningful when conscious minds are present.

9.3 Traditional support for physicalism Notwithstanding the almost wholesale elimination of traditional motivations for physicalism, related arguments in support thereof do still persist within the extant literature. Examples are undoubtedly numerous, therefore we will discuss just four adherents who I believe to be fairly representative of traditional support. In chronological order these are Gilbert Ryle (1949), Derek Parfit (1984), Victor Stenger (1993), and Richard Dawkins (2006). Additional support for traditional physicalism may also be found amongst those who would attempt to reinstate presentism in a relativistic setting. As mentioned in chapter 4, one such supporter of a growing block form of presentism is Rafael D Sorkin (2007a). The example I have in mind concerns a close colleague of Sorkin’s, Helen Fay Dowker who is a distinguished theoretical physicist in her own right. On one particular BBC Radio 4 broadcast Dowker mentioned that at some point she had converted from Christianity to atheism, while also changing her views of reality from the traditional block space-time model (eternalism) to favour the growing block (Dowker, 2017). This prompts the question as to whether there was a connection between atheism and a desire to model the universe according to a form of presentism. Of course atheists are not necessarily physicalists although the converse is true. Physicalism could be regarded as being at the extreme end of the atheism spectrum. One could therefore speculate that if Dowker’s atheism is physicalism, then to maintain consistency she converted from an eternalist view to one of presentism. If true this would

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be an example of presupposing physicalism, a charge that could possibly be laid at the door of all four of the authors to be considered here.

9.3.1 Ryle In his famous text, The Concept of Mind, Gilbert Ryle (1900-76) relies on a linguistic device referred to as a category mistake. Nowadays more commonly known as a type error, category mistakes are rooted in the idea that sentences can be grammatically sound while still asserting nonsense. According to Ryle dualism is one big type error. At the beginning of section 2 of chapter 1 he writes Such in outline is the official theory. I shall often speak of it, with deliberate abusiveness, as ‘the dogma of the Ghost in the Machine’. I hope to prove that it is entirely false, and false not in detail but in principle. It is not merely an assemblage of particular mistakes. It is one big mistake and a mistake of a special kind. It is, namely, a category-mistake. (Ryle, 1949, 5).

The origin of the category mistake lies with Descartes, a contemporary of Galileo, who being a man of science had no choice but to endorse Galileo’s findings of the omnipresence of mechanical laws. At the same time, being a moral and religious man, he could not accept that the human mind as just a more sophisticated clockwork mechanism. The differences between the mental and the physical were presented as differences within a class of categories, which include: things, material, attributes, processes, changes, causes and effects etc. Minds are things but different sorts of things from bodies. All of the processes governing minds are distinct from those controlling matter. So what form do type errors take? An example used by Ryle is the conjoining of two types in the same sentence such as She came home in a flood of tears and a sedan-chair. (Ryle, 1949, 11).

The types so conjoined are flood of tears and sedan chair. This is said to be a well-known joke illustrating the absurdity of conjoining terms of different types. However, inspection of the sentence reveals that with a slight modification it can be made to make logical sense even if the situation it describes is still ridiculous, that is She came home in a flood of tears and in a sedan-chair.

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Inserting the second “in” in the above sentence seems to do the trick. The corresponding disjunction that Ryle quotes is She came home either in a flood of tears or else in a sedan-chair. (Ryle, 1949, 11).

The either/or else clauses suggests that the disjunction is exclusive. According to Ryle this quote is still ridiculous but as far as I can see it does make logical sense. Perhaps she was happy that she came home in a sedan chair, or that she was so disappointed at not coming home in a sedan chair that she was in a flood of tears. Ridiculous? Yes. Logically possible? Also yes. This illustrates how perilous it can be to rely purely on the properties of language to draw conclusions about objects that the prose is being used to describe. Another thing that Ryle does is to presuppose physicalism. He begins with the example of a foreigner visiting the University of Oxford. Walking amongst the different colleges he asks: where is the University? as though the University were a separate and different type of edifice to each of colleges. The category mistake here is to assume that the University is a single separate building as opposed to a collection of colleges. But Ryle betrays his prior assumption of physicalism because in this analogy for the collection of colleges read the body, where the University represents the mind. By applying such an analogy we see that he is using physicalism as an axiom, he is not proving it. It would take us well off course to embark on a full analysis of Ryle’s work, but the subsequent text undoubtedly consists of many valid and sophisticated arguments. However, if his aim is to prove physicalism then his logic is on a very large circle. Another example that Ryle uses is that of a pair of gloves, this reads Thus a purchaser may say that he bought a left-hand glove and a right-hand glove, but not that he bought a left-hand glove, a right-hand glove and a pair of gloves. (Ryle, 1949, 11).

Here we see the same sort of mind-body analogy–the left hand glove and the right hand glove symbolizes the body whereas the pair of gloves represents the mind. But I could just as easily say that I have bought my wife a present consisting of a pair of gloves and a brooch. Ryle’s line of reasoning cannot now be used to deny the existence of the brooch. So type errors are mistakes in the language describing objects, they are not necessarily mistakes about the ontology of those objects.

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Lastly we should be aware that Ryle’s criticism of Cartesian dualism condemns the notion of a single ghost that is somehow harnessed to the machine. We have seen in the previous chapter that the form of dualism arrived at by Albert and Loewer, Bitbol, Squires, Hemmo and Pitwosky, and here is not of that sort. Indeed it might be interesting to speculate on how Ryle would respond to our localisation model. If he was so wedded to physicalism I suspect he would have followed a path similar to Lockwood and his supporters, but alas we will never know.

9.3.2 Parfit Another late twentieth and early twenty first century philosopher subscribing to personal identity reductionism was Derek Parfit (19422017) whose ideas are crystallized in his famous work Reasons and Persons (1984). Based on his reductionist approach Parfit was meticulous in his analysis of personal identity with a rigour rivalling mathematical standards. In the end Parfit concluded that personal identity could not be defined in any meaningful way. Individuals are nothing more than their constituent brains and bodies, and personal identity cannot be reduced to either. The upshot of this is that personal identity is not what matters, what matters is what he refers to as relation R: the psychological connectedness of memory and character in conjunction with overlapping chains of strong connectedness. In the absence of any evidence supporting a non-material component Parfit’s very impersonal analysis of our material selves is extremely persuasive. However, it seems that he did not reckon with the developments of modern physics and how they square with our everyday experience of life. Parfit makes use of science fiction plot devices, such as teleportation and replication to explore our intuitions concerning personal identity. Indeed we can devise a thought experiment involving an accidental replication where a teleportation scenario goes wrong by failing to disintegrate the subject at the departure point. Such a scenario is portrayed in an episode of Star Trek: the Next Generation entitled Second Chances. This concerns the character of Cmdr William T Riker who, eight years previously, had been accidentally and unknowingly duplicated on an earlier mission of the USS Potemkin to planet Nervala IV. Unaware of what had happened the copy of Riker that was transported off the planet continued his life and blossoming career in Star Fleet, whereas the duplicate Riker had been left marooned at the Nervala IV station for eight years until he was rediscovered during a later mission of the USS Enterprise, the crew of which included Riker. So what can we say about

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personal identity in such a case? Both copies are entitled to the same identity, the plot continued which included a conflict as to which copy was entitled to the identity, William Riker. Eventually both parties agree to the rescued Riker adopting the middle name Thomas, therefore taking on a new identity. This seems like a good description of relation R, and I think Parfit would have approved. The episode concluded with Thomas going on to live a separate and slightly more nefarious existence, much to the disdain of his more successful twin. So how does our model of the localisation of consciousness fit in with scenarios like this? This is similar to Everettian branching except that in the Star Trek episode the resulting two characters occupy the same timeline, whereas Everettian copies are isolated from each other on distinct timelines and therefore with no conflict of identity. Either way Lewis’ analogy of the road is perfectly apt. The scenario with the one Riker becoming two is analogous to a forked road. We can reconcile this with everyday experience by modelling a single mind travelling along that road. To be consistent with localisation that mind cannot travel on both routes beyond the fork, it is forced to take one or the other. So in that sense Parfit’s relation R model, which concerns the physical aspects only, may be subsumed into the localisation of consciousness theory. The Parfittian model can be modified slightly by adding a clause to the effect that it is not concerned with consciousness. Where we do include conscious minds we see that both individuals experience continuous lives, which are identical prior to the bifurcation point. This removes any conflict with the localised consciousness idea. Another interesting scenario, which Parfit explores and has taken place in reality, is what has become known as fission. This is briefly explored by Korfmacher (2006, section 4) and is a situation where a subject X is divided into Y1 and Y2 . Neither Y1 nor Y2 can be identical with X because Y1 and Y2 are not identical to each other and therefore to make such a claim would violate the transitivity property of identity. Therefore at the point of bifurcation X dies and two more Parfittian survivors emerge in its place. For Parfit this is not a bad thing because we are getting two individuals for the price of one. In the real world such cases involve patients that suffer an extreme form of epilepsy. The treatment involves a surgical procedure where the entire corpus callosum is severed, hence separating the two hemispheres of the brain. The result of this has become known as the divided mind. Physicalists often cite this scenario as empirical justification for their position. However, in this work we claim exactly the opposite, the results of these surgical procedures fit

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in very nicely with our localisation claims. To summarise, the first paragraph of Parfit’s section 87 reads Some recent medical cases provide striking evidence in favour of the Reductionist View. Human beings have a lower brain and two upper hemispheres, which are connected by a bundle of fibres (the corpus callosum). In treating a few people with severe epilepsy, surgeons have cut these fibres. The aim was to reduce the severity of epileptic fits, by confining their causes to a single hemisphere. This aim was achieved. But the operations had other unintended consequences. The effect, in the words of one surgeon, was the creation of ‘two separate spheres of consciousness’. (Parfit, 1984, 245).

In essence it was found that the left hemisphere received sensory input from sense organs on the right and controlled the right hand half of the body via motor function, the right hemisphere having a similar relationship with the left half of the body. During tests where the visual field was appropriately split it was found that the separate spheres of consciousness were completely unaware of each other, except maybe indirectly via other sensory input. In our model, the relationship between bodies and minds is not one-to-one. There is always the possibility of many minds taking a path through one body, hence Albert and Loewer’s position. To borrow Lewis’ road analogy once again, we can think of the surgical procedure described above as a single one-way road suddenly becoming a dual carriageway, where the severed corpus callosum is represented by the central reservation. Any non-physical mind can only take a route associated with one hemisphere only, and a mind confined to travelling the timeline of one hemisphere would always be unaware of the activity of the other. So again we can imagine this experience through the eyes of the subject. At this point we could entertain the hypothetical possibility that the corpus callosum could be reconnected. In such a case all of the memories associated with both hemispheres would become available to a mind that had traversed just one side of the carriageway. Therefore a token mind could never know whether it had taken the left-hand or the right-hand route post reconnection. The divided mind will be revisited again in subsection 11.1.2. Parfit considers many more such examples all of which can be subsumed into the localisation of consciousness model without generating a contradiction. We can read Parfit’s work without any reference to nonmaterial minds, and the whole thing makes perfect sense. When we apply our localisation model we can see Parfittian scenarios through the eyes of

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the subjects to which they are being applied and there is still no contradiction. Parfit’s presupposition of physicalism was an effective way for him to develop his ideas. But when we look at the entirety of his work in hindsight, it is easily seen that physicalism is unnecessary.

9.3.3 Stenger In the middle of 1993 an article appeared in The Humanist, which begins: A new myth is burrowing its way into modern thinking. (Stenger, 1993, 13). The myth referred to here is The Myth of Quantum Consciousness by Victor J Stenger (1935-2014). In this work Stenger protests strongly against the idea that the external classical world is not objective but is somehow brought into existence by the perception of minds. He is particularly vociferous towards another article that appeared in the same journal during the previous year (Lanza, 1992). One of the targets of Stenger’s condemnation, Robert Lanza’s The Wise Silence, does have some wisdom attached to it regarding the nature of the perceived classical world, most particularly at a microscopic scale. However, Lanza is not a quantum theorist, and his evident misunderstanding has left him wide open to attack by the mainstream. In this respect I believe Stenger is attacking a straw man. So what is it precisely that Stenger objects to in Lanza’s and other’s work? Personally I think it is best summarised by the statement In the laboratory, particles spring into existence as real objects only when we observe them. (Lanza, 1992, 24).

Here the italicised part is likely to mean that the act of observation, or measurement, is cast in the broadest possible context. The important question here however, relates to what we mean by existence. In this work we take it as given that existence is binary, that is imagined objects either exist or they do not, there is no third status. After the lengthy discussions of part I, we are throwing in our lot with eternalism and pure wave theories. In this model of the physical world we can represent a configuration of particles within a base space of a particular geometry, either as it is described here, or as a single point in a universal C-space of correspondingly large dimensionality. In the latter representation we do not see particles; instead this conjures up the image of a complex valued universal wave function over C-space. It is not that particles do not exist; rather it is that we are representing reality in terms of something more fundamental, the wave function. Particles are emergent, so when we choose a point in C-space then given that perspective, we observe

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particles, the positions of which are related directly to a particular subset of C-space coordinates. Alter the values of those coordinates and we change the positions of the particles. So to say that particles do not exist prior to measurement is tantamount to saying that a point in C-space does not exist until we arrive there. It makes no sense to say that parts of Cspace exist and others do not. Therefore the whole of C-space must exist and consequently so do the associated emergent entities such as particles. I believe that this is Lanza’s mistake, and is the nub of Stenger’s objection. Stenger is a realist in the sense that he takes the universe around us to be ontic and therefore independent of our minds. The final paragraph in his article begins with The overwhelming weight of evidence, from seven decade(sic) of experimentation, shows not a hint of a violation of reductionist, local, discrete, nonsuperluminal, nonholistic relativity and quantum mechanics— with no fundamental involvement of human consciousness other than in our own subjective perception of whatever reality is out there. (Stenger, 1993, 15).

Evidently Stenger is a physical reductionist, but I do not criticize him for that position because so am I, and I think most working physicists are too. It seems that Stenger does not necessarily deny the existence of conscious agents. He just keeps them out of physics, which as we have seen, is the correct thing to do. Stenger may or may not be a mind-body physicalist, it is difficult to be sure from this work. The one thing he could be criticized for is his claim that the Copenhagen interpretation of quantum theory has been finally confirmed (Stenger, 1993, 15), whereas in his later edition he only states that the Copenhagen interpretation is conventional (Stenger, 2002, 14). If this shows anything it is that Stenger has softened his position on interpretations of quantum theory, and is open to new ideas even though he may not like pure wave theories. In the end he comes across as a physicalist even though he may not be. If this shows anything, it is that we have to be careful not to read too much into the works of others.

9.3.4 Dawkins While commanding the role of a distinguished zoologist and evolutionary biologist Richard Dawkins (1941- ) is also well known for his atheistic views. Indeed he is the better known of three protagonists who have become known as the new atheists (Armstrong, 2009), the other two being Christopher Hitchins and Sam Harris. As mentioned earlier atheism is not

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necessarily physicalism. It is possible to hold the views of a dualist or mental monist without necessarily believing in an omnipotent universal mind. Given the modern physical theories that we endorse here, Squire’s universal mind, for example, maybe the same God referred to in the scriptures. However, as we have discussed already, Albert and Loewer’s metaphysics describes infinitely many minds each with an independent existence, there is no universal mind. Therefore on this basis Albert and Loewer could equally claim to be atheists while still promoting dualist views. As far as I can see Dawkins makes no claim to physicalism either in his arguably most famous text The God Delusion (2006), or in one of his earlier works, The Blind Watchmaker (1986). Dawkins has written many more books and articles, but at the risk of showing my ignorance, of these I can make no comment. Rather than promoting physicalism Dawkins main concern is his opposition to intelligent design in the context of universal creation. He focuses on evolution as a mechanism through which the diversity of life on Earth today has grown. Dawkins points to the discoveries of Charles Darwin (1809-1882) and Alfred Russel Wallace (1823-1913) as the dawn of a new age of enlightenment, which should mark the beginning of a gradual decline in religion generally and creationism in particular. Essentially he uses evolution as a platform to voice his opposition to religious fundamentalism. Dawkins is particularly antithetic to religion as a cause of so many ills in the world. The diversity of religious beliefs worldwide and over many millennia has resulted in humanity being divided into factions, which in turn is seen as the cause of so many conflicts. For this reason Dawkins describes religion as pernicious, a view with which I do have some sympathy. We only have to look back at the excesses of the sixteenth century Spanish Inquisition and its contemporary Protestant Reformation in Britain and Northern Europe, Nazi Germany in the 1930s and 1940s particularly with regard to the Jews, or the present conflicts in the Middle East, to appreciate the role of religion in conflict situations. However, I think we should beware of oversimplifying things too much. I am rather more inclined to the view that humanity will find any excuse for conflict, and pretty much any division will do including politics and race. If Dawkins is seriously looking for a cause for humanity’s violent tendencies then I believe he has missed the target by a substantial margin. Religious fundamentalism is merely an excuse not a cause. As a biologist, and a distinguished one at that, he should look no further than our own DNA and certain mutations of the MAO-A gene in particular. At this point I should make it clear that I am, most emphatically, no supporter of

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eugenics, I merely point out the possible role of certain genetic mutations or configurations, in our behaviour. I leave that one to those better informed than I to suggest possible courses of action in this matter. With regard to our main concern I should mention that that both evolution and creation are dynamic processes, and it appears that Dawkins makes no attempt to address the question of our dynamic experience and consciousness. It seems that Dawkins is thinking classically as opposed to quantum mechanically. Most of what Dawkins has to say on quantum theory describes the Copenhagen interpretation as preposterous and the many worlds interpretation as wasteful (Dawkins, 2006, 365). It is easy to see how someone with less than a full insight into quantum theory might adopt these views. As we have seen, the Copenhagen interpretation is not a precise as other interpretations. It has associated with it wave function collapse, which has problems of its own if it is considered ontic. We have already discussed difficulties associated with epistemic interpretations, so I do not consider this model credible. It is understandable to say that pure wave theories are wasteful if we are restricted to thinking in a classical context. However, once we accept the ontic status of the universal Cspace, then pure wave theories are the least wasteful of all because Cspace and the wave function it contains are the only ontic physical elements in it. Time is emergent from within this framework, so in this wider context the physical universe was neither created nor did it evolve, it just simply is. This does rather sidestep Dawkins’ argument against creation and for evolution.

9.4 Summary From the postulates on which we base our arguments we see that the theories offered by Lockwood and his many supporters lead inevitably to a contradiction. This denies our experience by revealing a static landscape completely devoid of anything dynamic. Michael Lockwood’s experiential manifold is a case in point. Simon Saunders’ attempt to address this problem using a relational theory of time, comes very close to a solution, but ultimately fails because there is no recognition of the perdurance of past events. The theory crumbles with any attempt to deny the existence of past events because that denies our first postulate by slipping into presentism. More traditional arguments for physicalism take no account of modern physical theories. A well-known example is Gilbert Ryle who deploys a linguistic device known as a category mistake. Notwithstanding the sophistication of Ryle’s arguments it was shown that properties of

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language describing objects have little bearing on the ontology of those objects. Moreover, it is seen that in his famous text, The Concept of Mind, he presupposes physicalism, which renders his whole argument circular. Derek Parfit, on the other hand, is a personal identity reductionist. Once we recognise that personal identity is attached solely to the physical body then his meticulous arguments pose no threat to the dualist or mental monist cause. Both Victor Stenger’s and Richard Dawkins’ views could easily be construed as physicalist. However, such a charge may be unjustified. If they are physicalists then they are playing their cards very close to their chests. Although their writing appears confrontational it is levelled against either misunderstandings of quantum theory, in Stenger’s case, or organised religion, from Dawkins’ viewpoint. Stenger is not necessarily implying that nonphysical minds do not exist. He appears to be confronting the misguided idea that minds can influence the physical world via a quantum mechanism. As we have seen already our postulates imply that minds have no influence whatsoever on anything physical. Although Dawkins claims to be an atheist this does not mean he is a physicalist. As we have seen Albert and Loewer could just as easily make the same claim. Whether they do or not is another question. Dawkins however, is very careful not to commit himself in any particular direction as far as consciousness is concerned. Having briefly mentioned Helen Fay Dowker, similar comments may apply to her views even though it appears that she is now committed to a form of presentism. This does deny our first postulate and hence leaves the way open to mind-body physicalism.

CHAPTER 10 PHENOMENAL CHANGE AND TIMELESS PHYSICS

In the previous chapter we argued against ideas proposed by some of the leading scholars favouring a physicalist solution to the mind-body problem while adhering to a timeless view of physics. Our arguments were part justified by a timeless interpretation that followed from part I. However, it is not inconceivable that we could be accused of relying on a model of timelessness that is considered too simplistic, unsophisticated, or just plain naïve, although I believe our interpretation is rather better described as pragmatic. The purpose of this chapter is to head off any such criticism by examining some of the leading alternative views of phenomenal (experienced) time, and to further clarify our position on the issue of timelessness in physics. Notwithstanding a comprehensive discussion of space-time in part I it may be argued that the discussion so far tells us everything we need to know about time. For physical time, adequately described by a B-theoretic account, this is true. However, nothing in part I comes close to clarifying the nature of phenomenal change. As we will see, phenomenal time is intimately associated with the mind-body problem. Much of the confusion surrounding interpretations of timelessness, and in particular the emergence of theories spearheaded by Lockwood (1996), seem to stem from the tendency of many philosophers to identify B-theoretic physical time with A-theoretic phenomenal change. It is our view that these two descriptions of temporality should be regarded as entirely distinct. In this regard we concur with Bergson’s (1910) view of time. Moreover, it may be regarded as questionable whether physical time should be regarded as time at all (Kroes, 1984, 441). Previously we have relied on the principle of localisation (chapter 8) to justify our experience of phenomenal time. Lockwood (1996) acknowledges that this approach in its current form (Albert and Loewer, 1988) solves the problem. However, as we have seen, Lockwood also sought to create another interpretation not requiring the existence nonmaterial conscious observers. We could speculate that Lockwood’s use

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of the phrase …a desperate expedient…, may indicate his revulsion for such a view. If this is the case then Lockwood’s position could not be considered legitimate. However, searching for alternatives to provide a stern test of our model certainly is a legitimate move. So, in the spirit of charity we should regard alternatives as tests for our own part solution to the mind-body problem. Our approach to the analysis of phenomenal time loosely mirrors the work of Ewing (2013). Ewing addresses the following leading theories of time and temporal change. 1. 2. 3. 4.

Presentism The moving spotlight theory Mind-dependent theories The irreducible fact theory

We discuss these in a slightly different order. Mind-dependent theories are discussed last and prior to that the irreducible fact theory is dealt with under the heading Objective theories. In the next section we deal with presentism by comparing a range of objections, including the need for a universal present contrary to relativity. In the order we present them, they are McTaggart’s (1908) original argument on which he bases his Unreality of time. This is followed by a discussion of Eichman’s (2007) objection, which is based on relativistic arguments and is therefore informed by empirical evidence, even though only indirectly. Following that we consider two further objections by Ewing, these are the contradictory nature of presentism (CNP) and the grounding objection. Of these two it is my contention that the latter is stronger than the former, and the reasons for this are examined. However, the strongest objections are those based on empirical evidence via tests of general relativity and the confirmed existence of macroscopic-Bell states. The other three theories listed are variations of eternalism, although, deep down it is my contention that there is only one interpretation of eternalism. Here the moving spotlight theory and objective theories generally should be thought of as hybrids, this is because in those theories, the present is endowed with special status. Ewing considers the moving spotlight theory to be incoherent. If by this she means that the mechanism for the spotlight is unspecified or illdefined then I would concur. Moreover, the whole idea of a moving now implicitly relies on another phenomenal time variable on which to base its movement. If time is considered to be one thing then the argument on which it is based, is circular. This is one possible source of incoherency to

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which Ewing refers. It also seems that the present illuminated by the spotlight is absolute as opposed to being relative, which is not in the spirit of relativity theory. This also has a bearing on the CNP, which considers the present to be absolute. Having examined the moving spotlight theory and presentism, we discuss why these, and theories of time in general, are considered substantive philosophical theses. That is theses that are debatable and not trivially true or false. Here we explore analogies to presentism and eternalism in a spatial context. The spatial analogies are respectively dynamic hereism and anywhereism (Figg, 2017). Remaining true to a pragmatic approach it would seem that any form of hereism is a nonstarter. If this is the case then can presentism be dismissed in a similar way, or is the fact that A-theoretic past, present, and future have no spatial equivalents, an impediment to this? The theory favoured by Ewing is the irreducible fact theory. This is objective because it indicates that the phenomenal passage of time is an …objective, irreducible fact about the spatio-temporal world. Here I assume that by objective she means outside of the mind and therefore physical. If so then this is where I would part company with Ewing. If the phenomenal passage of time is unanalysable then it must be considered a fundamental element of reality. This is an argument proposed by many presentists, for example Smolin (2013). In addition I would contend that it is a requirement for physicalists because for them, there is no nonmaterial mind on which to base the experience of phenomenal change. From the perspective of Ewing’s thesis the theory we favour is a mind-dependent theory. One thing that Ewing does here is to criticise mind-dependent theories for divorcing phenomenal change from the passage of time. In our form of mind-dependent theory this is absolutely not the case. If she is considering anything like the theory favoured here then she entirely misconstrues the nature of phenomenal change in a timeless physical landscape. Also depending on how it is interpreted Ewing’s irreducible fact theory may be seen as a mind-dependent theory in disguise. This and other issues are explored in detail in the penultimate section.

10.1 Presentism versus eternalism In chapter 4 we saw that the principle of relativity alone does not fundamentally change the common picture of our reality in the space and time context. Yes, we can construct an expression for the Galilean transformation (equation (4.7)) and pretend that we are working with a

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space-time continuum, but this is mere contrivance. In the reality of our immediate experience, we witness an instantaneous extended threedimensional present. As for the rest of time, the past has gone and the future is yet to come. We experience only the present–an instantaneous point in time. So we can be in no doubt that the present exists. But what can we say about the past or the future? This is where two schools of thought emerge dating back to the pre-Socratics. The debate began with two contemporary philosophers of the early fifth century BCE: Heraclitus arguing for the presentist’s cause and Parmenides for timelessness. Julian Barbour summed it up nicely in the passage, The Story in a Nutshell, just before the preface of his book, The End of Time, which begins Two views of the world clashed at the dawn of thought. In the great debate between the earliest Greek philosophers Heraclitus argued for perpetual change, but Parmenides maintained there was neither time nor motion. (Barbour, 1999)

Nowadays as far as the physics community is concerned it is the eternalist’s metaphysics of time that prevails, despite our language being replete with presentist terminology. Reasons for this become apparent in the remainder of this section, which includes arguments both from a purely philosophical standpoint (McTaggart, 1908; Ewing, 2013), and from a relativistic position (Eichman, 2007).

10.1.1 McTaggart’s objection McTaggart’s arguments are based on three classifications of time: Aseries, B-series, and C-series. Our experience of time is so ingrained that it has even insinuated itself into our own language in the form of tense. Indeed this represents one of the ways that McTaggart defines phenomenal time–the A-series, which classifies events in time as past, present, or future. So if we consider a specific event, E, in the future, then E is in the future up to a point in time when it is not. At this point E is in the present, and beyond this point E is forever in the past. In this way the A-series embodies change, hence the justification that the reality of time requires an A-series classification. For the B-series one event, E1 , may be regarded as earlier than another event, E2 , or said another way E2 is later than E1 ( E2 > E1 ). This is a permanent state of affairs, for example the year 2016 is always later than the year 2000. This series does not embody change therefore it cannot be used as a comprehensive description of time. It does however have one

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property in common with time–direction. This is evident from the “ > ” symbol used for the expression in the brackets. The C-series, like the B-series, has order, but it has no direction. To elucidate the meaning of order, consider the triplet of events ( X , Y , Z ) ,

this triplet is said to have the same order as ( Z , Y , X ) but with opposite

direction. However, neither of these has the same order as (Y , X , Z ) . That is because, in the first two examples Y is always between X and Z. It is this feature that defines the order. So a C-series is even more primitive than a B-series in that we have lost direction, but we have still preserved order. Like the B-series, the C-series is regarded as an inadequate description of time because it does not embody change. That is the preserve solely of the A-series. McTaggart’s argument for the unreality of time is in two parts: in the first part he argues that B-series and C-series descriptions of time are inadequate because they cannot represent change. In the second part he maintains that an A-series description leads to a contradiction. I have no quarrel with the first part, but for the second part I think his argument has a chink-in-the-armour, although it still arrives at the right conclusion. In this second part he argues that for every event, it will at one time be future, at another time be present and at a third time be (forever) past. In this way a particular event will instantiate all three temporal properties. But these properties are mutually exclusive therefore they cannot be co-instantiated. This is McTaggart’s contradiction–so far so good. If however, were we to say that an event can possess only one temporal property at a time. Then this is describing time in terms of itself, a manifestly circular argument. This is how McTaggart describes it Now we began by pointing out that there was such a contradiction in the case of time that the charasteristics of the A series are mutually incompatible and yet all true of every term. Unless this contradiction is removed, the idea of time must be rejected as invalid. (McTaggart, 1908)

So McTaggart establishes inconsistencies in the description of the Aseries. The next problem is to describe the three properties, not coinstantiated, without referring to any temporal property. It was to remove this contradiction that the explanation was suggested that the characteristics belong to the terms successively. When this explanation failed as being circular, the contradiction remained unremoved, and the idea of time must be rejected,… (McTaggart, 1908)

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So we arrive at a contradiction and then attempt to remove it by another argument that turns out to be circular. However, as we saw in chapter 2, circular arguments, although flawed, do not necessarily arrive at false conclusions. The conclusion of the circular argument may yet turn out to be true, in which case the contradiction is removed. In order to maintain the contradiction, McTaggart would have needed to replace the circular argument with a direct argument or one that generates another contradiction, this he fails to do. McTaggart’s arguments attempt to remove the temporal properties: past, present, and future. Since past and future are relative to the present, then McTaggart’s ideas, were they watertight, would represent a serious blow against presentism. This would leave only the permanent B-series (or possibly C-series) that eternalists would recognize.

10.1.2 Eichman’s objection The argument against presentism becomes significantly more powerful in a relativistic context. We have already provided an extensive description of relativity in chapter 4. Eichman (2007) provides a more powerful argument based on relativity theory in a form to which McTaggart had no access. To remind ourselves the special theory of relativity is derived from two postulates The principle of relativity: Physical laws are invariant for all inertial frames. Constancy of the speed of light: The speed of light in a vacuum is constant in all inertial frames. Here we see that the postulate of absolute time has been replaced by the postulate of the constancy of the speed of light, a modification that has very profound consequences for our view of space and time in a physical context. One of those consequences is that there is no notion of absolute simultaneity, see Figure 10-1.

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t

t

O

a

O

b

Fig 10-1: Two views of space-time where time, t, is shown on the vertical axis: a Galilean space-time showing all events simultaneous to the origin (in blue), O, and b Einsteinian space-time where the notion of simultaneity is replaced by trajectories of light (in red) passing through the origin, O.

In Figure 10-1, possible trajectories of material particles are illustrated for Galilean and Einsteinean space-times. Both exhibit chronology protection by forbidding closed loops in the trajectories. In the Galilean case we see that trajectories can only cross a time slice in one direction, while arbitrarily high velocities are still possible. This is illustrated in the part of the trajectory that is almost horizontal in Figure 10-1a. In the Einsteinian case the slope of the trajectory remains steeper than the light trajectories at every point, but there is no set of events marking simultaneity. The question is: can we define a present in the Einsteinian case? The definition of a present in the Galilean case is straightforward. Only the axis in the direction of motion is transformed with respect to a reference stationary observer. For all other coordinates, and most especially the temporal coordinate, any two observers will be in complete agreement about their values of particular events. That is, in time they always agree on which pairs of events are simultaneous. Not so in the Einsteinian case, here a pair of events can be timelike separated, where a material particle can reach the later event from the earlier one. They can be null separated where the later event can receive a light signal from the earlier one. Or two events can be spacelike separated where no signalling is possible between the two events in either direction. So how can we define a present in this context? Eichman (2007) focuses on three proposals to make presentism acceptable in a relativistic setting (Hinchliff,

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2000), these are: point presentism, null or cone presentism, and surface presentism. Point presentism relies on the fact that any event is always simultaneous with itself. This tautological definition of simultaneity is really the only one that makes sense in a relativistic context. According to Eichman, Hinchliff dismisses two objections to point presentism. The first is that the here-now present is isolated or lonely, although Eichman does not specify how Hinchliff rejects this objection. For me it is hard to dismiss on logical grounds. A present must contain more than one event otherwise it is trivial and can have no real utility when analysing the simultaneity status of a pair of distinct events. A second objection that Hinchliff dismisses is that for an event to be in the past it must have passed through the present. However, an event that is always at a non-zero spatial distance from a reference observer will never be in the present, yet it will certainly appear in the past after the reference observer crosses a corresponding light line passing through the event. However, it turned out that Hinchliff had little interest in defending point presentism, because he believed no one else in the philosophical community subscribed to it. On this point at least, I concur. Null or cone presentism, as Eichman calls it, defines a present that includes the here-now event and every point on its past null cone. So in this definition, when we look up at a clear night sky, the stars we see are in the present. Points that are timelike separated from the here-now but still in the body of the past null cone are regarded as being in the past. Eichman discusses a number of ways that Hinchliff defends null presentism, but the principal objection that Eichman makes can be summed up by the corresponding definition of simultaneity not satisfying an equivalence relation. In Galilean relativity, proof of this is a trivial exercise because any pair of simultaneous events must have equal values of t, whereas in special relativity the space of simultaneity is dependent on the inertial frame. In all forms of presentism the simultaneity relation is always reflexive. But the corresponding relation in null presentism fails to be either symmetric or transitive (Figure 10-2).

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E Y X Fig 10-2: Failure of symmetry and transitivity of simultaneity in null presentism. Let ~ say “is simultaneous with”, then X  E but E / X because all events  X lie on the past null cone centred on X. Also X  E and Y  E but X / Y . Failure of either of these conditions is sufficient to deny an equivalence relation between two simultaneous events in null presentism.

In Figure 10-2 we see ample reason for the failure of simultaneity applied to a pair of events in the same null present. This is important because presentism, amongst other things, is a metaphysical theory about what exists. When simultaneity fails to satisfy the conditions of an equivalence relation then we are opening the door to relativized existence. In physics, and in keeping with the principle of bivalence, objects, hypothetical or not, either exist or they do not. I would also extend this requirement to nonphysical objects as well. An object is something that by definition is objective, that is, it has an independent existence. Presentism asserts the existence of an isolated present subject to dynamic change. The classification of two distinct points in such a present at a specific instant is by definition, simultaneous. Simultaneity in point or null presentism fails the equivalence conditions, so a present cannot objectively exist for all observers residing within the corresponding hypersurface. The same arguments apply to null presentism when the future light cone is considered also. The requirement to satisfy an equivalence relation is perfectly reasonable when defining a present and therefore excludes point, pastcone, or future-cone based presentism. This leaves the third form of presentism that Hinchliff discusses, surface presentism which, by contrast, constitutes a significant challenge to anyone attempting to dismiss presentism generally. According to Eichman surface presentism seems to

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be based on anti-verificationism. In short this is where we can postulate the reality of some object without being, even in principle, able to verify its existence. In this case the object in question is a hidden inertial frame. To begin with it would seem, according to Eichman at least, that Hinchliff is proposing a modification to special relativity. I do not think this is quite the case. There a hidden inertial frame could exist without any modification to relativity theory because there are no adjustments to its equations. Therefore relativity would always yield the same results. We also saw that a similar situation exists regarding interpretations of quantum mechanics concerning in particular, hidden variables because the Schrodinger equation is not modified in anyway. Hidden variables were discussed at length in chapter 6, but it is mentioned again because the circumstances surrounding hidden variables are similar to those regarding a hidden inertial frame. Markosian (2004) has similar views to Hinchliff in that he tries to integrate presentism with special relativity. Although according to Eichman, Markosian’s views are somewhat stronger than Hinchliff’s. This is how Eichman describes it His stance on the issue, however, is somewhat stronger than Hinchliff’s, since Markosian accuses scientists of loading the special theory of relativity with unwarranted “philosophical baggage”. Markosian envisions two version of the special theory of relativity, which he terms STR+ and STR−. (Eichman, 2007)

Eichman goes on to say that it is the former (STR+) that contains the unwarranted “philosophical baggage”, presumably a form of verificationist principle, and because Markosian accuses scientists of loading special relativity with the aforementioned philosophical baggage, then STR+ represents the standard special relativity. Presumably then STR-, because it does not have a verificationist principle attached to it, may have a hidden inertial frame. This would not change its predictions but would require an external structure that exists outside of space or space-time in order to define a hidden inertial frame. If that is not philosophical baggage then I do not know what is. Moreover as per Laplace’s demon (section 7.1), if the past and future do not exist then in the deterministic theory of relativity it requires no more information to specify an entire space-time block than to describe a single time slice. If it is parsimony we are looking for in our theories then, like Eichman, I believe we should abandon any attachment we may have to an absolute present via some form of hidden inertial frame, or indeed any other form of presentism. Given a Galilean view of space-time, presentism

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may be a more natural view to take. On the other hand, even though we cannot disprove presentism outright, any form of presentism becomes a great deal more complicated in a relativistic setting than its alternatives.

10.1.3 Ewing’s objections Ewing bases her two objections respectively on the contradictory nature of presentism (CNP) and what she calls the grounding objection. We consider each of these in turn. The CNP may be stated as follows: (CNP) Relying on the renewal of the present for objective temporal passage results in the contradiction that the present must be both temporally absolute and non-absolute. (Ewing, 2013, 9).

The argument that Ewing proposes for presentism being self-contradictory is summarised as follows: P1.

P3.

Only the present exists, which means that there is an absolute present. (Presentism) In order for time to pass, the present must continually renew itself. (Objective temporal passage) P2 Ÿ The present must change .

P4. P5. C.

The present must change Ÿ the present is not absolute . The present is not absolute. P1 Ÿ ¬P1 .

P2.

Unfortunately there is a weakness in this argument. P4 asserting that a changing present implies that that present is not absolute can be equivalently written to say that an absolute present denies a changing one. If the word “absolute” is opposed to “relative” then this need not be the case. It is perfectly possible to have a changing present that all observers agree on at each instant. So we can have a changing, absolute present, or what I would call a universal present. This breaks the logical chain between P1 and C in the above argument. We further clarify these points in section 10.2 when we discuss the moving spotlight theory. In the discussion that follows Ewing begins by examining two questions: 1. 2.

What if time does not pass? What if the present is not absolute?

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In response to question 1 she posits a form of presentism called a genuine frozen presentism (GFP). Here she proposes an unchanging present that exists to the exclusion of all else. This denies not only presentism but just about all other possibilities as well. Therefore the cost of denying CNP by this strategy is too high. Ewing then considers an alternative proposal eternal frozen presentism (EFP), in which the present has a non-zero duration consisting of the now, earlier than now and later than now. These relational terms are equivalent to present past and future respectively (Horwich, 1987). This model is reminiscent of Smolin’s (2013) thick present. But being unchanging this looks more like a model of eternalism with bounded past and future. Ewing reasonably concludes that neither of these proposals is an effective way to avoid CNP. In discussing question 2 Ewing illustrates the problem with this proposal within the first few sentences. If the present is not absolute then there are no non-relative facts about the status of different instants, thereby transforming presentism completely into eternalism. As Ewing suggests this does offer a solution to CNP but at the price of presentism itself. And we can see that both EFP and GFP offer no mechanism for phenomenal change. However, as we have already seen there is a distinct weakness in CNP, which is proposition P4. The idea of an absolute, changing present is a perfectly reasonable and is likely to be what most presentists have in mind. Ewing’s second objection is, in my view, on more secure footing. This is the grounding objection which is based on three main ideas: S1. S2. S3.

Truth depends upon what exists. What exists in the present underdetermines truths about the past and the future. There are determinate truths about the past and the future. (Ewing, 2013, 14-5).

For details of this objection see Ewing (2013, 14). Before considering the grounding objection itself, Ewing first discusses reasons and implications for S1 then, based on this, goes on to consider the plausibility of S2 and S3. S1 is the statement that truth is grounded in what exists. As Ewing describes it there are no free-floating truths. I am not sure how a mathematician who is also a Platonist would respond to this. One could argue that the axioms of set theory are free-floating assumptions if not necessarily truths (see chapter 2). As a weak Platonist I would claim that truths are ideal versions that are abstracted from the real world. In this sense the Platonic world of ideas is grounded in the real world and hence

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does not causing any problem with S1. Another way of stating S1 is to say that for every truth there is a truthmaker in the world. This is the truthmaker principle, and this principle has two interpretations: the Strong Truthmaker Principle (STP), and the Weak Truthmaker Principle (WTP), which may be stated as follows: (STP) …for every true proposition there is a truthmaker that exists and grounds its truth. (Ewing, 2013, 15; Oaklander, 2002). (WTP) …no two worlds can differ in regard to what is true in them unless they differ in regard to what exists in them… (Ewing, 2013, 15).

According to Ewing the STP has problems with negative existentials while WTP does not. The objection to presentism is therefore grounded in the WTP, which has also been identified with a statement to the effect that truth supervenes on being (TSB). Stated another way this reads: (TSB) For any proposition P and any worlds W1 and W2, if P is true in W1 but not in W2, then either something exists in one world but not the other, or else some object instantiates a property or a relation in one world but not the other. (Keller, 2004, 342; Ewing, 2013, 16).

This is in essence S1 from our list above. Without delving deeper into Ewing’s discussion, she summarises the grounding objection as follows: P1. P2. P3. P4. P5. C.

Only the present exists. (Presentism) Truth must be grounded in what exists, and therefore truth supervenes on being. (TSB) What exists in the present underdetermines what is true of the past. (Partly S2) There are determinate truths about past events and things. (Partly S3) P2 ∧ P3 ∧ P4 Ÿ Past events and things must exist . P1 Ÿ ¬P1 .

Although I believe this argument to be stronger than CNP there is still a problem. If the passage of time is to be regarded as fundamental, a prerequisite for presentism, then the presentist can simply resort to causality to establish connections between the existing present and the non-existent past. In this way the truth may be grounded in memories of the past as opposed to an existent past, in a similar way to Barbour’s time capsules (Barbour, 1999, 30…). In addition the growing block

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interpretation, discussed at the end of chapter 4, can easily escape this objection because conditions in S2 and S3 as applied to the future are no longer there in P3 and P4. Because the growing block may be considered a hybrid it is still burdened with a universal present. Arguments like this would also apply to the moving spotlight theory assuming that it is mindindependent. Towards the beginning of her chapter on presentism Ewing states that the present must be of non-zero duration. From a physical standpoint this makes sense because the physical world is basically a second order system. That is fundamental propositions in physics are second order differential equations either in time or spatial coordinates. Numerical approximations of these equations are also second order and therefore require not only, the present time step but also the previous two. This is suggestive of a discrete spatialized time variable with only three time steps occupied at any instant. If presentism is the correct interpretation and time is spatialized in this way then it may be asked: what, if anything exists in the voids to the past and the future of the time steps presently occupied? This is where we need to briefly consider the spaces in which spatialized structures are embedded. With regard to embedding of structures into spaces large enough to accommodate them we may, in addition to the two main postulates on which this work is based, posit a third, (P III), continuity of existence, which would be applied to, what I call the non-substantive space. Before doing that I should be a little more precise about my intended meaning of substantive spaces and the non-substantive space. A substantive space may be thought of, either as a surface or hypersurface of some multidimensional body existing in a larger space, or as a lattice consisting of discrete locations embedded within a continuum. In other words substantive spaces can be considered to have structure, which cannot be assigned to the non-substantive space. The non-substantive space, on the other hand, is regarded as being structureless. It is unbounded in the quantitative properties of dimensionality, extent in any direction, and resolution, which is why it is regarded singularly. Its topology is succinctly described by \ℵ0 . This explains the use of the determiner, the, when introducing it. It is primitive and is capable of accommodating any real or imagined structure. The non-substantive space is the ultimate void. It makes no sense to assert its non-existence because any existing structure is accommodated in a space of larger volume or dimension. Finite resolution, extent, and dimensionality all imply structure therefore they must occupy spaces that are larger in some sense. We could meaningfully enquire about the space in which a particular structure is embedded. Upon

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obtaining finite dimensionality, extent, or resolution, we could repeat the enquiry, either ad-infinitum, or until we reach the ultimate void in a finite number of steps. This void also has the property that the more you take away from it the larger it gets. Such properties render the assertion of its non-existence absurd. The potential third postulate may be stated as: P III Continuity of existence: There are no closed boundaries to the non-substantive space. Here a closed boundary is one beyond which there are no points within the space. This contrasts an open boundary where points at the limit of the boundary are excluded from the space. This is equivalent to saying that irrespective of how close a point is to an open boundary there are always points closer to the boundary. But because closed boundary points are included in the space, then that space would be considered to possess a hard edge. For the non-substantive space this makes no sense. On the other hand a substantive space may, for example, be a sheet of rubber reminiscent of models describing gravitating objects in general relativity. Such a substance can have a distinct boundary. On the basis of the current discussion our three-dimensional base space is substantive, possibly as a hypersurface of some higher dimensional brane, whereas any structures associated with C-space are significantly more difficult to define. Because physical time is just an ordered sequence of configurations in the form of a path through C-space, then a present of non-zero but limited duration would not be meaningful. This would rule out a present consisting of three time steps mentioned earlier. However, if C-space is discrete then it must be substantive or at least have a substantive lattice within it. Therefore we could not rule out a present of limited duration. Moreover, its duration could be one time step with any previous time steps embedded into it as memories. This would not violate Ewing’s condition of being non-zero duration because in physics a likely lower limit for the duration of a single time step is one Planck time. If Cspace is non-substantive however, a present of limited duration would require a second substance apart from the wave function. But this is taking us too far off course. From the above discussion I would suggest that the strongest and most pragmatic objection to presentism is relativity theory especially when it is combined with the confirmed existence of macroscopic-Bell states. That is because these theories are grounded in empirical evidence, which in turn resonates with the implications of (TSB). Applying postulate (P III)

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to the patch of C-space covered by macroscopic-Bell states adds weight to the ontic nature of C-space in its entirety.

10.2 The moving spotlight theory This is a primitive interpretation of temporality that consists of elements common to later more refined theories. For this reason it is worthy of some consideration because it can still clarify much on the confusing nature of our temporal existence. Here we may regard the temporal continuum as a continuous line extending an unlimited distance in either direction. The present is defined as those events on a small interval of the line, which are illuminated by a spotlight and continuously moves in one direction. Past events have already been illuminated, present events are illuminated, and the future events are yet to be illuminated (Broad, 1923). It is not difficult to see that this is tantamount to defining time in terms of itself. However, we should dig a little deeper to see what might be going wrong. This theory consists of two main parts, which are often criticised as contradictory. The first is the spatialized, B-theoretic part, asserting that past, present and future exist equally. This is the eternalist component that may be seen as a backdrop to the second part asserting the existence of a moving now that is somehow brought into sharper focus by the spotlight. This second part is A-theoretic in nature and captures what it means to experience the flow of time. There is an implicit assumption that we are all in the present and move with it in a direction from past to future. As part of her analysis Ewing separates the A-theoretic part, asserting that there are non-relative facts about the tensed status of times, from the moving now. Ewing’s three components of the moving spotlight theory are: 1. 2. 3.

The eternalist component holds that all times exist. The A-theoretical component holds that there are non-relative facts about the tensed status of times. The moving “now” component holds that the “now” moves from the past to the future. (Ewing, 2013, 32).

Of these three components the first is entirely self consistent. The second is reasonable provided we hold to the notion of an absolute present. This divides the temporal manifold into three distinct sets: past, present, and future. However, there are two ways in which we can assert the absoluteness of the present. One way is to invoke the hidden frame idea but still allow the present to move from past to future. However, it seems that Ewing is suggesting a stronger definition of an absolute present,

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which includes the requirement that the location of the present along the temporal manifold could not be instantiated at more than one point, otherwise it could be argued that the position of the present depends on, or is relative to, the position of an observer in time. Therefore if the present is absolute then all observers are present at one time. If we then invoke the third component of the theory and allow the present to move in some sense, then each observer is instantiated at more than one location in time. This would destroy the absoluteness of the present as defined by Ewing, thereby rendering the whole moving spotlight theory incoherent. That said it is not quite clear to me what Ewing means when she uses the word absolute, does she mean non-relative, non-changing, or possibly both? If not both then the two certainly have different meanings. Ewing claims that the incoherence of this theory has similarities with the contradictory nature of presentism. Again for me this is debatable, in the form of presentism described by Ewing the present cannot move anywhere because there is no future to move to and no past to move from. The present consists of a single physical entity, the spatial universe in its entirety, which is dynamically changing. This model harks back to Newton who thought of time as being irreducible and somehow outside of physics, or put another way, physics is within time. This description of time is a purely A-theoretic and therefore a non-spatialized form. The spatialized description of time in equation (4.7) describing the Galilean transformation, may be regarded by the philosophers of the day as mere mathematical contrivance. Given the prevailing view in Newton’s time all observers would agree on a present because that is all that exists, therefore the present can be defined as absolute. I agree with Ewing that her two objections to presentism are related, but the CNP is weakened by its claim that equates an absolute present with a non-changing one, if indeed she describes absolute to mean non-changing. A similar objection applied to the moving spotlight theory is significantly stronger, because of the incoherent notion of moving in time. Adhering to Bergson’s (1910) view that treats physical time and phenomenal duration as distinct concepts, this problem is addressed. We will revisit this issue again in section 10.4 where we consider objective theories.

10.3 Are current theories of time substantive? To summarise theories of time so far, we see that they fit broadly into the three categories, presentism, eternalism, and hybrids. Hybrids consist of elements from the pure forms of the other two and include, for example, the moving spotlight theory and growing/shrinking block models. Hybrids

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however, still include a privileged universal present. In pure presentism neither the past nor the future can be said to exist. Opposed to this is pure eternalism in which neither past, present, nor future can claim any kind of special status, all times exist and are on a par. In this work we take a pragmatic view by assuming pure eternalism, this is because the current empirical evidence grounded in modern physics points us in that direction. However, for now it is to be expected that debates regarding the nature of time will persist, and in consequence so will related theories of mind. The main question being asked in this section is: how debatable are the current theories of time? In the title of this section we use the word substantive to mean debatable. A substantive philosophical thesis is one which is neither trivially true nor obviously false. As soon as a particular thesis is regarded as obviously true or false then any and all debate ceases. Figg (2017) goes into great detail to justify the status of presentism and eternalism as substantive philosophical theses, but it would take us too far off course to pursue a similarly lengthy analysis. However, there is one chapter from Figg’s work that is particularly enlightening as far as our purposes are concerned. In his chapter 7 Figg considers spatial analogies of eternalism and presentism. These are labelled respectively as anywhereism and dynamic hereism. For his discourse Figg uses a fictional scenario in a similar way to Parfit’s approach to personal identity over time. He considers a fictional island situated at a location from which no other land masses are visible. The islanders have no knowledge of seafaring and have no record of ever being visited by sailors from over the horizon. The surrounding waters are treacherous so they never swim far. They believe their island to be the centre of the world and the horizon to be its limit. The various celestial bodies traversing the sky during the cycle of the day are destroyed as they disappear over the Western horizon and are recreated later as they emerge from the Eastern skyline. As far as the islanders are concerned nothing exists beyond the horizon in any direction. On one particular day a ship emerges from over the horizon, traverses the region of hazardous waters and drops anchor at a suitable distance from shore. When the sailors are greeted by the locals the captain informs them that they are from a faraway land beyond the horizon. The islanders promptly reject their story as pure fantasy and suggest that they were created at the limit of view, complete with false memories of the land they say they were from. In the story so far the sailors can be classed as anywhereists whereas the islanders as static hereists. The islanders believe that everything exists here and within the visual limit of the horizon. But

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the story continues. In response to the islander’s views of reality the captain offers any islander courageous enough to travel with them aboard ship to visit people and places beyond the horizon. One of the islanders responds and they set sail. Upon reaching a destination out of sight of the islander’s homeland, one would expect that the islander would concede that the captain is correct and acknowledge the existence of other places beyond the world that he/she knew. With the ship anchored in a dock of an unfamiliar destination, Figg then considers the consequences of the islander responding in the following way: I was wrong that everything exists within sight of the island. But you are wrong to think that things exist elsewhere than here. I see now that everything there is exists within sight of this dock. (Figg, 2017, 182).

The dumbfounded captain, asking whether the islander’s homeland still exists receives the response, “No!” The captain then invites the islander back aboard ship for the return journey. Upon reaching the island once more the captain suggests that the existence of the island is proven and that not everything exists within the sight of the dock. Places, things, and people do exist out of sight. But then the islander responds: You are right that the island exists, and you are right again that not everything exists within sight of the dock. But you are wrong that there exist things elsewhere than here. Everything that exists is within sight of this island. (Figg, 2017, 183).

The flabbergasted captain then asks: but what about the dock? To which the islander responds, “There is no dock, it does not exist.” The interpretation of location adopted by the islander is dynamic hereism, which is a more appropriate analogy for presentism in time. The reason for this is that our location in time changes by virtue of the fact that we experience change, just as the islander aboard ship experienced a change in spatial location. However, for the rest of the islanders that remained behind their location did not change beyond the sight of their own land. This is not a suitable analogy for time because, as we perceive, in time we are always on the move. For the islander that travelled only the place he/she could see, exists. So it is for presentists in time, only the events, things, and people in the present, exist. However, this analogy is not as strong as it first appears. Adopting an eternalist’s view for a moment, although time is strongly spatialized, in a globally hyperbolic space-time the time direction is strictly one-way, whereas the journey of the islander in the above fictional account is analogous to a time loop.

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There is currently no known way to create time loops. That said, in the spatial analogy I suspect that no one is seriously going to question the existence of places, people, and things that are simply too far away to be seen. In this work it is assumed that existence is an absolute property not relative one. In other words objects do not cease to exist just because they happen to be out of sight, and if we could revisit earlier times, we would say the same about times not in the present. Classical general relativity in a globally hyperbolic space-time does not, by definition, admit time loops, and this is the source of the impasse as far as time is concerned. However, doubts still exist as to whether space-time really is globally hyperbolic (Ori, 2007). Although Ori’s time machine is hidden behind an event horizon and this will be accompanied by similar problems to those discussed in section 4.5. But the supermetric in canonical quantum gravity is not globally hyperbolic and therefore, although the probability is vanishingly small that an observer can visit the same configuration more than once, it may still be non-zero. Here we accept that this does not completely dislodge Figg’s conclusion that eternalism and presentism are substantive metaphysical theses. However, the above discussion should raise serious doubts that presentism is in anyway as strong as eternalism in the current state of the art. In the remaining sections we address questions regarding the distinction between phenomenal and physical time. We put forward the argument that inconsistencies, particularly in a mind-body context, arise in the eternalist’s camp because of a failure to recognise this distinction.

10.4 Objective theories These models provide descriptions of dynamic change that do not require the presence of conscious observers. For objective dynamics to be a real feature of nature we cannot rely on the timelessness that is promoted in this work. Instead we would need to fall back on either pure presentism or some other hybrid theory that elevates a set of events on a spacelike hypersurface to some special status. Moreover, that hypersurface would need to move in a future direction with respect to phenomenal time.

10.4.1 McCall’s objective dynamic theory A particularly interesting attempt in this direction was provided by Storrs McCall (McCall, 1976). McCall essentially begins with descriptions of four theories, A, B, C, and D, of temporal becoming, which ostensibly

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represents a progression of ideas ultimately leading to an objective dynamic model (theory D). Theory A is the traditional Minkowskian view where the whole of physical reality is framed within a single space-time manifold. This is a purely eternalist model, which was fully described in chapter 4 that makes no appeal to quantum mechanics. Its disadvantage is that it is entirely deterministic in a classical sense irrespective of whatever knowledge humans may have about past, present, and future events. This theory could be represented by a single timeline extending without bounds in opposite directions. The first of the classically nondeterministic models that McCall discusses (theory B) is what he calls the distinguished branch theory. Here the universe has a branched structure in which each branch is represented by a four dimensional Minkowski manifold. The main branch or stem could be pictured as a vertical timeline, as in theory A, with other branches representing histories, which could have happened but did not. For me this model is reminiscent of the nonlocal hidden variables theory described in section 6.4.3. Its disadvantage is that a second physical substance is required in order to distinguish the main stem as special, and as such this theory is rooted in a form of substance dualism. The existence of a distinguished branch is a feature that is absent from theory C. This is what McCall refers to as the multiple reality theory, and can be identified with the pure wave function theories described in section 6.4.4. In this model we see similar branching as in theory B, but with the exception that no branches are distinguished. The branching structure extends all the way back to the origin of the universe, but the entire structure is unchanging. Indeed all of the theories described so far in this section may be described as unchanging although it could be argued that the end point of the distinguished branch in theory B moves and would be an indicator of where the present lies. The only theory of these four that is overtly dynamic is theory D. McCall’s objective dynamic theory is one in which the state of a tree structure similar to those in theories B and C is seen to change in phenomenal time. Theory D closely resembles the objective collapse theories described in section 6.4.2. In this theory the tree structure consists of a single trunk with no root system. Ascending the trunk we eventually reach the beginning of the crown above which we see a rich branching structure described in a similar way to the previous two theories. In Figure 10-3a we see a particular tree, whose state is represented by a snapshot of the universe at time, t1 . The trunk is the unique history of the universe

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earlier than the present, t1 , whereas the branches in the crown represent the multitude of alternative futures beyond t1 . As time progresses the tree evolves by the elimination of all but one of the branches projecting from t1 , with the remaining branch becoming part of the trunk. The present is therefore represented by the moving point on the trunk where the crown starts. The tree at later times is shown in Figures 10-3b and 10-3c, where the present times are respectively t2 and t3 .

t3 t2

t1 a

b

c

Fig 10-3: McCall’s (1976) objective dynamic theory in which the present moves away from the past (below) towards the future (above). The present,

tk

(a k = 1 , b k = 2 , c k = 3 ), increases with phenomenal time and

marks the boundary between a unique past and an open future.

In Figure 10-3 we see that the trees to the future of that at

t1 are sub-trees

of Figure 10-3a. Similarly Figure 10-3c is a sub-tree of the tree in Figure 10-3b. In this way the tree is reduced in size as the present moves from the past towards the future. This theory nicely instantiates objective phenomenal change but it is not without its problems. This theory is heavily criticised by Kroes (1984, 431). However, to me Kroes’ objections are not entirely clear. When compared with pure presentism, in which no time coordinate is defined and time is regarded as purely phenomenal, this theory does seem a little extravagant. Both the objective dynamic theory and pure presentism require phenomenal time as an irreducible element that exists outside of physics. Moreover the objective dynamic theory also requires a physical time coordinate. It seems that someone originally persuaded by presentism then converting to the objective dynamic theory would be accepting a more convoluted single

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dynamic entity than that described by presentists. All that would be required is the pure presentist’s model to describe our experience of the physical world. Therefore it is difficult to see its advantage over presentism apart from its ability to explain quantum superpositions. There is one thing that Kroes says that he does not do that I find rather unexpected. He makes no attempt to address the problems with this theory in a relativistic setting. This is what he says of McCall’s objective dynamic theory: He also describes how his theory can be adapted so as to give a viable account of ‘relativistic time flow’; within the present context, however the details of these modifications are not of interest. (Kroes, 1984, 433-4).

For me the main problem with this theory is the difficulties encountered when applying it in a relativistic context. This is because the state of the tree at any given instant is not just dependent on the location of the present in time, t, but also on the inertial frame, f, of a particular observer. The state of the tree for observer, O1 , at ( t1 , f1 ) say, is not the same as it is for

a second observer, O2 , at ( t1 , f 2 ) , even though they observe the world from the same position and instant. This is because they move relative to each other and, as can be gleaned from chapter 4, branches emanating from past, spacelike separated events for one observer no longer exist, whereas those branches do exist for the other observer because the same events that they stem from are in the spacelike future. In another example an observer could move at high speed in a tight circle, in this case there will be branches oscillating in and out of existence. This makes nonsense out of the notion that imagined objects either exist or they do not. To connect this with classical logic the proposition, object A exists, is the logical negation of, object A does not exist. If we are asserting that a particular branch exists for one observer but not another then that branch, or indeed any branch stemming from a spacelike separated event resides in some hinterland between existence and non-existence. In his abstract McCall openly admits that he sacrifices the principle of bivalence in classical logic (McCall, 1976, 337), and in chapter 2 warnings were issued that there would be theorists who do exactly that. In this work we stick rigidly to the principle of bivalence, which renders McCall’s theory entirely flawed in branches constituting a hyperbolic space-time. McCall’s dynamic theory is a manifestation of objective collapse theories. In chapter 6 we engaged in extensive discussions of the reasons why these theories make no sense in relativistic setting. In a nutshell they require a hidden inertial frame, as is the case for all theories proposing a

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universal present, and the only practical way to define an observer independent present is to define it in terms of a hidden frame. Therefore for our co-moving observers O1 and O2 the present at t1 would be defined by

( t1 , f h )

where f h is the hidden frame common to all observers.

Alternatively McCall seeks to allow not only dependence on time but also on frame. This is what he says: This does not make the universe observer-dependent, but it does make it frame-time-dependent. (McCall, 1976, 345).

However, specific instances where frame-time dependence implies observer dependence are in situations where distinct observers occupy distinct frame-times. The existence of branches stemming from spacelike separated events in objective collapse/dynamic theory is ambiguous. In a reality respecting the principle of bivalence such a situation is untenable.

10.4.2 The irreducible fact theory This model describing the passage of time is the one favoured by Ewing (2013). Its title is likely to be due to Ewing herself although she seems to be influenced mostly by Maudlin (2007) whose approach to phenomenal change is one of establishing a distinction between the B-series properties of time (order and direction) and the A-series (the experience of time passing). One of Maudlin’s conclusions, with which I fully concur may be summarised by Its passage is not a myth. The passing of time may be correlated with, but does not consist in, the positive gradient of entropy in the universe. (Maudlin, 2007, 142).

This is entirely consistent with our postulate, P II. Maudlin’s final word on the passage of time may be stated as …it is not to be reduced to, or analyzed in terms of, anything else. (Maudlin, 2007, 142).

Therefore it seems Maudlin echoes Bergson’s views. For Ewing therefore it is the passage of time itself which is the irreducible fact. Other aspects of time, as part of a space-time manifold or as an ordered sequence of configurations, are completely analysable in mathematical terms and are

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therefore objective. The question is: can the passage of time be considered objective? Ewing’s first sentence describing this theory is The irreducible fact theory combines eternalism with the objective passage of time. (Ewing, 2013, 50).

In order to address the question of objectivity we need to be clear as to its meaning. If we mean entirely mind independent then we are asserting that change takes place even if there are no conscious observers to witness it. This is what I would describe as material objectivity, which if applied in this case would contradict the assertion of timelessness on the physical side of reality and therefore deny postulate, P I. However, P I only asserts that the physical world is timeless. Another question that arises is: does the meaning of objectivity only apply to physical things? In section 8.3.1 we implied that minds themselves are objective, only their experiences are considered subjective. This is because we can view minds as independent of each other. If we adhere to this definition then there is no quarrel with Ewing. Disagreement only arises if she insists on material objectivity. If therefore, she does not stick rigidly to material objectivity then there is no reason why we cannot consider the source of time’s passage to be, for example, Squires’ universal mind. This will be revisited again in section 10.5 when we consider mind-dependent theories, particularly in relation to the distinction between the experience of time passing and the B-theory aspects. However, if Ewing considers objectivity to be material, in our sense, then she needs to explain how this does not reduce to the concept of a moving present–universal or otherwise. She certainly acknowledges a timeless background, and a materially objective passage of time directly implies a moving present. If, in turn, she is claiming a moving present to be incoherent, as in the moving spotlight theory, then she is falling into her own trap. A likely cause of this is the tendency to identify physical time with phenomenal time. This is a mistake because, again echoing Bergson, as concepts they could not be more distinct. Indeed with regard to coordinate time, as Kroes’ description is perfectly apt, in which he says …the distinction between past, present and future does not apply to coordinate time;… But since these notions are essential for our conception of time, it is dubious whether coordinate time deserves to be called 'time' at all. (Kroes, 1984, 441).

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In other words coordinate time, or physical time as we prefer to call it, is a path through a timeless landscape, whereas phenomenal time is something that is experienced. They are on opposite sides of the explanatory gap. Ewing’s favoured theory is not timeless because phenomenal time is included in it as the irreducible fact. We will consider phenomenal time in more detail in section 10.5. But before that we consider parametric time and conclude that it is closely related to coordinate time, while remaining a distinct from experienced time.

10.4.3 The role of parametric time as distinct from phenomenal change In this short section I hope to clear up any misconceptions regarding the roles of coordinate time, parametric time, and their relation to phenomenal change. Kroes (1984) does make some interesting points regarding parametric time in a relativistic context. However, he seems to exaggerate the distinction between parametric and coordinate forms of time. He (1984, 440) begins a line of argument that tempts us to regard parametric time as being a source of time passing, whether that is his intention however, is another question. His argument is introduced by In the following, I shall argue that, if we take due account of the role of parameter time in relativity theory, there is no need to disqualify the flow of time as an illusion within the context of this theory. (Kroes, 1984, 440).

This is entirely in keeping with our postulate, P II. But it would be a mistake to suggest that the status of parametric time, in anyway, implies dynamics within the physical realm. In Galilean relativity we can describe the path of a particle, in terms of three-velocity, through space as a curve with absolute time parameterised onto it. Similarly in a relativistic context we can describe a path of a particle through space-time in terms of its fourvelocity with proper (parametric) time as a parameter–see equation (4.16). The similarity between these models is striking. However, we should also be aware of a distinction wherein the three-dimensional version, the test particle exists at one point only and we see its trajectory across a nonzero interval of absolute time. Whereas in the relativistic case the test particle trajectory is embedded into space-time–there is no time parameter independent of the coordinates. Parametric time, or proper time as relativists prefer to call it, is a variable monotonically related to coordinate time that is mapped onto an essentially static world-line of some test particle. Coordinate time, being part of a system of coordinates, is entirely contrived. Coordinate systems

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are human inventions, they are not natural. This is why physicists generally and relativists in particular go to great lengths to make their mathematics coordinate invariant. We choose our coordinates, and there is nothing preventing us from designating a time parameter on some test particle as our coordinate time. However, this in no way denies our experience of time passing, because that is an entirely separate concept.

10.5 Mind-dependent theories In chapter 8 we saw that there are a variety of mind-dependent theories differing only in the way minds relate to each other and to physical reality. In keeping with the current state of the art and our pragmatic approach, it can be said that we are not in any position to determine which of these models is closest to the truth, assuming of course that a mind-dependent theory is the correct one. As has already been suggested, trying to combine physical time with phenomenal time is like trying to mix oil and water, and this is where a great many problems arise. Therefore it is necessary to regard these concepts of time on entirely different levels. Other scholars take this idea a little further by extending it to metaphysics as a whole. For example, Mulder (2014) stratifies metaphysics into conceptual levels that he refers to as gears. The first gear relates to general concepts within the physical realm that are amenable to mathematical analysis. In the context of time this describes views such as eternalism, B-theory, and perdurantism that treat time as a geometric sub-manifold of space-time (or space-C). Second gear concepts are a little more specific and familiar. In the context of time the corresponding views are presentism, A-theory, and endurantism. Mulder also explores a third gear relating to life and biological systems, but does not seem to commit himself to either a monist or a dualist stance. He also hints at the possibility of stratifying metaphysics further by suggesting the possibility of lower/higher gears and maybe more in between as required for a particular study. However, it is not our intention to delve too deeply in this direction as it would take us too far away from our intended objective. Where this kind of demarcation appears necessary, it may be an indication that we are dealing with materially different irreducible objects within reality as a whole. In this work we suggest that the mind and the wave function are two such irreducible objects. In this section we address a number of objections against minddependent theories and to show where they fail when applied to ours. These are as follows: it is a common mistake to assume that minddependent theories require a universal present, both Kroes (1984, 438) and

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Ewing (2013, 41) do seem to fall into this trap. It is also claimed that these theories are incoherent because the process of generating temporal passage within the mind cannot be clearly described (Ewing, 2013, 43). Further, it is also suggested that the passage of time is divorced from phenomenal change in mind-dependent theories (Ewing, 2013, 47-50), whereas in our theory, phenomenal change and the experience of time passing are shown to be identical. We see that these objections vanish in the context of spaceC discussed in chapter 6. Kroes also has difficulty with Grünbaum’s (1971; 1974) description of temporal ordering and parallelism. Kroes makes some valid points, but these complications vanish in a dualistic context. In the following we consider Kroes’ analysis before finally addressing Ewing’s objections.

10.5.1 Reply to Kroes A characteristic of our mind-dependent theory is that from a physical standpoint there is no presumed universal now. Pre-empting further discussions in chapter 12 neither you nor I can know whether or not all other minds are contemporary with ours, you can only know about yourself. Grünbaum (1971, p210) comments that …the claim that the now is minddependent does not assert that the nowness of an event is arbitrary… (Kroes, 1984, 438).

This is to say that Grünbaum is not necessarily adhering rigidly to a universal present, nor is he denying its existence. In section 12.1 we revisit the various mind-dependent theories consistent with the principle of localisation. There we see that Squires’ UMV does define a universal present whereas it is not a general feature for the other models. Kroes remarks that Here we touch upon the first serious shortcoming of Grünbaum's theory, for any theory about the flow of time, which does full justice to the role of time in our experience, has to account for the fact that just one privileged moment of time is called 'the present'. (Kroes, 1984, 438).

So Kroes assumes that in any theory of time it is essential that a universal present is defined. There is absolutely no evidence for this. Each of us as individuals can only know about our own experiences, including that of time passing. Others that we interact with in this life are the avatars possessing identities with which we are familiar. Those avatars are extended in physical time, and each of us is located at a specific point

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along the timeline of our body at any given instant of phenomenal time. Of course Kroes would be correct in the case of Squires UMV, moreover this would define a hidden frame. However, we are not yet in a position to decide whether this is the case. In general we assert that the present is relative, and this is in keeping with the spirit of relativisation of space-time and the quantum state. This should be sufficient to address Kroes’ first objection to mind-dependent theories generally. If we are to assert the existence of a universal present we must have empirical evidence for it. It is certainly not necessary to account for our individual experiences. Kroes second problem with Grünbaum’s theory relates to its claimed inability to reconcile the causal relationship between mental events, which do become, and physical events that are without becoming. In our C-space context involving branching timelines this is not difficult. Here we may use the road analogy borrowed from Lewis (2007) again, but instead of considering a measurement of an electron’s spin state as in section 9.2.3 of the previous chapter, let us suppose that the token mind in question is faced with two choices over which it has full control and complete knowledge of the consequences of each. Upon approaching the junction the becoming of alertness to the consequences of either choice is a mental event within the token mind. The physical situation perceived, on the other hand, does not become because it is always there in C-space. The physical situation at the junction triggers the mental event of alertness to the situation. There is no problem here. The direction of causality is from the physical to the mental and never the reverse. If you are assuming physicalism however, mental events just reduce to physical events within the brain of the perceiver. In these circumstances all events are physical and causality reduces to a correlation with high conditional probability. Of course the material events within the brain are there always and they have the same relationship with the perceived external events whether or not a dualist or physicalist interpretation is adopted. Of course the physicalist is burdened with the demand for an explanation of mental events becoming in a timeless context. Kroes takes specific issue with Grünbaum’s assertion that although the passage of time is not a property of the physical world, physical events are nevertheless temporally ordered. Up to a point I concur with Kroes, temporal ordering of points in a timeless landscape makes no sense. However, such points can possess order of a kind. As we see in section 10.1.1, C-series can possess order and be without direction. Further, we can define a direction to a sequence of points based on some natural arrow, for example the increase in entropy in appropriate cross

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sections. It then becomes a B-series. I have to agree with Kroes that this does not constitute temporal ordering. A-series properties can only be added when a mind perceives the events one after the other in a defined order and direction. However, this is no detriment to our principle of localisation. Another question that Kroes asks is …why is it not possible for a certain event to be in my past and in the future of somebody else, and why are opposite time flows excluded? (Kroes, 1984, 444).

I do not propose to comment on the text that follows this, instead I will respond to this question directly and show that it is partly addressed by our model. The first thing to note is that there are actually two questions here, can an event… be in my past and in the future of somebody else… and …why are opposite time flows excluded? The fact that Kroes used the term somebody rather than ‘someone’ may indicate that he is assuming physicalism. However, putting that to one side the answer to the first part is definitely yes and that is because our mind-dependent model allows for individual minds to be non-contemporary. That is one mind can be ahead of another, but in normal circumstances this cannot be perceived because we only interact with the physical avatars of others and not directly with their minds. To answer the second part is more difficult because it requires us to fully understand the mechanism by which the point of perspective is forced to move with respect to the associated mind’s own phenomenal time. In broad terms this mechanism is likely to be related to a direction of increasing knowledge along the timeline of the brain, and may be related to a mind’s unconscious desire to learn more. A first step would be for models like this to become more generally accepted, only then would researchers be sufficiently motivated to focus more on any associated specific mechanism. This in turn would offer a deeper mind-dependent explanation for dynamic physical models as we understand them, which is underpinned by the best universal physical law that we have at present–the Wheeler-DeWitt equation. At present it seems reasonable to suggest common factors for various minds and avatars that dictate a particular direction for phenomenal time at specific points in space-C. This would provide our universally agreed direction for experienced time.

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10.5.2 Reply to Ewing Ewing’s response to the moving spotlight theory and to models that require the notion of a moving present generally is to describe them as incoherent. As we have seen in section 10.4.2 insistence on material objectivity, which is implied by mind-body physicalism, precisely infers a moving present of some kind in the irreducible fact theory. However, she escapes her own incoherency argument by considering the passage of time to be an irreducible element of reality. However, for mind-dependent theories, particularly ours, a universal present is entirely redundant. And yet she does seem to insert a universal present into mind-dependent models. For example consider the statement The general idea behind the mind-dependent theory, then, is that the objective passing of time requires the present to move – flow in some way – along the B-theoretic temporal continuum. (Ewing, 2013, 41).

Note that she uses the determiner, the, when referring to the present infers that it is universal. We can take at least one step towards escaping the incoherency accusation by insisting that there is no universal present in our model, with the exception of Squires’ UMV. And yet even in the UMV each individual mind still only experiences its own changing present. For mind-dependent theories Ewing asserts that the flow of time is an incoherent process because it is an aspect of the mind that cannot be described. But before providing our explanation I shall differentiate between two forms of mind-dependent thesis. In broad terms these are 1. 2.

The first (weak) mind-dependent thesis: The flow of time is a real aspect of the mind. The second (strong) mind-dependent thesis: The flow of time is an illusion.

To reiterate our position we discount the second, because strong minddependent models deny postulate, P II. Therefore in what follows we will not address any arguments against strong mind-dependent models. As was discussed in chapter 8 our model does not rely on a universal present. In Squires’ UMV a universal present may form naturally due to the close proximity of individual minds in C-space but aside from that there is no such requirement. Each individual mind is localised in Cspace. Phenomenal time is internal to each token mind just as proper time is parameterised onto distinct space-time trajectories of individual material

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test particles. The main difference though is that phenomenal time is experienced, it cannot be directly measured with any physical instrument. We may map specific events that we experience onto phenomenal time, which gives us the sense of before and after. For example event A happens then event B happens. Events A and B are distinct locations in space-C. Moreover, because these events are timelike separated they are distinct locations in C-space irrespective of what coordinates we choose. By definition B-series time is an ordered sequence of physical configurations and therefore can be viewed as a path through C-space. An individual mind may pass through event A first then later, in its own internal phenomenal time, it passes through event B. In other words a nonmaterial mind’s point of perspective within the physical world moves along a path of events in Cspace. This is the mechanism by which we experience the passage of time–there is no incoherency. Granted phenomenal time being internal to the mind is, by definition, not reducible to anything material, and Ewing does describe the passage of time as an irreducible fact within the spatiotemporal world. In our model the passage of time is an irreducible fact of the mind. So here I would posit that Ewing’s irreducible fact theory is a mind-dependent one in disguise. Ewing (2013, 47-50) objects to mind-dependent theories also because of a perceived difference between change and the passage of time. I agree with Ewing that there would be grounds for objection if it were true. In what follows we will examine whether this claim can be substantiated and if so whether it has any adverse consequences for our model. Ewing considers two possibilities 1. 2.

Change occurs mind-dependently. Change occurs mind-independently in a world wherein time in and of itself does not genuinely pass. (Ewing, 2013, p48).

In our model change does not occur mind-independently therefore we do not need to address option 2, so in what follows we consider only objections to option 1. According to Ewing this would mean that …time would not pass and nothing would change outside of the mind. (Ewing, 2013, 48).

However, it seems that Ewing is imaging this in the context of our familiar three-dimensional world. It is imagined that in

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However, when this is applied to the extreme-dimensional space-C as a whole, the idea of a totally timeless world becomes a lot more palatable. Moreover, as we have seen, particularly from the discussions of chapter 6, this is where current scientific evidence is pointing. Imagining the branched structures being completely frozen within an almost infinite-dimensional space-C manifold is easier to deal with. We can then conceive of a single point of perspective moving along a high probability path. Due to the phenomenal time intrinsic to it, the mind being in contact with the space-C manifold at a point of perspective, it would perceive the familiar world dynamically changing solely as a result of its motion through this higher-dimensional landscape. In this way we may view the passage of time and phenomenal change to be one and the same as the movement of the point of perspective through space-C. Therefore in line with Ewing’s perfectly reasonable requirement we may argue that the passage of time and phenomenal change are identical, and that there is nothing wrong with our mind-dependent theories on this account.

10.6 Discussion and summary This chapter has focused in particular on the nature of time and our perception of it. We have seen that theories of time fall into three types: eternalism (timelessness), presentism, and hybrids which combine elements of the other two. Eternalism and presentism in their purest forms are single theories they are not larger categories consisting of more than one theory. This point is made because of the possibility that we are accused of adopting a naïve viewpoint. In this work we adhere to pure timelessness of the physical world. There is only one theory of this type, and those who subscribe to it may be accused of relying on a naïve view of timelessness by those who claim to be eternalists but actually subscribe to one of the many hybrids. The hybrid theories mentioned here are: the moving spotlight theory, the irreducible fact theory and McCall’s (1976) objective dynamic theory. In addition we may include mind-dependent theories where mindbody physicalism is assumed. I have no doubt that there are many other potential hybrid models since there are likely to be a multitude of ways to combine eternalism and presentism without arriving at a contradiction. However, the body of scientific knowledge accumulated to date contains not a hint of evidence for an absolute or universal present as is required by

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both presentism and the hybrids. Therefore minimising our ontology in this way leaves us with the only viable theory time–eternalism, and this is the reason for adopting our first postulate, P I. To reiterate, one feature required of all hybrids is a universal present, which moves with respect to phenomenal time. Because of the principle of localisation the hybrids are particularly attractive for physicalists because at any instant world-lines in space-C intersect with the universal present at, if not a specific point, a small volume cross section occupied by the brains of individuals, and this would enable physicalist models to satisfy the principle of localisation in conjunction with eternalism. The problem facing physicalists however is that worldlines in space-C are branched and therefore the question arises as to how a single individual mind before a bifurcation can become two after it. That said, intersections between world-lines of brains and the universal present does provide instantaneous small neighbourhoods with which physicalists can pin supervened material minds. If this is the case then can a potentially infinite number of non-interacting minds exist in one brain at an instant that can divide into roughly equal numbers at a bifurcation point? If the answer is yes then this raises the question of what constitutes each mind within one brain at any given instant. If the number of minds is large enough then it can be argued that mind substance has radically different properties to matter, and thus we have fallen back onto dualism. If, on the other hand, we are taking a dualist position then could the universal present be defined by where, for example, Squires’ universal mind actually is? If so then this raises the question as to whether token minds are actually created at bifurcation points. This is a possibility that cannot be ruled out, but I believe that it is premature to speculate about the possible creation or termination of minds. In this work, taking one step at a time, we just show their independent existence based on the available evidence. Personally I prefer to think of the universal mind as being a connected cluster of many perduring individual minds. Another important point raised by the discussions in this chapter is the apparent dualist nature of time. On the physical side we have spatialized time variously known as coordinate time, parametric or proper time, or indeed in its most general form, any likely path through C-space. And because we have argued for mind-dependent theories, we regard the quale of phenomenal time as a purely mental feature, making it irreducible in material terms. This dual aspect of time is similar to that of colour for example. Both time and colour have quantitative attributes on the physical side, geometric properties in the case of physical time and wavelength of light in the case of colour. Likewise we have corresponding qualia on the

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mental side, and there exists an isomorphism in each case between these and their physical counterparts. Just as in the case of colour and wavelength, during life in the physical world there is a monotonic relationship between physical and phenomenal time. Throughout this chapter we have argued against a universal present, specifically one that emerges from the physical/spatio-temporal world. However, it is an empirical fact that we experience a changing present, but this is a matter of perspective. In the moving spotlight analogy it is as if we each carry our own little spotlights. I use the word little here to indicate that it only illuminates your own present and no one else’s. So the moving spotlight theory does have its uses even though we have argued against it for requiring a universal present. Ewing’s irreducible fact theory combines the timeless physical landscape that is space-C with the passage of time as an irreducible fact. In this work we take irreducible to mean not reducible in physical terms. Therefore the passage of time is not physical. On this point we entirely concur with Ewing’s position. Ewing also deplores the prospect of divorcing the passage of time from change, another point on which we agree. However, she claims that the irreducible fact is objective, in other words not part of the mind. So she has a dualist ontology consisting of the physical world and the irreducible passage of time. Presumably then the mind is reducible to the physical, in which case she subscribes to mindbody physicalism, but not to total physicalism. Personally I find this a very odd state of affairs. In the interests of parsimony I would prefer to reinterpret Ewing’s theory by changing the meaning of the word objective to include the presence of minds. In this way we can place the passage of time firmly within the mind. Moreover, if the passage of time is a property of all minds collectively then surely this is all the independence one needs. Therefore Ewing’s irreducible fact theory nicely converges to Squires’ UMV, but it is a viewpoint from the inside so to speak.

CHAPTER 11 BIO-NEURAL SYSTEMS AND ARTIFICIAL CONSCIOUSNESS

In the first sentence of their abstract Tononi and Balduzzi (2009) refer to the neural substrate of consciousness. Such a phrase is typical of researchers in the brain sciences generally and cognitive neuroscience in particular. If we are slack in its reading then we can easily fall into the trap of thinking that all researchers in this field are presupposing full supervenience of the mind on the neural substrate. In some cases this is true, but it need not necessarily be the case. A good neuroscientist will hedge his or her bets by not being too committed one way or the other. This is where we need to be clear about what we mean by consciousness and what we mean by the mind. A similar statement by Koch (2009a) reads The working hypothesis is that consciousness emerges from neuronal features of the brain. (Kock, 2009a, 1108).

Once we are clear that the mind is a distinct entity and that consciousness is merely a property of that entity, we can appreciate that Kock, for example, is not necessarily saying that the mind emerges from the neuronal features of the brain. This is because consciousness must necessarily emerge from the brain by virtue of the mind’s interaction with the physical world. In general therefore we can say that such statements are of no threat to our endeavours to persuade the reader in a direction away from physicalism. In the context of this chapter, we regard consciousness as being interchangeable with awareness. And when we talk of awareness, more often than not we refer to awareness of the physical world. So Kock’s statement above could be rephrased to read The working hypothesis is that consciousness of the physical world emerges from neuronal features of the brain.

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Whether this was Kock’s meaning however, we can only speculate, but reading his statement this way, and so treating all similar statements, provides us with the freedom to view the mind as an objective entity independent from the physical, a claim that we are beginning to show has a strong scientific basis. Therefore when we refer to consciousness we are not speaking of the substance of the mind, we are discussing the property of the mind to be conscious, or to be aware. And in order to be aware of the physical world our consciousness of it must emerge from that physical world. In one of his best known works, Who’s in charge?, Michael S Gazzaniga (2012) shows himself to be one of those rare neuroscientists who argue for the reality of free will. However, from its reading it is likely that Gazzaniga still subscribes to the theory that the mind fully supervenes on the physical, in the same way that traffic supervenes on vehicles. This echoes Gilbert Ryle’s claimed relation between mind and body, which as we have seen, is based on a presupposition of physicalism. In the sections to follow we will see that such a presupposition is unnecessary, both in Gazzaniga’s work and in the collection of studies, The Cognitive Neurosciences (Fourth edition, 2009), edited also by Gazzaniga. The emergence of consciousness of the physical world is mysterious, not because we cannot define the mind in terms of anything else, but because we do not know how the nonmaterial mind interfaces with the neural circuits of the brain, and it is this interface that is intimately connected to consciousness. So a research question for neuroscience relates to which neural circuits are involved in consciousness and which are not. It may seem surprising that an unequivocal answer is beginning to emerge. We will also see that Gazzaniga (2012, 121) also considers what he calls the quantum hornet’s nest. However, Hameroff and Penrose (2014) take the idea of quantum neuroscience much further. They suggest the possibility, notwithstanding the warm (37°C) wet environment of the brain, that coherent quantum states can exist for a sufficiently long period, allowing the cortex to be considered as a classical-quantum hybrid, hence enabling what we think of as the mechanism of choice. After a brief sojourn through the complexities of quantum neuroscience we discuss the possibility of machine consciousness. We will see why no purely classical machine can ever host a conscious agent, at least not one capable of free choice. In short this is because such a mechanism is purely deterministic. However, recent research into quantum computing is highly suggestive of the possibility of artificial classicalquantum hybrids possessing similar properties to those of the natural

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cortex. Once we know the required architectures of certain corticothalamic circuits, we will in principle be able to construct those architectures from manufactured classical-quantum hybrid components the exact details of which we can only presently guess at.

11.1 Consciousness through cognitive neuroscience To illustrate the distinctness of consciousness with that which is conscious (the mind), we may use the example of a sleeping subject whose level of consciousness varies throughout the sleep period. Throughout any sleep period lasting typically for several hours, there are periods of rapid eye movement (REM) interspersed with a deeper unconscious state more akin to coma. During deep sleep periods our level of consciousness is essentially zero. But this absence of consciousness does not mean that the mind has ceased to exist, it is more likely to indicate a temporary disconnection in some sense from the physical world. So consciousness varies while the existence of the mind is constant.

11.1.1 The explanatory gap I think very few would disagree that the brain enables the mind’s consciousness of the physical world. But this is not the issue. The debate amongst philosophers of consciousness has centred on what has become known as the explanatory gap (Nagel, 1974; Levine, 1983), which entails that phenomenal conscious experience (qualia) cannot be reduced to mere brain states. The problem of closing the explanatory gap has come to be known as the hard problem (Chalmers, 1995; 1996). This book is intended to provide reasons, amongst others, why the hard problem can never be resolved, and further that it is unnecessary. However, detractors claim that we will reach a point when science is sufficiently advanced to allow the bridging of explanatory gap. Our response to this claim would be that, while neuroscientist’s may be able to prove an isomorphism between qualia and brain states, this does not demonstrate identity. Indeed we can point to such an isomorphism between the colour spectrum from red to violet and the electromagnetic wavelengths 700-400 nm, for example. But I would defy anyone to describe the experience of seeing the colour red purely with reference to the wavelength of light exciting retinal L-cones, or anything else that might exist on the objective side of the isomorphism. Block (2009, 1115) points to those who assert the idea that subjectivity and objectivity are properties of concepts rather than properties of states. In this way the claims of ontological dualism are deflected by the

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counter claim that such dualists are really arguing for conceptual dualism. Here Block uses the example of Mary, the neuroscientist of the distant future, mentioned in section 7.3, who is confined from birth to a monochrome environment. When Mary is released from her old environment she, for the first time, has the experience of seeing, for example, the colour red. Prior to this she was fully acquainted with all of the objective facts of seeing red. However, experience is subjective, and Mary’s, notwithstanding her total knowledge of the relevant mechanisms, is entirely new. Block uses the analogy of someone who knows that Lake Michigan is full of H2O but learns something new when told that it is full of water. Granted this person has not learnt any new material fact but has learnt that liquid water is H2O. However, this analogy says nothing about mental perception because neither water nor H2O are experiences. They are merely different designations for the same substance. The fact remains that it is not possible to describe the experience of seeing red without some reference to the colour red. It is a primary concept, a mental state distinct from the brain state that corresponds to it in the isomorphism. Again detractors will argue that light in the approximate wavelength range of 660-700 nm exciting the L-cones of the retina, is seeing red, but this only describes the physical facts. Describing these facts in ultimate detail will not inform someone blind from birth what it is like to see red. So the explanatory gap remains. Block arrives at this conclusion but in a slightly different way. He summarises this in the following single sentence paragraph Importantly, this line of reasoning does not do away with the explanatory gap but rather reconceives it as a failure to understand how a subjective and an objective concept can pick out the same thing. (Block, 2009, 1115).

If I read this correctly then Block is saying that subjective and objective concepts can refer to the same thing, and that failure to appreciate this is erroneously interpreted as the explanatory gap. If my interpretation of this is correct then I concur. Block then goes on to argue for conscious experience to be a faculty of biological systems only. While it is not our objective to become embroiled in such arguments, I would point out that any physical process, including biological, could in principle be simulated electronically. Where quantum biological elements exist, for example possibly in photosynthesis or within microtubules, then appropriate quantum hardware would be required. This is an essential feature of any physical system to host a conscious mind.

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11.1.2 The interface with the mind It seems that most scientists with any interest in consciousness studies or brain science would agree that either the brain is conscious (physicalism), or that the brain represents an interface between the physical world and nonmaterial minds (dualism/mental monism). For neuroscientists the question of interest is: which circuits or subsystems of the brain are involved in consciousness, and which are not? Recent studies by Kock (2009b) suggest that it is the thalamocortical (TC) system and its satellites within the forebrain. These studies include electrophysiological, psychophysical, and functional imagining techniques, which have allowed neuroscientists to narrow down associations with conscious experience to these areas (Kock, 2009b, 1137). In addition Tanoni and Balduzzi (2009) point to certain corticothalamic (CT) circuits as being essential for conscious experience. These circuits are part of the same system but conduct signals in the opposite direction to TC fibres. Moreover some theories of consciousness have been linked with thalamocortical oscillations in TC-CT pathway activity (Ward, 2011), which might suggests some form of feedback more akin to closed-loop control. Very tight regulation of overall system gain would yield extreme sensitivity of the type that may be required for consciousness to manifest itself. It must be said however that these ideas are highly speculative. Integrated information theory (Tanoni and Balduzzi, 2009 and references therein) may also offer a pointer to which circuits are involved with consciousness. The circuits of the TC-CT system possess a comparable number of neurons to the cerebellum. However, the level of connectivity within the TC-CT system is significantly higher than it is in the cerebellum, so it would make intuitive sense for the regions of higher connectivity to be more closely associated with consciousness. I have to admit that my knowledge in this area is limited, but I can get some feel for the architecture through the knowledge that the brain consists of around 1011 neurons with a total number of connections approximating 1015 . From this we can say that each neuron has, on average, 104 dendrites (inputs) connecting to other neurons. However, we should not expect this to be uniform. At one extreme, 1015 dendrites are possible between as little as 30million neurons, assuming every neuron connected to every other. Regions like this, of high connectivity, are called complexes, and are more prevalent in the TC-CT system. Whereas the cerebellum consists of a large number of small semi-independent complexes, its connectivity is therefore comparatively low. We could speculate that regions of high connectivity within the cortex provide a window onto the physical world

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for nonmaterial minds. For a more detailed analysis of the connectivity in various regions of the brain in connection with integrated information theory, the reader is referred to Tononi and Balduzzi (2009). Another aspect of the connection between the mind and the physical world was discovered as a result of experiments on divided-brain patients in the early 1970s. There is a notional module found to exist in the left hemisphere known as the left-brain interpreter. Divided-brain patients were discussed in chapter 9 when considering Derek Parfit’s thought experiments. It was Michael S Gazzaniga and Roger Sperry who carried out these experiments, and the latter received the 1981 Nobel Prize in medicine for his contribution to this area. The function of the interpreter is to make sense of the world by providing an appropriate interpretation of the data that enters via sensory inputs. Neuroscientists cite this as the reason why we feel unified. At this point I would caution the reader that physicalists might be tempted to regard the interpreter as the centre of consciousness. This would be a mistake, the interpreter is so called for good reason and it carries out its interpretations for something else. That said it does have a close association with conscious thought. In one example (Gazzaniga, 2012, 76) we may consider someone walking through long grass suddenly being confronted by a rattlesnake. An almost instantaneous reaction is to jump back to a safe distance. A mistaken interpretation of this event is to suggest that this was a conscious response. However, the conscious interpretation of such an event is only presented to the subject at least several hundred milliseconds after the fact. The reaction was generated by a direct link from sensory inputs via the amygdala, just below the thalamus, to the appropriate motor nerves. In short evolution has provided us with the faculty to react significantly faster than conscious processing would allow. Such processing through the TC-CT system and the interpreter is just far too slow. The discovery of the interpreter was realised through experiments of the type given in the following example (Gazzaniga, 2012, 82). A divided-brain patient was shown an image of a chicken claw in the right hand visual field only. Due to the architecture of the central nervous system it is the left hemisphere that receives this image. At the same time a picture of a snow scene was presented to the left visual field, which is received in the right hemisphere. When asked what was seen, the subject replied that they had seen a chicken claw and nothing else. The patient was then shown images of a range of objects including a snow shovel and a chicken. When asked which was most appropriate the patient pointed to the chicken with their right hand and the snow shovel with their left. When

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asked why he/she responded the way that they did, the reply was: oh that’s simple, the chicken claw goes with the chicken… and you need a shovel to clean out the chicken shed. The point is that the patient’s interpreter module made up a story about his/her actions that best fitted what it was aware of, the chicken claw but not the snow scene. The subject had completely misinterpreted the action of his/her own left hand. In chapter 9 the road analogy (Lewis, 2007) was adapted to consider divided-brain patients. Given the hypothetical possibility of reconnecting the corpus callosum, it was reasoned that a conscious mind could access memories of an intact brain but not be aware of whether it had taken the right or left carriageway. However, the fact of the interpreter being located on the left side might indicate that minds always adhere to the left hemisphere as a result of this surgical procedure. It is very likely that future studies will clarify the situation further. However, if this possibility is verified, the subject’s mind will still be unaware that it had been associated with the left hemisphere during the period of disconnection. On the other hand it could be that any nonmaterial mind associated with the right hemisphere would be incapable of reporting a coherent narrative until reconnection is established. At this point it is useful to recall that our physical aspect can be modelled according to Lewis’ road analogy. That is the road behind us represents our unique past (provided that the road branches in a forward direction only), and the branching road ahead corresponds to our many possible futures. The idea that we can refer to our past and possible futures as though they exist now, stems initially from relativity theory and its concurrence with an eternalist metaphysics, a model at least partially verified by the existence of macro-Bell states. In other words our minds exist at one point along the road at any given instant, a state of affairs that is verified by our common experience. This kind of reasoning is not generally considered by neuroscientists simply because, for them it is not necessary. Physicalism is therefore a natural starting point for the neuroscientist, and they are happy as long as they are performing experiments and obtaining consistent results. If this is the case then there exists a situation where the scientific community has been collectively guilty of begging the question. Neuroscientists, who may assume physicalism in a limited context, can obtain results, which apparently verify physicalism in the eyes of other scholars who then go on to proclaim the fact. In this way, although no one intends to deploy a circular argument, other workers outside the field of neuroscience misinterpret their results as verification of physicalism, while not appreciating that it was one of their original assumptions. Therefore this collective behaviour

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can lead to an entirely flawed argument. One such example immediately comes to mind.

11.1.3 The readiness potential In the late 1970s Benjamin Libet and colleagues conducted an experiment on a patient undergoing a neurosurgical procedure, which showed that there is a delay between stimulation of the cortical surface representing the patient’s hand and conscious awareness of a corresponding sensation in the hand (Libet et al, 1979; Gazzaniga, 2012, 128). Later experiments, in which the intention to perform an action was monitored, indicated a buildup of neural activity approximately 500 milliseconds before the action to which the activity corresponded. By itself this made sense and did not present any problem. What did shake things up however was the realisation that this neural activity preceded any conscious intention to act by a good 300 milliseconds. This neural activity became known as the bereitschaftspotential or readiness potential (Libet et al, 1983). Moreover, these techniques have been refined to the point where invasive surgery is unnecessary. Using fMRI it can be shown that outcomes of an intended action can be encoded in brain activity up to ten seconds before the subject is aware of its intention (Soon et al, 2008). This would seem like manna from heaven for hard neuroscience determinists and physicalists in general. However, this is collective circular argument and is therefore flawed. Those holding such views are neglecting two important points: (i) signal paths in the brain are rarely open-ended but are by and large part of closed loop systems, and (ii) brains are physical systems which are at root governed by quantum laws. At the beginning of the previous subsection it was mentioned that complexes in the brain consist of signal paths within feedback architectures. In any complex system it is an intractable problem to determine the cause of any particular activity or event where feedback encompasses large parts of the system. I know from my own experience how frustrating it can be to diagnose causes of pathological behaviour in factory machinery that is controlled by two or three nested feedback loops. With its 1011 neurons the human brain is many orders of magnitude more complex than such simple systems. So I believe it to be premature to link certain neural activity exactly to specific causes of intentions, whether conscious or not. Further studies have shown that not only are the, what and when of intention need to be considered, but also so should the whether (Brass and Haggard, 2008; Gazzaniga, 2012, 200), the whether being whether or

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not to implement a particular intention. Brass and Haggard’s study clearly indicates that other parts of the brain can put the brakes on an intention encoded in an already established readiness potential ramp. Nowadays the dorsal fronto-median cortex (dFMC) is known to implement such regulation, and it is likely that this system is in a state of fine balance with other generators of intention. Such a fine balance is indicative of a high gain system in which intentions can be controlled by signals of sufficiently low amplitude that quantum effects can play a significant role. It is not uncommon to find oneself travelling along a bumpy road. We feel the bumps as they are encountered. The profile of the irregular features (the state of the road at one point) exactly corresponds to what we feel as such features are encountered. This is our isomorphism again, except that the road is the timeline of our brain and its irregular features represents the neural activity. Our experience of encountering those features is our phenomenal consciousness of those objective states. It is therefore not surprising that there is an exact one-one correspondence, even though there is no identity. In the next section we will consider how quantum effects may hold the balance of power between systems generating intention laden readiness potentials and the regulatory action of the dFMC. It is also suggested where we might look for coherent quantum states.

11.2 Quantum aspects As neuroscience develops it is only a matter of time before we reach the stage of resolving subsystems sufficiently small for quantum effects to become significant. Many critics argue that the wet, 37o C environment of the brain is too noisy to detect of any coherent quantum state. However, pioneers in this field, Stuart Hameroff and Roger Penrose have, since the early 1990s, originated and developed a theory whereby sub-neuron structures can support discernible quantum states (Hameroff, 1998; Hameroff and Penrose, 2014 and references therein). These consist of 2D polymers of the protein dimer, α/β-tubulin, which form themselves into tube-like structures known to molecular biologists as microtubules. Each microtubule has an outer diameter of ~25 nm, an inner diameter of ~12 nm, and a length of typically a few microns. Each tubulin dimmer is roughly peanut-shaped with a length of approximately 8 nm and a diameter of around 4 nm. A microtubule is built from tubulin dimmers that are aligned in the same direction as its axis, with its circumference made up of 13 dimers (Penrose, 1997, 131).

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The theory concerns the tubulin dimers themselves, and posits that each dimer potentially has several eigenstates in the classical basis, and can therefore exist in a quantum superposition of these states (Srivastava et al, 2015). However, for simplicity each dimer is generally modelled as a two-state system. The theory further suggests that transitions in the dimer’s eigenstates are related to morphological changes that, even due to their small mass, represent shifts in gravitational energy. The theory adopted by Hameroff and Penrose is based on an objective reduction of a superposition to a classical eigenstate based on the difference in gravitational energy between the two states. The DiósiPenrose criterion for gravitationally induced objective collapse predicts a collapse time approximated by

τc =

= . ΔEG

(11.1)

The gravitational energy difference, ΔEG , of a mass, m, within a radius, r, is approximated by the energy required to pull m out of its own gravitational well. In the Hameroff-Penrose case, the two lobes of a tubulin dimmer are either close together or stretched apart, where these represent two eigenstates. One can then appreciate ΔEG being the energy difference between the low-energy state (lobes close together), and the high-energy state (lobes further apart). Penrose notes that both the processes of classical physics and pure unitary evolution of quantum theory, U, are deterministic and computational. Therefore there can be no room for conscious volition within either of these domains. It is also noted that traditional interpretations of quantum theory, as discussed in chapter 6, rely mostly on some form of reduction, R, from the quantum level to the classical level, which is regarded as a stochastic process. The implication being that there is no room for conscious action here either. Therefore Hameroff and Penrose posit that the R process is an approximation to what they term orchestrated objective reduction (Orch OR), and that this process takes place at the level and sites of the α/β-tubulin dimmers. Orch OR is also considered to be a gravitationally induced collapse, which in some special way is non-computational. The non-computability feature of this process is essential in order to model conscious volition. Given their physicalist bias it is understandable that Hameroff and Penrose should search for a physical process that is non-computational (Penrose, 1997, 84).

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When it comes to the search for sites in the brain where a conscious mind can interface with the physical world, in my view there can be no better place to concentrate our efforts than the substructure of neural microtubules. However, Orch OR is still a form of objective collapse. Penrose acknowledges that there are problems with OR generally and he cites those connected with energy conservation (Penrose, 1997, 88). He then goes on to suggest the possibility that energy conservation problems can be resolved by consideration of reduction as a gravitationally induced phenomenon. However, further problems have been suggested by Shan Gao, in which the discrete nature of space-time precludes a collapse duration given by equation (11.1), where it is shown that collapse events may have considerably shorter duration (Gao, 2010 and references therein). However, although equation (11.1) is considered provisional, Gao does not dismiss the OR process altogether, he merely suggests appropriate modifications. For me however, these are not the most serious problems with OR. In chapter 6 we discussed the problems with collapse theories generally which have to do with their incompatibility with relativity. Objective collapse of any kind is a nonlocal physical process requiring the introduction of a privileged hidden frame in order for it to fit in with a relativistic scheme. In my view this problem is insurmountable. The way to resolve this problem is to do away with any kind of collapse process altogether. In Figure 11-1 we are presented with similar figures to those used by Hameroff and Penrose (2014) to illustrate their arguments. However, we suggest a different interpretation in line with pure wave theories. Figure 11-1a shows the superposition of two appropriate classical eigenstates. The gravitational energy difference between these states reaches the threshold, ΔEG = = τ c , whereupon one is selected and the other ceases to exist (Hameroff and Penrose, 2014, 53). My question is: by what mechanism does the state, not chosen, cease to exist? And another related question is: what does the choosing? I do not think these questions can be ignored. Equally I do not think that the answer can be found within physics. Penrose himself notes that known physics cannot answer these questions either. So, on this point we all concur. However, whereas Hameroff and Penrose address this problem by searching for new physics (Penrose, 1997, 83-4), our approach is to search for a solution completely outside of physics. By ruling out a collapse mechanism I propose to call the process orchestrated decoherence, or Orch D for short.

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Orch OR

a Time

Space

Time

Space

b

Fig 11-1: a The Hameroff-Penrose proposal, Orch OR, where of two superimposed states the right hand one is selected and the other ceases to exist. b Our proposal in which both states continue to exist but become isolated from each other when the gravitational energy difference exceeds an appropriate threshold. Figures adapted from Hameroff and Penrose (2014, 52-3).

The alternative is to adopt the literal interpretation of quantum theory and to accept that superpositions of states continue to evolve under the U process only. Instead of one or several states simply ceasing to exist, they can be considered isolated from each other when associated gravitational energy differences exceed the necessary threshold. When decoherence is seen in the laboratory, the apparent reduction to one classical state is stochastic. However, in our interpretation decoherence

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taking place within the microtubule sub-neuronal structures of the brain, will be in close proximity to any nonmaterial mind passing through that point in space-C. Therefore there exists the possibility that this proximity is so close that a nonmaterial mind can effectively feel its way through the branching structure of the wave function associated with the superposition of tubulin states, and hence to choose its path. This is the Orch D process from the viewpoint of the subject, which is a nonmaterial mind endowed with the ability to choose its actions. At the present state of the art Hameroff and Penrose appear to have progressed from the physical side further than most in the study of conscious systems. From the mental side we have already discussed such names as Albert and Loewer, Hemmo and Pitowski, Squires and Bitbol. It is like two teams constructing a railway tunnel from both sides, where success is dependent on both teams digging in the right direction. The last stage of the construction where they break through would constitute the discovery of a seamless join between the physical and mental domains, resulting in an improved understanding of the interface between mind and matter.

11.3 Artificial consciousness The title of this section may seem very odd given that we are arguing for the existence of minds independent of anything material. However, as we have previously remarked, consciousness is the property of minds being aware of the physical world, and this requires what may be described at the familiar level as a suitable configuration of matter persisting through time. As far as is known the only places that such configurations exist is in the brains of humans and higher animals. And the only way to understand the interface between mind and matter is to treat it on an entirely abstract level while at the same time, building an appreciation of how neurons work at the level of raw inputs and outputs. In this way it should in principle be possible to construct such a configuration artificially.

11.3.1 Top-down As alluded to here, the best way, in my view, to approach the construction of an artificial conscious system is to consider the problem from both topdown and bottom-up perspectives. One of the most significant contributions to the top-down strategy of understanding conscious systems on an abstract level is the 20th-anniversary edition of Douglas R Hofstadter’s Godel, Escher, Bach: an Eternal Golden Braid (Hofstadter,

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1999). At the time of its original publication 20 years earlier, the crystallization of understanding of the relationship between consciousness and quantum systems (Albert and Loewer, 1988) had not yet happened, even though such a relationship may have been strongly suspected. The central concept of this monumental work is what Hofstadter refers to as the strange loop. Briefly a strange loop constitutes a set of statements distributed over many levels, which form a loop of reference from one statement to each in turn leading back to the original. Strange loops are therefore indirectly self-referential, a topic that we originally touched on way back in chapter 2, where we provided an example of a directly selfreferential statement in equation (2.10). A simple example of indirect selfreferential statements is as follows (Hofstadter, 1999, 21)

The following sentence is false The preceding sentence is true.

(11.2)

We see that the behaviour of the loop formed by these statements is pathological in a similar way to the liar paradox, illustrated in equation (2.10). The difference here is that there is nothing wrong with either of the statements in isolation. So when designing systems involving such loops, we need to be aware of the potential for distributed pathological properties. However, this simple example does not illustrate statements on distinct levels. By levels we may consider that images on the screen of a video game are on a higher level than the logic gates controlling them. Statements on distinct levels like this may form a strange loop by influencing each other in a concrete way via references in both directions between levels. This is what Hofstadter refers to as a tangled hierarchy, and it is in such hierarchies operating in a dynamic way that consciousness is thought to manifest itself. An excellent visual example of a tangled hierarchy is Escher’s Drawing Hands, see Figure 11-2 (Hofstadter, 1999, 690).

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Fig 11-2: Drawing Hands by MC Escher. (With kind permission © MC Escher Company.)

Notwithstanding the rigour of his work Hofstadter still believes that the mind is simply the highest level of the brain’s functioning capacity. For example consider his statement, which reads My belief is that the explanations of “emergent” phenomena in our brains – for instance, ideas, hopes, images, analogies, and finally consciousness and free will – are based on a kind of Strange Loop, an interaction between levels in which the top level reaches back down towards the bottom level and influences it, while at the same time being itself determined by the bottom level. (Hoftstadter, 1999, 709).

Notice that he classifies complexes of thoughts such as ideas, hopes, images, and analogies, in the same vein as consciousness and free will. Whereas the former can be expressed either in writing, symbols or images, the latter cannot. This is because the latter are properties that I would call capacities of the mind, which cannot be separated from it in symbolic form. His position is however not surprising given that he does not delve too deeply into the foundations of physics. Had he done so he might have

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realized the ultimately static nature of physical reality, and hence concluded differently. In one example Hofstadter considers the case of a robot negotiating a T-maze. Here the robot decides whether to turn left or right at each T-junction of the maze. Its decisions are based on the result of a subroutine that calculates the next digit in the string constituting 2 in decimal form, going left when the digit is even and right when it is odd. The highest level of the robot’s core processes, its self-symbol, is unaware of the reason why it wants to go left or right. This by its self does not constitute making a choice because it would not view its choices as its own. If, on the other hand, the self-symbol had some measure of influence over its eventual choice via some form of higher-level recursive process (a strange loop) for example, then it may have a deeper appreciation of the essence of choice (Hofstadter, 1999, 712). However, I would argue that this is still computable, and that choice is ultimately a non-computable process. Moreover we can point to the self-symbol, like the rest of the robot’s physical structure, being distributed in time, whereas, as we have already seen, minds have the distinct feature of being instantaneous in time. With all of that said, what Hofstadter calls the self-symbol, along with its surrounding high-level programming, would very likely provide a platform for a conscious mind to interface with the rest of the physical world. That is the self-symbol represents the rails on which the wheels of the mind run on. Notwithstanding my disagreement with Hofstadter, I and, I assume, many others would argue that that is only a matter of interpretation. In my view Hofstadter’s contribution to an understanding of the interface between mind and matter, in a top-down sense, is one of par excellence.

11.3.2 Bottom-up In the above subsection I have argued that Hofstadter’s top-down approach can be placed entirely within the envelope of computable systems. It is most likely that all physical systems are computable, and it is a matter of fact that during our lives we occupy the physical world. This leads to two possibilities, either (i) we are computable entities or (ii) we are noncomputable entities with a connection to the physical world. Either way our physical aspect must be computable. There is therefore an imperative to understand physical systems at the lowest level. As we have discussed Hameroff and Penrose have made considerable progress towards this understanding at a microscopic level. At this point it is illuminating to

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investigate the next level above the microtubule structure, and to consider neurons as little more than sophisticated logic gates. The Von Neumann architecture of any post WWII digital computer is considered to be a universal machine in the context of data and information processing. It is also true that the ultimate building blocks of any such machine are logic gates of three basic types: AND, OR, and NOT (Figure 11-3a, b, c). However, in engineering circles it is well known that these three functions can be constructed from one gate type called NAND (Figure 11.3d) which is an AND gate followed by a NOT. We have already stated in chapter 2 that the two functions from which the NAND is constructed, is universal. This is very useful for engineers because the requirement to construct one gate type only represents a considerable cost reduction. A B

A∧ B

A

a A B b

¬A

c A A∨ B

B

¬( A ∧ B)

d

Fig 11-3: Logic gates with inputs on the left and outputs on the right: a AND, b OR, c NOT and d NAND.

As we can see here these logic gates are very well defined and it is possible to design very specific circuits to carry out particular functions. However, strategies like this are regarded as very inflexible and are only useful for well-used standard designs. Larger digital designs are often implemented using either programmable gate arrays or in firmware using standard non-volatile memory chips (Figure 11-4). In these cases the address bus of a memory device acts as its input where, for a particular address, the required output byte (eight bits) appears at the output. The deployment of such devices provides for greater flexibility because of the ease of re-programming.

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a1 a2

d1 d2

d8

an R/W

Fig 11-4: Typical memory device architecture. An n-bit address bus (inputs) is shown on the left with a single byte data bus (output) on the right. On reprogrammable devices there is a read/write (R/W) pin by which the device can be put into write mode allowing circuits to be modified without expensive hardware changes.

Such devices in combination with discrete logic gates are constituents of the Von Neumann stored program machine, which is the basis of all electronic computers from the SSEM (Manchester Baby), a thermionic valve based machine that came online in 1948, right through to the present day. However, from the 1980s onwards researchers in AI became aware of requirements that are considered intractable if we restrict ourselves to the traditional design of systems from the lowest levels. Pattern recognition within in large bodies of data is just such an example. In order to solve pattern recognition type problems, engineers drew lessons from neuroscience and began to implement what have become known as artificial neural networks. Over recent decades this has evolved into a highly sophisticated branch of computer science. Neural networks consist of devices, which can be implemented in hardware using amplifiers and resistors. However, because this approach is very inflexible they are, in almost all cases realised in software and run on stored program machines already briefly described. Artificial neural cells consist of a single binary output representing the axon of its biological counterpart, accompanied by any number of binary inputs (dendrites). Each input is

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programmed with an analogue weight, which uniquely determines the functioning of that particular cell (Figure 11-5). 1

w a1 1 w a2 2

3

b

A B

a

-2 -2

¬( A ∧ B)

b

an

wn

Fig 11-5: Neural network cell architecture: a General cell with binary inputs, ai , analogue weights, wi , and binary output, b. b Three-input neural cell designed to operate as a NAND gate using two of its inputs. Because the NAND function can be realised, this shows the universality of neural networks in the context of binary logic.

For the general neural cell shown in Figure 11-5a, the output is given by

­1, ° b=® °0, ¯

¦a w > 0 ¦a w ≤ 0 i

i

i

i

i

b, ai ∈ {0,1}

wi ∈ \ .

(11.3)

i

For the example in Figure 11-5b we can see that the only way for a “0” output to be realised is if both of the free inputs are in a “1” state, hence defining the NAND function. This alone shows the universality of artificial neural networks as presented here. For application of neural networks in a commercial context we rarely see feedback introduced. Most commercial neural networks employ three layers of cells where the inputs are fed into layer 1 and outputs taken from layer 3. The second, hidden, layer takes inputs from layer 1 and supplies its outputs to layer 3. The weights on the inputs to all of the cells are programmed in training mode, using a specific algorithm and a set of pre-designed training data. Such systems are highly contrived and do not represent anything like the kind of organic architecture one might expect from natural circuits in the brains of humans or animals. However, the fact

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that appropriately trained neural networks can be very effective provides some insight into the way our own brains actually work. I can offer two such examples, both co-authored by myself (Day et al, 2009; Butcher et al, 2014). In the first example (Day et al, 2009) a neural network was trained with absorption spectra from a standard polychromatic X-ray source that had been filtered through a small number of single element layers. From this it was demonstrated that the network could determine the compositions of larger numbers of layers, and therefore could, in principle, establish elemental compositions of natural samples employing a relatively cheap polychromatic X-ray source. In the second example (Butcher et al, 2014) a neural network was trained with magnetic responses of magnetized reinforcing bars in a concrete bed where intact and artificially corroded bars existed. The network subsequently demonstrated its ability to detect corrosion in a variety of reinforced concrete structures, a capability that has drawn considerable interest from the construction industry. An equivalent but alternative approach to both of these examples would be to provide people, appropriately trained, with visual representations of such data and ask them to spot certain features. With the examples above, we have removed the subjective element but retained the advantage of neural networks that exist naturally in our own brains. This shows that mindless neural networks can still demonstrate some measure of autonomy, a feature that aids unconscious or autonomic behaviour in humans and animals. At the other end of the neural network research spectrum, scientists who investigate human or animal behaviour are motivated by the desire to design systems capable of passing the Turing test. This is considered to be the Holy Grail of AI research. The kind of neural architectures in these more esoteric applications are expected to be more organic with potentially, any output fed to any input. Obviously without due consideration such systems are likely to be either unstable or to have a limited spectrum of behaviours. In principle however, there is no reason why such systems, given sufficient complexity and development, cannot behave in a way similar to ourselves. Such networks would need to be in both training and operational modes simultaneously instead of these modes being separated as in commercial systems. Therefore the adjustment of the weights throughout the network would be an ongoing process. In more organic networks it would be expected that long-term memories are stored in the weights of the cell inputs, since these define their overall functioning. Moreover, the weights will be the largest determinant of network behaviour. How the weights are determined is a

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matter of ongoing research, but in biological systems I would suspect that such weights and behaviour are determined, at least in part, by factors from below. These factors are likely to be the entangled quantum states of the α/β-tubulin dimmers discussed earlier. In this work we have argued that quantum states are necessary for consciousness to manifest itself through the mechanism of orchestrated decoherence (Orch D). Therefore neural networks with behaviour and weights not based on a quantum coherent state at some lower level can never be considered conscious. This may seem like a bold statement, and I have to admit that should I encounter an artificial intelligence that is deemed to have passed the Turing test and behaves in a rather human-like way, I would be inclined to hedge my bets and treat it with similar respect that I afford my fellow humans. However, if I knew that such a system is based on a purely classical neural network, deep down I could be fairly confident that there is no consciousness present. In order for the system to possess the capacity to host a conscious mind, it would need to consist of neural cells whose input weights are controlled by coherent quantum states associated with the corresponding cell. If implemented in hardware, these quantum states would likely exist within the bodies of the cells, just as they seem to do in natural biological neural networks.

11.4 Summary In this chapter we have briefly explored the interface between mind and matter from three distinct perspectives: (i) traditional neuroscience, (ii) more recent quantum aspects of neuroscience, and (iii) the prospects of machine consciousness. Most neuroscientists traditionally assume that all mental functions originate in the brain, and that all cortical processes are deterministic. This is an important reason why physicalism is so prevalent throughout many other sciences. Statements by neuroscientists to the effect that consciousness is emergent from brain functions may also be misinterpreted in a physicalist way. However, to be conscious is to be aware of the physical world, and this is emergent from the brain even though the mind, which is the conscious element, need not be a physical entity. As we have discussed in chapter 6, quantum mechanics plays a central role in the study of the mind-body problem. This is because quantum laws govern the body side of the problem. This fact did not escape the attention of Hameroff and Penrose who, as far as I am aware, have probed deeper into the quantum properties of sub-neuron structures than any other group of researchers. Some may consider their work

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controversial, but I for one believe they are looking in the right place, particularly on the body side of the problem. Based on the discussion so far however, I would propose a modification of their Orch OR mechanism to Orch D, which allows for a literal interpretation of quantum theory based on unitary evolution alone. This would allow the consideration of material systems able to host nonmaterial conscious minds without themselves being conscious. This adequately removes all of the problems associated with the explanatory gap. This leads quite naturally to the prospect of machine consciousness. Nowadays we know that algorithms can simulate all physical processes. Moreover, it was the difficulty of simulating quantum processes that led Richard Feynman to suggest using quantum systems for computation. However, in principle quantum systems can be simulated by classical systems of sufficient capacity. It is this fact that renders all physical processes computational. Any act of conscious volition is not subject to such computation. That is part of the explanatory gap. A physical system hosting conscious minds may allow the freedom of those minds to make choices by virtue of rendering their macroscopic states dependent on lower level quantum states. And the aforementioned quantum states would provide the necessary bifurcations in the wave function that allow minds to make the corresponding choices. The neuroscientists of the near future will likely draw lessons from the α/β-tubulin structures forming the cytoskeleton of neural cells. If this is where the lower level quantum states reside, as Hameroff and Penrose propose, then processes at the sub-neuron level will provide clues to the direction that engineers must take in the development of artificial consciousness.

CHAPTER 12 SELECTED CONSEQUENCES OF LOCALISATION

I stand barefoot on the beach. I walk across the sand until the waves start to gently wash over my feet. As I continue to walk my feet become submerged and yet they remain firmly in contact with the ground. In this analogy the land represents all that is known and consequently the shore is the boundary of our knowledge. Continuing with the analogy we may paddle out to sea a limited distance without our feet leaving the ground. In a similar way we are able to explore aspects of the unknown without loosing contact with established science. The more we explore the more likely it is that we will discover new facts consistent with established knowledge. But there are undoubtedly those who would impatiently allow their feet to leave the ground by swimming out further with all of the hazards that that entails. In this chapter we must take great care not to succumb to such temptation. This is said at the risk of those readers who will accuse me of doing exactly that. In the introduction we quoted two postulates, P I and P II, from which the principle of localisation can be deduced. Part I of this book is dedicated to the justification of postulate, P I, and postulate, P II, is taken as self-evident. In part II mental aspects are discussed. Chapter 7 is dedicated to motivations for physicalism, while chapter 8 describes the principle of localisation in detail. The discourse continues in chapter 9 where we consider competing theories consistent with postulate, P I. It is shown that these theories fail by implicitly or overtly denying P II. In chapter 10 we briefly analyse some of the popular theories of experienced duration in a Bergsonian sense and how this relates to physical time. It is shown how mind-dependent theories of phenomenal change, on which our approach relies, can be misconstrued. These issues are addressed and further, it is demonstrated that the notion of a universal present is entirely redundant. Moreover the idea that phenomenal change is best described as an irreducible fact of the mind is reinforced. The chapter following this discusses neuroscience and the prospects of artificial consciousness in the context of its localisation in C-space.

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In this chapter we will explore some consequences of our localisation model and consider whether any of these might be testable. As far as it is known there are no repeatable experiments allowing for the demonstration of phenomena that may be considered paranormal. The discovery of such an experiment would surely mark a new dawn in scientific enquiry. And given the current state of the art this looks unlikely in the foreseeable future. Therefore in this work we are sure to raise more questions than answers. That said it is hoped that any new questions will be more probing and targeted than those that have hitherto been posed. Here we begin by using the localisation model to more clearly define the bounds within which phenomena is considered normal. These directly relate to normal sequences of experienced configurations. We begin with a discussion of relationships between minds to each other and the physical world in the context of normal sequences. Here minds, or more accurately distinct points of conscious contact with the physical world, do not directly relate to each other, they can only interact with the physical world itself. In addition the movement of those conscious contact points are such that they experience the world in accordance with normal sequences that are consistent with the laws of physics as briefly outlined in chapters 4-6. This hints at phenomena beyond normal sequences where the wave function still behaves in accordance with the Wheeler-DeWitt equation, or some successor of it, over C-space. In other words the laws of physics are still satisfied, but mental contact points temporarily move along trajectories not consistent with normal sequences. This may be akin to a stylus jumping the groove of a vinyl disc. The laws of physics are not violated but because of their non-physical status, minds may temporarily experience sequences of events that may only be described as bizarre.

12.1 Normal sequences These are sequences of events of the type that dominate our lives, and by definition most of life’s experiences consist of events that would not be regarded as out of the ordinary. There are however, a smaller class of normal sequences that, over a given time period would be considered unusual. In technical parlance we say these events have a low probability density over their C-space neighbourhood. For example witnessing the Northern Lights as far south as Cornwall in the UK would be considered improbable over a 10-year period. Over a 100-year period however, a similar event is significantly more probable. Such macroscopic events are entirely accounted for by the laws of physics and their probabilities are

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dependent on prior conditions, that is to say all probabilities are conditional. Events that are so improbable that they would be unlikely to occur anywhere in the known universe and within its lifetime may also be regarded as normal because there is no apparent violation of physical laws. An example mentioned in chapter 7 considers shards of glass on a kitchen floor spontaneously reassembling themselves into a pristine wine glass and flying up onto the edge of a table. This event requires a large number of elementary particles, mediating required forces, to be individually and precisely directed towards the shards, and to other materials that impact on them, in such a way that, not only are they directed towards each other, but also that conditions are met that allow the cracks in the glass to instantly heal. As was mentioned originally, taking the usual scenario of a glass falling off a table and precisely negating the momentum of every elementary particle involved, would achieve the required result. There is no violation of physical laws, just a highly improbable sequence of configurations. In quantum physics, given prior conditions specified as precisely as the theory allows at time, t0 , a closed system may be predicted to be within a precisely defined distribution over a range of eigenstates at a later time, t1 > t0 . A sequence of events leading to an eigenstate as improbable as the reassembled wine glass is no less valid as a normal sequence than any other more probable chain of events. This incredibly improbable eigenstate still has a nonzero probability. Technically we may say that it is within the envelope of the wave function. Any normal sequence may be defined by the following two conditions, Definition 12.1: Normal sequence (i) Given any time ordered sequence of eigenstates, {Ei : i = 0," , n} , the conditional probabilities satisfy

P ( Ei Ei −1 ) > 0 , ∀i ∈ ` ∩ [ 0, n ] .

(ii)

In the limit of a continuous C-space the topological metric d ( Ei −1 , Ei ) → 0 .

The first condition states that for the realisation of one eigenstate, the next eigenstate has a nonzero probability, and the second says that the pair of eigenstates, ( Ei −1 , Ei ) , are adjacent in C-space. Eigenstates, Ei , are effectively locations in C-space and the wave function is distributed over many eigenstates. In our model only minds are localised. Therefore only

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minds can meaningfully follow normal sequences. It is in the context of normal sequences that we now discuss mental configurations allowed by the principle of localisation.

12.1.1 The SMV and solipsism In the single minds view (SMV) you the reader should view reality from your own perspective only. In other words, when you interact with others all you encounter are their physical aspects. You may infer their conscious status from your observation that their behaviour is in some sense qualitatively similar to your own, but apart from that you have no evidence of other’s minds, they could just as easily be automata as far as you are concerned. This is also referred to as the instantaneous minds view (IMV) because minds exist at isolated instants (Loewer, 1996), and so the IMV and SMV may be used interchangeably. This is not the same thing as solipsism because in the SMV you would not deny the existence of other minds. They are just not part of your consideration since there is no direct interaction. The solipsist on the other hand believes that his/her mind really is the only one in existence and this can be considered a special case of the SMV. Therefore solipsism is allowed by the principle of localisation and this is the reason that it is mentioned here. However, I think that many, like me, would find such a view most unpalatable, so there will be no further discussion of it here. To recap Albert and Loewer (1988) object to the SMV on the grounds that it violates certain supervenience conditions. However, as was discussed in chapter 8, there is no evidence whatsoever for such supervenience in a quantum context. The notion of supervenience was not originated with quantum theory in mind, so it should only apply in a classical framework. Keeping to within a classical context implies what I have called local supervenience (LSU), local that is in a C-space context. This says that where minds are present, mental states supervene on brain eigenstates, and are precisely pinpointed in C-space. The SMV is one possible consequence of localisation of consciousness and is illustrated in Figure 12-1.

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Fig 12-1: Schematic of a region of C-space showing branched normal sequences and instantaneous positions of individual minds (red dots). It is possible to focus on a region of nonzero measure containing one mind only (SMV). Details of where and how minds are located in the 3D base space are not shown.

In Figure 12-1 we immediately see two possible consequences of localisation. In cases where one mind exists on a small enough section of a branch, it may converse with another physical avatar where there is no other contemporary mind. This is the mindless hulk scenario discussed in previous chapters. The second consequence is that physical aspects of individuals act like time corridors. We see an example of this where the branch on the extreme right (Figure 12-1) may be regarded as the world line of a single body. In that case there are two non-contemporary minds occupying the same body. Situations like this effectively remove the rather distasteful nature of the mindless hulk problem as viewed by some, by making it temporary. Minds may exist in the same body but separated in physical time. Avatars’ worldlines therefore, can in principle be traversed as time corridors again and again, since they represent nothing more than locations in the space-C product manifold. However, this possibility is not the only one.

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12.1.2 The many minds view (MMV) As discussed in chapter 8 the view favoured by Albert and Loewer (1988) is their version of the many minds view (MMV). This is sometimes known as the continuous minds view (CMV) because an infinite number of individual minds exist throughout the branching structure of the wave function (Loewer, 1996). However, I reiterate my objection to the word continuous on the grounds that minds are countable discrete units, so the density of minds is regarded as being no greater than the density of rational numbers on the real line. Beyond that minor objection, the picture that this conjures up is one of minds densely packed throughout the paths of every normal sequence in C-space. For Albert and Loewer this rescues the sought after quantum level supervenience (Figure 12-2).

Fig 12-2: Schematic showing the same region of C-space as in Figure 11.1 but where a countable infinity of individual minds (in red) exist as atoms of a fluid confined by the wave function. Each mind however can act independently of any other; even adjacent minds do not interact (Albert and Loewer, 1988).

Even though minds on the scale shown in Figure 12-2 appear as a single continuous fluid body, each individual within it acts independently of any others. This rescues the stronger supervenience as previously discussed, thereby removing the mindless hulk problem. For this possibility the conscious status of any physical avatar with the appropriate

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brain structure, at any point in time, is guaranteed. Despite their independence, individual minds are still confined to travel the paths of normal sequences.

12.1.3 The universal mind view (UMV) In this possibility there is a single universal mind, which at any instant exists at one point in C-space. The universal mind possesses a multitude of distinct tokens, which exist in separate physical bodies distributed throughout the base space. As far as I am aware it was Euan J Squires (1991; 1993) who first proposed this hypothesis. Without any reference to base space locations a schematic of this model is shown in Figure 12-3. Perhaps an easier way to visualise this is to postulate a single cluster of independently acting minds that always en masse occupy a single branch of the wave function. Note however, the emphasis on the word independently. Up to this point we have considered individual minds to be entirely independent in a C-space context. In the UMV minds remain independent of each other in the base space context only. In C-space they all occupy the same branch of the wave function. Therefore full independence of individual minds is illusory. This can be illustrated by a simple example of an entirely contrived computer game.

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Fig 12-3: An illustration of the universal mind model. A single universal mind navigates one branch of the wave function. Individual minds are merely token minds of the universal mind, which occupy their own bodies at distinct locations in the base space. The choices of any one token mind will force all of the others to experience those choices by virtue of being confined to the same branch. This interaction between minds is subtle and is only apparent via the physical interface (Squires, 1993).

Suppose you are playing this game with your partner. The object is to navigate a single dot on the screen around various obstacles. This is a game of cooperation, not competition. You have control of the horizontal position of the dot while your partner has independent control only of the vertical. The position of the dot represents the position of the universal mind in C-space. The horizontal axis represents all of the factors in this contrived world over which you have full control, likewise for the vertical axis in relation to your partner. In the same way we each have control of factors in our immediate environment, e.g. body position etc., whereas our position in C-space is literally the matter and base space geometry configuration over a naturally preferred, but not privileged, Cauchy hypersurface. The computer game analogy clarifies the general picture quite well in that it illuminates the idea of independent variables of action. However, what we have described here has some naïve aspects as well. Its most obvious shortcoming is that the two players have complete control of

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their collective position on the screen, whereas in the real world minds are generally confined to travel along reasonably high probability normal sequences in one direction only, reflecting our forward time experience. Obviously such a game could be enhanced to make the analogy more realistic. However, the basic description given here serves our purpose. The most notable feature of this model is that the individual minds are always in contact with other minds via the universal mind. This contact is unconscious and must necessarily be nonlocal in a hidden variables sense. In Figure 12-3 we see that the universal mind exists at one time. As discussed in chapter 8 this necessarily invokes a preferred hidden frame with which to define a universal present. However, the corresponding hypersurface is in no way obligated to be orthogonal to the energy flux vector in space-time. This frame is the choice of the universal mind and has nothing to do with the physics. This further clarifies the nonlocal character of the UMV. Individual events at the quantum level will have an associated probability, which is not apparent unless the corresponding experiment is repeated many times. Notwithstanding the physically dependent probabilities of possible events, the universal mind can shepherd its individual tokens in particular directions so that they all remain on one branch. We could speculate that this is one of the mysterious way in which God works. However, keeping our feet firmly on the ground we would have to concede that the whole debate surrounding possible mental configurations in C-space is likely to remain speculative for the foreseeable future.

12.1.4 Weak nonlocality in the MMV As discussed in chapter 8 the claim of weak nonlocality (Hemmo and Pitowsky, 2003) represents a model that is intermediate between the two extremes of the MMV and the UMV. However, Hemmo and Pitowsky’s claim is derived purely from the physics of any relevant situation, and it has been shown in more recent work (Felline and Bacciagaluppi, 2013) that Albert and Loewer’s MMV is not invalidated. A likely reason for this is that the correlations cited by Hemmo and Pitowsky do not necessarily imply a causal relationship between minds. However, Felline and Bacciagaluppi also state that weak nonlocality cannot be ruled out. So, what is weak nonlocality in the mental context? The short answer is that, unlike the UMV where connections between minds are strong enough to keep them all on the same branch and in a universal present, any connection is weak enough to allow minds to navigate independently

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through the branches of the wave function. This is succinctly illustrated in Figure 12-4. Hemmo and Pitowsky’s claim for weak nonlocality originates with a study of Bell and GHZ states. However, the correlations cited appear to be between the minds of one physical avatar and the reports of another, and not directly between minds. We therefore take issue with Hemmo and Pitowsky on this point, which was also discussed to some extent in chapter 8. An example statement dealing with Alice and Bob performing an experiment involving a Bell-state of an electron pair given in equation (8.3), clarifies the crux of the problem. In the following quotation we suppose that there are only correlations between the minds of Alice and Bob’s reports (and vice versa) in line with Albert and Loewer’s model …but not between Alice’s sets of minds and Bob’s sets of minds. This means there will be a Bob +mind which witnesses Alice reporting a – result, while the latter report is associated with a +mind of Alice. (Hemmo and Pitwosky, 2003, 237).

The final sentence certainly entails a contradiction. A +mind of Alice would not be on a branch where Alice reports a –state, and for simplicity we are assuming honesty. The report of Alice is a physical matter only, and an Alice +mind will only exist on a branch where Alice reports +states. Since we are assuming honesty then there is no choice in this matter. The probability associated with Alice witnessing + or –states is a function solely of the physics governing the experiment. Therefore minds will be distributed through the branches according to the probabilities with which they are associated. Any correlations are en masse consistent with those probabilities. There are no correlations between individual minds over and above what we expect from the physics. In this way Hemmo and Pitowsky cannot claim a correlation based on direct links between minds (Figure 12-4). However, such a scenario cannot be ruled out either. Any effect due to direct interactions between minds would rightly be regarded as beyond normal experience. In Figure 12-4 we are shown a distribution of minds across a region of C-space where there is a branching wave function. This is identical to that illustrated in Figure 12-1, but with the addition of mental links, which are assumed real. In this model individual minds normally meander through the branches in a way similar to that in the SMV and MMV models. However, the inherent links between minds (shown in red) would account for communication via a non-physical channel. Such communication may influence choices of minds, or enable the realisation

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of events with extreme low probability. Such events being sufficiently sparse throughout space-time would be seen as bizarre, but on closer inspection could still be accounted for by the normal laws of physics. As discussed already this model is not suggested by Hemmo and Pitowsky, because they do not overtly assert the existence of direct mental links. A mind being redirected as a result of communication via a nonmaterial channel may view such an event as paranormal, whereas another, not directly having that experience, would offer explanations within the laws of physics based on a rigorous analysis of available facts. In this model both parties would be correct in their evaluations. A classic example would be a neuroscientist or psychologist offering explanations of claimed paranormal events experienced by usually one individual. Any conflict would just be a matter of perspective.

Fig 12-4 A similar situation to that shown in Figure 11.1 including direct but loose connections (in red) between individual minds. In an SMV context the discreteness of minds is manifest. Such links cannot be deduced from the physics. The physics only allows us to calculate conditional probabilities at branch points. Where branch points are initiated in the brain, an associated individual mind can make a choice. In this model any choice made can be influenced from elsewhere via the links. If the subject were conscious of such influence then an associated event would be regarded as paranormal.

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12.1.5 Normal sequences in context In this section we have based our experience of normal life in terms of a formally defined sequence of configurations. All of the models summarised here fully explain normal experience because they all satisfy definition (12.1). This is the first constraint shown below. There are also two additional constraints (2 and 3) that define our normal experience of life. The second constraint comes in two forms, which we designate strong and weak. The weak form forbids direct conscious interaction between minds, whereas the strong version denies any kind of communication at all. The third constraint denies the ability of nonmaterial minds to carry their own memories. Essentially these constraints deny any faculty of nonmaterial minds that are not required for the experience of normal life. Therefore all of the following constraints are sufficient to define normal life as we experience it. Definition 12.2: Normal constraints 1. Minds follow normal sequences in C-space (definition (12.1)). 2a. Minds do not interact with each other directly (strong form, SMV, MMV) or 2b. minds do not interact consciously with each other directly (weak form, UMV, weak nonlocality). 3. Minds do not carry memories independently of matter. Once accepted, then we can say that any violation of these constraints would be regarded as paranormal. As an example time-slips violate constraint 1 because, in such a case, a nonmaterial mind experiences a discontinuous sequence. Andrew MacKenzie (1997), for example, provides a good representative list of time-slip cases. However, any further discussion of paranormal events remains beyond the scope of this work. In what follows we consider some further consequences of constraint 2.

12.2 Isolation of nonmaterial minds? Of the four models just considered the last two exhibit some kind of coupling between nonmaterial minds. It is very difficult to describe what this means in terms of the more familiar description of a communication channel, the implication being that nonmaterial minds can influence each other in ways that cannot be consciously perceived. If such influence is possible then this raises a question as to whether it is, or can be two-way. For the purpose of the discussion to follow let us suppose that it is. In

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chapter 10 we argued the case that it makes no sense to identify physical and phenomenal concepts of time–conceptually they are entirely distinct. So if it is in principle possible to acquire even a crude measure of phenomenal time, does it make sense to discuss objective delays in this context? Related to that we can ask: is there a sense in which mental events can somehow be regarded as simultaneous? The other possibility is that no such channels exist and that nonmaterial minds are entirely independent and isolated from each other as in the SMV or Albert and Loewer’s MMV. In these circumstances whether it makes sense to consider an instantaneous objective configuration of minds across spaceC, is questionable. Here we may use the analogies of pre and post relativistic treatments of physical time. In the late seventeenth century Newton and his contemporaries would have regarded the world in presentist’s terms. Time would have been seen as purely phenomenal with the additional property of being accurately measurable via oscillating mechanisms– clocks. In this model of reality the definition of simultaneity is straightforward due to its representation as an equivalence relation. As we have seen mainly in chapters 4 and 10, simultaneity in a relativistic context is not absolute. Simultaneity may be defined in terms of a single reference observer, or in terms of a hidden frame. In this way simultaneity still satisfies an equivalence relation. However, in a relativistic context there is no such thing as mutual simultaneity between two observers occupying different inertial frames. In relativity theory therefore, each observer carries their own proper time, which depends wholly on the local physical circumstances–relative velocity and gravitational potential. Moreover, because the nature of time is so dependent on the local physical situation, it is itself regarded as a physical object, and this leaves us with the question of where A-series experienced time fits into the scheme. These issues were addressed up to a point in chapter 10. However, if we can in principle know the local circumstances over a large patch of spacetime then proper time for all observers therein can be related, even if a definition of absolute simultaneity is lacking. In other words we can take any two observers and express time for one observer as a function of the other. If we now apply the same reasoning to phenomenal time for nonmaterial observers over a large region of space-C, then could their respective phenomenal times be similarly related? In the physical analogy, although there is no absolute simultaneity there is still the possibility of communication. In this way proper time for each of a large number of observers can be related. For nonmaterial observers however, if

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we are insisting on constraint 2a then no communication is possible between them. Therefore although each observer carries their own phenomenal time, not only is there no absolute simultaneity but also they cannot be related even in a relativistic sense. This implies a further consequence that it makes no sense to talk about a configuration of minds throughout space-C. Essentially no such configuration exists. The establishment of a configuration of minds is only possible if communication potentially exists between nonmaterial minds. Configurations of minds depicted in Figures 12.1-2 are therefore only for the convenience of the reader, they are not to be taken too literally. In the models of Squires and, Hemmo and Pitowsky however, some form of communication between minds does exist even if it is only at an unconscious level. In these theories constraint 2b is used to restrict direct communication between nonmaterial minds. Note however, that this does not deny all communication, only that which is consciously perceived. This would allow in principle, the phenomenal time as experienced by one nonmaterial observer to be expressed in terms of another. Whether this extends to a definition of simultaneity however, is still questionable. If for example, a phenomenal delay can in principle be measured between a message and a reply from observer A to observer B and back again, then a notional event at a point bisecting the interval between observer A sending its message and receiving a reply would be regarded as simultaneous with the event where observer B receives the message and immediately replies. This is certainly true both for absolute time and in the relativistic case with respect to observer A. However, as we have seen in the latter case, simultaneity can only be defined for a specified observer, and we have no reason to expect anything different for phenomenal time in relation to nonmaterial observers. But moreover, it would also seem that phenomenal time has no geometric interpretation–it is not extended. So if we consider space-C to be analogous to threedimensional space in a Newtonian model, then phenomenal time would either be absolute or, if relativistic in some sense then we would require the analogy of a hidden frame. Therefore in these models it is, at least in principle, possible to define a configuration of nonmaterial minds. And whether a configuration of nonmaterial minds exists depends on which of the above models is correct, again assuming that one of them is. As a final remark on configurations of minds in space-C, it seems that the four possible models allowed by the principle of localisation may not be mutually exclusive. Moreover weak nonlocality may be viewed as a combination of the UMV and MMV. When we combine these with the SMV it suggests that minds may reside in many clusters of varying size

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with some as isolated individuals. For example the density of minds could vary across space-C where large clusters that we have been referring to as the universal mind, may be many. Likewise there may be variation in the intensity of unconscious links between minds allowed by constraint 2b between pairs of minds. On top of that such configurations could change dynamically, but only in the context of phenomenal time. The problem is that there is no established scientific approach allowing us to probe mental configurations and therefore this remains a task for future generations. Addressing the more immediate concern, it is still not known whether the principle of localisation in C-space will become generally accepted, but given the trajectory indicated by the evidence so far accumulated, it is my feeling that it will.

12.3 Controversial topics: a final word Based on deeper analysis of reality and on current scientific theories, it is concluded that mind-body physicalism is untenable. This however, hinges on certain other conclusions at the foundations of physics and the metaphysics of time that are still intensely debated. The two main issues within physics having a bearing on our conclusion are the interpretations of quantum mechanics and the information loss paradox in relation to gravitational collapse. Additionally, on a metaphysical level, there is still fierce debate as to the nature of phenomenal time and how it relates to the way it is treated within mainstream physics. It was the appearance and subsequent development of relativity theory at the beginning of the twentieth century that fundamentally changed the way we think of time. Prior to this the debate between presentism and eternalism was fairly balanced and very old–dating back to the fifth century BCE. As was seen in chapter 10, during the twentieth century, not only did the debate shift markedly in favour of eternalism but, because this could not explain the experience of phenomenal change, a number of popular hybrid theories, consisting of elements from timelessness and presentism, emerged. The debate revolved around the question of experienced change in the face of physical time being viewed as an extra spatial dimension. Backed by strong evidence, the new theory asserted space-time to be little more than a four-dimensional static set of events. No absolute present could be discerned and there was no explanation for experienced change. Philosophically this was unacceptable and so the debate persisted. Coupled with substantive debates within the foundations of physics, things became even more complicated after the emergence of

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quantum theory. Since experimentation to test quantum theory began with the earliest double-slit experiment involving electrons (Davisson and Germer, 1928) there has been a growing suspicion that a deeper substrate of reality existed in which the classical world as we experience it, is nothing more than a façade. For example, when we divorce the passage of time from a timelike direction in space-time we observe an ordered set of configurations. Moreover, it is found that the observed configurations constituting space-time are but a single thread woven through an almost infinite-dimensional manifold that is the universal C-space. This is what we arrive at when we take quantum theory at face value and apply only unitary evolution along this thread. Notwithstanding the, so far one hundred percent success of this theory, there have been many learned academics who found it difficult to let go of the idea that the classical world we experience is the basis of all reality. Indeed to some extent this remains the case today. The response was the ascent of many so-called interpretations of quantum theory that sought to combine the predictions of quantum mechanics with our experience of the classical world. In the various interpretations, with one exception, it is the classical world that holds sway, with quantum effects being significant only at microscopic scales. Two groups of interpretations posited some form of wave function collapse. In one group the collapse was objective because, in these theories the wave function itself is objective. In the other, collapse is not objective because the wave function is regarded as a purely epistemic entity. All empirical tests to date have pointed to the wave function being an objective entity. These so-called objective collapse theories posit two mechanisms for wave function evolution, unitary transformations and a nonlocal discontinuous collapse process. However, as was argued in chapter 6 there are significant problems with the collapse process in a relativistic setting. Two other groups of interpretations discussed were nonlocal hidden variables and pure wave theories. Hidden variables theories are realist in the context of the classical world. In other words they consist of two components, the wave function and the sequence of actual particle configurations that is the history of the universe. In addition it may be that they can also be local or nonlocal. However, from the work of John Bell (1964) all of the local versions are firmly ruled out. Moreover nonlocal hidden variables theories may be regarded as ontologically more extravagant than pure wave models because of the requirement for two components to reality. In a purely physical context then this is probably justified.

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However, in section 8.3.2 a connection was made between nonlocal hidden variables and Squires’ UMV where the universal mind is considered to occupy a single point in C-space. Given that any point in Cspace has a unique history then there would be an actual history as far as the universal mind is concerned. This unique history and the universal mind it is associated with, plays the role of the hidden variables. Moreover, in sections 10.4-5 Ewing’s (2013) irreducible fact theory explaining the universal passage of time was also connected with Squires’ UMV. Borrowing the elephant and blind men analogy again, it is as though DeBroglie, Bohm, Hiley, Kaloyerou, and Ewing, represent the proverbial blind men encountering an elephant for the first time. With no disrespect intended, this analogy shows that ostensibly competing or unconnected theories can develop in a way that converges towards a more general overview (the elephant), which in this case is Squires’ UMV. This does not say that the UMV is the correct model, it is only indicative of the direction that current research seems to be taking. Although we are promoting pure wave theories in this work, that is only on the physical side of reality. There is nothing to suggest that any hidden variables have to be physical, they could just as easily be associated with a universal mind as suggested above. Moreover, it could be that theories converging to or concurring with the UMV are entirely incorrect and that the truth is closer to the SMV, MMV, weak nonlocality, or some combination thereof. In this work it is acknowledged that we are far from being in a position to resolve this issue. Therefore it seems that in a debate between those favouring pure wave theories and promoters of nonlocal hidden variables, physics alone is unable to decide the issue. Deciding factors must include the status and distribution of nonmaterial minds. The remaining controversial issue in this work, which is still a subject of considerable debate, is the information loss paradox associated with gravitational collapse. In section 4.5 we considered two possible scenarios for this, which are identical up to a point arbitrarily close to the event horizon of a black hole. Before this point the process is rather uncontroversial. However, there is disagreement about what takes place beyond that point, and opinions fall broadly into two camps. In one the material surface of a collapsing star falls through the horizon and continues to a singularity at its centre. Evaporation via the Hawking process takes place once the surface is beyond the horizon, wherein all information about material that had previously fallen through the critical radius, is lost. Supporters of this scenario may consider this to be a somewhat academic exercise because it can only take place beyond t = ∞ .

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In practice this information remains permanently inside a trapped surface, so in principle it still exists. In section 4.5 however, we provided reasons why this description is unsatisfactory. The other interpretation, endorsed in this work, is that the Hawking process begins before the material surface reaches the horizon, and analogue acoustic simulations indicate that this is the case (Barcelo et al, 2006). Once the material is close to the horizon, like in the previous interpretation, it is subject to unlimited time dilation at the horizon. However, the evaporation time (coordinate time) is finite, but the radiation is not purely thermal. Encoded within it is all the information about everything that fell into the gravitational well at and after the collapse. This information is therefore recovered. The reason this is important is because it has implications for unitary evolution, which is preserved in this process. And on the physical side of reality we are promoting the unitary transformation to be the basis of all change. That is, the wave function at one point in C-space is related to it at any other by a unitary transformation. Another issue related to the information loss paradox is so called time machines. These are regions of space-time exhibiting time loops, which are allowed by solutions of Einstein’s equations. However, we already know that these violate unitary evolution (Deutsch, 1991), and this is why they are briefly mentioned in section 4.5 and here. It is conjectured that timelike loops are not admissible in a space-time that: is asymptotically flat, topologically simple, contains no event horizons, and consists only of mass-energy fields satisfying the 4-volume average null energy condition. This is based on theorem 4.1 that rules out the formation of causality violating regions classically, Moreover it is conjectured that expectation values of fields at relatively small scales will still satisfy the corresponding energy conditions (Hawking, 1992). Since closed null geodesics must exist at the boundary of any causality violating set, then their absence implies no such violation. Moreover, the restriction to singly connected topology precludes event horizons therefore any closed causal loops must be naked. A time machine solution recently published (Ori, 2007) is still hidden behind an event horizon and therefore the time loops within are not naked, so for our purposes unitary evolution remains on safe ground. While our localisation of consciousness model appears consistent throughout, there are those who would regard it as ontologically extravagant. Yes space-C has a high dimensionality and is large. Moreover, the reality we can directly perceive, even with twenty first century instrumentation, barely scratches the surface in space-C terms.

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However, for any manifold having some kind of structure one can always enquire as to the space in which it is embedded, and in principle such an enquiry is indefinitely iterative. These issues were briefly explored in section 10.1.3. If it is not considered extravagant to posit large manifolds like space-C therefore, then we are likely to question the number of irreducible elements in our ontology. In our model there are just two: the mind and the wave function, and these complement each other in certain ways, for example the mind is localised and the wave function is widely distributed across space-C. Moreover, it is possible that the wave function resides in the non-substantive space and thereby generates a structure for space-C. However, this will depend on which single complete theory of quantum gravity remains empirically consistent. Many philosophers may consider two irreducible elements to be extravagant, hence the push towards some form of monism. However, there are many examples from diverse philosophies and cultures where duality injects richness far beyond what is possible otherwise. For example a large register consisting of “0”s and “1”s has a vast number of possible combinations compared to just one value to the exclusion of the other. The world presents many examples of duality, for example polarity–positive and negative–yin and yang, which contributes a rich diversity to the world. Take away one of the polarities and the world is bland. Similarly take away the mind and this leaves only the wave function with nothing to experience it. Assume the mind is part of the wave function and it is without dynamics–there is no prime mover, no experienced duration, and no change, in short there is no passage of time. Therefore, it is my contention that two irreducible elements is a minimum requirement for an acceptable ontology of reality, while three or more may be considered too extravagant.

12.4 Conclusions and future direction In this chapter it has been shown how we may model our normal experience of reality in terms of a sequence of material configurations. Moreover, it can be clearly seen how this is embedded into the wider context of a static universal manifold (space-C). At any instant your nonmaterial mind is localised in space-C. What we call the wave function is defined over space-C, unlike classical gravity or electromagnetic fields that are only defined over the base space. This is what makes the wave function difficult to visualise and to work with generally. Taking the square modulus of the wave function provides a probability for each configuration relative to earlier configurations. And the most probable

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future configurations are dictated by paths of high probability moving in a direction of increasing entropy or knowledge. In our C-space locality, these high probability paths appear to have a fractal branching structure, and it is most likely that our normal sequence of events lies along one of these paths. Normal sequences and constraints are more clearly defined in definitions, 12.1 and 12.2. The principle of localisation states that only nonmaterial minds are localised in C-space. This is all we can gain from our direct experience. It is not currently possible to decide on particular relationships between minds. Therefore our localisation model must be regarded as part of a larger scheme, which remains incomplete. The single (SMV) or instantaneous (IMV) minds view, the many minds view (MMV) of Albert and Loewer, the universal mind view (UMV) of Squires, the weak nonlocality model of Hemmo and Pitowsky, or any combination of these are all possibilities describing normal reality as we experience it. We are not yet in a position to eliminate these possibilities or any others that may be proposed in the future. If reality is ultimately dualist in a mind-material context, then this leaves us with the question of how science can penetrate the mental side. Psychologists can surely offer some contribution here. However, psychology is increasingly probing areas of neuroscience and there may be a tendency to focus too much on the material side. Traditionally science focusses on what can be materially measured–instruments are physical objects and as such can only probe the material side of reality. The mental side remains entirely cut off from the enquiries of traditional science. So it appears that we need to expand the definition of science to include purely mental areas, while still adhering rigidly to its methods. Alternatively, still following scientific methods, a new discipline may be initiated under a different name. Either way the approach would be the same. Progress in this direction is being made already by the Mind and Life Institute whose biennial conference is hosted by His Holiness the Dali Lama. Application of scientific methods to reports of defined experiences by adept Buddhist practitioners is one possible route to take. In a dualist reality the mental side of life can only be investigated in its own terms. Attempting to define mental properties in material terms is futile. What is needed is a twin track approach, and it is my belief that interested readers should pay particular attention to the proceedings of the Mind and Life Institute.

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12.5 A final thought: where is all the data? If we consider a presentist/physicalist model for a moment, which is straightforward to visualise, the data specifying the fields and positions of every particle is distributed throughout our base space at any instant. This includes data stored in the brains of animate organisms. However, we have argued for an eternalist/dualist model in which nonmaterial minds, ostensibly unable to carry memories, are localised in a universal space-C manifold with C-space as a subset. Notwithstanding its extreme dimensionality an empty C-space itself carries no data, therefore the data specifying the experience of a mind at a given instant, assuming that it carries no memories of its own, must be encoded in its position relative to some α-point and other possible landmarks within the C-space. So in a perverse way it is nonmaterial minds that are responsible for the existence of the data specifying their experiences. A model involving independent minds densely packed in C-space according to the Born rule (Albert and Loewer’s MMV) would require minimal data overall for its specification. Large quantities of data would only be required to specify the location of a single individual mind within space-C. Similarly a single universal mind (Squires’ UMV) exists at one point in Cspace, even though its many tokens may be distributed throughout a lowdimensional base space. Therefore the UMV model also requires minimal data for its specification. Models involving a low density of discrete minds navigating independently or semi-independently through C-space according to the Born rule, would require significantly more data for their specification simply because each individual mind is less constrained and the data required to specify all of their locations is cumulative. Such models include the SMV/IMV, and Hemmo and Pitowsky’s weak nonlocality. The question is: do models requiring lower quantities of data for their specification make them more likely? It is like asking: is God a minimalist? This is certainly one question for future generations to ponder over.

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INDEX

abduction 10,76, 98, 99. AdS-CFT correspondence 140, 141, 143. ADM formalism 151, 152, 158, 159, 184, 232, 244. Albert, David xiii, 17, 18, 20, 25, 282, 286-295, 297-303, 320, 326, 328, 331, 333, 334, 381, 382, 394, 396, 399, 400, 403, 410, 411. Al-Biruni 86, 87. Al-Kindi (Alkindus) 84, 89. Antikythera mechanism 83, 84. Arab scholars 10, 78, 83-86, 93. Archimedes 83. artificial consciousness 8, 23, 24, 369-391. logic gates 382, 385, 386. neural networks 24, 305, 306, 386-389. strange loops 24, 382-384. Turing test xxiv, 24, 388, 389. Avicenna 86, 87, 93, 94. axioms 9, 11, 13, 31, 32, 40, 4346, 53-59, 61, 67, 68, 70, 72, 73, 76, 86, 92, 97, 148, 182, 208, 209, 219, 221, 256, 305, 345. Zermelo-Fraenkel (ZF) 53, 61, 70. ZFC (with axiom of choice) 53, 59, 61, 68, 70, 72, 73, 209. intuitionistic 70.

von-Neumann Bernays (VNB) 61, 62, 73, 209. Bacon, Roger 78, 88-90. Francis 75, 78, 90-96. Barbour, Julian x, xii, xiv, 158160, 184, 300, 312, 337, 346. Barcelo C, xii, 110, 139, 144, 408. Bekenstein-Hawking formula 143, 251. Bitbol, Michael 17, 18, 293, 294, 300, 303, 319, 326, 381. Black holes xii, 110, 131-146, 244, 251, 253, 407. evaporation, see Hawking process. Boole, George 31, 35, 38. Brouwer, LEJ 9, 27, 67, 70. Cantor, George 47, 49. diagonal procedure 49, 50, 52, 56, 60. cardinality of the continuum 6466. category mistake xix, 20, 324, 325, 332. causal sets 149, 150, 154, 250. superposition of 250. Chalmers, David J 1, 23, 371. choice xxii, 75, 308, 384. axiom of 53, 59, 70. mechanism of 24, 293, 295, 296, 362, 370, 390, 398-401. Chrysippus 9, 31, 33, 34, 36-39, 41, 46. completeness of formal systems 38, 47, 52, 55-57, 60, 70, 74. connectives (logical) xvi, 33-39, 46, 53, 54, 71, 72. conjunction (logical) xvi, 8, 17, 27, 33-35, 38, 49, 56, 69, 84, 99.

426

conscious basis 305, 306. consistency of formal systems 57, 68, 69. contradiction, principle of 33, 39, 41, 42, 46-48, 51, 52, 56, 60-65, 73, 74, 227, 236, 281, 300, 338, 339, 344, 366, 400. proof by 41, 42, 46, 63, 64. continuum hypothesis (CH), see generalised continuum hypothesis configuration space (also Cspace) xix, xxi, xxiii, xxiv, 57, 12-20, 23, 25-27, 123, 149, 157-161, 163, 166, 167, 173, 179, 185, 209, 212-214, 234, 243, 281, 184, 185, 196, 197, 201, 202, 206, 210, 211, 214, 218, 222-224, 229, 237, 241, 244, 245, 249, 252-257, 282, 285-287, 291, 292, 294, 296, 297, 300, 306, 308-310, 312, 313, 318, 319 321, 329, 365, 367, 391-400, 402, 405408, 410, 411. absolute 158. relative 158, 159, 161, 163, 179, 255. corticothalamic (CT) system 371, 373, 374. Dali Lama 25, 410. Dawkins, Richard 20, 323, 330333. deduction 10, 31, 34, 38, 76, 7982, 84, 87, 88, 92-94, 96, 98, 108. Descartes, René 15, 90, 280, 282, 324. Deutsch, David 12, 110, 184, 221, 239, 240, 256, 311-314, 317, 319, 320, 408.

Index

Diósi-Penrose criterion 378. disjunction 33, 34, 37, 99, 325. divided-brain patients 327, 328, 374, 375. Dowker, Helen Fay 323, 333. dualism xi, xiv, xx, 2, 3, 8, 17, 18, 280, 281, 296, 303, 324, 326, 354, 367, 371-373. Cartesian 20, 280, 326. Eichman, Peter 21, 335, 337, 339-344. Einstein’s field equations 128, 131. Einstein summation convention 127, 248. equivalence, principle of 110, 126, 209. eternalism xx, 3, 6, 8, 19, 21, 109-111, 114, 146, 153, 281, 310, 317, 323, 329, 335-351, 353, 358, 360, 366, 367, 405. Euler-Lagrange equations 164, 166, 169, 171, 172, 175-177. event horizons xii, 110, 131133, 135, 138-140, 143-145, 251, 407, 408. Everett, Hugh III 4, 5, 184, 240, 241, 282, 287, 293. Ewing, Kyley 21-23, 335-337, 344-350, 357-359, 361, 364366, 368, 407. excluded middle, principle of 32, 33, 39, 55, 70, 71, 73, 77. experiential manifold 19, 306309, 312, 313, 317, 332. explanatory gap 23, 359, 371, 372, 390. feedback systems 23, 373, 376, 387. sensitivity of 373.

The Disembodied Mind

Fayerabend, Paul 79, 99, 104107. Figg, Travis Matthew 351, 352. formalism 32, 67, 68-70, 73. Frege, Gottlob 31, 35, 38, 67. Gazzaniga, Michael S xv, 23, 370, 374, 376. generalised continuum hypothesis xvii, 67, 72. generalised coordinates xxiv, 12, 157-163. Gödel, Kurt 32, 47, 52, 53, 5557, 60, 61, 63, 67, 70, 73, 74. gravitational collapse 11, 25, 131-146, 405, 407. Greaves, Hilary 19, 311, 317, 319-323. Grosseteste, Robert 87-89. growing block 11, 111, 146-150, 153, 154, 185, 250, 255, 346, 347. Hameroff, Stuart 23, 24, 27, 370, 377-381, 384, 389, 390. Hamilton, William R 12, 155, 170. Hamiltonian formalism 167, 169-175, 177, 179. Hamilton-Jacobi equation 12, 157, 175-179, 182, 238, 248. Hamilton’s equations 172, 174, 176. Hawking process 139, 145, 407, 408. Hawking, Stephen W xiv, 109, 110, 135, 136, 138-141, 143145, 152, 408. Heraclitus x, 337. Heyting algebra 71. hidden inertial frame 110, 146, 151, 343, 356. Hilbert, David 67-70, 74.

427

Ibn al-Haytham (Alhazen) 78, 84, 85, 89. Ibn Sahl 89. identity, logical 33, 39. identity personal 18, 20, 312, 318-321, 326, 327, 333, 351. indemonstrables 33-38, 46. induction 10, 76, 78, 80-82, 84, 86-88, 91, 92, 94, 95-100, 105, 108. mathematical 69, 162. infinity xvii, 47, 50, 57-67. countable, ℵ0 xvii, 47, 49, 51, 69, 160, 396. of the continuum 50, 57-67. intuitionism 31, 32, 62, 67-71, 73. Ismael, Jenann 19, 311, 317319. JƗbir ibn HayyƗn (Geber) 84, 86, 93. Kroes, Peter 22, 334, 355, 356, 358-363. Kuhn, Thomas S 79, 99, 101103, 107. Lagrange, Joseph-Louis 12, 155, 179. Lagrangian formalism 164-169, 175. Lakatos, Imre 79, 99, 102, 103, 107. left-brain interpreter 23, 374. Lewis, Peter J 19, 20, 311, 317, 322, 323, 327, 328. localisation of consciousness, principle of 7-9, 16-20, 22, 2426, 280-301, 310, 326-328, 334, 361, 363, 367, 391, 392, 394, 395, 404, 405, 408, 410. consciousness causes collapse 17, 282-285.

428

many minds view (MMV) 17, 18, 26, 287, 288, 290, 291, 292, 295-297, 299-301, 305, 396, 397, 399, 400, 402-404, 407, 410, 411. single minds view (SMV, also instantaneous…IMV) 25, 287, 289-292, 297, 315, 394, 395, 400-404, 407, 410, 411. perspectival realism 17, 293, 294. selected consequences of 391-411. universal mind view (UMV) 17, 18, 26, 295-297, 301, 361, 362, 364, 368, 397399, 402, 404, 407, 410, 411. weak nonlocality 18, 25, 297, 299, 300, 399-402, 404, 407, 410, 411. Wigner’s friend xiv, 13, 216, 217, 282, 284, 290, 291. Lockwood, Michael J 18, 19, 27, 300, 303-313, 317, 319, 326, 332, 334, 335. Loewer, Barry xiii, 17, 18, 20, 25, 282, 285-295, 297-303, 309, 313, 315, 320, 326, 328, 331, 333, 334, 381, 382, 394, 396, 399, 400, 403, 410, 411. logic, classical xvi, 9, 10, 24, 27, 31-57, 62, 64, 71, 73, 76. alternatives to classical logic 67-72. Maxwell, James C xi, 115, 126. Maxwell’s equations 110, 114, 115, 249.

Index

McCall, Storrs 9, 22, 27, 353357, 366. McTaggart, John ME 21, 281, 335, 337-339. Mersini-Houghton, Laura xii, 110, 144. metric: C-space 14, 252, 253, 393. differential 14, 127. FWR 152. Kerr-Newman 131, 132. Minkowski 110, 123, 126, 127, 131. Scharzschild 131, 135, 138, 143. deSitter 141-143. space-C 253. space-time 14, 123, 128, 131, 135, 136, 151, 152, 232, 244, 247. superspace 247, 353. topological xviii, 123, 393. Michelson, AA 115, 117. Michelson-Morley experiment 110, 116, 117. microtubules xxi, xxii, 24, 372, 377, 379, 381, 385. α/β-tubulin, dimmer 24, 377, 378, 389, 390. Mill, John Stuart 78, 95-97. Mind and Life Institute 25, 410. Minkowski, Hermann xi, 110, 123, 153. mixed states 182, 303, 304. Morley, EW 115, 117. motivations for physicalism 15, 16, 260-279, 293, 323, 391. argument from physiology 267, 268.

The Disembodied Mind

causal closure 261-264, 269, 272, 277-279. epistemic physicalism 262, 263, 272-277. knowledge argument 262, 270-272. methodological maturalism 262, 268-272, 274, 277, 278. naïve set theory (NST) 59, 60. von Neumann-Bernays (VNB) axioms 61. Newton, Isaac 11, 12, 78, 88, 90, 94, 100, 111, 128, 129, 153, 155, 156, 167, 169, 175, 181, 350, 403. Newtonian mechanics 12, 100, 102, 103, 110, 114, 124, 156, 157, 179. laws of motion and universal gravitation 11. neuroscience 3, 7, 8, 369, 370377, 386, 389, 391, 410. neural substrate 369. Noether’s theorem 156, 169, 173, 174. normal sequences 392-402, 410. definition 393. orchestrated objective reduction (Orch OR) xxii, 24, 378-380, 390. orchestrated decoherence xxii, 24, 379, 381, 389, 390. Panineau, David 15, 261, 263268, 277, 278, 280. paradigm shifts 3, 101, 102, 107. paranormal xiii, xiv, 7, 25, 392, 401, 402. time-slips 402. Parfit, Derek 20, 323, 326-329, 333.

429

Parmenides x, 337. Penrose, Roger xv, 22-24, 27, 50, 52, 100, 101, 109, 184, 218, 222, 231, 242, 243, 249, 370, 377-381, 384, 389, 390. phenomenal time see time. physicalism see also motivations for xi, xiii, xxi, xxii, 2-4, 6, 8, 15-17, 20, 22, 26, 27, 74, 109, 180, 183, 260-281, 283, 284, 287, 289, 293, 300, 302, 303, 317, 322-333, 362-364, 366, 368, 369, 370, 373, 375, 389, 391, 405. Plato 9, 31-33, 57, 84, 89, 93, 103. Plesch, Peter H xiii, 89, 313. Popper, Karl R 78, 99-104, 106, 107. postulates 9-11, 57, 91, 239. on which this book is based 6, 8, 16, 17, 26, 27, 302, 332, 333, 349, 391. of quantum mechanics 13, 209-221. of relativity 109, 110, 114, 126, 339. presentism xxii, 3, 8, 11, 19, 21, 114, 146, 151, 153, 185, 250, 281, 310, 316, 323, 332, 333, 335-353, 355, 356, 360, 366, 367, 405. Pythagoras’ theorem 42-46. quantum consciousness 20, 329. quantum mechanics xiii, xiv, xxi, xxiii, xxiv, 12-15, 181-257, 405, 406. axioms of 208-221. Bell inequalities 235, 237. blackbody radiation 181, 182, 185-189.

430

Index

Bohmian mechanics 184, 234, 237, 238, 241. Bohr, Neils 181, 222. Born rule xix, xx, 201-206, 212, 227, 239, 292, 293, 411. Compton effect 190, 191. commutation relations 182, 194-196, 199. DeBroglie, Louis 13, 192, 407. hypothesis 181, 182, 184, 191, 192, 193, 206, 207. density matrix formalism 182, 219-221, 303, 304. double slit diffraction 182, 183, 206-208, 224, 227. early theory 13, 182, 185192, 256. Einstein, Albert 13, 181, 188, 189, 222. entangled states 183, 214, 216, 217, 230, 231, 235. epistemological theories 183, 221-228, 240. EPR systems 18, 230. full theory 13, 181, 191208, 221. Heisenberg’s matrix theory 13, 182. hidden variables (also pilot wave theories) 18, 22, 183, 184, 221, 222, 231, 233240, 256, 281, 296, 301, 343, 354, 399, 406, 407. nonlocal 18, 22, 221, 231, 233-240, 281, 296, 301, 354, 399, 406, 407.

interpretations of xii-xiv, xx, xxi, 12, 13, 18, 22, 25, 27, 182-184, 205, 211, 221243, 256, 343, 405, 406. Copenhagen 4,13, 183, 222, 228, 233, 241. macroscopic-Bell states 14, 185, 224, 253-255, 257, 300, 335, 348, 349. Seth Lloyd threshold 254. objective collapse 22, 24, 183, 184, 221, 222, 228234, 240, 256, 354, 356, 357, 406. photoelectric effect 189, 190. pure wave theories (also many worlds) xiii, 13, 17, 22, 24, 27, 183, 184, 218, 221, 222, 231, 233, 240243, 249, 256, 354, 406, 407. quantum binary digit (qubit) xxiii, 14, 15, 183, 212-217, 254, 291, 299. quantum gravity xiv, xv, xxi, xxiii, 5, 11, 13, 14, 123, 146, 147, 153, 155, 159, 163, 183, 184, 242-253, 256, 409. canonical theory (CQG) 13, 184, 242, 244-249, 256, 303, 308, 353. superstrings and Mtheory 249, 250. LQG 11, 14, 153, 242, 250-252, 256. causal set theory 147, 250-252.

The Disembodied Mind

Schrodinger’s cat xiv, 13, 214-216, 218, 219, 284, 291. Schrodinger’s wave equation 12, 182, 184, 194, 203, 222, 229, 234, 240. superpositions xxi, 19, 24, 182, 183, 201, 208-210, 214, 215, 218, 221, 224, 227, 229, 234, 250, 282, 286, 288, 291-294, 306-309, 315, 317, 321. uncertainty relations xxiv, 182, 196-201, 211, 224-227. Wheeler-DeWitt equation 13, 14, 244, 247-249, 252, 363, 392. Wigner’s friend xiv, 13, 216, 217, 282, 284, 290, 291. readiness potential 23, 376, 377. reductio ad absurdum 46-48. reductionism 20, 261, 267, 320, 326, 328, 330, 333. relativity xi, xii, xiv, xxii, 3-5, 7, 11, 16, 21, 27, 30, 76, 100, 103, 109-154, 184, 209, 231, 233, 239, 242, 251, 255, 256, 279, 280, 306, 335, 336, 339, 343, 348, 359, 375, 379, 403, 405. special theory 11, 106, 114126, 341, 343. general theory xii, xiv, 5, 7, 14, 76, 100, 103, 126-154, 159, 184, 209, 240, 242, 247, 249, 252, 281, 288, 303, 314, 335, 348, 353. Galilean, principle of 11, 111-114, 341, 343, 359. Rindler, W 109.

431

Russell, Bertrand 32, 60-62, 67, 73. Ryle, Gilbert 20, 323-326, 332. set theory xvi, 10, 47, 57-67, 70, 73. causal set theory see causal sets. Schwarzschild solution 131-143. Schutz, Bernard 109, 117, 124, 126, 127, 134, 135, 139, 232. Seager, William 15, 16, 261, 262, 272-278, 282. simultaneity 14, 110, 133, 136, 153, 154, 339-342, 403, 404. Smolin, Lee 255, 336, 345. solipsism 283, 394, 395. Sorkin, Rafael 11, 27, 111, 147, 149, 153, 252, 323. space-C xxiii, 14, 25, 185, 252254, 257, 285, 288, 319, 360, 361, 363, 365-368, 381, 395, 403-405, 408, 409, 411. space-time xxiii, 4, 5, 11, 14, 16, 27, 109-154, 158, 163, 184, 185, 206, 231, 232, 242-254, 256, 257, 281, 282, 285, 314, 323, 334, 337, 340, 343, 352-354, 356, 357, 359, 360, 362, 364, 379, 399, 401, 403, 405, 406, 408. Minkowski (flat) 121-126. curved 126-131. speed of light, constancy of 110, 115, 117, 118, 126, 209, 339. Squires, Euan J xiii, 17, 18, 22, 25, 293, 295-297, 300, 301, 320, 326, 358, 361, 362, 364, 367, 368, 397, 398, 404, 407, 410, 411. statements xv, 9, 31, 32, 45, 55, 59, 69, 72, 73, 82, 99, 105.

432

self referential 60, 382, 383. stationary action, principle of 12, 164, 165, 169, 179. Stenger, Victor 20, 323, 329, 330, 333. Stoljar, Daniel 15, 261, 262, 265, 266, 268-272, 274, 277, 278. Suggett, GJ xii, 110, 135, 136, 144. superpositional dimension 19, 306-308. manifold 19, 308. supervenience xxiv, 291, 292, 320, 322, 323, 369, 394, 396. local (LSU) 292, 320, 394. tensor product xviii, 210, 216, 220. tensors 127, 129, 130, 152, 247, 252, 253. thalamocortical (TC) system 373. thought, laws of 31, 32, 39. time x, xii, xiv, xxiii, 1, 5-8, 11, 12, 14, 18, 19, 21-23, 25, 27, 109-154, 334-368, 393, 395, 399, 403-405, 408. as A, B or C-series 337-339, 357, 362, 363, 365, 403. capsules 312, 346. irreducible fact theory 21, 22, 335, 336, 357-359, 364366, 368, 407. mind-dependent theories of 22, 23, 335, 336, 358, 360362, 364-367, 391. moving spotlight theory of 21, 335, 336, 344, 347, 349, 350, 358, 364, 366, 368. objective dynamic theory of 22, 27, 353-357, 366.

Index

objective theories of 22, 335, 350, 353-360. phenomenal xxii, 25, 334, 335, 355, 359, 360, 363367, 403-405. problem of 184, 244-249, 281. relational 19, 310, 314-317, 332. substantivalism 314. travel 145, 146, 353, 402, 408. transformation, conformal 135, 136. coordinate 109, 120, 245. Galilean 113-115, 117, 119, 336, 350. Lorentz 117, 119-122. topological 133. Vachaspati, Tanmay xii, 110, 143, 144. Weyl, Hermann x, 4, 16, 67, 281, 300, 315, 318. Wheeler, John A 5, 109, 245. Whewell, William 78, 95-97, 100, 106. Zeh, H Dieter xii, xv, 16, 151, 152, 155, 184, 239, 242, 244, 248, 249, 282, 285, 300. Zermelo, E 32, 58.