Knowledge and the Philosophy of Number: What Numbers Are and How They Are Known 9781350102903, 9781350102934, 9781350102910

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Knowledge and the Philosophy of Number

Mind, Meaning and Metaphysics

Series Editors: Johannes Brandl, University of Salzburg, Austria Christopher Gauker, University of Salzburg, Austria Max Kölbel, University of Vienna, Austria Mark Textor, King’s College London, UK The Mind, Meaning and Metaphysics series publishes cutting-edge research in philosophy of mind, philosophy of language, metaphysics and epistemology. The basic questions in this area are wide-ranging and complex: What is thinking and how does it manage to represent the world? How does language facilitate interpersonal cooperation and shape our thinking? What are the fundamental building blocks of reality, and how do we come to know what reality is? These are long-standing philosophical questions but new and exciting answers continue to be invented, in part due to the input of the empirical sciences. Volumes in the series address such questions, with a view to both contemporary debates and the history of philosophy. Each volume reflects the state of the art in theoretical philosophy, but also makes a significant original contribution to it. Editorial Board Annalisa Coliva, University of California, Irvine, USA Paul Egré, Institut Jean-Nicod, France Olav Gjelsvik, University of Oslo, Norway Thomas Grundmann, University of Cologne, Germany Katherine Hawley, University of St. Andrews, United Kingdom Øystein Linnebo, University of Oslo, Norway Teresa Marques, University of Barcelona, Spain Anna-Sophia Maurin, University of Gothenburg, Sweden Bence Nanay, University of Antwerp, Belgium Martine Nida-Rümelin, University of Freiburg, Switzerland Jaroslav Peregrin, Czech Academy of Sciences, Czech Republic Tobias Rosefeldt, Humboldt University of Berlin, Germany Anders Schoubye, University of Edinburgh, United Kingdom Camilla Serck-Hanssen, University of Oslo, Norway Emily Thomas, Durham University, United Kingdom Amie Lynn Thomasson, Dartmouth College, USA Giuliano Torrengo, University of Milan, Italy Barbara Vetter, Humboldt University of Berlin, Germany Heinrich Wansing, Ruhr University of Bochum, Germany

Knowledge and the Philosophy of Number What Numbers Are and How They Are Known Keith Hossack

BLOOMSBURY ACADEMIC Bloomsbury Publishing Plc 50 Bedford Square, London, WC1B 3DP, UK 1385 Broadway, New York, NY 10018, USA BLOOMSBURY, BLOOMSBURY ACADEMIC and the Diana logo are trademarks of Bloomsbury Publishing Plc First published in Great Britain 2020 Copyright © Keith Hossack, 2020 Keith Hossack has asserted his right under the Copyright, Designs and Patents Act, 1988, to be identified as Author of this work. Series design by Louise Dugdale Cover image © shuoshu / Getty Images All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without prior permission in writing from the publishers. Bloomsbury Publishing Plc does not have any control over, or responsibility for, any third-party websites referred to or in this book. All internet addresses given in this book were correct at the time of going to press. The author and publisher regret any inconvenience caused if addresses have changed or sites have ceased to exist, but can accept no responsibility for any such changes. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. ISBN: HB: 978-1-3501-0290-3 ePDF: 978-1-3501-0291-0 eBook: 978-1-3501-0292-7 Series: Mind, Meaning and Metaphysics Typeset by Deanta Global Publishing Services, Chennai, India To find out more about our authors and books visit www.bloomsbury.com and sign up for our newsletters.

To my wife Mary

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Contents Preface

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Introduction 1  1 Properties 9   2 Frege’s Theory of Concepts 21   3 The Logic of Quantity 33  4 Mereology 43   5 The Homomorphism Theorem 67   6 The Natural Numbers 85  7 Multiplication 105  8 Ratio 115  9 Geometry 131 10 The Ordinals 151 Notes References Index

191 194 199

Preface Number is central in the philosophy of mathematics; less obviously so, knowledge of number is central in the philosophy of mind. How is it possible for the human mind to grasp the truths of mathematics? What must the human mind be, if it can discern the mathematical realm? A mind is that which knows. There is much we do not know about knowledge, so there is much we do not know about mind. Consciousness is knowledge, but we do not know what consciousness is. We delight in a beautiful scene, and like the impressionists we delight in the experience of the scene. The experience is conscious: we are aware of the experience, but we do not know what consciousness is. The first time we look into Euclid’s Elements, we are filled with wonder at the beauty of the reasoning. We understand the axioms and see they are true, and we follow the astonishing proofs. We are using our reason, we know that we are, but we do not know what reason is. Consciousness and reason are deep philosophical problems where we know that we do not know. Even to Socrates himself, the oracle found it necessary to explain that sometimes knowledge of ignorance is philosophical progress. In this book I have set out to explore just how much of mathematics is knowledge that can be obtained by reason alone. It has been a long project that I started twelve years ago. I feared the project had come to a premature end when I became ill in the summer of 2015. I wish here to record my profound gratitude to Mr Richard Gullan, consultant neurosurgeon at King’s College Hospital in London, and to everyone in the Haematology Department at King’s. They not only saved my life but restored me to the excellent health that has enabled me to complete my book. This is also the place to thank the many people who have helped me in my philosophical journey. I first thank my teacher Dorothy Edgington and my mentor Mark Sainsbury, who helped me both at the outset and along the way. I thank my former colleagues at King’s College London, and especially David Galloway, Jim Hopkins, Andrew Jack, Chris Hughes, M. M. McCabe, David Papineau, Anthony Savile, Gabriel Segal and Mark Textor. I thank my current and former colleagues at Birkbeck, especially Salvatore Florio, Alex Grzankowski, Hallvard Lillehammer, Nils Kürbis, Øystein Linnebo and Ian Rumfitt, and I

Preface

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was much helped by conversations and correspondence with Bahram Assadian, Neil Barton, Long Chen, Julien Dutant, Marcus Giaquinto, Simon Hewitt, Peter Jackson, Jonathan Nassim, Michael Potter and the late and sadly missed Bob Hale. I must also thank the two anonymous referees for exceptionally helpful and detailed comments. And heartfelt thanks to my beloved Mary, who faithfully saw me through my illness, and to Alan and Janice, friends in need.

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Introduction

There is knowledge of number. That fact is better known than the premises of any sceptical argument to the contrary. So if a metaphysics of number is to be credible, it must make room for our knowledge of number. On the widely held metaphysical theory that numbers are sets, it is notoriously difficult to avoid making knowledge of number seem fantastical. The metaphysics proposed here is that numbers are properties, not sets. Magnitudes are a kind of property, and numbers are magnitudes. Mass, length and time are fundamental magnitudes discovered by the science of physics, and the natural, real and ordinal numbers are fundamental magnitudes discovered by the science of mathematics. There really are numbers, and we really do have knowledge of them. An initial hint that numbers are magnitudes comes from their algebra. The natural numbers, the positive real numbers and the ordinal numbers each have an associative operation of addition which defines a linear order. I call a system with that precise algebraic structure a positive semigroup. We find positive semigroups cropping up not only in mathematics but in the physical sciences as well. The fundamental physical magnitudes have this same algebraic structure: for example, mass is a positive semigroup because addition of masses is associative, and the addition defines a linear order. The same is true for length, area, volume, angle size, time and electric charge: these and other physical magnitudes all have the structure of a positive semigroup. This book proposes a theory of number that accounts for the common algebraic structure of numbers and the physical magnitudes.

1  Mathematical Knowledge According to Hume, mathematical truths are ‘discoverable by the mere operation of thought’. Whatever the empirical facts turn out to be, Hume tells us, ‘the truths demonstrated by Euclid would for ever retain their certainty and evidence’ (2007: Part 1, §20). But many philosophers have disputed this. According to

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Knowledge and the Philosophy of Number

Quine (1951), current total science is only a working hypothesis, and its parts are all equally subject to revision in the light of future experience. Mathematics is justified only qua part of total science, so mathematics too is revisable, and thus it is neither certain nor evident. Quine does, at least, advise us to believe mathematics for the time being, since he says we should believe our currently best theory. But Field (1980) says, ‘We should withhold belief: arithmetic is not an inseparable part of total science, so it derives no support from the empirical success of physical science.’ According to Field, for all we know, numbers are just a useful fiction. This book argues that views such as those of Quine and Field rest on a category mistake. The phrase ‘category mistake’ was introduced by Gilbert Ryle to label the error of taking a term to refer to an entity of a certain ‘logical type or category’, when in fact the term refers to an entity of another logical type. Ryle’s famous example was the confused tourist, who has visited all the Oxford colleges, and then asks directions to Oxford University, which he wishes to visit also: the category mistake is taking the term ‘Oxford University’ to refer to an entity of the same logical type as the colleges (1963: 17). The concept of a category originates not with Ryle but with Aristotle, who proposed a metaphysical taxonomy to divide all the ‘things that are’ into their highest natural kinds. Aristotle begins his taxonomy with the distinction between subject and predicate: a thing that can be predicated of a subject he calls predicable (1941a: 1a20). The items that can be the subject of a judgement are then divided, by Aristotle, into the categories of individual and quantity. Examples of quantity are a plurality, such as some leaves, a continuum, such as a stretch of time or space, or a series, such as Plato and Socrates in that order.1 Quine and Field both take numbers to be objects, i.e. impredicables; I will argue that this is a category mistake, for in fact numbers are properties and so belong in the category of the predicable. Frege too takes numbers not to be predicable. He gives the following argument that numbers are not properties: In language, numbers most commonly appear in adjectival form and attributive construction in the same sort of way as the words ‘hard’ or ‘heavy’ or ‘red’, which have for their meanings properties of external things. It is natural to ask whether we must think of the individual numbers too as such properties, and whether, accordingly, the concept of Number can be classed along with that, say, of colour. Is it not in totally different senses that we speak of a tree as having 1000 leaves, and again as having green leaves? The green colour we ascribe to each single leaf, but not the number 1000. If we call all the leaves taken together its foliage, then the foliage too is green, but it is not 1000. (1968: 27e–28e)

Introduction

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If numbers are not properties, they can never be predicated in a judgement, which is why Frege concludes they can only be objects. But his reasoning rests on the category mistake of taking the term ‘the foliage’ to refer to an individual, when in fact it refers to something in the category of quantity, namely the plurality of 1000 leaves. The correct observation that the number 1000 is not a property of an individual does not exclude the possibility that numbers are properties of pluralities. But Frege jumped too quickly to the conclusion that a number is an object, not a property.

2  The Sceptical Consequence Taking numbers to be objects ensnares us in the sceptical consequence that we have no knowledge of number. For if numbers are objects, metaphysics cannot offer us any conception of their nature which makes epistemological sense. We cannot suppose that numbers are physical objects: the suggestion that the number 1000 is located somewhere in space and time is absurd. Nor can numbers be mental objects constructed by our own minds: there exists at most a potential infinity of mental constructions, whereas the axioms of mathematics require an actual infinity of numbers. So if numbers are objects, they are neither physical nor mental. Quine and Field conclude that numbers are ‘abstract’ objects, if they exist at all. It’s unclear what an abstract object is supposed to be, except that it’s not physical and not mental either. Perhaps ‘abstract’ objects are timeless objects that are not spatiotemporal. If so, abstract objects would be causally isolated from the spatiotemporal realm, so there would be no information channel connecting us and them, so our beliefs about numbers would not be knowledge. Frege attempted to do without an information channel by proving the axioms of arithmetic by logic alone, but his proof foundered on Russell’s paradox of the set of all sets that are not elements of themselves. Gödel (1964: 483–4) put forward the theory that the mind has a kind of intellectual perception of the abstract world, analogous to our perception by the senses of the physical world, but this suggestion has seemed too fanciful to be taken seriously. The theory that numbers are objects, therefore, traps us in the sceptical conclusion that we lack knowledge of number. But according to Moore, every valid argument for scepticism is a reductio ad absurdum proof that at least one of its premisses is false. I take the false premiss here to be the assumption that numbers are objects. Frege’s argument shows that numbers are not properties of objects, but it does not show that numbers are not properties of Aristotelian

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quantities. I will be suggesting that that is exactly what numbers actually are. The natural numbers are properties of pluralities, the positive real numbers are properties of continua, and the ordinal numbers are properties of series. The reason there are these three different kinds of number is because there are these three different categories of quantity.

3  The Logic of Quantity The ‘pure’ predicate calculus has the quantifiers and the propositional connectives as its only logical constants. There are laws of logic about identity: for example, there is the law that everything is identical to itself, so we need the identity predicate ‘=’ as an additional logical constant. On an intended interpretation, the quantifiers of the predicate calculus range over only individuals. A more general logic should be able to deal with inferences about any kind of item that can be the subject of a judgement, so its quantifiers need to range over all the Aristotelian categories, including quantities. Aristotle gives the following definition: ‘“Quantity” means that which is divisible into two or more constituent parts’ (1020a7, 6a27). So a more general logic needs a predicate to express the relation of part to whole. There are laws of logic about part and whole: for example, there is the law that a part of a part of something is a part of it. So we shall need a new logical constant for the part relation, just as we did for the identity relation.

4 Equality According to Aristotle, ‘The most distinctive mark of quantity is that equality and inequality are predicated of it’ (1941: 6a27). Aristotle was aware of the remarkable axioms, already known to the mathematicians of his day, that connect the equality of quantities with their structure of part and whole. These axioms were subsequently systematized by Euclid (Heath 1956: Vol.1, 155) as his ‘Common Notions’: 1. Quantities which are equal to the same quantity are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Quantities which coincide with one another are equal to one another. 5. The whole is greater than the part.

Introduction

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Equality comes in many varieties, each with its own standard of equality. In mathematics, pluralities are numerically equal if they tally, continua are geometrically equal if they coincide when brought into superposition and series are ordinally equal if they are in order-preserving one-one correspondence. In the physical sciences, two items are gravitationally equal if they are in equilibrium in the beam balance, electrostatically equal if they are in equilibrium in the torsion balance, and so on. Each variety of equality has its own ‘standard of equality’, a test procedure that determines whether items are equal. Items will be disposed to give the same result when tested if and only if they are equal. The philosophical theory of sparse property realism says that every disposition has some property as its basis: thus if two items have the same disposition, their shared disposition can have a shared basis. For example, we recognize sameness of mass as the basis of the disposition of gravitationally equal items to balance. Mass is a family of properties (i.e. some properties) such that gravitationally equal things instantiate the same property of the family. As with gravitational equality, so with other standards of equality: recognition of each kind of equality allows us to recognize a corresponding family of properties.

5  The Homomorphism Theorem Any relation that is an equality in the sense of the Common Notions is an equivalence relation. It therefore partitions its domain into equivalence classes of equal quantities. Everything in a given equivalence class is equal to everything else in the same class, and the classes do not overlap. Each equivalence class is the extension of a disposition to respond the same way to the same test. So if quantities with the same disposition share the same basis property, each equivalence class is also the extension of a basis property. Call the basis property shared by the quantities in the same equivalence class a ‘magnitude’, and call the family of all the basis properties for the same equality relation the corresponding ‘system of magnitudes’. Since the equivalence classes are disjoint, the magnitudes of a given system have disjoint extensions and are, therefore, the determinates of a determinable. We arrive at the following: Magnitudes Thesis Whenever there is a standard of equality that satisfies the Common Notions, there is a corresponding system of magnitudes such that quantities are equal if and only if they instantiate the same magnitude of the system.

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Knowledge and the Philosophy of Number

Applied to the various categories of quantities, the Magnitudes Thesis tells us of the existence of various system of magnitudes. Pluralities are numerically equal if they tally, so the Magnitudes Thesis tells us that numerically equal pluralities instantiate a single magnitude from some system of magnitudes, which we recognize as the natural numbers. In the same way, we discover that geometrically equal lines have the same length and that ordinally equal series have the same ordinal number. In the physical sciences the Magnitudes Thesis tells us that gravitationally equal things have the same mass, and electrostatically equal things have the same charge. It is in this way that we become aware of the various kinds of magnitudes that underlie the various kinds of equality. We can equip a system of magnitudes with an algebra by defining an operation of addition by ‘addition of representatives’ as follows. Given magnitudes a and b of the same system, to find their sum a + b, choose a quantity x that instantiates a and a disjoint quantity y that instantiates b. (It will be shown in subsequent chapters that this is always possible for pluralities, continua and series.) Denote by ‘x&y’ the whole whose parts are x and y, in other words, the quantity that is the logical sum of x and y. Then the sum of the magnitudes a + b is defined to be the magnitude of the logical sum x&y. (The second Common Notion assures that this definition is independent of the choice x and y of representatives.) Because we define addition of magnitudes by logical addition of quantities, a structural connection arises between the algebra of the logic of quantities and the algebra of magnitudes. It is proved in Chapter 5 that addition of magnitudes so defined is associative and defines a linear order on the system of magnitudes. In other words, the system of magnitudes is a positive semigroup. The argument is completely general and applies to any species of equality: if the properties that are the bases of an equality are a system of magnitudes, then that system of magnitudes is a positive semigroup. The relation between a quantity and its magnitude projects onto the system of magnitudes an algebraic structure derived from the algebra of the logic of quantity. Because the projection happens in the same way for every equality that conforms to the Common Notions, the same sort of algebra crops up in every case: that is why natural number, length, ordinal number, mass and charge are all positive semigroups. The structure of the algebra of a system of magnitudes is a direct projection of the underlying algebra of the system of quantities they measure. From a knowledge of the respective systems of quantities we can derive the usual mathematical axioms for the natural numbers, real numbers and ordinal numbers, and we see why these axioms must be true.

Introduction

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That is the strategy of the present book, the plan of which is as follows. Chapter 1 reviews the arguments for property realism and criticizes nominalist attempts to do without properties. The argument for property realism is especially strong for magnitude properties, without which quantitative causal explanation would be impossible. Chapter 2 criticizes Frege’s attempt to dispense with properties in favour of his theory of ‘Concepts’. It is argued that Frege’s theory is unsatisfactory because it fails to tell us which are the natural classes. His theory also obliges us to work in second-order logic, a logic with many inconveniences and no expressive advantages over a first-order logic, the variables of which are permitted to range unrestrictedly over all the ‘things that are’, including properties and quantities. Chapter 3 endorses Aristotle’s statement in Categories that not only individuals but also quantities can be the logical subject of judgement. Reference to quantities and quantification over quantities is a prominent feature of natural language, and gives rise to the intuitively evident laws of the logic of quantity. The chapter highlights the striking resemblance between these laws and the algebra of a positive semigroup. Chapter 4 discusses Leśniewski’s mereology, which is often misinterpreted as an ontological theory about individuals. The chapter gives nine axioms of mereology each of which is a priori when interpreted as laws of the logic of  pluralities, and nine axioms which are each a priori when interpreted as laws of the logic of continua. Eight of the nine axioms are common to pluralities and continua, and the eight Common Axioms are proved to be deductively equivalent to the two familiar axiom systems of Tarski and of Simons. This mandates an interpretation of mereology as the pure a priori logic of quantity. Chapter 5 presents the Magnitudes Thesis that equal quantities have a common magnitude. The Homomorphism Theorem, which states that the equivalence classes of equal quantities inherit an algebraic structure from the mereology of the quantities, is proved. Given the Magnitudes Thesis, this theorem entails that a system of magnitudes is a positive semigroup. Chapter 6 defines numerical equality of pluralities by means of the tallying algorithm. It is shown that pluralities that tally are equal in the sense of the Common Notions, so we deduce from the Magnitudes Thesis that a natural number is the common magnitude of pluralities that tally. It follows by the Homomorphism Theorem that the natural numbers are a positive semigroup. Every plurality has a mereologically least part, so the numbers are a positive semigroup with a least member. From this all the axioms of Peano Arithmetic can be deduced, except the axioms for multiplication.

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Chapter 7 notes that Peano Arithmetic without multiplication is a decidable theory, but full Peano Arithmetic, including the axioms for multiplication, is an undecidable theory. Thus multiplication introduces something new. Multiplication is defined by a second algorithm – Euclid’s axiom of ‘repeated addition’. By reasoning a priori about this algorithm, we deduce the axioms for multiplication of natural numbers, thus completing the deduction of all the axioms of Peano Arithmetic. Chapter 8 develops the theory of the real numbers out of the theory of multiplication of a quantity by a natural number. A third Euclidean algorithm, this time to find the greatest common measure by repeated subtraction, leads to the positive rational numbers, and hence to Eudoxus’s theory of ratio and proportion. It is proved that the ratios are a model, unique up to isomorphism, of the usual axioms for the positive real numbers. The sole new assumption needed for this proof is that there does exist a system of magnitudes that is complete, in the sense that it contains representatives of every ratio. Chapter 9 says that geometry underwrites the completeness assumption, for Euclidean geometry entails that the lengths are a system of magnitudes that contains representatives of every ratio. Therefore, we know that the axioms for the positive real numbers have a model. The real numbers thus find an adequate a priori foundation in Euclidean geometry. Against this it is commonly objected that physics has discovered that physical space is non-Euclidean. The objection is misconceived, for ‘Euclidean space’ means not a physical thing but a shape, whose properties can indeed be discovered a priori, regardless of the actual shape of physical space. Chapter 10 says an ordinal number is the magnitude of a series. A series has a non-commutative mereology, and every series is well-ordered, so the Homomorphism Theorem tells us that ordinals are a well-ordered positive semigroup with non-commutative addition. The existence of the infinite series of natural numbers entails the existence of an infinite ordinal, and this together with the reasoning of the Burali-Forti paradox suffices to prove the axioms for the constructive ordinals. The upshot is that we need to rely neither on empirical evidence nor on the dubious authority of set theory for our knowledge of number. The laws of mereology and Euclid’s Common Notions are evident a priori, and the Magnitudes Thesis is the product of philosophical reflection, so by reason alone we can arrive at knowledge of number.

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Properties

The thesis of this book is that numbers are magnitudes. A system of magnitudes is by definition a plurality of properties that are determinates of a determinable and that have a characteristic algebra. The different types of number are different systems of magnitudes. Magnitudes are properties, and the present chapter makes the case for property realism – the metaphysical theory that properties are entities that really exist. Section 1.1 recalls Aristotle’s doctrine that in a judgement there is a part of reality that is the subject, and another part of reality which is predicated. Section 1.2 lists the principal theories of predication, including property realism and various alternatives. Section 1.3 discusses Davidson’s theory that there are no predicables and that no entities at all need be mentioned in the semantics of predicates. His theory is criticized on the grounds that it entangles us in the semantic paradoxes. Section 1.4 makes the case for property realism and criticizes two versions of nominalism on the grounds that they cannot explain the difference between a natural class and a merely miscellaneous collection. Section 1.5 argues that there are many different types of properties. Section 1.6 introduces the type of properties that are magnitudes. Section 1.7 says that extensive magnitudes have a common algebraic structure and are needed in the theory of ratio and proportion, which is indispensable in the physical sciences. Section 1.8 says that many kinds of extensive magnitudes are known to exist and that numbers are extensive magnitudes too.

1.1 Predicables If numbers are a part of reality, and numbers are properties, then properties are a part of reality. But does it even make sense to speak of parts of reality? Parmenides thought not – he taught that reality is One: [The One] is now, all at once, one and continuous. … Nor is it divisible, since it is all alike; nor is there any more or less of it in one place which might prevent

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Knowledge and the Philosophy of Number it from holding together, but all is full of what is. (Frag. B 8.5–6, 8.22–24. In Cornford 1939: 35–9)

But the doctrine that there is only one thing is absurd. Reality is a many, namely the totality of what Aristotle calls ‘things themselves’(1941a: 1a20). Aristotle sees classification as a central task of science. Each special science taxonomizes the things in the part of reality that fall within its special province; Aristotle thought that metaphysics, the most general of all sciences, must taxonomize at the very highest level of generality. He first divides all the ‘things that are’ into two categories: the predicables ‘such as man or horse’ and the impredicables, ‘such as the individual man or horse’ (1941a: 1b4). Definition. An entity is called a predicable if it can be predicated of a subject. Aristotle’s doctrine is that reality divides into two highest kinds: the category of things that can be predicated and the category of things that cannot be predicated.

1.2  Different Accounts of Predication Property realism agrees with Aristotle that the category of predicables is needed in the account of what it is for something to fall under or satisfy a predicate. For example, according to property realism, the predicable wisdom, ‘wisdom itself ’ as Plato would say, is a being of a fundamentally different kind from the individual beings such as Socrates who are instances of wisdom. There are three versions of property realism: (i) the extremely abundant theory says that for every class, there is a property which ‘unites the class’ – every member of the class instantiates the property (Lewis 1983); (ii) the moderately abundant theory says that for every possible predicate there exists a corresponding property: Every meaningful predicate stands for a property or relation, and it is sufficient for the actual existence of a property or relation that there could be a predicate with appropriate satisfaction conditions. (Hale 2013: 133)

(iii) the sparse theory says that properties are in much shorter supply, for it is only the natural classes that have members united by a common property, so only those predicates express a property whose extension is a natural class. All realist theories agree that if ‘wise’ predicates the property wisdom, then y satisfies

Properties

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‘x is wise’ if y instantiates wisdom. Sparse property realism is the theory I shall be advocating. Alternative theories of predication may be classified as follows.1 1. Davidson’s theory is a nominalist theory that denies the existence of properties. According to Davidson, the theory of satisfaction does not need to mention any entities whatsoever: the semantics of a predicate is given simply by stating the satisfaction condition of the predicate. When we have given the satisfaction condition for each predicate, we have given the whole semantics of predication for the language, without needing to mention any non-linguistic entities. 2. The extension theory is another nominalist theory. It says that there is indeed a specific part of reality associated with each predicate, but this is just its extension, in other words the plurality (alternatively, the set) of things that fall under the predicate. On the extension theory, y satisfies ‘x is wise’ if y is an element of the set of wise persons: there is no need to invoke a separate category of predicables. To deal with counterfactual situations, the extension theory may invoke the predicate’s intension, the supposed function that maps each possible world to the extension the predicate would have were that world actual. 3. The paradigms theory is a nominalist theory that says the specific part of reality associated with a predicate is its collection of paradigms, which can be just a few archetypal instances of the predicate, not its whole extension. On this account, y satisfies ‘x is wise’ if y relevantly resembles paradigm wise people, such as Socrates. 4. On Frege’s theory, the entities that can be predicated are categorially different from other things. Frege calls them Concepts: they are not properties, but functions from objects to the truth values – the True and the False. On Frege’s theory, y satisfies ‘x is wise’ if the value of the Concept presented by ‘x is wise’ is the True for the argument y.

1.3  Criticism of Davidson According to Davidson, there is no need to postulate predicable entities, for a semantic theory that conforms to Tarski’s ‘Convention T’ can be ‘enough for an interpreter to go on’ (1984: xiv). ‘Convention T’ is the requirement that a semantic theory for a language ℒ must entail for each sentence s a theorem of the following form: s is true in ℒ if and only if p.

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where p is the sentence of the metalanguage that translates s. If the semantic theory of ℒ is given in ℒ itself, Convention T reduces to the disquotational requirement that the theory prove every instance of the following schema: ‘s’ is true in ℒ if and only if s. Such a semantic theory should be finitely axiomatizable if it is to serve its interpretative purpose. A theory might typically include axioms of three types: axioms of the first type say for each referring term in ℒ what its referent is, axioms of the second type say for each predicate what its satisfaction condition is and axioms of the third type are compositional. Such a semantic theory for English, if given in English, could include the following axioms: ( 1) ‘Snow’ refers in English to snow. (2) ∀y (y satisfies ‘x is white’ in English ↔ y is white.) (3) A subject–predicate sentence is true in English if and only if the referent of the subject term satisfies the predicate. These three axioms allow the theory to prove the following familiar theorem: ‘Snow is white’ is true in English if and only if snow is white. Axioms of the first type look outside language: axiom (1) is ontologically committed to the existence of snow. But axioms of the second type do not commit us to the existence of any entity outside language. According to Davidson, the satisfaction relation is fully characterized by the collection of all disquotational axioms like (2) – one axiom for each predicate of the language: ‘The recursive definition of satisfaction must run through every primitive predicate in turn’ (1984: 47). Beyond the totality of these axioms there is nothing more that can be said, or that needs to be said, about what satisfaction is. But the collection of axioms like (2) cannot be the whole explanation of what it is for a thing to satisfy a predicate. According to the disquotational theory, for each one-place predicate A(x) of English, the satisfaction relation must meet the following condition: (4) ∀y (y satisfies ‘A(x)’ in ℒ ↔ A(y)) For example, Socrates satisfies ‘x is wise’ in English if and only if he is wise. But this account of satisfaction cannot be correct, for ‘x does not satisfy x’ is itself a one-place predicate of English, so substituting it for ‘A(x)’ in (4) yields: ∀y (y satisfies ‘x does not satisfy x’ ↔ y does not satisfy y)

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But this entails, absurdly, that ‘x does not satisfy x’ satisfies ‘x does not satisfy x’ ↔  ‘x does not satisfy x’ does not satisfy ‘x does not satisfy x’. This contradiction is Grelling’s Paradox, as expounded by McGee (1991: 31). The paradox is closely related to Tarski’s Theorem about the undefinability of truth, which may be stated as follows: A theory in a language ℒ is inconsistent with arithmetic if it entails all instances of the schema (T) ‘Tr(“ϕ”) ↔ ϕ’, where “ϕ” is a structurally descriptive name (or Gödel number) of the sentence ϕ of ℒ and Tr is a predicate whose extension on the intended interpretation is the set of true sentences of ℒ.2

McGee comments as follows: For some purposes, a sufficient response will be simply to insist that the theory of truth for a language must never be developed within the language itself, but rather within an essentially richer metalanguage. But for other purposes one would like to be able to develop a semantic theory of a language within that very language; most urgently one would like to be able to talk consistently in English about the semantics of English. (1992: 235)

McGee goes on to show that we cannot develop a satisfactory account of truth simply by taking schema (T) as our starting point and restricting it only just enough to restore consistency. He gives a proof that there are ‘a great many’ mutually incompatible maximal consistent sets of sentences satisfying schema (T), many of which assert things about truth that are obviously untrue. Without a more substantive theory to guide us as to which maximal consistent set of sentences to prefer, reliance on schema (T) cannot issue in a satisfactory theory of truth: some part of non-semantic reality must, therefore, be brought into the account of satisfaction. This refutes the following uniqueness claim by Davidson: Sentences like ‘“Snow is white” is true in English if and only if snow is white’ are trivially true. … [T]he totality of such sentences uniquely determines the extension of a truth predicate for English. (1984: xv)

1.4  Property Realism The extension of a predicate is the multitude of things that satisfy the predicate. According to some nominalists, the extension is the only part of reality that

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needs to be mentioned in the theory of predication. Other nominalists say we do not even need the whole of the extension: a few well-chosen paradigms of the predicate will suffice – a thing satisfies the predicate if it resembles the paradigms. But property realism says that the theory of predication must start with properties: the fundamental way for a thing to satisfy a predicate is for it to instantiate the property which the predicate expresses. There are two classical arguments for property realism: the semantic argument and the causal argument. The semantic argument says some predicates are instantiated by a great many things: for example, the extension of the predicate ‘wise’ is the multitude of all wise persons, past, present and future. Every English speaker uses the word ‘wise’ with exactly the same extension. How is this remarkable feat of semantic coordination achieved? The semantic argument concludes that speakers can use ‘wise’ with a common extension because of their common familiarity with the property wisdom – it is the property not the speakers that determines the extension. The semantic argument does not require that every predicate attributes a property. For example, consider the predicate ‘grue’, which by definition something instantiates if it is green and examined, or blue and unexamined. There is no need to postulate a property grue, for our shared grasp of the public meaning of ‘grue’ is adequately explained by our grasp of the defining properties green, blue and examined. Indeed, a second argument for realism, the causal argument, leads to the conclusion that very few predicates express properties. This argument starts from Plato’s observation that not every predicate is of equal scientific value. The good scientist, according to Plato, classifies in ways that respect natural boundaries, by ‘carving nature at the joints’ and bringing ‘a dispersed plurality under a single Form’ (1961: 265d). Plato’s doctrine is that some things are a natural class only if they fall ‘under a single Form’, in other words, if there is a single property they all share. One of Hume’s ‘Rules by which to judge of causes and effects’ is the following: The same cause always produces the same effect, and the same effect never arises but from the same cause. (Hume 1738: Book I, Part III, Section XV)

The doctrine ‘same cause, same effect’ is central in the philosophy of causation and requires the theory of natural classes. Events are ‘the same’ for purposes of causal explanation if they fall in the same natural class, and Hume’s ‘Rules’ are a first approximation to the process whereby science discovers what natural classes there are. The causal argument says that we must postulate properties to explain what distinguishes the natural classes from the rest: a class is natural only

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if its members have a property in common. Since there are innumerably many classes, of which only a few are natural classes, properties must be few if they are to explain naturalness. Thus the causal argument leads to sparse property realism, and rules out both extremely abundant and moderately abundant realism. The existence of laws of nature gives a further argument for property realism. Consider the law ‘All humans are mortal.’ According to the property realist, this law is about two properties, namely humanity and mortality, and the nomic connection between them. If we say with the nominalists that there are no such entities as humanity and mortality, we shall then be in a difficulty. The truth stated by the law is not about the extension, the humans who actually exist, all of whom are indeed mortal. For even if none of the world’s actual humans had ever been born, and a completely different population had been born in their place, it would still have been a truth that all humans are mortal. The law states a truth about properties, not a truth about individuals. The law that all humans are mortal supports counterfactuals: for example, the god Apollo is not a human being, but if he had been human, he would have been mortal. But the exceptionless general truth that all the coins in my pocket are bronze is not a law, for it does not support counterfactuals. This silver coin is not a coin in my pocket, but even if it had been a coin in my pocket it would not have been bronze. Thus not every predicate can feature in a law. How may we distinguish those that can from those that cannot? Property realism says that only those predicates can figure in laws that have a property as their semantic value: there is such a property as humanity, but there are no such properties as being grue or being a coin in my pocket. Property realism does better than competing nominalist analyses of causation. Suppose it is a law of nature that red rags annoy bulls. You wave a red rag at a passing bull and the bull is annoyed. The annoyance of the bull was caused by the rag’s being red: all would have been tranquillity had the rag been a different colour. The property realist says that the rag’s being red is constituted by the rag’s having the property red. The nominalist’s alternative analysis is that the rag’s being red is constituted by the rag’s being in the extension of ‘red’. But quantum entanglement aside, causation is a local matter. Suppose one of the many actually red things had ceased to be red, perhaps a certain postbox a thousand miles away had been painted green. Then the extension of ‘red’ would have been different, but the rag would still have annoyed the bull. That distant postbox is causally irrelevant, so replacing the property with its extension does not preserve the correctness of causal explanation.

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On the paradigms theory, x is red if and only if x resembles a paradigm of redness, for example that red postbox. But the bull’s annoyance was caused by the rag’s being red; the rag’s resembling that postbox is causally irrelevant. If the postbox had been painted green, the rag would not have resembled it, but it would still have annoyed the bull. The paradigms are irrelevant to the explanation of the bull’s annoyance. Predicates with the same extension can differ in their explanatory power. The property realist says this is because coextensive predicates can express different properties. For example, the cordates (animals with a heart) are all and only the renates (animals with a kidney) (Quine 1986: 9–10). The predicates ‘cordate’ and ‘renate’ have the same extension, but they have different explanatory power. The circulation of the blood of a certain animal is explained by its being a cordate: its being a renate is irrelevant. The realist argues that this shows that cordate and renate invoke different properties.

1.5  Kinds of Property Individuals can be arranged in a structure of genus and species, so if Aristotle’s taxonomic project is on the right track, we should expect his predicables also to have a structure of genus and species. So if the predicables are properties, we should expect there to be many genera of properties. And this is exactly what we do find. For example, there is the genus of sensory properties, the species of which include colour, musical tone, odour and flavour, and there is the genus of ethical properties, whose species include the virtues and the vices. True statements about the genus to which a property belongs pose problems for the nominalist. Consider the following truths: ( 1) Red is a colour. (2) Patience is a virtue. In sentence (1) the property red looks like the subject, and the property of being a colour looks like the predicate, so the logical form appears to assign the property red to its species. If nominalists say there are no properties, they need to give truth conditions for (1) that do not refer to properties. Jackson (1977) discusses the attempted nominalist paraphrase of (1) as: (1*) Everything red is coloured.

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This has the form ∀x  (red(x)  →  coloured(x)), so ‘red’ occur here only as a predicate, not as a name of a property. But Jackson observes that sentences that are to each other as (1) is to (1*) need not be paraphrases. For example: (1a) Everything red is extended. is not equivalent to (1a*) Red is an extension. Similarly (2) is not equivalent to: (2*) Everyone patient is virtuous. (2) is true, but (2*) is false: a person needs more than a single virtue to be virtuous. Because nominalists deny the existence of properties, they are unable to parse the truths that assign properties to their kinds.

1.6 Magnitudes Many properties are ‘determinates of a determinable’. That is to say, they comprise a plurality of mutually exclusive contraries – if something has one of the properties of the plurality, it cannot simultaneously have any of the others. Thus colour is a determinable, for all the colours are contraries. The same surface cannot be red and green, so red and green are determinates of the determinable colour. Similarly the same surface cannot be round and square, so round and square are determinates of the determinable shape. By contrast, flavour is not a determinable, for the same dish can be both sweet and sour. The intensity of pain is a determinable: the same pain cannot be both excruciating and just a twinge. But this determinable has additional algebraic structure, for some pains are worse than others, so pains have a linear order of intensity. The intensity of pain is an example of an important kind of property, the magnitudes. Definition. A plurality of properties is said to be a system of magnitudes if they are a determinable and the determinates are linearly ordered. Definition. A property is said to be a magnitude if it is a determinate of some system of magnitudes. Length, area, volume, mass and temporal duration are determinables that are systems of magnitudes: for example, any two different lengths are contraries, and

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the lengths have a linear order. Not every determinable is a system of magnitudes in the present sense: for example, colour is a determinable, but not a system of magnitudes, because the order of the colours in the colour wheel is cyclic, not linear. Kant distinguishes two sorts of magnitudes: the extensive and the intensive (Kant 1929: B202–8). A system of intensive magnitudes is one for which addition of its determinates does not make sense. For example, the intensities of pain are a system of intensive magnitudes. Suppose A has toothache and B has toothache too. Because painfulness is a magnitude, either the toothaches are equally painful, or A’s is worse than B’s or B’s is worse than A’s. Indeed, it is possible to have an ‘ordinal scale’ to report the intensity of pain, say on a scale of 1 to 10. But despite such use of numbers, there is no arithmetic of pain, for there is no way to add the intensity of two pains. A system of magnitudes is extensive if the determinables not only have an intrinsic order but are also capable of addition: indeed, their intrinsic order must arise out of their additive structure. The extensive magnitudes reflect the mereological ‘extent’ of a quantity – if you extend a quantity by adding a further quantity, thereby you add to its magnitude. Many species of extensive magnitudes have already been discovered by science, and more will no doubt continue to be discovered. A typical example of an extensive magnitude is mass: the different possible masses are determinates of a determinable, for a given quantity of matter has at most one mass. Mass is a system of magnitudes, because the masses have a definite natural order. And mass is an extensive magnitude because as you add more matter, you add to the mass. Other familiar examples of extensive magnitude are length, area, volume and temporal duration. We can add lengths to lengths, areas to areas, volumes to volumes, and so on. For all these systems of magnitudes we have the following axioms of addition. If a, b and c are variables ranging over the magnitudes in some one system of magnitudes, then: M1. Closure: given a and b, there always exist a magnitude d such that d = a + b. M2. Associativity: a + (b + c) = (a + b) + c. M3. Restricted subtraction: a ≠ b if and only if there exists a magnitude d such that either a = b + d or b = a + d. Axiom M1 says that any two magnitudes of the same system have a sum. Axiom M2 says that given any three magnitudes, summing the first with the sum of the second and third gives the same result as summing the sum of the first and

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second with the third. Axiom M3 says that magnitudes are different if and only if one of them is the sum of the other plus some third magnitude. This third magnitude can be thought of as the result of ‘subtracting’ the smaller magnitude from the larger. On any such system of magnitudes the addition operation is consistent with the linear order, for the order is defined in terms of the addition as follows: Definition. A magnitude a is said to be less than a magnitude b, written ‘a  q·b but ¬ p·c > q·d. Since p·a > q·b, there exists a magnitude e such that p·a = q·b + e. If p·c < q·d then by assumption p·a < q·b. Contradiction, so ¬ p·c < q·d. So p·c = q·d. Let N be any natural number. Then Np·c = Nq·d, so Np·c < (Nq + 1)·d, so by assumption Np·a < (Nq + 1)·b, so Np·a < Nq·b + b. But p·a = q·b + e, so Np·a = N(q·b + e) = Nq·b + N·e < Nq·b + b so N·e < b. But since N was any number, this contradicts the assumption that S is Archimedean. So m·a > n·b ↔ m·c > n·d. Since m·a < n·b ↔ m·c < n·d, it follows that m·a = n·b ↔ m·c = n·d. So a:b = c:d. Theorem 8.9 (After Scott 1974: 49–50). Let S' be a commutative system of magnitudes that satisfies the Density axiom and the Dedekind completeness axiom. Then S' is complete. Proof. Let S' be such a system of magnitudes. Then S' is Archimedean by 8.6. Let S be any Archimedean system of magnitudes. Let a and c be given in S and let c' be given in S'. We must show there is a fourth proportional a' in S' such that a':c' = a:c. By 8.8 it suffices to find some magnitude a' in S' such that for every m and n, m·a < n·c ↔ m·a' < n·c'. If a' is in S', then the following cases are possible. It may be that (i) for some m and n, m·a' < n·c' but ¬ m·a < n·c. In that case a' is too small to be the fourth proportional. Or it may be that (ii) for some m and n, m·a < n·c but ¬ m·a' < n·c', in which case a' is too big to be the fourth proportional. Or it may be that (iii) for every m and n, m·a < n·c ↔ m·a' < n·c', in which case a' is just right. Let x be the plurality of all those members b' of S' which are too small, that is x = [b': ∃m ∃n m·b' < n·c' ∧ ¬ m·a < n·c]. We prove that x has a least upper bound a' which is neither too big nor too small, and which is, therefore, the required fourth proportional. (1) We prove that x, the plurality of magnitudes that are too small, has an upper bound. Let b' ε x. Then b' is too small, so there are m and n such that m·b' < n·c' but ¬ m·a < n·c. Since S is Archimedean, we can choose a natural number p such that a < p·c Then n·c ≤ m·a ≤ mp·c, so n·c < mp·c, so n < mp. Since m·b' < n·c', m·b' < mp·c', so b' < p·c'. But b' was any element of x, so p·c' is an upper bound of x, so x is bounded above. (2) Let a' be the least upper bound of x. Suppose a' is too small. Then for some m and n, m·a' < n·c' but ¬ m·a < n·c. Then a' ε x. Since m·a' < n·c',

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there exists some d' such that m·a' + d' = n·c'. By 8.7, choose e' such that m·e' < d'. Let b' = a' + e'. Then m·b' = m·(a' + e') = m·a' + m·e', so since m·e' < d', m·b' < m·a' + d' = n·c', so m·b' < n·c'. But ¬ m·a < n·c, so b' is too small, so b' ε x. But a' is the least upper bound of x, so b' ≤ a'. But b' = a' + e'. Contradiction. So a' is not too small. So for every m and n, m·a' < n·c' → m·a < n·c. Suppose a' is too large. Then for some m and n, m·a < n·c but ¬ m·a' < n·c'. Since m·a < n·c, for some d, m·a + d = n·c. Choose a natural number q such that c < q·d. Then qm·a + q·d = qn·c. Since c < q·d, n·c < nq·d, so qm·a = qn·c - q·d < qn·c - c, so qm·a < (qn - 1)·c. Now n·c' ≤ m·a', so qn·c' ≤ qm·a'. So (qn - 1)·c' < qn⋅c' < qm·a'. So for some e', (qn - 1)·c' + e' = qm·a'. Choose d' such that qm·d' < e'. Then qm·d' < qm·a', so d' < a', so for some a'', a'' + d' = a'. So (qn - 1)·c' + qm·d' < qm·a'. But qm·a' = qm (a'' +d'), so (qn - 1)·c' + qm·d' < qm·(a'' + d'), so (qn - 1)·c' < qm·a''. Now let b' ε x. Then there are natural numbers M and N such that M·b' < N·c' but ¬ M·a < N·c. We have established that (qn - 1)·c' < qm·a'' and qm·a ≤ (qn - 1)·c. It follows that Nqm·a < N(qn - 1)·c ≤ M(qn - 1)·a, so Nqm ≤ M(qn - 1). So since M·b' < N·c', it follows that Mqm·b' < Nqm·c' < M(qn - 1)·c' < Mqm·a'', so Mqm·b' < Mqm·a'', so b' < a''. So a'' is an upper bound for x. But a'' < a', the least upper bound of x. Contradiction. So a' is not too large. So for every m and n, m·a < n·c → m·a' < n·c'. So a' is just right, because for every m and n, m·a < n·c ↔ m·a' < n·c'. So for every a, c in S and c' in S', there is some a' in S' such that a:c = a':c'. So S' is complete.

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The previous chapter showed that, given the assumption that there exists a complete system of magnitudes, the axioms for the real numbers can be obtained. The present chapter shows that a complete system of magnitudes does indeed exist. Geometrical congruence is an equality in the sense of the Common Notions, so we deduce by the Magnitudes Thesis that since equal lines are the same size, they are the same length. The Homomorphism Theorem then tells us that the lengths are a commutative system of magnitudes that obey the Density axiom. The Appendix proves they also obey the Dedekind completeness axiom, so by Theorem 8.9 of the previous chapter the lengths are a complete system of magnitudes. What is more, we can define multiplication and division of lengths by the elementary methods of Descartes, and hence prove that all the axioms for the real numbers hold for the lengths. The theory of the real numbers can thus be founded on arithmetical and geometrical intuition alone, without dependence on set theory. This account of the reals presupposes that geometrical intuition is sound. But few would now accept Kant’s doctrine that we know the properties of physical space a priori, for the General Theory of Relativity (GTR) shows that Euclid’s Parallels Postulate is not true of actual physical space-time. But geometry is the science of shape, not the science of the shape of space-time. So the GTR does not call in question Hume’s doctrine that geometry deals with a priori ‘relations of ideas’. The chapter ends with the suggestion that geometrical intuition may be the real motivation for the Power Set axiom of set theory. The chapter is divided as follows. Section 9.1 discusses congruence, the standard of geometric equality. Section 9.2 shows congruence is an equality in the sense of the Common Notions. Section 9.3 deduces by the Homomorphism Theorem that the lengths are a system of magnitudes: the Appendix proves that the lengths are complete and contain representatives of every possible ratio. Section 9.4 shows that multiplication and division of lengths can be defined

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by Descartes’s methods, so the lengths are a model of the axioms for the real numbers. Section 9.5 discusses the abundance of the real numbers and why ‘almost all’ are ‘transcendental’ like π and e. Section 9.6 considers whether the discovery that physical space is non-Euclidean stymies a geometric foundation for the real numbers. Section 9.7 argues that since it is a presupposition of the GTR that space-time is locally Euclidean, the GTR is not inconsistent with Euclid. Section 9.8 says that the Parallels Postulate is not about the shape of space, but about a relation between two properties, namely the property a line has if it is straight and the property a surface has if it is plane. Section 9.9 discusses the problems set theory has with the Continuum Hypothesis and suggests that the geometrical intuition of the continuum is mathematically indispensable.

9.1  Geometrical Equality The fourth of Euclid’s ‘Common Notions’ says that ‘quantities that coincide are equal’. The other Common Notions are axioms that give information about equality, but the fourth seems concerned more with defining equality than with giving information about it. Euclid is saying that the ‘standard of equality’ is coincidence. This standard cannot be applied immediately to quantities that are separated in space. Euclid, therefore, speaks of ‘applying’ one figure to another, by which he means moving it to superpose it on the other: the figures are congruent, and hence geometrically equal, if they coincide upon application. Euclid uses this proof technique in the congruence proof of Book I, Proposition 4. Some commentators have suggested that since congruence is a specifically geometrical standard of equality, the ‘coincidence’ spoken of in the fourth Common Notion is not a genuinely common standard of equality.1 But if movement of items is permitted in the definition of ‘coinciding’, then the fourth Common Notion can be interpreted as applying to pluralities also, since tallying can be implemented by an algorithm for bringing corresponding members of a plurality into coincidence, for example, by aligning units side by side. The theory of several species of magnitudes can be derived from the definition of geometrical equality: equal lines have the same length, equal angles have the same angular measure, equal surfaces have the same area, and equal solids have the same volume. Of these, the straight line and its length are by far the simplest case. A straight line is a linearly ordered continuum, which may have two endpoints, or one, or none at all. The unterminated infinite straight line is important for our theory, as is the infinite ‘half line’, a straight line with

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just one endpoint. But the Axiom of Archimedes fails for infinite lines: given a terminated line, it cannot ‘when multiplied’ exceed an infinite line, no matter how often it is ‘added to itself ’. Thus it is only terminated lines that can have a ratio. Definition. A straight line x is called finite if it is terminated, that is if there are points A and B such that x is the line AB. Definition. Finite lines x and y are said to be equal, written ‘x ≈ y’, if x and y coincide when brought into superposition. Definition. x is said to be greater than y, written ‘x ≻ y’, if y is equal to a proper part of x. Definition. x is said to be less than y, written ‘x ≺ y’, if x is equal to a proper part of y.

9.2  Congruence Is an Equality Two figures are congruent if one can be superposed on the other, i.e. if one can be moved so as to coincide with the other. The practical procedure of bringing shapes into coincidence is an algorithm that we understand how to carry out. In Chapter 6 we confirmed the Equality Axioms E1–E4 for the case of numerical equality by imagining carrying out the tallying algorithm. By imagining carrying out the ‘application’ algorithm, we can similarly confirm E1–E4 for the case of geometrical equality. E1. ‘Quantities equal to the same quantity are equal to one another.’ (Figure 9.1) x y z

Figure 9.1  Things equal to the same thing.

Justification: If x, y and z are lines, then if y equals x then y can be applied to x, so it can be moved to coincide with x. If z equals x then z can be applied to x, so it can be moved to also coincide with x, and hence to coincide with y. So y and z coincide upon superposition, so y equals z.

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E2. ‘If disjoint equals are added to equals, the wholes are equal.’ (Figure 9.2) x x’ y y’

Figure 9.2  Equals added to equals.

Justification: if x and y are lines, then x&y is the region that is their mereological sum. So if x' equals x then x' can be moved up to coincide with x. And if y' equals y then y' can be moved up to coincide with y. Since x and y are disjoint and x' and y' are disjoint, then since y' equals y, one can apply y' to y without disturbing the coincidence of x and x' just effected. But then this pair of applications has brought the sums x&y and x'&y' into coincidence, so x&y equals x'&y'. E3. ‘Two quantities are unequal if and only if either the first is equal to a proper part of the second or the second is equal to a proper part of the first.’ Justification: Let AB and CD be any two terminated lines. Apply CD to AB so that A and C coincide: then three mutually exclusive cases exhaust the possibilities (Figure 9.3). A

B

C

D

A

B

C

D

A

B

C

D

Case 1: AB = CD, B = D

Case 2: AB < CD, B between A and D

Case 3: CD < AB, D between A and B

Figure 9.3  Trichotomy – the three cases.

Either (case 1) B and D coincide, in which case AB equals CD; or (case 2) D lies to the right of B, in which case AB coincides with a part of CD, so AB is equal to a part of CD, so CD is greater than AB; or (case 3) B lies beyond D, in which case CD coincides with a part of AB, so CD is equal to a part of AB, so AB is greater than CD.

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E4. ‘The whole is not equal to the part.’ (Figure 9.4) A

B

C

Figure 9.4  Whole greater than part.

Justification: If B is a point within the line AC, then AB cannot be brought into coincidence with AC, so the whole AC is not equal to the proper part AB. That the whole is greater than the part is justified as follows. If B is a point within the line AC, then AB is a proper part of AC, and the part AB coincides with itself, so AB is equal to a proper part of AC, so the whole AC is greater than the part AB.

9.3  The Lengths Are a Complete System of Magnitudes The four axioms of equality E1–E4 are a priori laws when the equality is congruence. It was argued in section 5.8 that a class of items all of which are equal in the sense of the axioms of equality is a natural class. The Magnitudes Thesis says that a magnitude is the kind of property that unites a natural class of equal items: the magnitude property that geometrically equal lines have in common is their length. Thus there will be a one-one correspondence between the equivalence classes of congruent lines and the lengths – each equivalence class is the extension of exactly one length, and each length has exactly one equivalence class as its extension. Definition. A property is called a length if it is the magnitude which unites a virtual class of congruent finite lines. Notation. ‘|x|’ for the length of the finite line x. From our considerations about natural classes and property realism in section 5.8, we arrived at the following two generic Magnitude Axioms: Equality. Quantities are equal if and only if they have the same magnitude. Representatives. Every magnitude has some quantity as a representative. When we applied these generic axioms in Chapter 6 to pluralities, with tallying as the equality relation, we arrived at the Axioms of Number NN1 and NN2. We can now apply the generic axioms to finite lines, with congruence as the equality relation, to arrive at a corresponding pair of Axioms of Length: L1. |x| = |y| ↔ x ≈ y

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(The length of x is identical with the length of y if and only if x and y are equal.) L2. ∀a ∃x a = |x| (Every length has a line as a representative.) The Appendix deduces by the Homomorphism Theorem 5.9 that the lengths are a positive semigroup. Let L be this semigroup. By Theorem 5.10 it follows that L is commutative and satisfies the Density axiom. It is proved in the Appendix to this chapter that L is a complete system of magnitudes. The proof relies on Aristotle’s characterization of continuous quantity: A line is a continuous quantity, for it is possible to find a common boundary at which its parts join. In the case of the line, this common boundary is the point. (1941a: 5a1-2)

Aristotle’s insight into the nature of continuity was rediscovered by Dedekind: The problem is to indicate a precise characteristic of continuity that can serve as the basis for valid deductions. For a long time I pondered over this in vain, but finally I found what I was seeking. This discovery will, perhaps, be differently estimated by different people. The majority may find its substance very commonplace. It consists of the following. In the preceding section attention was called to the fact that every point p of the straight-line produces a separation of the same into two portions such that every point of one portion lies to the left of every point of the other. I find the essence of continuity in the converse, i.e., in the following principle. ‘If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.’ As already said I think I shall not err in assuming that everyone will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed. To this I may say that I am glad if everyone finds the above principle so obvious and so in harmony with his own ideas of a line. For I am utterly unable to adduce any proof of its correctness, nor has anyone the power. The assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we find continuity in the line. (Dedekind 1901: 11–12)

I think we must agree with Dedekind that his continuity axiom is indeed a primitively evident first principle. Theorem 9.5 of the Appendix uses Dedekind’s

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continuity axiom to prove that the lengths satisfy the Dedekind completeness axiom, from which it follows by Theorem 8.9 that L is a complete system of magnitudes, in the sense of Chapter 8.

9.4  Multiplication and Division of Lengths Can we just go ahead and identify the positive real numbers with this system L of the lengths? One reason to hesitate is that the positive reals are a semifield with multiplication and division, whereas multiplication of magnitudes is not in general well-defined. As we noted in Chapter 8, it makes no sense to multiply a volume by a volume, so how can it make sense to multiply a length by a length? A second difficulty is that the field of real numbers has a natural unit, the real number 1, which is the multiplicative identity. But there seems to be nothing to pick out a natural unit when we are dealing with continuous magnitudes: there is no natural unit of length. However, Descartes long ago showed how to overcome these difficulties. The unit element of a field is usually defined to be that element e such that given any element x, e.x = x.e = x. Thus we are accustomed to define the unit in terms of multiplication. But instead we could reverse the usual order of definition, and define multiplication in terms of the unit e. We can arbitrarily choose some particular magnitude e as the unit, then relative to e we use proportionality to define an operation of multiplication: the product of a and b is the unique ‘fourth proportional’ c, such that e:a = b:c. Since the fourth proportional always exists in the complete system L we can indeed extend this positive semigroup of lengths to a positive semifield. This approach to multiplication was adopted by Descartes to define multiplication of lines by lines. He chose one line arbitrarily to be the ‘unity’ or multiplicative identity, and then defined the product by the fourth proportional. Just as arithmetic consists of only four or five operations, namely addition, subtraction, multiplication and division … so in geometry to find required lines it is merely necessary to add or subtract other lines; or else taking one line which I shall call unity in order to relate it as closely as possible to numbers, and having given two other lines, to find a fourth proportional which shall be to one of the given lines as the other is to unity (which is the same as multiplication); or again to find a fourth line which is to one of the given lines as unity is to the other (which is the same as division) … I shall not hesitate to introduce these arithmetical terms into geometry for the sake of greater clearness (Figure 9.5).

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D

A

B

Figure 9.5  Multiplication of lines. For example, let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA. Then BE is the product of BD and BC. If it be required to divide BE by BD, I join E and D, and draw AC parallel to DE: then BC is the result of the division.2

Descartes’s method of multiplying and dividing lines will allow us to define multiplication and division of lengths by representatives. Thus to find the product of lengths a and b, choose an arbitrary length e as the unit, draw AB such that |AB| = e. On the line BA extended choose D such that |BD| = a. Choose a point C not in BD such that |BC| = b. Join AC. Construct E on BC extended such that DE and AC are parallel. Then c = |BE| is the required product of the lengths a and b, because BA:BD = BC:BE, so e:a = b:c. Descartes was able to use the Parallels Postulate as a handy practical way of computing products of real numbers. But we don’t need the Parallels Postulate to prove that multiplication in L is always well-defined. According to Theorem 8.9 of the previous chapter, any commutative system of magnitudes that satisfies the Density axiom and the Dedekind completeness axiom is complete. Theorems 9.4 and 9.5 of the Appendix tell us that L is a complete system of magnitudes. By definition of a complete system of magnitudes, given any two magnitudes a and c in an Archimedean system of magnitudes S, and any magnitude c' in a complete system S', there is a fourth proportional a' in the complete system such that a:c = a':c'. So here let S = S' = L. Then since L is complete, it is Archimedean (Lemma 8.6). So by Theorem 8.8, given any three lengths e, a and b of L there always exists a fourth length c such that a:e =c:b. So now we can define multiplication and division in L by Descartes’s technique. Choose arbitrarily some length to be the unit length e: for example, we could define e to be the current length of the standard metre in Paris. Write ‘1’ for the selected unit length, and define a.b as the fourth proportional c of 1, a and b. Definition. A length c is called the product of a and b if c is to b as a is to 1.

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For the inverse of a, we have that 1 is to a as the inverse of a is to 1. So 1:a = a-1:1, so a:1 = 1:a-1, so we are assured of the existence of a-1 provided it is guaranteed that for any length a there always exists a length b such that a:1 = 1:b. But this is guaranteed by the completeness of L. So we can now routinely verify that L satisfies all the axioms for a complete positive semifield, and hence that it satisfies the axioms for the positive real numbers. Any other such semifield is not only isomorphic to L but the isomorphism is unique (Scott 1974: 35). We thus have a categorical axiom system for the real numbers: if we know by philosophical reasoning that the Magnitudes Thesis is true, then we know that the lengths are a system of magnitudes that really exist.

9.5  Transcendental Real Numbers The proof that the lengths are a complete system of magnitudes assures us that our axioms for the real numbers are true. But it leaves important questions unsettled. We know that the side and diagonal of a square are incommensurable, so since the real numbers will include every ratio, the positive semifield L must include not only the whole positive semifield of the positive rationals Q+, but some extension of them. But there are a great many positive semifields that extend the positive rationals. For example, the constructible numbers are the rational numbers plus all the real numbers of the form m + n√2 that can be constructed as Descartes does with straight edge and compass as instruments. Descartes was also willing to allow the use of more complex instruments based on conic sections. This gives a great many more numbers, but takes us no further than the algebraic numbers, i.e. the numbers that are solutions of polynomial equations of the form anxn + an1xn-1 + … + a1x + a0 = 0. But even the algebraic numbers do not begin to exhaust all the real numbers there are. The problem of squaring the circle is the problem of finding √π, and this cannot be solved by Descartes’s methods of constructing ingenious mechanical instruments, because √π is not an algebraic number. Numbers like √π, which are not algebraic, are called ‘transcendental’. It turns out that nearly all real numbers are transcendental. But how many real numbers are there altogether? Eudoxus has given us the way to find out. If we wish to know the ratio of real numbers x and y, we imagine asking for every pair of natural numbers m and n whether m.x is less than n.y. This infinite sequence of questions generates an infinite sequence of Yes–No answers. If we code the answer ‘Yes’ by 1 and the answer ‘No’ by 0, then each ratio x:y will generate an infinite sequence {vp} of 1s and 0s. Then working within set

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theory we could regard {vp} as an infinite binimal fraction for the ratio x:y. (In base 2 arithmetic, a ‘binimal fraction’ is the binary equivalent of a decimal fraction in base 10.) Corresponding to each binimal will be the set of natural numbers {p: vp = 1}, which is an infinite subset of the set of natural numbers. Thus every ratio is paired with an infinite subset of the natural numbers, and every subset of the natural numbers is paired with a ratio: so the ratios are in one-one correspondence with the set of sets of natural numbers. We arrive at Cantor’s epoch-making conclusion that the real numbers are equinumerous with the power set of the natural numbers. Cantor’s Theorem says that no set is in one-one correspondence with its own power set. An infinite set is uncountable if it is not in one-one correspondence with the natural numbers. So the ratios are uncountable, and, therefore, the real numbers are uncountable. Since the rationals are countable, only countably many reals are rational: so ‘nearly all’ real numbers are not rationals. The constructible and algebraic numbers are countable. Therefore, ‘almost all’ real numbers are not algebraic but transcendental.

9.6  Doubts about Euclidean Geometry The axioms for the real numbers can be proved from the axioms of geometry, but the axioms of geometry are no longer universally accepted as a priori truths. The discovery that non-Euclidean geometry is not the true geometry of the physical world has led some to conclude that geometry is an empirical science which discovers contingent truths about physical space. A common contemporary suggestion is that the reals should, therefore, be founded in set theory, rather than Euclidean geometry. A model for the axioms of the real numbers is standardly obtained by identifying the real numbers with Dedekind cuts of the rationals (Dedekind 1901: 12–13). A cut is the earlier half of any partition of the rationals into two sets, the elements of the first of which precede every element of the second. We can add and multiply cuts by appropriate addition and multiplication of representatives. It can be shown that the system of cuts satisfies the Dedekind completeness axiom, so they have the least upper bound property. Thus the cuts with the defined operations are a complete dense commutative positive semigroup, so they provide representatives of every ratio in the sense of Eudoxus. So if we were leery of overreliance on Euclidean geometry we could propose instead to identify the set of real numbers with the set of Dedekind cuts, supposing such a set exists. With the real numbers available from set theory, any desired n-dimensional geometry can be realized as a set of n-tuples of reals, on which is defined a

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real-valued distance function, the metric. We regard the n-tuples of reals as coordinates, and by varying the metric we vary the geometry accordingly; on the empiricist view, the true geometry is the one whose metric best fits actual physical measurements. For three-dimensional Euclidean space, the metric is given everywhere by Pythagoras Theorem: ds2 = dx2 + dy2 + dz2. However, the GTR says that the true metric of space-time takes a different form, and varies locally from place to place. According to Wheeler, space-time is a dynamic physical entity: ‘Spacetime tells matter how to move; matter tells spacetime how to curve’ (2000: 235). On cosmic scales space-time is not correctly described by Euclidean geometry, so some conclude that Euclidean geometry is not true, hence not known, and therefore not known a priori. But we find traces of Euclidean geometry alive and kicking inside GTR, for the very topology of the space-time presupposed by GTR is inherited from the topology of Euclidean space. This can be illustrated by considering the Euler characteristic of polyhedra, which is a topological invariant. According to Book XIII of the Elements, the five Platonic solids – the tetrahedron, cube, octahedron, dodecahedron and isocahedron – are the only regular solids in Euclidean space. That is because the Parallels Postulate forces the sum of the angles of a triangle in Euclidean space to be 180 degrees. The Euler characteristic of a polyhedron is the number F + V – E, where F is the number of faces of the polyhedron, V is the number of vertices and E is the number of edges.3 In Euclidean space F + V – E = 2 for every regular solid. So if we take something that has the shape of a cube in the local Euclidean space of our home planet, and bend and stretch it a bit, it will be misshapen and no longer be a cube, but it will still be a polyhedron, and its Euler characteristic F + V – E will still be 2. If instead we take the cube to a region near a moderately massive star where the local space-time has a moderate curvature that similarly bends and stretches the cube, its Euler characteristic will be unchanged. The thought experiment suggests that although the local geometry near the star is different from Euclidean space, the topology is the same. The reason it is the same is because the non-Euclidean geometry of GTR is a construction out of Euclidean geometry, as the next section shows.

9.7  Euclid Presupposed in Non-Euclidean Geometry What mathematicians call ‘Euclidean space’ is in one dimension an infinite line that is straight, in two dimensions an infinite surface that is plane, and in three or more dimensions an infinite solid that is ‘flat’. These so-called spaces are better

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regarded as shapes, and each is the simplest of its dimension. Now as well as the straight line, there are infinitely many other one-dimensional shapes. How are we to study these other shapes? The differential calculus is the great mathematical discovery that allows the systematic description of complex shapes in terms of simpler shapes. The key observation is that any ‘smooth’ curve has at each of its points a tangent line, which has the same direction as the curve itself has at that point. So such a curve is fully described by giving the tangents at each of its points. If we draw enough of the tangents we see that they form an outline of the curve itself. Thus the discovery is this: we can describe the shape of an arbitrary smooth curve in terms of just one particularly simple shape, the straight line. We can adopt the same idea with surfaces. Just as a smooth curve can be described by giving at each point the Euclidean straight line that is tangent to it, a smooth surface can be described by giving its Euclidean tangent plane at each point. Just as the tangent lines twist and turn in the ambient space to follow the space curve, so the tangent planes in the ambient space twist and turn to follow the surface. The twisting and turning of the tangent plane in the ambient space describes the shape of the smooth surface (Figure 9.6).

Figure 9.6  Tangent plane.

The differential calculus can cope with surfaces as well as curves. It teaches us how to calculate the ‘differential’ of the function that describes the surface. The differential ‘knows’ all the tangent planes and thus embodies a description

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of the shape of the surface. But Gauss taught us how to calculate the shape as an intrinsic feature of a space, without reference to any ambient Euclidean space and the tangent planes that live within it. Indeed, there are three-dimensional shapes which can never be found in Euclidean three-space, but whose shape can be adequately described by the methods of Gauss (O’Neill 1966: 316). Does this mean that the mathematical description of the surface has broken free of its origins in Euclidean geometry, which can now be discarded? But we are still not rid of Euclidean space! For before we can obtain the shape by Gauss’s methods we need the real-number coordinates of the points of the surface. But just as we used the flat Euclidean plane as a tangent plane to help us describe how a surface was tilting and slanting, we use it also to put coordinates onto parts of the curved surface. We bend the plane to fit the surface; if it doesn’t fit perfectly at first, we stretch it a bit here, and shrink it a bit there, until it finally does fit perfectly. Labelling the original plane ‘D’ and the fitted plane ‘f(D)’, O’Neill writes: ‘If we think of D as a thin sheet of rubber, we can get f(D) by bending and stretching D in a not too violent fashion’ (1966: 125). We thus can give coordinates to every neighbourhood of every point on the surface, for each point simply inherits the coordinates of the point on the plane that we have fitted it to. We use the Euclidean plane as a chart of a part of the surface (Figure 9.7).

Figure 9.7  Coordinate chart.

Some surfaces can be given coordinates from a single chart, but not every surface can be fitted to the plane in this way. For example, there is no way to smoothly distort a plane so that it covers an entire sphere. The solution is to use

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another Euclidean plane. We fit one plane to the north hemisphere and a bit of the south, and a second plane to the south hemisphere and a bit of the north. We glue the two copies smoothly together where they overlap, and now every point on the sphere has coordinates. Points near enough to the north pole have coordinates only from the first plane; points near the south pole have coordinates only from the second. Points sufficiently near the equator have coordinates from both planes, but because the two planes have been glued together smoothly, no problems of inconsistency arise. Our pair of charts are an atlas of the surface. A shape that can be given an atlas in this way is called a differentiable manifold (Bröcker and Janich 1982: 4). To calculate the shape by Gauss’s methods and describe the intrinsic shape of our manifold, we need to know how to determine the distance between pairs of points from their coordinates. If we are given the coordinates of two points in Euclidean space, it is straightforward to apply Pythagoras Theorem to obtain the distance between them. In a differentiable manifold, however, we must refer to our atlas. Each chart in the atlas has been fitted to a piece of the manifold by bending and twisting a Euclidean space to match the manifold. Thus the coordinate mesh has been bent and stretched, and we cannot read off the distances from the coordinates, as we can in Euclidean space. To obtain the shape of the manifold, we need to know the functions that give the ‘warping functions’ of our chart.4 Since we then know exactly how the mesh has been warped at each point, we can calculate distances on the surface by calculating the distance on the Euclidean chart, and then correcting to take account of the warping. Thus a differentiable manifold incorporates a Euclidean geometry three times over. First, we need Euclid to provide the coordinate charts to fit to the surface. Second, we need Euclid in order to define the ‘smooth’ gluing together of the charts by the chart transformations. Bröcker and Janich write: If one were to consider the whole manifold as being formed by a gluing process from the chart domains, which one knows as well as one knows the open subsets of Euclidean space, then it is precisely the chart transformations that show how different chart domains are to be glued together. (1982: 2)

Third, we need Euclid in order to define local distances on the manifold. Euclidean space has not gone away when we move to a manifold, for every manifold is a construct out of Euclidean spaces. The theory of the differentiable manifold is founded in the theory of Euclidean space, and in no way replaces Euclidean geometry, much less falsifies it.

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9.8  What Is a priori in Euclid? A shape is a kind of property, the kind a spatial continuum can have. I suggest that Euclidean geometry is best regarded as the a priori mathematical science of certain particularly simple shape properties, and the laws that connect them. A two-dimensional continuum can have the property of being plane, and a boundary within that continuum can have the property of being straight. These shapes allow us to define distance: if two points lie in the same plane, the distance between them is the length of the straight line connecting them. If no straight line in the plane connects the points, the definition allows the distance to be determined by the methods of the differential calculus. Thus Euclid’s Parallels Postulate is best regarded as a statement of one of the laws that connect these shape properties. The following proposition is equivalent to the Parallels Postulate (Heath 1956: Vol.1, 220): In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. This does not assert that anything actually instantiates the property plane or the property straight. Even if nothing actually instantiated them, the properties would still exist, and the general laws that connect them would still hold good. We cannot tell a priori whether a two-dimensional continuum exists: but we do know that if it did, then if it were plane, then given any straight line in it and any point of the continuum not lying in the line, there would exist at most one parallel through the point. We can interpret Euclid as being concerned with the relations between the shapes, not with the question whether these shapes have instances in nature. This is the way that Hume interprets Euclid in the Enquiry. He says that the business of geometry is the a priori determination of the relation of shapes, which he calls ‘figures’: All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra and Arithmetic; and in short every affirmation which is either intuitively or demonstratively certain. That the square of the hypothenuse is equal to the square of the two sides, is a proposition which expresses a relation between these figures. … Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle

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in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence. (Hume 2007: 20, emphasis original)

I conclude that the a priori status of geometry is not undermined by the GTR.

9.9  Should We Base the Reals on Set Theory? Set theory purports to provide a structure, as discussed in section 7.2, within which we can find the real numbers and also the real-valued functions of several real variables. So set theory can define a variety of distance functions that can be interpreted as the metrics of a variety of ‘geometries’. Thus set theory offers to supplant geometrical intuition as the foundation of geometry. But so far from being a foundation for geometry, set theory may itself need a geometric foundation. We do not seem to be in possession of an agreed single conception of set that can meet all the needs of mathematical theory. Naive set theory says that whenever there is a many, a plurality of things, there exists the set of these things: the theory collapses when dealing with the Russell set of the sets that are not members of themselves. The leading alternatives to the naive theory are ZFC and NBG. (ZFC is named after its inventors Zermelo and Fraenkel, with ‘C’ for the Axiom of Choice, and NBG is named after its inventors von Neumann, Bernays and Gödel.) ZFC is based on the iterative conception of set, but NBG is based on the ‘limitation of size’ conception.5 The iterative conception says that only those sets exist which are reached by an iterative process of successively taking power sets and unions: the paradoxes are avoided because it cannot be proved in ZFC that the Russell set is ever reached. The limitation of size conception says that sets are classes, and that classes exist without having to be reached by any iterative process: however, not all classes are sets, but only those that are not ‘too big’. A class is too big if it has as many elements as there are ordinal numbers, in which case it is called a ‘proper class’. According to NBG, a proper class cannot be an element of a set: the Russell class of sets that are not members of themselves exists in NBG, but it is not a set, so paradox is prevented. These two conceptions of set are inconsistent with each other: the iterative conception conceives of sets as being ‘formed’ from other sets stage by stage, whereas the limitation of size conception of NBG pictures classes as pre-existing en masse, one class for every many: the sets are simply those of the classes whose elements happen to be not too many. The iterative conception justifies the Axiom of the Power Set but cannot justify a satisfactory theory of the von

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Neumann ordinals, so ZFC appropriates the Axiom of Replacement from NBG. The ‘limitation of size’ conception justifies Replacement but cannot justify Power Set, so NBG appropriates the Power Set axiom from ZFC. Neither the iterative conception nor the ‘limitation of size’ conception is able simultaneously to justify both Power Set and Replacement, so we have no single conception of set that underwrites all the axioms of standard set theory. According to Gödel: Cantor’s continuum problem is simply the question: How many points are there on a straight line in Euclidean space? An equivalent question is: How many different sets of integers do there exist? (1964: 470)

The Continuum Hypothesis (CH) states that the cardinal number of the continuum is aleph1 – the cardinal number of the first uncountable ordinal. As we have seen, the continuum is equinumerous with the set of real numbers, which is equinumerous with the set of subsets of the natural numbers. Thus, CH says how the set of real numbers compares in size with sets arrived at by the processes that generate the sets in the iterative hierarchy. It is now known that neither CH nor its negation can be proved from the current axioms of set theory. The proof that the negation of CH is unprovable was discovered by Cohen, who writes: A point of view which the author [Cohen] feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now aleph1 is the set of countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set C is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach C. This point of view regards C as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. (2008: 151)

According to Cohen, the continuum is an incredibly rich set which cannot be given to us by iterating our way up through the ordinals. However, as we have seen, the real numbers are given to us with certainty by Dedekind’s continuity axiom and Eudoxus’s theory of proportion. Thus so far from geometry resting on set theory and empirical physical facts, we might conjecture that the postulation of the axiom of the power set in set theory is motivated by our geometrical intuition of the continuum.

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Appendix to Chapter 9 Notation. In this Appendix, variables from the end of the alphabet such as ‘x’, ‘y’ and ‘z’ range over finite lines. Variables from the beginning of the alphabet such as ‘a’, ‘b’ and ‘c’ range over lengths. The symbol ‘ 0 then φ(n) is a successor; if a > 0 and b = 0 then φ(n) is a limit. (2) If φ(n) is a successor, p(n) = 2a3b-1, which is the notation for the predecessor of φ(n). (3) If φ(n) is a limit, q(n) is the Gödel number of a computer program that outputs the sequence 2a–131, 2a-132, 2a-133, … . By definition, an ordinal is constructive if there is a system of notation S in which it receives a notation. Only ordinals less than ω2 receive a notation in the 2a3b system. But there are more complex systems of notations that extend the system far beyond ω2 and still conform to Kleene’s definition (Rogers 1987: 205). A system of notation S' is said to be maximal if every ordinal that receives a notation in some system S receives a notation also in the maximal system S'. Kleene (1938) proved that a maximal system exists. So an ordinal is constructive if some system assigns it a notation, and hence if a maximal system does. But a maximal system maps a subset of the natural numbers onto the set of all constructive ordinals, which is therefore countable. So since set theory says there are uncountably many countable ordinals, it says that there exist some countable ordinals that are not constructive. The least of these is the Church– Kleene ordinal ω1CK. Clearly ω1CK is not a successor, else its predecessor would receive a notation, and hence ω1CK would receive a notation too. But ω1CK is the von Neumann ordinal of the set of its predecessors, which are an increasing ω-sequence, so there exists an increasing ω-sequence the upper limit of which is ω1CK. But no computer program outputs a fundamental sequence for ω1CK: if one did, we could use it to supply a notation for ω1CK. Now it can be proved that a von Neumann ordinal is constructive if and only if it is the ordinal of a recursive well-ordering (Rogers 1987: 212). Since ω1CK is not constructive, we deduce that the relation that orders the constructive von Neumann ordinals is not recursive. Our theory of the ordinals as magnitudes is quite different from the theory of the constructive von Neumann ordinals, but the theory of ordinal notations sets up a correspondence between them. Every magnitude ordinal has a successor: this corresponds to Church’s first ‘process of generation’. Every increasing ω-sequence of magnitude ordinals has a limit: this corresponds to his second ‘process of generation’. Church saw that deducing the existence of a von Neumann ordinal and supplying it with a notation go hand in hand. Just so, deducing the existence of a magnitude ordinal and supplying it with a notation go hand in hand. Therefore, the magnitude ordinals that can receive a notation are all and only the magnitude ordinals whose existence can be deduced from the Magnitudes Thesis, so every such magnitude ordinal is constructive.

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10.9  Stopping at the Constructive Ordinals The Magnitudes Thesis entails the existence of the collection of all the ordinals that are constructive in the sense of Church and Kleene. But are the constructive magnitude ordinals all the magnitude ordinals there are? The Magnitudes Thesis is silent on this question, so it may look as if we could consistently extend the collection by postulating a non-constructive ordinal. Then our two generating principles would allow us to infer non-constructively that every countable series of countable ordinals has a countable limit. We arrive at a collection of magnitude ordinals analogous to and no less numerous than Cantor’s ‘second number class’ of countable infinite ordinals. But once we begin the process of postulating further ordinals, there seems no natural stopping place. We could continue now beyond the countable ordinals by postulating an uncountable magnitude ordinal, which would lead to a collection of magnitude ordinals no less numerous than Cantor’s ‘third number class’ and so on. Each time we expand the collection of ordinals by postulation, there seems nothing to stop a further expansion by a subsequent postulation. We would thus replicate within the theory of the magnitude ordinals the same kind of indefinite extensibility which according to Dummett characterizes set theory and the theory of the von Neumann ordinals. (1991: 317). Is there a way to remove the indefiniteness and arrive at a determinate totality of all the magnitude ordinals? We can argue as follows that the constructive magnitude ordinals are all the magnitude ordinals there are. The existence of a non-constructive ordinal would entail, by axiom O2, that there is a series of nonconstructive length. Now it is known that a von Neumann ordinal is constructive if and only if it is a recursive ordinal, that is, if it is the ordinal of a recursive wellordering (Rogers 1987: 211–12). By adapting the proof, we could show that a magnitude ordinal is constructive if and only if it is the ordinal of a series whose precedence relation is recursive. Therefore, if a non-constructive magnitude ordinal did exist, by O2 it would have some series as a representative, and the precedence relation of this series would not be recursive. A relation is decidable if and only if it is recursive, so this series would have an undecidable precedence relation. But that is not consistent with the conception of ‘series’ on which the present account rests. We have defined a series as the kind of item that can be classified by a relational predicate. The classification is effected by means of a sentence with a serially referring subject term – it is not sufficient for the subject term merely to refer plurally to the members of the series. Our theory of series envisaged an infinite language. We can describe such a language, though we

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cannot use it. Thus the conception of series requires us to abstract away from the finiteness of language, but it cannot require us to abstract away from language altogether. Every language must have a syntax, and every syntactic property must be decidable. So an expression is a serial term only if for all symbols t and t' that occur in the expression, it is decidable in what order they occur, i.e. whether t precedes t': if we removed this restriction, the notion of a serially referring term would lose all connection with the theory of reference, and could only be some set-theoretic construct. Therefore, no series in our sense can have an undecidable order of precedence. Since a representative of a non-constructive magnitude ordinal would have an undecidable order of precedence, there can be no such magnitude ordinal. These considerations about serial reference are one reason to think that the constructive magnitude ordinals are all there are. A second reason to think so is that the Magnitudes Thesis treats the ordinal numbers as magnitudes. Property realism infers the existence of magnitudes from the dispositions and casual powers of physical things. That is how in previous chapters we inferred the existence of the natural numbers and the geometrical lengths. Therefore, if the ordinal numbers are a system of magnitudes, they too should have some connection with the dispositions of physical things. Computers are physical things, but only the constructive magnitude ordinals are connected with the dispositions of computers. The ordinal length of a series is a measure of its computational complexity, which determines the physical resources of time, hardware and memory storage that have to be devoted to computation of the order of its terms. Since higher ordinals seem to have no tangible connection with computers or other physical things, the property realist will see no reason to postulate them into existence.

10.10  The Existence of Sets This book has argued that numbers are not sets and that set theory is not needed to justify the axioms for the numbers. I do not, however, mean to suggest that set theory is not needed in mathematics at all. On the contrary: Set theory pervades modern mathematics. Some special branches and some special styles of mathematics can perhaps do without, but most of mathematics is into set theory up to its ears. If there are no classes, then there are no Dedekind cuts, there are no homeomorphisms, there are no complemented lattices, there are no probability distributions, … for all these things are standardly defined as one or another sort of class. (Lewis 1991: 58)

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We need sets in mathematics, but how do we know that sets exist? The intuitive conception of sets is the ‘naive’ theory that whenever there are a plurality x of things, there exists a single individual σ(x), the set of those things. The things x are said to be the elements of the set σ(x), and the set is said to combine its elements. In view of the paradoxes, the iterative conception more cautiously says that whenever some things have been formed, there is next formed a set that combines them. The limitation of size conception says that whenever there are not too many things, there exists a set of those things. What these various conceptions have in common is that they think of a set as a complex unity, namely a single individual σ(x) standing in a distinctive constitutive relation to a plurality x. We can arrive at knowledge that sets exist if philosophical reflection can assure us that suitable complex unities exist. Sparse property realism supplies suitable complex unities. It explains resemblance in terms of instantiation: two items resemble, according to sparse realism, if they instantiate the same property. But what is instantiation? It cannot be explained by postulating an instantiation relation, on pain of Bradley’s regress (1893: chapter 3). Sparse property realism connects instantiation with truth: if an item instantiates a property then a belief that attributes the property to the item is true. A theory of truth that is substantive and not merely disquotational requires the existence of complex unities, and so supplies entities that accord with the conception of set. For example, Russell’s correspondence theory of truth says that a belief is true if there exists the corresponding fact. According to Russell, the fact that corresponds to a true belief is a complex unity, which ‘knits together into one complex whole’ the things the belief is about. Russell writes: If we take such a belief as ‘Othello believes that Desdemona loves Cassio’, we will call Desdemona and Cassio the object-terms and loving the object relation. If there is a complex unity ‘Desdemona’s love for Cassio’ consisting of the object terms related by the object-relation in the same order as they have in the belief then this complex unity is called the fact corresponding to the belief. (1967: 75)

Applied to a sentence that expresses a belief, Russell’s theory entails that a sentence is true if there exists the corresponding fact, the complex unity which ‘knits together’ the things the sentence is about. In Hossack (2013), I used Russell’s correspondence theory to give the following account of instantiation: some things instantiate a predicable if there exists the Russellian fact whose constituents are the things and the predicable.

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Of course the correspondence theory of truth is not the only possible account of sentential truth. But alternative accounts, if they are substantive, also require the existence of complex unities. Thus according to the theory of propositions, the proposition expressed by a sentence is a complex unity composed of the things mentioned in the sentence: a sentence is true if it expresses a true proposition. According to the theory of states of affairs, the state of affairs represented by a sentence is a complex unity composed of the things mentioned in the sentence: a sentence is true if the state of affairs it represents is actual. From each of these substantive theories of sentential truth we can deduce the existence of complex unities, be they facts, propositions, or states of affairs. This immediately gives a model for the axioms of a theory of sets, as follows. In plural logic just as in singular logic, the law of identity is a logical truth: whenever there are some things, they are identical with themselves. So given some individuals x, it is true that x are identical to themselves, so a substantive theory of truth tells us that there exists a complex unity σ(x) standing in a distinctive constitutive relation to the plurality x, namely the Russellian fact, proposition or state of affairs that x = x. This complex unity σ(x) consists of the plurality x and the identity relation. We can now identify the set whose elements are x with the complex unity σ(x), and we say an individual u is an element of the set σ(x) if u is one of x. Given the nuanced assumptions in the set theory NFU7 about when one may consistently assert the existence of the set of things that fit a given description, it can be proved that the axioms of NFU are true when ‘set’ and ‘elementhood’ are so defined (Hossack 2014). Thus if facts exist or if propositions exist or if possibilities (states of affairs) exist, then there exists a model of the axioms of NFU, which can be seen as a sophisticated version of naive set theory. If we then import the constructive ordinals of this chapter into NFU, we obtain sets of every constructive rank. Those are quite enough sets for all the needs of ordinary mathematics, which outside set theory itself has no need for very large ordinal numbers. We noted in section 10.2 that Boolos writes concerning κ, the least cardinal λ such that λ = ℵλ, that to the best of his knowledge there is nothing in the rest of mathematics that needs so large a number. Michael Potter states that ‘the overwhelming majority of 20th century mathematics is straightforwardly representable by sets of fairly low infinite ranks, certainly less than ω + 20’ (Potter 2004: 220; Potter n.d.). Thus given the theory of numbers of this book, and given the set theory NFU, we obtain all the sets we need for the whole of ordinary contemporary mathematics, which is all of mathematics apart from those branches of the subject that explore the consequences of axioms as ontologically unrestrained as Replacement.

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Appendix to Chapter 10 In this Appendix, the notation ‘{x:Fx}’ means the virtual class of satisfiers of ‘F’, where F is any predicate free for x. Upper case letters from the beginning of the alphabet abbreviate notations for virtual classes, upper case variables from the end of the alphabet range over pluralities, lower case variables from the end of the alphabet range over series and lower case variables from the beginning of the Greek alphabet range over ordinals. The sign ‘