The Mechanics of Meaning: Propositional Content and the Logical Space of Wittgenstein's Tractatus 9783110889130, 9783110172188

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David Hyder The Mechanics of Meaning

W DE G

Quellen und Studien zur Philosophie Herausgegeben von Jürgen Mittelstraß, Dominik Perler, Wolfgang Wieland

Band 57

Walter de Gruyter · Berlin · New York 2002

The Mechanics of Meaning Propositional Content and the Logical Space of Wittgenstein's Tractatus

by David Hyder

Walter de Gruyter · Berlin · New York

2002

© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

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A CIP catalogue record for this book is available from the Library of Congress

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Hyder, David: The mechanics of meaning : prepositional content and the logical space of Wittgenstein's Tractatus / by David Hyder. — Berlin ; New York : de Gruyter, 2002 (Quellen und Studien zur Philosophie ; Bd. 57) ISBN 3-11-017218-6

© Copyright 2002 by Walter de Gruyter GmbH & Co. KG, D-10785 Berlin All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. Cover design: Christopher Schneider, Berlin. Printed in Germany.

To my parents

Preface Wittgenstein's Tractatus Logico-philosophicus still enjoys its reputation as one of the most obscure works in the philosophical canon. Yet despite its obscurity, or perhaps indeed because of it, it continues to be taught in the classroom. One reason for this is undoubtedly the openness of the book: most people can find something there that interests them, and the text is so spare that there is plenty of room to lodge one's own interests. So in deciding to publish a fulllength study, one courts disaster. After all, the joy of the book, at least for those who do enjoy it, lies in doing the interpretive work oneself, and someone who presumes to tell us what it all means is obviously something of a spoilsport. Nevertheless, my aim in the following pages has been to give a concrete interpretation of a major part of Wittgenstein's early work. Few would be so foolish as to claim to give an exhaustive reading of the Tractatus, and I do not imagine I have. But I do propose one that applies to the book as a whole. In other words, I hope that the reader will gain insight into the meanings of terms and passages of the text whose central importance has never been in question, even though their meaning has remained opaque. And I advocate a particular view of the main theme of the book, that is to say of the problems that motivate it, which has received little attention in the literature. My method has not been, however, to engage in a close reading of the Tractatus itself, rather I have concentrated much of my effort on earlier sources: Wittgenstein's letters and notebooks, Russell's Theory of Knowledge, Helmholtz's "Facts in Perception", Hertz's Mechanics. My aim has been to reconstruct as fully as possible the set of problems that Wittgenstein took himself to be working on in the years 1912-1918, in order to explain the solutions we see him give in the Tractatus. I am aware that this approach is vulnerable to a serious objection: Why should we not take the Tractatus as a self-contained philosophical work, and interpret, so to speak, within the text? How can one hope to determine what an elementary fact is, for instance, by looking to sources outside the text itself? To this I would reply first that should the reading I am proposing directly conflict with a passage in the Tractatus, then I would of course have to revise it. But if such objections are directed against the use of such external sources generally, I can only reply that esoteric readings of Wittgenstein do not get us very far. First, sticking to the one text is a good classroom exercise, but it cannot set methodological bounds on scholarly interpretation. Second, although it is hard

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Preface

when reading Wittgenstein to hook up what he says to other lines within the philosophical tradition, I think that we must make an effort to do so. If we fail to, we end up with a philosophy that cannot be applied. We need more exoteric work, and that means scholarship that establishes systematic links to other philosophers and philosophies. That is what I have tried to do. It is in part for these reasons that I have adopted two conventions in the text that some may find inconvenient. The first is to retain, for the most part, the notation of Principia Mathematica when citing or commenting on Russell's and Wittgenstein's work. Explanations of the notation I do use in my text (and it is not that much) are given on p. XIII, and the reader is referred to Principia Mathematica itself should more detail be required. This approach seemed preferable because it may help others avoid a problem I encountered when I began research in this area, namely the difficulty of understanding all that strange notation in Principia Mathematica, and in the Notebooks 1914-16. My hope is that readers of my comments will immediately be able to understand the relevant passages in the Tractatus, in Russell's and Whitehead's works, and in the Notebooks should they decide turn to the originals. A further reason is that both Russell's notation and Wittgenstein's changes to it are not mere additions to their properly philosophical work. The two go hand in hand, and transliteration into modern notation is not always possible without distorting the original sense. This is particularly evident when one considers, for instance, the role of free and bound variables in Principia Mathematica, but it also proves to be the case when we look at Wittgenstein's "copula-theory". The second convention I have adopted is that of always quoting Wittgenstein in German, and providing translations in the footnotes. My thinking here is similar: Wittgenstein wrote in German, and the German text is the primary source. Translation into English is not always possible without distorting the original sense. As a rule, I quote other sources in the original language, and provide translations in the body text. Fragmentary passages, or passages quoted at length elsewhere in the text are, however, quoted in translation. In almost all cases, the translations are my own. I often render words that connect to the sciences with greater emphasis on that meaning: Abbildung as mapping, Mannigfaltigkeit as manifold, etc. This last term, which is central to my interpretation, is particularly nettlesome, since its two meanings can only be rendered in English by the distinct terms "multiplicity" and "manifold". It does not help that these two meanings are not always truly distinct in German. My translations are often less elegant than the established ones, and indeed I intend mine as glosses, not as improvements on the latter, which often better capture the natural sense of an expression. The only exceptions are those passages from Wittgenstein's notebooks where the agreement between my translation and Anscombe's was so close that is was pointless to preserve the slight differences. In these cases, I have simply used her translation.

Preface

IX

Many people have helped me during my work in the last ten years, some without knowing it. Without the patient intelligence of Ian Hacking and Alasdair Urquhart, whose supervised and advised me while I wrote the thesis on which this book is based, my early research would never have come to fruition. There are several people I barely know, or indeed have never met, without whom this work could never have been written: Nicholas Griffin, whose work has been invaluable to my understanding of Russell's judgment-theory; Brian McGuinness and Joachim Schulte, whose critical edition of the Tractatus has changed the ground-rules for work in this area by linking together all the early sources. Using their edition can seduce one into thinking that these links were evident all along. A similar debt is owed to the editors of Russell's Collected Works. I have profited over the years from discussions with the late Lorenz Krüger, and with Ulrich Majer, both in Göttingen, where I spent the years 1993-1995 on a scholarship from the German Academic Exchange Service (DAAD). Robert Tully and Jack Canfield in Toronto were the first to direct my attention to many of the texts I discusss, and they later made numerous helpful comments on the thesis. In the past few years I have learned much on the subject of wissenschaftliche Erkenntnistheorie from friends and colleagues at the Max Planck Institute for the History of Science and the Humboldt University in Berlin, above all Michael Heidelberger, Jutta Shickore, Matthias Neuber and Torsten Wilholt. Correspondence over the last two years with Jesper Lützen has been invaluable to my understanding of Hertz's Mechanics. During a research term in Bloomington, Indiana, I profited from Michael Friedman's and Daniel Sutherland's knowledge of Kant's philosophy of science. Conversations with EvaMaria Engelen, Holger Sturm and Jaroslav Peregrin at the University of Constance helped me during the preparation of the final manuscript, as did the critical comments of the series editors. Johannes Wienand and Sven Schulz helped get the manuscript in its final form, making numerous insightful comments on the content as well. My students in a joint seminar with Dr. Engelen on Wittgenstein's Philosophical Investigations forced me to think more carefully about the connection between Wittgenstein's early and late work. Above all, I am grateful to Uta Matthies for her support of my labour. My research over the period in question has been funded by the DAAD, the Social Science and Humanities Research Council of Canada, and the Max Planck Institute. The Deutsche Forschungsgemeinschaft generously supported me in preparing the text for publication. To all of these organisations, and to those who had to read my grant applications, my warmest thanks.

Table of Contents

Preface

VII

Notation

XIII

1. Introduction 1.1 Russell' s Theory of Judgment 1.2 Scientific Epistemology and the Theory of Manifolds 1.3 Interpreting Wittgenstein's Later Philosophy 2. Helmholtz' s Perceptual Manifold 2.1 Manifolds and Perception 2.2 "The Facts in Perception" 2.3 "The Application of the Axioms to the Physical World" 2.4 Pre-established Harmony 3. Russell' s Theory of Judgment 3.1 The Origins and Purpose of Russell's Judgment-theory 3.2 Wittgenstein's Objections 3.3 Constraints on Judgment 4. The Breakdown of Wittgenstein's Copula-theory 4.1 The Copula-theory 4.2 The ε-copula and the Propositional Form 4.3 The Early Picture Theory 4.4 Wittgenstein's "Correct Theory of Propositions" 5. Logical Space

1 7 12 15 19 19 24 29 42 49 55 63 67 76 81 87 91 108 113

5.1 Truth-functions and the "Core Logical Space" 5.2 Manifolds and Quantification: 4.04-4.0411

116 131

6. The Picture-theories of Helmholtz, Hertz and Wittgenstein

152

6.1 The No-man's-land 6.2 Physical Science 6.3 Models and Manifolds

157 167 188

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Table of Contents

7. Conclusion

192

Bibliography

209

Index

214

Concordance of Passages from the Tractatus

228

Concordance of Entries in the Notebooks: 1914-1916

229

Notation

Dot-notation and modern equivalents pz>p.z).pv~p

(p/>)D(pv ~p)

T0{a).TXF).eXa,F)-.z>.Fav ~ Fa

(T Q (a)& 7J(F)& ε,(α,,Ρ>) ^ (Fav ~ Fa)

Notation from Principia used by Russell and Wittgenstein in 1912-1916 T0x xT0x Τ0χ (x).T0x

The function formed from a particular proposition, such as T0a. The class satisfying TQx, i.e. {x: T0x} An arbitrary value of T0x. The assertion of all significant values of T0x.

Wittgenstein's "epsilon-copula": it is the function defining the class of all pairs of elementary predicates and individuals that form existing facts. ε, (a, F)

When F and a are related by the copula, it follows that Fa, and conversely.

J(S,a,R,b)

Russell's "judgment-relation" in its early and late versions. In the early version, the components of a possible fact R(a,b) are related to a subject 5 in the judgment-relation J. The later version adds the "form" y, which corresponds to Wittgenstein's "copula".

J(S a R b γ)

x~et'y

The function "is not a member of t'y", where t'y is the type of y.

1. Introduction Readers of the later Wittgenstein will be familiar with a regress argument of the following form. Suppose that in order to utter a meaningful sentence, I must have mastered rules governing the use of the words in that sentence. Suppose furthermore that these rules are something I have learned, either by being told what they are, or by being shown the objects to which the words refer. In the latter case, I will have derived rules for using the words from knowledge of their referents, for instance from knowing what kinds of things they are. In the former one, I will have done so by drawing on my mastery of other words, namely those used to express the rules themselves. Either way, Wittgenstein argues, we run into problems: every ostensive definition, every statement of a rule can be misunderstood. The misuse of words can never wholly be forestalled in this manner, from which Wittgenstein concludes that learning how to use words correctly is not the same as learning how to distinguish between different kinds of things, nor is it like memorising a book of rules. Knowledge, if it is to be stated, must be expressed in a language whose words have meanings. But if the meanings of words always depend on further knowledge, we could never get started with the business of speaking meaningfully. I will refer to such a regress as a "sense-truth regress" in what follows. Such arguments are targeted against a particular conception of intentionality. On this view, intentions mark the division between meaningful and meaningless speech—only if I have an intention when uttering a phrase can I be said to have meant something, to have asserted something that counts as right or wrong. Wittgenstein challenges us to explain what this "having an intention" consists in. He argues that so long as we think of intentions as involving a peculiar kind of linguistic symbol (a rule or a sample), we shall beg the question, for the doubt concerning the correct application of these symbols will simply recur. What distinguishes meaningful from meaningless language cannot itself be open to doubt concerning its meaning. In short, it cannot itself be language, in so far as language is something that, by its very nature, can be misunderstood. The space of meaning is bounded at its periphery neither by true statements concerning the make-up of the world, such as those made in traditional metaphysical theories, nor by rules that are themselves expressed in language, even if these are imagined to be the private mental languages of idea-theories.

2

1. Introduction

This line of argument is evident at a number of points in the Philosophical Investigations, where it is taken to show that there must be rules that are somehow followed blindly, without further interpretation. But it also motivates much of the theory of language and logic advanced in the Tractatus. Indeed, it is present in nuce in Wittgenstein's earliest philosophical writings, where it is marshalled against Russell's "theory of judgment". This theory was explicitly targeted against theories of propositional judgment like those of Meinong and Frege, which relied on intentional objects or senses in order to explain how the judging subject could be directed toward a determinate state of affairs. Such intentional objects allow one to finesse the problem of explaining how judgments retain a definite meaning when they are false, in other words when the intended state of affairs does not obtain. On Russell's view, however, such intentional entities were dangerously idealistic, and he aimed to eliminate them by means of his judgment-theory. Thus the problem of accounting for false judgments was a central concern of this theory. And Wittgenstein, during his first stay at Cambridge with Russell, seized on just this difficulty. He argued that on Russell's approach one could not distinguish false judgments from nonsense without assuming that other judgments were true. In his view, Russell's theory of judgment made meaning dependent on truth, and thus it had to be wrong. In letters from the period 1912-13, in the "Notes on Logic" and the "Notes Dictated to Moore" of the same period, and finally in the Tractatus itself, Wittgenstein insists that theories of propositional judgment must show that it is "impossible to judge nonsense". And he quickly comes to the conclusion that the impossibility of nonsense judgments can be secured only if we postulate a logical language whose very structure prevents the formation of nonsensical propositional signs. He does not explain in the Tractatus why this is the only alternative, although we do find there one specific application of a general regress argument to the case of non-denoting names. This argument is implicit in the principle that the significance of a proposition (which for Wittgenstein meant its bivalence) cannot depend on the truth of another." For if all propositions depended on the truth of others for their meaning, it would follow that the entire structure of meaningful language depended on the contingent truth of some set of propositions, which seems absurd. Alternatively, there might be some propositions which were necessarily true, from whose truth the meanings of the rest depended. But then they would not be bivalent, and would therefore not count as significant. The connection of this principle to Wittgenstein's 1

See Tractatus 2.0211, and the "Notes Dictated to Moore". In L. Wittgenstein. Notebooks 1914-1916. 2nd edition, eds. G. H. von Wright and G. Ε. M. Anscombe. Oxford: Basil Blackwell. 1979, p. 117. In the following, references to the Tractatus will be made only with the number of the relevant section. References to the "Prototractatus" will be prefaced with the abbreviation "PT".

1. Introduction

3

claim that it is, or ought to be, impossible to judge nonsense is clear enough. If the same act of judgment sometimes counts as significant and sometimes as nonsensical, this must be because in the second case certain conditions on significance are not fulfilled. So we might seek to guard against this eventuality by postulating that these conditions be met, meaning that the judgment act would count as legitimate only under certain constraints. Wittgenstein contended that the notion of such postulates, of constraints on the scope of judgment, is incoherent. Propositions and judgment do not and cannot depend on the truth of other propositions for their significance. Although these two postulates (that one cannot judge nonsense, and that no proposition depends on another for its sense) are explicitly stated in the Tractatus, their link to another central thesis of the book, namely the doctrine of showing and saying, is not immediately obvious. Wittgenstein suggests that the logical syntax of language cannot be "said", although it is "shown" by the propositions of logic. This logical syntax somehow points at the shared structure of language and world, and is in consequence a condition for the existence of significant propositions. Thus it follows immediately that logical propositions do not make true statements in Wittgenstein's usual sense: if they did, then we would clearly have a case where the truth of some propositions determined the sense of others, and Wittgenstein will not allow this. If a sense-truth regress can be avoided, it is by denying that such boundary conditions on meaning can ever be false, or perversely enough, that they are ever really true. So if we grant Wittgenstein his two postulates, and we accept the claim that logical propositions reflect the essential structure of language, we can infer that logical propositions cannot be true propositions, at least in Wittgenstein's strict sense. But this is to interpret entirely within the context of the Tractates' s own, obscure doctrines. What is missing here, and I would suggest that this is so in the literature generally, is a unified, positive explanation of these postulates. Why did Wittgenstein first claim that one's theory of propositional judgment must make it impossible to judge nonsense? What sorts of supplementary conditions on judgment were he (and Russell) considering? And why exactly do these constraints get demoted (or elevated, depending on your point of view) to the status of the unsayable? By giving specific historical and systematic answers to these questions, we can arrive at a better understanding not only of the Tractatus, but of Wittgenstein's philosophy in general. For the regress argument I derived above from the Investigations is anticipated in the Tractatus not just in its general outlines, but in many of its details. Seeing how this is so allows one to see that the function of the Tractatus's logical space closely parallels that of a language-game in the later work. The connection between these various Tractarian theses is more readily grasped in modern terminology. As I just observed, Wittgenstein's first work in this area took the form of a criticism of Russell. He argued that on Russell's

4

1. Introduction

theory of propositional judgment, judgments were guaranteed to be meaningful only under certain conditions. For instance, the objects involved in the judgment had to exist, and, furthermore, they had to belong to the appropriate ontological types. However, such constraints could be expressed only metalinguistically if one was to avoid either contradictions or mere redundancy. Suppose that in order to be sure that my judgment that "Your hat is brown" is meaningful, I must eliminate the possibility that your hat does not exist. I may try to do this by means of a constraint such as "there is an χ such that χ = your hat", an expression that Russell took to mean "your hat exists". In fact, if this constraint is to achieve anything, it must mean something like '"your hat' denotes a". For otherwise the constraint might be meaningless for exactly the same reason that "Your hat is brown" threatened to be so. If you don't have a hat, then "your hat is brown" and "your hat exists" are equally meaningless, because the phrase "your hat" lacks a denotation in both cases. On the other hand, if you do have one, the constraint is redundant. Adding significanceconstraints on judgments at the object level therefore achieves nothing. On the other hand, adding them at the meta-level leads to a regress. Suppose that to ensure that I speak meaningfully in an object language, I must first have fixed the syntax and semantics of that language at the meta-level. Then the difficulties will simply recur, for all the judgments at the meta-level will be open to the same ambiguities. In the meta-constraint '"your hat' denotes a", we have made use of the terms "a" and '"your hat'". What is to ensure that these terms have denotations?2 The first main interpretational strand in this book will therefore be to explain how Wittgenstein arrived at his diagnosis of this problem in Russell's theory of judgment, and how he thought he resolved it in the Tractatus by introducing the notion of a "logical space" of elementary facts and propositions. I maintain that any adequate interpretation of Wittgenstein's early work must take this route, for the simple reason that Russell's theory was the point of origin of Wittgenstein's own work on a propositional theory.3 Still, such an analysis gives us 2

3

For the purposes of my exposition here, I am using the example of non-denoting names, which is the one sort of constraint on significance that has received substantial attention in the literature. The bulk of my analysis in this book, however, concerns other sorts of constraints (on the ontological type membership of objects, on the existence of forms), which have rarely been discussed, and which are, in my view, of greater significance for our understanding of Wittgenstein's philosophy, both early and late. The regress argument gets its bite from the problems generated by such constraints, and it is they, far more than indestructibility of names, which still play an active role in the Investigations. I do not mean that such a genetic analysis is always required of interpretative work. However, the obscurity of Wittgenstein's text demands that we provide as detailed an account of the problems Wittgenstein sought to solve as we do of the answers. Above all, researchers did not know of the existence of Russell's Theory of Knowledge manuscript until the 1970's, and it has only been available in print only since 1984. So we are obliged, in my

1. Introduction

5

only half of the picture, because Wittgenstein went on to give this result a quite particular twist. He took it to show that our meaningful use of language presupposed that both experience and language had spatial structures, and, furthermore, that these structures could not be meaningfully described, that they could perhaps be shown, but certainly not said. In adopting this position, he drew on a neo-Kantian tradition in German-language philosophy of science that ascribed a central role to the concept of a manifold both in theories of perception as well as in the philosophy of science. It should be emphasised that Russell was aware of much work in this tradition, as a glance at his Essay on the Foundations of Geometry will confirm. But Russell was ultimately hostile to any argument that assumed the existence of a priori structures of experience. For instance we see him arguing, in his 1913 Theory of Knowledge, that Kant's understanding of space has "suffered a series of shattering blows", with the result that "the space of actual experience is appropriated by psychology, the space of geometry is appropriated by logic, and the space of physics is left halting between them in the humbled garb of a working hypothesis".4 Russell's logico-epistemological project at this point aimed to show how scientific notions like that of space were logically constructed by the subject, instead of being inherent phenomenal structures of an idealist consciousness. Furthermore, the truth of these notions when applied to experience was to consist in their corresponding to objective states of affairs, and not to a priori conditions of experience. His theory of judgment, in turn, was to explain how logic could emerge in a world consisting only of subjects, objects, relations and forms—an atomised and monistic world bereft of ideas or representations, let alone structures of experience. So Wittgenstein's rejection of Russell's approach, which was influenced by two distinct forms of neo-Kantianism (that of Frege, and that of scientific epistemologists such as Helmholtz, Hertz and Boltzmann) was not just a rejection of specific theses within the theory of propositional judgment. Consciously or not, Wittgenstein was reasserting the Kantian doctrines that there cannot be logically determinate judgments without intuitive structures of experience, and, more strongly, that logic is not-—as Russell maintained—the most general form of natural science, but is rather the by-product of our structures of cognitive representation. The second strand of my interpretation concerns this neo-Kantian tradition of "scientific epistemology". By examining the philosophical writings of Hermann von Helmholtz and Heinrich Hertz, I show how Kant's a priori intuitions of space and time were extended in their work into what I call "manifold-

4

opinion, to fill in some basic historiographical gaps at this point if we are to make interpretative progress in the future. B. Russell. Theory of Knowledge. The 1913 Manuscript. The Collected Papers of Bertrand Russell vol. 7. ed. E.R. Eames and K. Blackwell. London: Allen & Unwin. 1984, p. 22.

6

1. Introduction

theories". These are theories whose fundamental structures are derived from the theory of manifolds that developed in the second half of the nineteenth century, beginning with the work of Riemann and Graßmann. I will argue that Wittgenstein's notion of a "logical space" is one more instance of such a theory. The Tractatus's theory of language supposes that both elementary propositions and the elementary facts to which they refer (when true) are organised in isomorphic structures. Each elementary proposition points to what Wittgenstein calls a "logical place" in the space of elementary facts. The dimensions of these manifolds correspond to sets of intersubstitutable objects and names, so that the symbol that results when one of these names is replaced by a variable selects a cut through the field of elementary propositions. Significant propositions in the strict sense always assert something about connections between points (logical places) in the space of elementary propositions, and can therefore be true or false depending on whether these connections obtain.5 In contrast, logical propositions pick out invariant structural properties of the space itself, and are for this reason always true. They stand in the same relation to the logical space as geometrical propositions do to Kant's pure intuition of space. They are true a priori, in that they pick out invariant properties of the fundamental structure of experience. But for this very reason, Wittgenstein argues, they are devoid of empirical content, and cannot be viewed as making significant assertions in a strict sense. In the opening chapters I keep these two strands distinct. Chapter 2 deals with Helmholtz's manifold-theory of perception and its relation to later German-language theories of knowledge and science. Chapter 3 gives a detailed account of Russell's theory of judgment and of the problems it raised. By separating these two themes of the Tractatus, I want to emphasise that Wittgenstein's reasons for moving to such a spatial model lay within the philosophy of logic. Although he came to interpret his theory more and more from the point of view of such manifold-theories (both phenomenological and physical), this should not blind us to the central connection to the philosophy of logic and the theory of intentional judgment. Briefly put, Wittgenstein transcendentally deduces the existence of a logical space, and the premisses of this deduction are found in the theory of meaning. In Chapters 4 and 5,1 describe the programme of research that culminated in the Tractatus's, theory of logical space. Finally, in Chapter 6, the background in scientific epistemology is connected with the logicist one, and I move from there to a more general account of the philosophy of science in the Tractatus. The Conclusion considers the significance of the reading I propose for our understanding of Wittgenstein's later work. In order to give the reader a sense of how these quite diverse topics condense in what 5

In the case of an elementary proposition on its own, what is asserted is only that the elementary fact that belongs at that place in fact obtains.

1. Introduction

7

became the Tractatus's theory of language, I will give a brief overview of my discussion in the following pages, beginning with some background on Russell's theory of judgment. 1.1 Russell's Theory of Judgment In the years after Whitehead and he completed Principia Mathematica, Russell turned his attention to epistemology, and to the unfinished theory of judgment that he had presented in the introduction to that book. As I suggested above, this theory was intended to eliminate propositions or intentions as independently subsisting entities, and to demonstrate instead how they were abstracted out of acts of judgment. Russell hoped thereby to construct a theory of intentionality within a monistic universe, in which the subject, the objects with which he was acquainted, and his cognitive acts would all have same ontological status. In this monistic world, propositional judgment consisted in an unmediated relation between a subject and the objects of his judgment—no thoughts, senses or ideas would intervene. The truth of a judgment would then be defined as a relation between such a "judgment-complex" and the fact (also a complex, in Russell's terminology) whose existence the judgment asserted. I should emphasise that this project was not merely part of Russell's and Moore's anti-idealistic backlash—it was fundamentally connected to Russell's understanding of the logicist project, that is to say to the status that logical and mathematical propositions were to be given now that the analysis of Principia Mathematica was in place. By reducing propositional judgment to the plane of things, Russell aimed to give every logico-mathematical proposition a strictly objective interpretation. They would not refer to ideal entities, and there would be, in consequence, no epistemological mystery concerning the justification or objective significance of mathematics in the sciences. The truth of logical and mathematical propositions would consist quite simply in correspondence relations between (highly abstract) states of affairs, some of which would be the judgment-complexes, and others the facts to which they referred. For example, the mathematical theory of manifolds, although undoubtedly a theoretical construction of human mathematicians, would consist, on Russell's theory, in true propositions whose variables ranged over real objects. Like Husserl, Russell aims to explain how mathematical theories can be simultaneously human constructions (for the theory of manifolds is undoubtedly the product of mathematical investigations), and at the same time objectively true (for they can be referred to objective features of the world). But in sharp distinction to Husserl, Frege and other idealist philosophers of mathematics, Russell wanted to avoid concluding that because mathematics is an intellectual construct, its subject matter is in any sense the human mind.

8

1. Introduction

Despite these long-term aims, Russell quickly bogged down in his efforts, not least because of Wittgenstein's attacks. I have already indicated the thrust of Wittgenstein's objections. On Russell's theory, he objected in the Tractatus, it would be possible to "judge nonsense", by which he meant that the theory did not ensure that the propositional content of a given judgment or assertion was uniquely correlated with a possible state of affairs. Its failure to do so meant that there was no principled distinction between judgments and propositions that could be true, and pseudo-judgments and pseudo-propositions that could never be so. In short, his objection reduced to insisting that to be termed "false", a judgment must be possibly true, for if this were not the case, there would be no distinction between sense and nonsense.6 In a letter to Russell from the summer of 1913, Wittgenstein insisted that the significance of judgments in the sense just outlined must be secured "without the need for any other premisses". His notes from the years following, both written and dictated, repeatedly address the following dilemma: either the elements composing a (supposed) propositional judgment are so typed and ordered that they unambiguously describe a unique and possible state of affairs, or they are not. Suppose they are not—that is, suppose that we require additional premisses (I shall call them "riders") to ensure that the propositional judgment has the right types of elements and an unambiguous structure. These premisses would include assertions concerning the semantic categories of the elements of the propositional judgment, as well as assertions that the objects about which I judge exist, as in the example I gave above. But assertions are judgments, and in consequence the same ambiguities that were to be shored up by the riders on the original judgment will recur in the riders themselves. If I need to know that "Your hat is brown" is a legitimate judgment only if a hat is a physical object, and brown is a colour, then I also need to know that "a hat is a physical object" and "brown is a colour" are legitimate judgments. So it might seem that the reason I know that, for instance, "your brown is brown" is nonsense is that I know that "brown is a physical object" is false. But then we would have at least one propositional function that is significant for both hats and colours, namely "x is a physical object". If we assume that the categorical distinction between colours and their bearers is absolute, then this is a contradiction. If we allow that it is not absolute, then there is no longer any reason to regard "Your brown is brown" as nonsense. Lastly, if we maintain that it is absolute, but that there is no contradiction here because the function "χ is a physical object" has a truth-value on 6

More precisely, in order to be either true or false in the strict sense, a proposition must be bivalent. In order to be bivalent, it must specify a possible state of affairs. In consequence, the unambiguous correlation of a proposition with a possible state of affairs is the basic requirement for meaningful judgment.

1.1 Russell's Theory of Judgment

9

the argument "hat", but not on "brown", then our rider will do no work for us. As in the case of the existential constraint I outlined previously, this rider is either absurd or redundant. Wittgenstein concluded that, whatever account one was to give of prepositional judgment, it could not be the case that the significance of prepositional judgments depended on knowledge of the truth of riders such as these.7 Once again, either they would lead to contradictions or they would fail in the job assigned them. In making these objections, Wittgenstein was invoking a difficulty that Russell and Whitehead had recognised themselves in the late stages of their work on Principia Mathematica. Realising that functions straddling types (functions they fortunately rarely needed) would undermine the distinctions between the types, they replied that all those propositions in the book that used such functions should be regarded as statements concerning symbolism. Today, we would say that all such statements were expressed in a meta-language. In those propositions of the book where no such functions occur, but where there is what they called "typical ambiguity" (as in propositions containing only free variables), the appropriate matching of symbols was taken to be implicitly secured. Such expressions could traverse the hierarchy of types and functions precisely because they were not, strictly speaking, propositions, but only schemata thereof. Once one pinned such a schema down to a particular level in the hierarchy by fixing the type of one of the variables, the appropriate types of the other variables were supposed to be fixed implicitly by their syntactic connections in the schema. From a modem point of view, some propositions in Principia Mathematica are in the object-language, some are in the meta-language, and some are indeed mixed. For instance, the antecedent to a conditional containing type-restrictions is to be interpreted at the meta-level, whereas the consequent is to be interpreted at the object-level. It seems, however, that Russell and Whitehead were not overly troubled by the peculiar status of the type-theory, nor by the various axioms (of reducibility, of infinity) which were also required to secure the significance of large parts of the work by guaranteeing that various objects and functions existed. Wittgen7

In order to forestall an obvious objection, I should emphasise that the argument I present here is a step in a regress argument. As in the Investigations, the point is not that we cannot have rules for making distinctions between kinds of things, nor statements asserting that certain things exist, nor that such rules cannot be used to narrow the scope of statements to a particular domain. The point is rather that such rules, in so far as they are themselves expressions (or, in the early work, judgments), cannot cap an interpretative regress. In the limit, there cannot be such rules. Thus in the example above, we might want to follow Frege in allowing that both "Your hat is brown" and "Your brown is brown" are significant because both "brown" and "hat" are proper-names, and such a category is more fundamental than those of physical objects or colours. But then the argument could be applied again with regard to the logical categories of proper-names and function-names. In Chapters 3 and 4 we will see why this difficulty was so troublesome for Russell's judgment-theory.

10

1. Introduction

stein, by contrast, inveighed against all of these prerequisites for the significance of the book in his letters to Russell in the period of 1912-1913. It is not easy for us today to appreciate why these criticisms were so serious for Russell's project. But if we recall the dependencies I mentioned above—logical concepts are defined in terms of ontology by means of a theory of propositional judgment, and scientific and mathematical concepts are then developed within logic—we will see that these were indeed grave problems for Russell. If we need an axiom securing the existence of an infinite (or indeed of a finite) number of objects to be true in order for other propositions to be significant, then parts of what Russell regarded as logic depend on empirical requirements not just for their truth but for their very meaning. If the significant application of every proposition in Principia depends on a meta-linguistic check that all its names denote, and that they all denote things of the right types, then the metalanguage itself had better be in order. But a meta-language contains terms referring to both the object language and its referents, so that using one will push us in the wrong direction. Russell wanted to reduce logical concepts to ontological ones, so that truth, meaning and mathematics would be objective features of reality, features inhering in the correspondence between judgment-complexes and judged-complexes. If significant judgments cannot be formulated without meta-linguistic constraints, the whole project will founder. According to Russell, the symbolism of Principia, indeed language in general, depends for its meaning on the underlying intentional judgments. These are, in turn, objective states of affairs. So it is absurd to assume that restrictions implemented in a meta-language could in any sense be used to ensure that judgments be significant. On the contrary, it is the significance of the judgments that grounds intentionality, and thus the meanings of signs. In Chapter 3,1 show how this sort of a sense-truth regress developed out of Russell's theory of judgment. The "riders" required by such a theory are of three sorts: riders on the types of the elements entering into Russell's judgmentrelation, riders securing the existence of what Russell called "logical forms", and riders securing the existence of the objects involved in the judgment. I concentrate in my analysis on the first two of these, since the regress argument ensuing from the second, which leads to the doctrine of the indestructibility of objects, is well-known in the literature as a result of Anscombe's work. 8 In Chapter 4, I reconstruct Wittgenstein's version of this theory (his "copulatheory"), and show how his ultimate break with Russell led him to the theory embodied in the Tractatus. On this view, the possibility of significant elementary propositions depends on the existence of two isomorphic spatial structures, the one consisting of the field of elementary facts, and the second of the field of Above all in G.E.M. Anscombe. An Introduction to Wittgenstein's »Tractatus«. Hutchinson. 1959.

London:

1.1 Russell's Theory of Judgment

11

elementary propositional signs. This postulate, I argue, represents Wittgenstein's response to the sense-truth regress. Because the significance of elementary propositions cannot be secured by means of supplementary premisses or riders, Wittgenstein assumed that the internal structure of these signs reflects those of the possible facts that they pick out. Only on this assumption, thought Wittgenstein, can we secure the distinction between significant propositions and nonsense without getting embroiled in a regress or in contradictions. These two parallel structures are what Wittgenstein came to call the "logical space". It is this notion that links the properly logicist arguments of Wittgenstein's early work with those that are directly concerned with the philosophy of science, and which draw on the work of scientific epistemologists such as Helmholtz and Hertz. In Chapter 5, I discuss its role in the Tractatus's theory of logic. The first part of this chapter concerns the theory of truth-functions. I then examine Wittgenstein's critique of Russell's and Frege's axiomatic demonstrations of logical propositions. This critique hinges on distinguishing propositions that are absolutely general and empirically true, in other words scientific principles, from those that are logically true. Both of his predecessors had failed, on Wittgenstein's view, to give a binding distinction between these two kinds of propositions, because their axiomatic method derived logical laws from higher ones without adequately justifying the latter. Russell had attempted to do so on ontological grounds, but this meant using a language that contravened (or presupposed) the very logical features of reality it was supposed to specify. Frege's approach avoided such difficulties by means of syntactic definitions and the introduction of both senses and truth-values. Wittgenstein rejected both of these approaches. On the Tractatus's theory, the distinction between the two sorts of maximally general proposition is grounded in their differing relations to the logical space. The treatment of truth-functional propositions is extended in the second half of Chapter 5 to cover quantification. Here I argue that the notion of a logical space first does real work for Wittgenstein only in the context of the theory of quantification. Wittgenstein conceives of quantified propositions as signs that select subsets of the manifold of elementary propositions on the basis of their common features. Thus logical propositions that contain quantifiers make statements about the inherent class structure, that is to say the internal properties of the logical space. These a priori classes are in turn the basis for contingent general propositions that are used as basic axioms in the various sciences. Logical propositions describe inherent, one might say topological properties of that space, whereas scientific laws are concerned with appearances within the space. From a strictly logical point of view, Wittgenstein was already committed to this view once he rejected Russell's and his earlier reductive theories. His fundamental objection had been that these theories destroyed the propositional

12

1. Introduction

content of judgments by dissociating their denoting elements. The structure of the propositional thought, in other words Frege's sense, was therefore irrecoverable in the case that the proposition was false. But he was equally unwilling to admit Frege's third Reich of senses, and since he never shared Frege's indifference regarding the source of truth-value assignments, he continued to hold to a correspondence theory of truth. The need to mediate between these two requirements—senses are required for us to understand false propositions, but neither senses nor truth-values can be introduced as distinct kinds of things—forced the Tractatus's solution. The sense of an elementary proposition is the possibility that its corresponding fact might occur. That of a truthfunctional or quantified proposition consists in those truth-possibilities that are consistent with its being true. The possibilities referred to here are defined as the combinatorial possibilities of the sets of elementary objects. And these compose, for the reasons I have outlined above, a space of elementary facts whose dimensions are basic ontological types.

1.2 Scientific Epistemology and the Theory of Manifolds The second major part of my interpretation involves connecting the notion of a logical space to a distinct intellectual tradition, namely that of early German philosophy of science (in Hermann von Helmholtz's terminology, "scientific epistemology"). I begin by considering what I call the "manifold theory of perception", which developed out of Helmholtz's research in acoustics and physiological optics, where he drew heavily on the work of other researchers such as Hermann Graßmann and James Clerk Maxwell. This account of human perception was taken up and expanded by other researchers such as Hering, as well as students of Helmholtz's such as von Kries and Lipps. By the end of the 19lh century, it was a scientific commonplace, so that we find authors such as Poincare, Boltzmann, Weyl and Carnap assuming it as given in their writings on epistemology. The theory was only ever scientifically well-grounded in the cases of colour-analysis, and to a lesser extent, acoustics. Helmholtz, Graßmann and Maxwell discovered that colours form a three-dimensional continuum that resembles physical space in its having dimensions and ordering relations, and which as a result can be characterised as a "manifold". Even in Helmholtz's early philosophical writings, we find him extending this theory to cover the entire range of human perception. In his later lectures, he claims that the existence of space-like structures in human perception show that there are a priori truths concerning the internal structure of all our various sense-faculties, and which correspond to those Kant discovered in our pure intuitions of time and space.

1.2 Scientific Epistemology and the Theory of Manifolds

13

In fact, no one ever made good on this general claim—no one has ever seriously believed in the existence of a "manifold" of odours. But it had an undeniable appeal to both scientifically inclined philosophers and philosophically inclined scientists, for it suggested that—to speak somewhat anachronistically—all of our possible experiences come packaged in data-spaces. The idea is attractive because, when it is coupled to the belief that mechanics, and perhaps all scientific theories are ultimately manifold-theories, as Hertz, Boltzmann and others argued, it opens the possibility of precisely defining the goal of scientific theory, and offers hope of reaching that goal. If the entire field of perception has an implicit mathematical structure, then the aim of physical theories will be two-fold: phenomenologically, science will seek to correlate the mathematical relations implicit in perceptions with the quantities involved in fundamental physical theory; whereas realistically, it will offer a physical and physiological explanation of why this correlation exists. For instance, if we observe a gradation of differences in our perceptions of various colourintensities and colour-tones, we can attempt to correlate these differences with different objective properties of the hypothetical (hypothetical, because only mediately perceptible) causes of these colour-perceptions, namely the wavelength and amplitude of physical light. Conversely, once we are in possession of such correlations, we can begin to investigate the physiological mechanisms whereby such physical properties of light are transformed into states of excitation of retinal receptors and nerves. Such an epistemological theory views the physiology of the human organism as embodying mathematical transformations of one sort of manifold (defined by the variable parameters of light) onto another one (the three-dimensional manifold of colour-sensibilia). The term for such a transformation is a mapping, in German an Abbildung, which is also the term that Wittgenstein uses to describe the relation of picturing as it exists, for instance, between the grooves in a gramophone record, the sound waves in the air, and the acoustic image that we form of these. Far from being an invention of Wittgenstein's, this is a view of the nature of perception widely held by German-speaking scientific epistemologists of his day. In Chapter 2, I give an outline of Helmholtz's major philosophical monograph, "The Facts in Perception", in which I show how he applied his research on sense-physiological manifolds to problems in the neo-Kantian philosophy of science. I use Helmholtz as my principal exponent of this epistemological tradition for two reasons. The first is quite simply his central role in its development. There is no evidence that Wittgenstein had read Helmholtz himself, but I do not think that this is of great consequence to the interpretational background. Arguments employing the notion of perceptual spaces of this general form can be found at the end of the 19th century in the writings of Fechner, Mach, Boltzmann, Poincare, Lipps and others. In order to make a start at understanding what the contribution of such philosophy to the Tractatus was, I have chosen to

14

1. Introduction

take one representative from this tradition, namely its inaugurator. The other reason to look at Helmholtz's work is its connection to Hertz's theory of science,9 which we know to have had a direct influence on Wittgenstein. In Chapter 6,1 discuss the Tractatus's account of scientific knowledge by looking in detail at Hertz's arguments. My concern is not so much the "picture-theories" of the two authors, which form an evident point of connection between Hertz's Mechanics and the Tractatus, as their shared understanding of scientific and linguistic systems. Both authors think that there is a sharp distinction between a priori and a posteriori elements of theories, and furthermore that this distinction is entirely captured on the semantic side by a distinction between structural properties of a representation space and the contingent events that occur in that space. Hertz's theory is an extreme form of Kant's theory of science, which has been modified in two basic aspects, both of which concern the theory of manifolds. As is well known, Kant views the task of science to consist in tying contingent appearances in space and time back to models in pure intuition. These models employ a purely mathematical concept of matter (a "pure empirical" concept, in Kant's terminology), and they thereby replace implicit mathematical relations between the intensional magnitudes of sensations, what Kant calls the matter of appearances, with relations between the external magnitudes of time and space defined by the movement of purely mathematical particles. Hertz follows Kant's approach quite closely, but he seeks to eliminate completely the non-geometrical components of Kant's pure scientific models, for instance the concepts of force and cause. Secondly, he holds, in the light of Helmholtz's work, that Kant's intentional magnitudes are themselves organised in manifolds. Thus the semantic relation at the heart of scientific models consists in a mapping relation between purely geometric theories of matter and appearances in a perceptual manifold. This relation is indeed a form of "picturing" or Abbildung, and I claim that this is the interpretation that Wittgenstein gave to his theory of logical space once he extended it beyond its logicist origins. But this term and the notion of "picturing" in general is far less important to our understanding of the theories in question than is that of a mapping within spaces of representation. And indeed, when we turn to Wittgenstein's subsequent development, we find that this notion plays a critical role that extends, as I have already suggested, to the Philosophical Investigations.

9

Buchwald and Heidelberger have argued that both in his experimental practice and his philosophical outlook, Hertz was a "Helmholtzian". See for instance J. Buchwald. "Reflections on Hertz and the Hertzian Dipole". In Heinrich Hertz: Classical Physicist, Modern Philosopher. ed. D. Baird, R.I.G. Hughes and A. Nordmann. Dordrecht: Kluwer. 1998. pp. 269280, and M. Heidelberger. "From Helmholtz's Philosophy of Science to Hertz's Picture Theory". In Baird et al. 1998. pp. 9-24.

1.3 Interpreting Wittgenstein's Later Philosophy

15

1.3 Interpreting Wittgenstein's Later Philosophy In the Conclusion, I discuss Wittgenstein's later modification of the notion of a logical space, and the corresponding turn in his understanding of judgment and intentionality. It is not possible to trace out this development up to and beyond the Philosophical Investigations within the confines of this book; however, it is worth stating here the directions such an analysis would take. The connections between the early philosophy I am concerned with and the later notion of a philosophical "grammar" are evident already in the late 1920's, when Wittgenstein wrote his Philosophical Remarks. At this stage, he is still concerned with explaining the "internal connection" that must hold between an intention and its object. He continues to use the notion of a space of possibilities within which the intentional object is situated. And he still holds that there is an essential connection between the dimensions of this space and something he calls "grammar", which he identifies in one passage with the "Theory of logical types". But he now considers all these notions under their temporal aspect: the space in question is a space of actions and events; linguistic statements are instructions for carrying out such actions and for recognising such events when they occur; and the grammatical categories relate not just to ontological ones, but to the varying methods involved in such intentional projection. This temporal extension of the framework of the Tractatus is, in my view, quite faithfully reflected in the shopkeeper language-game that opens the Philosophical Investigations. Here we are shown how an instruction—"Five red apples"—composed of three distinct types of word is mapped by means of operations with charts and tables onto the intended object, the five red apples. The famous questions with which the section concludes—How does the shopkeeper know how what he is to do with the words on the slip, and with his charts, drawers and apples?—is a question concerning the fundamental constraints, or background conditions on the language-game. What ensures that the intention recorded on the slip of paper will be faithfully interpreted by the shopkeeper? What does he need to know in order for the intention to be determinate? In both the early and the late work, Wittgenstein insists that such questions are, at the limit, unanswerable. There is no body of knowledge that will ensure, once I am possessed of it, that I am able to project my intentions onto their intended objects. Of course despite this deep thematic similarity, the reasons offered for this claim in the early and late work are quite different. As we have seen, according the Tractatus, we cannot state the background conditions ensuring that our meaning is determinate, because such statements cannot reach their target. Either they overshoot it by employing functions that undermine the very distinctions they are supposed to establish. Or, if they stay within the bounds of these distinctions, they are revealed to be redundant. Since in the early as well as the late work, Wittgenstein assumes that sense is determinate,

16

1. Introduction

he concludes in the Tractatus that we do indeed have knowledge of the relevant background conditions, but only in a degenerate sense, for this knowledge cannot be stated. What is it then that we know but cannot say? The simplest answer is that we have metaphysical knowledge. We know what kinds of objects make up our world, and we know how they combine with one another to form objective states of affairs. Still, we would be overlooking the central concern of the Tractatus if we took the point to be merely that metaphysical knowledge is transcendental knowledge. For as in the contemporary work of Russell, Meinong, Husserl and Frege, this claim is essentially tied to a more modern concern, which is the problem of intentionality. For Wittgenstein, not only must we have metaphysical knowledge of the structure of the world, but we must also have a language that mirrors this structure perfectly. When I know a thing, I also know its name. I know its form, so that I know how to recognise other things as being of the same form. And I know how to recognise the names of these things as being of the same form as its name. These rigid connections between the elements of the world and their names ensure that I can make unambiguous judgments concerning states of affairs that do not and indeed may never obtain. That is to say, the problem of forming and projecting intentions has a temporal aspect even in the Tractatus, for thinking and intending consist in operating with signs. But since Wittgenstein insists that the rules governing these operations must be inviolable (in that I literally cannot think illogically) it ensues that the signs themselves must hinder their own misapplication. Language must have an absolutely rigid structure precisely to ensure that our intentions are not diverted during our manipulation of the symbolism. Just as the connections holding between physical systems are time-independent in a strictly mechanistic physics, so in a strictly mechanistic language (as we have today in our computer-languages) there can be no slippage in the symbolism between the moment that a thought is formed, and the moment it is verified. Wittgenstein insists on the necessity of a rigid syntax, because he sees it as the only way of ensuring that intentions could be determinate. Now, as I suggested above, there was an alternative to this unusual conclusion on the table already in Principia Mathematica, namely to suppose that the rules I need to follow in order to speak meaningfully are expressed in a metalanguage. Even if it is true that we cannot state the constraints on legitimate judgments by means of propositions concerning the kinds of things that make up our world, still it would seem that we could implement these constraints at the symbolic level. What we need is a "theory of correct symbolism", in Russell's words.10 But this response is unsatisfactory, for it merely shunts the problem up a level. If the meta-linguistic expressions are themselves to be in10

Russell quoted by Wittgenstein in a letter addressed to the former on August 19,1919. Wittgenstein. Notebooks, p. 130.

1.3 Interpreting Wittgenstein's Later Philosophy

17

terpreted as making objective statements concerning things and their names, it is no solution at all. And if we prescind from all reference to the objective domain, than we can do so only by using terms like "true", "significant" or "denotes" without analysing them. That is why Wittgenstein insists that the picturing relation cannot be said: only by assuming that some signs are significant, and then referring to their structural properties can we show what their meaning consists in. But such a practice is clearly dependent on there being such signs, and is useless for the purpose of establishing assertorically the boundary between meaningful and meaningless symbols. In other words, both in the Tractatus and in his work from the thirties, Wittgenstein had acknowledged the importance of grammatical categories. They determine how intentional projection takes place, and they "steer my hand", as he puts it. But the rules that derive from them are either "unsayable" or "unjustifiable", because any attempt to establish them post facto would render them contingent, and sever the internal link that he called "the essence of language". According to the Tractatus, such rules attempt to discern the fundamental classstructure of reality, that is to say the type-structure of the logical space. If we could state them, they would amount to normative prescriptions concerning the use of language that derived their authority from that class-structure. But there cannot be such a normative science according to Wittgenstein. On the other hand, already in the 1914 "Notes Dictated to Moore", Wittgenstein allowed that one might describe the syntactical rules involved in language. The prohibition is only on metaphysical justifications of these rules. For in order to justify the rules metaphysically, we would have to violate them first. This is the paradox with which we began: if the statements embodying normative justifications were true, then they would have to be significant. But if they are significant, then they can also be false. And that means, in the opinion of the young Wittgenstein, that the structure they describe could be otherwise. So it would not be the necessary structure of reality, meaning that the rules with which we began were not the ultimate normative rules. This regress is blocked by the postulate of isomorphism, which is later rendered temporally as the requirement that the Satzform, our actions, and the intended event be "internally connected". But in the Investigations, the notion of an internal connection is systematically dismantled. Wittgenstein denies that we can coherently imagine that there are signs, mental or otherwise, that secure their own determinate projection. Appeals to metaphysical justifications, for instance to ostensive definitions, are rejected as well, because they can always be misunderstood. Similarly, explicit rules of grammar, although they may remove misunderstandings, can never close the justificatory regress. How then is it blocked? Wittgenstein argues that we can appeal ultimately only to the actual usage of others. The regress is capped not by a theory of grammatical and ontological categories, such as the theory of types, but by agreement in forms of

18

1. Introduction

life. This agreement, precisely because it defines the Spielraum within which disagreements are negotiable, is not open to discussion. Rather, it is something that shows itself in our being able to disagree about the things we say.

2. Helmholtz's Perceptual Manifold 2.1 Manifolds and Perception Before turning to the genetic analysis of the Tractatus's theory of logical space, I want to introduce the reader to one strand in the tradition of German "scientific epistemology" from which Wittgenstein derived his notion of a logical space. I will do by analysing two texts of Hermann von Helmholtz's: his widely disseminated philosophical lecture entitled "The Facts in Perception", and its technical appendix. In making this choice, I am not suggesting that Wittgenstein was directly familiar with this work, as he doubtless was with the work of Helmholtz's student, Heinrich Hertz. My reasons for doing so are more modest. Helmholtz laid the scientific groundwork for what I call the "manifoldtheory of perception", for it was his research in the physiology of perception, above all his two mammoth works on physiological optics and acoustics, which controlled the development of this field in the scientific context. And he was also the first person to use these results to develop a modified form of Kantian epistemology of the sciences that drew on these scientific results. An essential characteristic of this epistemology is its claim that all elements of human perception come structured in manifolds: colours, aural tones, sensations of hardness, warmth, etc. are all supposed to be organised in space-like structures. The elementary perceptions of the human subject consist in concatenations of these primitive elements. Corresponding to this spatialisation of sensibilia was what one might call a "reification" of spatial points: on this model, the perception of a colour in space is a concatenation of a spatial point with a colour point, where each of these "points" belongs to its respective manifold. This model of perception became so widely disseminated in the period around the turn of the century, that most scientists and philosophers working in scientific epistemology draw on it to some degree.1 To figure out how exactly Wittgenstein became Within the "psychophysical" tradition inaugurated by Fechner, this theory becomes so selfevident that we find it expounded in 1899 in an issue of the "Sammlung Göschen" (a series of pamphlets on various fields in science and mathematics) by G.F. Lipps as a basic premise of the field. Cf. G.F. Lipps. Grundriss der Psychophysik. Leipzig: Göschen. 1899. Examples of prominent thinkers before and after the turn of the century who discuss the notion of sensation-spaces and their relation to the philosophy of space and perception include: L. Boltzmann. Principien der Naturfllosofl: Lectures on Natural Philosophy 1903 -1906. ed. I.M. Fasol-Boltzmann. Berlin: Springer. 1990. R. Carnap. Der Raum: Ein Beitragzur Wis-

20

2. Helmholtz's Perceptual Manifold

acquainted with such epistemological theories is quite probably impossible, and, for my purposes here, unnecessary. I will content myself with giving an exposition of the views of its first exponent in order to set the background for my discussion o f Wittgenstein's adaptation of this epistemology. When I return to this topic in Chapter 6, it will be to link the discussion of the notion of a logical space with the philosophy o f science of Heinrich Hertz's Principles of Mechanics, in which context an understanding of the basic epistemological approach outlined here will be important.2 The essential characteristics of Helmholtz's epistemology may be summarised as follows: (1) It has a realist and an idealist interpretation, the first of which corresponds to the objective perspective of an observer, the second to the solipsistic perspective of a transcendental subject. Thus it is a monistic world-view with two interpretations, as I explain below. (2) It distinguishes between internal representations, which are "signs" [Zeichen] of external objects or systems, and the external entities themselves, whose actual nature is unknown. (3) It accords no particular significance to our being conscious of internal representations. Consequently it extends the notion of thinking to cover both conscious and unconscious, high- and low-level cognitive processes. (4) It uses an expanded notion of Anschauung or intuition to characterise the field of "possible signs" that may make up our experience. To conceive of something is to conceive of all the possible representations which that thing might bring about in us. Colours, sounds and other sensibilia are structured in manifolds'. they, along with properly spatial intuitions, determine a multi-dimensional manifold of possible primitive sense-data.

2

senschaftslehre. Berlin: Reuther & Reichard. 1922. H.Poincare. La Science et l'Hypothese: Bibliotheque de Philosophie scientiflque. Paris: Flammarion. 1909. Β. Russell. An Essay on the Foundations of Geometry. London: Routledge. 1996. H. Weyl. Philosophie der Mathematik und Naturwissenschaft. Munich: Oldenbourg. 1927. Recently, a full-length study on Wittgenstein and spatial theories of sense-data has been published by Lampert. It gives a far more detailed account of the history and various incarnations of such theories than that I give here. Cf. T. Lampert. Wittgensteins Physikalismus: Die Sinnesdatenanalyse des »Tractatus logico-philosophicus« in ihrem historischen Kontext. Paderborn: Mentis. 2000. This chapter itself appeared in a shortened form, without reference to Wittgenstein, as D.J. Hyder. "Helmholtz's Naturalised Conception of Geometry and his Spatial Theory of Signs". Philosophy of Science. Suppl. 66 (1999), pp. 273-286. I have expanded on the connection between Helmholtz's work on geometry and the colourspace in D.J. Hyder. "Physiological Optics and Physical Geometry". Science in Context. 14 (2001), pp. 419-456.

2.1 Manifolds and Perception

21

As in Kant's analysis of perception, Helmholtz's theory assigns a central role to intuitive manifolds in which the effects of unknown objects are arrayed. Unlike Kant, however, Helmholtz views all aspects of perception as having the structure of a manifold. These manifolds are therefore a condition for the representation of any knowledge of the external (or noumenal world). We can visualise his model as follows: Systems in World cause primitive appearances in perceptual manifold

Primitive appearances are aggregated into high-level representations

World

Mind

Primitive appearances map onto states of unknown complexity in the external world

High-level representations refer to primitive appearances in the manifold

I said in point (1) that this picture allows two interpretations, two kinds of monism. The first is the realist monism of the natural scientist. It says that the mind in the above diagram is not distinct from the world, but part of it: it is one physical system embedded within another. From this realistic perspective, the system in question is a physical one, and the science which investigates it is that of sense-physiology or psychophysics: it investigates the mental lite of the organism from the standpoint of physics. The connections between organism and environment are, however, structured: the perceptual organs and the nervous system of the organism determine how external causes map onto internal states; the boundary layer of nervous system and world (for instance the retina) determines how external states in the world get represented in terms of internal states. The external systems are transduced by this boundary layer in a double sense. On the one hand the internal signs capture real aspects of, and relations among, the external entities which are their causes, for they translate the exter-

22

2. Helmholtz's Perceptual Manifold

nal causes into internal signs. On the other hand the sense organs, in transducing, say, air vibrations into motions of bones in the inner ear and, in turn, into nerve-impulses, not only alter the medium of representation, but also discard many aspects and properties of these external systems. The inner systems are only ever partial copies of the external ones, in that they faithfully replicate some, but by no means all aspects of the systems they represent. But the realist monistic interpretation leaves room for a complementary, idealist one. Having given a realistic description of the perceiving and thinking organism, we can now ask how the world would appear from the point of view of the subject. On the realistic picture, the threshold of the subject's consciousness—the lowest-level class of cognitive acts of which the subject is aware—is of no particular ontological significance. From the point of view of the subject, however, this threshold will be the limit of experiential reality. We think of colours as primitive perceptual acts (or aspects of such) while at the same time acknowledging the existence of physiological processes of which we are unaware, which bring about the subjective experience of colour. But a given consciousness has no internal evidence for the existence and operation of these processes. It will experience colours as fundamental constituents of external reality; whereby "external" is no longer to be understood as "outside the internal world", but, instead, as "situated in space and independent of my will". For this consciousness too there is only one world, although it is divided into portions called "external" and "internal". This consciousness can hypothesise an outside world whose laws are the causes of the regularities in the internal perceptual manifold. But all such hypotheses and theories will get their plausibility and validity only to the extent that they connect to the field of possible experience within which any conceivable entity must make itself known. What is novel in Helmholtz's approach is his expanded notion of a manifold. Unlike in Kant, where we have two a priori manifolds of space and time in which sensations are situated, Helmholtz introduces whole new sets of a priori truths into epistemology—one for each sense-faculty. Thus it may seem puzzling that Helmholtz set this theory to empiricist ends.3 He rejected the doctrine of psychophysical parallelism, which accounted for the aprioricity of geometry by grounding it in a pre-established harmony between mind and world, and insisted instead that what we perceive as the necessary structure of experience was in large measure learned. Thus the title to the second appendix to "The Facts in Perception" reads: "Space can be transcendental, without the axioms being so". That is to say, some aspects of the way we organise experience must be given transcendentally, in other words that portion which is built into our 3

Cf. G.C. Hatfield. The Natural and the Normative. Theories of Spatial Perception from Kant to Helmholtz. Cambridge: MIT Press. 1990, Chapter 5, pp. 165-234, for a detailed account of Helmholtz's empiricism.

2.1 Manifolds and Perception

23

physical natures. But we don't know at what level these aspects are to be found, and conscious experience is a poor indicator. Helmholtz claimed that Euclidean space, as well as the intuitive relations between colours and other primitive sensibilia needn't be among those aspects. He postulates his sensory manifolds in order to reconcile the demand that there be some structure to experience with his own, thoroughgoing empiricism. For it is hard to understand how something (geometry) can appear to us as an unshakeable condition of all possible experience, and yet at the same time be contingent. If the doctrine of psychophysical parallelism were true, then geometry would appear transcendentally true because it was doubly true: true of the external world, and true of the internal one as well. Helmholtz's argues that it works the other way round: there are some unknown fundamental constraints on how we perceive, and these do yield us a primitive network of experience; we construct higher-level structures by observing regularities and relationships within the lowest-level sense-data, which structures we concretise at an early age in the form of perceptual manifolds. We then mistakenly take the properties of these manifolds to be necessary properties of both mind and world. The contingent properties of these manifolds go transcendental on us, without our being aware that this has happened. But these transcendental properties are by no means necessary. They are the deductively necessary consequence of premisses arrived at inductively.4 The connection to Wittgenstein, for which I will argue more extensively in Chapter 6, can be summarised as follows: The Tractatus claims that all significant language must refer to contingent data in a manifold of possible occurrences, which Wittgenstein calls a "logical space". But he is evasive when it comes to telling us what this space consists in. Some remarks made at the beginning of the book (for instance 2.0251, and also PT 2.0252) strongly indicate that the space in question is a phenomenal space, in other words one very much like that first described by Helmholtz in the papers we shall be discussing. But Wittgenstein also talks about spaces of representation in the physical sciences, and about the depictive relationship holding between physical systems and perceptions, which would indicate that the logical space is not phenomenal at all. I assume that Wittgenstein's working model was a phenomenal space, but that he declined to commit himself to this interpretation because he felt that the logicist arguments that we will explore in Chapter 3-5 freed him of this obligation. He had a priori arguments for the existence of the logical space that were better than any intuitive one could be. But in addition, he likely shared the opinion of Helmholtz, which was that high-level perceptual manifolds were caused by physical systems which also involved manifold-structures. In other words, the Helmholtz does believe in a logic which is binding on any thought; however it would appear to be a simple identity logic. See the geometrical constructions at the end of this chapter for an illustration of this view.

24

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relation between high-level conscious experience, and the physical systems of which we are not conscious is a mapping between manifolds. 5 Furthermore, Wittgenstein thought that the fundamental properties of any possible manifold were the very properties that logic would show: since we don't know what the logical space looks like, the most we can do in "completing logic", as he once described his aim, is to anticipate those properties that any such space would have to have. So logic is, in a sense, a general theory of manifolds. As I have already suggested in the Introduction, I think that this epistemological model is only half the story. But I think that it is one that has been largely neglected in interpretations of the Tractatus, in part because commentators have been unfamiliar with the corpus of writings in German that form the necessary background. So my purpose in this chapter is to provide some of that background, even if the choice I make is to some extent arbitrary, and even if it means ignoring Wittgenstein for a while. Let me then turn to two texts of Helmoltz's in which the epistemological theory that concerns us is first presented fully.

2.2 "The Facts in Perception " The two papers I discuss have an intricate publication history. The one is Helmholtz's lecture "Die Tatsachen in der Wahrnehmung" ("The Facts in Perception"), a Rektoratsrede held at the Friedrich-Wilhelms-Universität in Berlin on 3 August 1878. That paper was calculated, as he told his wife, to make even his colleagues "uncomfortable", for "it contained new thoughts ... and it is after all always better that they think me too educated than trivial".6 The second paper took its final form as the third technical appendix to the talk when it was published. The appendix was not, however, a simple addendum to the talk, for it had been published in a longer form and in English, in Mind in April 1878. In fact, the appendix is the technical basis on which the talk rests. The talk presents an onto- and epistemological theory revolving around the analysis of perception, whereas the appendix sets this theory out in greater detail, showing how it may be realistically or idealistically interpreted, as well as how it overturns the thesis that the axioms of Euclidean geometry are given a priori. This last concern drove the whole project: the appendix originally appeared as a re-

5

6

In German, the word for a mapping in this sense is "Abbildung'. "Mapping" is often a better translation of this term in the Tractatus than the term "depiction" or "picturing". L. Königsberger. Hermann vonHelmholtz. vol. 2. Braunschweig: Vieweg. 1903, p. 247.

2.2 "The Facts in Perception"

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ply to one Professor Land of Leyden,7 who had objected to Helmholtz's contention in his "The Origin and Meaning of the Geometrical Axioms" 8 that the question of whether space was Euclidean or not was an empirical one. Land argued that the notion of "imaginability" used by Helmholtz in his "The Origin and Meaning" was "stretched far beyond what Kantians and others understand by the word". 9 Helmholtz responded with a definition of "intuitively imaginable" that not only leaves room for the possibility that our experience of space may be non-Euclidean, but also undermines the utility of any geometry that does not rest on empirical axioms—axioms derived inductively from measurements with rigid bodies. The theory of perceptual manifolds gives us a way of defining "rigid body" as a pure regularity among elementary sensations. According to Helmholtz, a rigid body is an aggregate of appearances in the sensory manifold that we have, for whatever reason, designated as rigid, and by means of which we have imposed a metric on space. The talk and appendix were influential: the concept of physical geometry is one essential part of Einstein's arguments in favour of relativity; the manifold theory of perception diffused into the work of later psychophysicists; the signtheory plays an important role in Hertz's The Principles of Mechanics. And, as I argue here, these ideas determined to a great extent, whether mediately or immediately, the ontology and epistemology of Wittgenstein's Tractatus. My discussion is naturally skewed toward the latter connection, and so I begin by looking at Helmholtz's notion of substance in order to see how he distinguished between objects, of which we are consciously aware, and of which we make "finished pictures", and the elements of primitive sensations, which are not objects in any conventional sense. These elements correspond, in essence, to coordinate values of the representational space. Wittgenstein later put expressed this conception as follows: "That a point in the plane can be represented by means of a pair of numbers ... shows that the represented object is not the point, but the point-fabric [Punktgewebe]".10 I should emphasise that Helmholtz's use of the term manifold was not metaphorical. He appealed in other writings to Riemann's original definition of the concept11 to emphasise the legitimacy of applying the notion to various forms of perceptual structures: 7

J.P.N. Land. "Kant's Space and Modern Mathematics". Mind. 2 (1877), pp. 38-46. H. von Helmholtz. "The Origin and Meaning of Geometrical Axioms". Mind. 3 (1878), pp. 212-225. 9 Land. "Kant's Space and Modern Mathematics", p. 41. 10 L. Wittgenstein. Wiener Ausgabe, Band 1, Philosophische Bemerkungen, ed. Μ. Nedo. Vienna: Springer. 1994, p. 19. " B. Riemann, 1854. "Über die Hypothesen, welche der Geometrie zu Grunde liegen". In B. Riemann. Gesammelte mathematische Werke, ed. H. Weber and R. Dedekind. Leipzig: Teubner. 1876, p. 273. 8

26

2. Helmholtz's Perceptual Manifold Riemann nennt ein System von Unterschieden, in welchem das Einzelne durch η Abmessungen bestimmt werden kann, eine nfach ausgedehnte Mannigfaltigkeit oder eine Mannigfaltigkeit von η Dimensionen. Somit ist also der uns bekannte Raum, in dem wir leben, eine dreifach ausgedehnte Manigfaltigkeit von Punkten, eine Fläche eine zweifache, eine Linie eine einfache, die Zeit ebenso eine einfache. Auch das System der Farben bildet eine dreifache Mannigfaltigkeit, insofern jede Farbe ... dargestellt werden kann, als die Mischung dreier Grundfarben. ... Ebenso können wir das Reich der einfachen Töne als eine Mannigfaltigkeit von zwei Dimensionen betrachten, wenn wir sie nur nach Tonhöhe und Tonstärke verschieden nehmen.... Riemann calls a system of differences in which the individual element can be determined by η measurements, an η -fold manifold, or a manifold of η dimensions. Thus the space that we know and in which we live is a three-fold extended manifold, a plane a two-fold, and a line a one-fold one, as is indeed time. The system of colours also constitutes a three-fold manifold, in that each colour can be represented ... as the mixture of three elementary colours, of each of which a definite quantum is to be chosen. ... We could just as well describe the domain of simple tones as a manifold of two dimensions, if we take them to be differentiated only by pitch and volume ... .' 2

Riemann, by the same token, calls his generalised notion of a manifold a "magnitudinal concept" [Größenbegrifjß and goes on to explain that such concepts are only possible when there are general concepts at hand that allow of distinct determinations. He concedes that there are few general concepts that correspond to his notion o f a continuous mathematical manifold, but he also names physical space and colours as examples. Both writers were in other words careful to emphasise that physical or intuitive space was only one sort of manifold with which w e are familiar in our experience. And both of them conceived of a manifold as a concept with dimensions. Like a conventional concept, which is "determined" by giving its variables values, the concept of a manifold is also "determined" in this manner. But it admits of, so to speak, orthogonal determinations, which select regions of the complex object it describes. The term is in other words applied in a quite general sense at its inception. According to Helmholtz, the components o f our primitive sensations constitute our internal manifold, but they are not objects in the conventional sense of that word: they are like addresses in that space or point-fabric. What we experience as macroscopic objects are regularities in that space: Was wir aber unzweideutig und als Tatsache ohne hypothetische Unterschiebung finden können, ist das Gesetzliche in der Erscheinung. ... Alles, was in der Anschauung zu dem rohen Materiale der Empfindungen hinzukommt, kann in Denken 12

Helmholtz, Hermann von. "Über den Ursprung und die Bedeutung der geometrischen Axiome". In H. von Helmholtz. Vorträge und Reden, vol. 2. Braunschweig: Vieweg. 1903, pp. 16-17.

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27

aufgelöst werden, wenn wir den Begriff des Denkens so erweitert nehmen, wie es oben geschehen ist. Denn wenn "begreifen" heißt: Begriffe bilden, und wir im Begriff einer Klasse von Objekten zusammensuchen und zusammenfassen, was sie von gleichen Merkmalen an sich tragen: so ergibt sich ganz analog, daß der Begriff einer in der Zeit wechselnden Reihe von Erscheinungen das zusammenfassen suchen muß, was in allen ihren Stadien gleich bleibt. ... Wir nennen, was ohne Abhängigkeit von Anderem gleich bleibt in allem Wechsel der Zeit: die Substanz; wir nennen das gleichbleibende Verhältnis zwischen veränderlichen Größen: das sie verbindende Gesetz. Was wir direkt wahrnehmen, ist nur das letztere. But that which we may unambiguously, and may without any hypothetical supposition take to be a fact, is regularity in appearances.... Everything added in intuition to the raw materials of sensation can be resolved in thought, if we take the notion of thought to be so expanded as it was above. For if "conceptualising" means: forming concepts, and we in this conceptualising seek out and group together in a class of objects, those things which in themselves possess the same characteristics: so it follows analogously that the concept must seek to unify that which remains the same at all stages of a sequence of appearances changing in time ... . We call that which remains the same in a change in time without any dependence on other things: substance·, we call the constant relation between variable quantities: the law binding them together. That which we perceive directly is only the latter.13 H e l m h o l t z g o e s on to explain that entities s u c h as light and h e a t w e r e p r e v i ously designated as substances in this sense, w h e r e a s w e n o w k n o w t h e m to b e species of m o t i o n — r e g u l a r i t i e s in s o m e m o r e primitive substrate. Aristotle, for e x a m p l e , a n a l y s e d s e e m i n g s u b s t a n c e s like s u g a r c u b e s into u n i f i c a t i o n s o f qualities such as whiteness, sweetness and hardness, and h e g a v e these qualities a real correlate: the s e n s i b l e f o r m . But f r o m H e l m h o l t z ' s p o i n t o f v i e w , h e w r o n g l y hypostatized the qualities in doing so: h e c o n f u s e d potentialities o f o u r sense o r g a n s w i t h real entities. Helmholtz sees this progression, in w h i c h subs t a n c e gets a n a l y z e d into t e m p o r a r y c o n n e c t i o n s o f , or r e g u l a r i t i e s w i t h i n , deeper substances, as ongoing: in analyzing colours and sounds into regularities of m o t i o n in the ether and in the air, h e is simply carrying the project o n e step further. Since his definition o f thought includes sensory operations of the l o w est order, n o t h i n g h i n d e r s u s in repeatedly a p p l y i n g o u r h i g h - l e v e l s c i e n t i f i c r e a s o n i n g to t h e p r o b l e m o f d e c o m p o s i n g t h e low-level c o g n i t i v e o p e r a t i o n s that yield us the apparent data of our perceptions. F o r " e v e n the m o s t e l e m e n tary representations contain in themselves thinking, and p r o c e e d a c c o r d i n g to

13

H. von Helmholtz. "Die Thatsachen in der Wahrnehmung". In Η. von Helmholtz. Vorträge und Reden, vol. 2. Braunschweig: Vieweg. 1903, p. 240.

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the laws of thought".' 4 We cannot know in advance how deep this analysis may go, nor what the fundamental constituents, the fundamental manifolds, might look like. Nonetheless, at some point our nervous system makes contact with the world, yielding us signs of external objects. Helmholtz of course restricts his discussion to those signs of which he had a description: colours, tones and tactilia. These are differentiated from each other (1) as modalities15 of sensation, where each modality corresponds to a given sense—hearing, sight, etc., and within each modality, (2) by having some value within a scale of possibilities. We are not usually conscious of these primitive perceptions, and certainly not of the conjunctions which make up even a simple perceptual judgment such as "that surface is hot". Helmholtz refers to sensory experience of the deepest sort as the sensation [Empfindung] of sensation-modes [.Empfindungsweisen]. Among these he mentions colours, primitive tones, but also spatial locations—hence the quarrel with the Kantians. But he is clearly open to the possibility that these too may themselves be the objects of further analysis. Such analysis would, however, go beyond the domain of psychology, to the neuro-mechanical level, of which we are never conscious. The term "perception" [ Wahrnehmung] is reserved for the perception of facts [Tatsachen], which are regularities holding among the sensations. Thus the title of the talk "The Facts in Perception" plays on the idea that what we perceive as objects are temporal regularities playing out within the space of sensations. Only such temporal regularities deserve the name picture [Bild], whereas the sensations are merely signs that have no similarity to systems in the outside world. Even when he fails to observe these terminological distinctions, Helmholtz is clear that our traditional conception of an object is illsuited to the elements of our fundamental sensations: Wir haben in unserer Sprache eine sehr glückliche Bezeichnung für dieses, was hinter dem Wechsel der Erscheinungen stehend auf uns einwirkt, nämlich: "das Wirkliche". Hierin ist nur das Wirken ausgesagt; es fehlt die Nebenbeziehung auf das Bestehen als Substanz, welche der Begriff des Reellen, d.h. des Sachlichen, einschließt. In den Begriff des Objektiven andererseits schiebt sich meist der Begriff des fertigen Bildes eines Gegenstandes ein, welcher nicht auf die ursprünglichen Wahrnehmungen paßt. We have in our language a fortunate expression for that which constantly affects us from behind the alternation of appearance, namely: "the actual". This expresses only the action; missing is the concomitant relation to existing-as-substance, which the concept of the real, that is of the factual, includes. In the notion of the objective, 14

15

"Schon die ersten elementaren Vorstellungen enthalten also in sich ein Denken und gehen nach den Gesetzen des Denkens vor sich." Helmholtz. "Die Thatsachen in der Wahrnehmung", p. 240. Helmholtz. "Die Thatsachen in der Wahrnehmung", p. 219.

2.2 "The Facts in Perception"

29

on the other hand, there intrudes generally a notion of 'the finished picture of an object,' which does not fit the most fundamental perceptions.16 The "objects" of our most fundamental sensations do not differ from the objects of our everyday language because of some absolute distinction between perception and cognition. The concept "object" does not fit the elements of primitive perceptions because it invokes an entity, a real correlate of a representation, to which we ascribe attributes. The object unifies the attributes, and is thus conceived of as a thing existing independently of our perceptions, a thing in itself. We are not inclined to talk that way about colours, spatial coordinates or aural tones, nor indeed about simple aggregates of them. It is the regular conjunction of these that we perceive. As will be increasingly important when we come to Wittgenstein, the perception of regularities among elementary perceptions depends on our experience having a basic class-structure: it must be possible to perceive similarities and difference among our most basic experiences. This class structure is equivalent to the topological relations among the elementary perceptions.

2.3 "The Application of the Axioms to the Physical World" So much for the "popular" part of Helmholtz's talk. The paper in Mind, later the third appendix to the talk, takes this line of reasoning considerably further. In "The Facts of Perception", the comparison of modalities of perceptions (colours, tones) to spaces has a familiar ring: we all know about musical scales and colour wheels. But Helmholtz runs his comparison both ways, and it is the second bit which is unfamiliar. Kantian spatial intuition, like the Aristotelian common sense it replaced, is a special mode of perception: in space objects are individuated and external; other external perceptible qualities attach to these things only as they are spatially situated. For Aristotle, the common sense unified the senses and was at the same time present in them all, and likewise for Kant, space and time are a priori conditions of all external experience (although other kinds of pure intuition are logically conceivable). It seems natural for us to think of space as a vessel existing prior to its contents. They must have a cause for being where they are; however, we do not conceive of the space itself, or consequently the geometric properties we may ascribe to it, as requiring a cause. Helmholtz objects that his opponents have misunderstood the significance of these facts: Eben deshalb sind Schopenhauer und viele Anhänger von Kant zu der unrichtigen Folgerung gekommen, daß in unseren Wahrnehmungen räumlicher Verhältnisse 16

Helmholtz. "Die Thatsachen in der Wahrnehmung", p. 241.

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überhaupt kein realer Inhalt ist, daß der Raum und seine Verhältnisse nur transzendentaler Schein seien, ohne daß irgend etwas Wirkliches ihnen entspreche. For just this reason Schopenhauer and many followers of Kant came to the incorrect conclusion that there is absolutely no real content in our perceptions of spatial relations, that space and its relations are only a transcendental appearance, to which nothing actually corresponds17

But colours are, Helmholtz has argued, arranged in a manifold of possibilities, and we do not have visual perceptions without their being coloured. Therefore it makes just as much sense to say that—at least within the visual field—colours are an a priori condition for spatial perceptions as it does to say the converse. The spatial location of an object may be fixed by means of some arbitrary set of three independent coordinates, just as its colour may be fixed by means of its coordinates in a colour chart: Wir sind aber jedenfalls berechtigt, auf unsere räumlichen Wahrnehmungen dieselben Betrachtungen anzuwenden, wie auf andere sinnliche Zeichen, z.B. die Farben. Blau ist nur eine Empfindungsweise; daß wir aber zu einer gewissen Zeit in einer bestimmten Richtung Blau sehen, muß einen realen Grund haben. Sehen wir zu [anderer] Zeit dort Rot, so muß dieser reale Grund verändert sein. But we are in any case as justified in bringing such considerations to bear on our spatial perceptions as on other sensible symbols18 e.g. colours. Blue is only a way of sensing; but that we see blue at a particular time and in a particular direction must have a real ground. If at another time we see there red, then this real must ground have changed.19

As far as individual elementary perceptions are concerned, spatial coordinates are no more and no less properties than colour determinations; however, conceived of in this way, the thing as bearer of properties becomes superfluous. If a fundamental visual perception is taken to be of form "Blue at location x,y,z", then we are just as entitled to regard ( x , y , z ) as predicates of Blue as we are to think of the point ( x , y , z ) as having the predicate Blue. And it is all the more natural to do so once we have admitted that Blue itself belongs to what Helmholtz calls a Qualitätskreis, which is a spatial structure of sense-qualities. A more neutral expression of the fact would read (Blue, x,y,z), where the first slot of the expression contained a colour coordinate, and the last three spatial coordinates. If anything corresponds to the notion of object here, it is the de17

18 19

H. von Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt". In Helmholtz. Vorträge und Reden, vol. 2, p. 403. Cf. Tractatus 3.32. "Das Zeichen ist das sinnlich Wahrnehmbare am Symbol." Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt", p. 403. The last sentence has einer instead of anderer. I have emended it in comformity with the German text of the reply to Land cited in footnote 22 below.

2.3 "The Application of the Axioms to the Physical World"

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terminate values of the coordinates; but they do not, considered individually, look much like pictures of independent entities. That notion forces itself on us only when we begin to assemble such sensations into complex perceptions, by means of shared elements. I stress this point partly because of its connection to the Tractatus terms Sachverhalt and Tatsache, the first being an elementary fact, the second a group of the latter, 20 and partly to highlight Helmholtz's fusion of the notions of space and quality. On the one hand, qualities are arranged in manifolds, they are determinations of concepts, or modalities of perception; on the other hand, spatial properties (locations) are treated as contingent predicates, or merely concomitants, of these other modalities. There are no bearers of any of these properties—spatial, chromatic, acoustic—beyond their elementary concatenations: [Blue, 2,3] is perhaps the bearer of Blue, 2, and 3; but none of the latter has an obvious claim to being treated as a thing. It is perhaps worth noting in passing, that Helmholtz, like Kant and, indeed, most philosophers, shows no interest in taste and smell, and, as far as I am aware, conducted no physiological research on these sense faculties. These two kinds of sensibilia have always been regarded as devoid of conceptual content. When Kant sought, in the third Critique, to devalue instrumental music, the worst he could think of was to compare it to perfume. Helmholtz, living in the century of Schopenhauer, would never dream of such a thing; however it is not a trivial objection to his undertaking that he takes no interest in the other two faculties. Helmholtz's manifold theory makes little sense when applied there, and this makes the general validity of the enterprise seem doubtful, and least in the aggressively philosophical form it takes here. For one must remember that Helmholtz was above all a natural scientist. In his publications on optics and acoustics he described the composition of colours and sounds as physical systems (waves and their properties) and, simultaneously, he carried out physiological investigations of the structure of the retina and of the cochlea, as well as of the optical and mechanical properties of these structures. When he talks of internal signs and pictures, saying that they depict functionally systems in the external world, he is referring to the pathways described in this work. For instance, a vibrating string is a physical system with mechanical properties we can describe mathematically. The string sets air in motion, and the pressure waves in the air map some, although not all, properties of the motion of the string. The bones in the inner ear replicate these motions yet again, they set fluid in the cochlea in motion, and its spiral structure decomposes the waves in the liquid (essentially performing a Fourier transform) into their component frequencies. But Helmholtz had no equivalent theories for taste and smell. Since Helmholtz gave up physiological research after his move to Berlin, that is 20

Α Tatsache in the Tractatus is a "positive" fact, that is to say it is a set of existing elementary facts.

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immediately after the publication of "The Facts in Perception", he never put this unified manifold theory to the test. Except for a sequence of papers on the colour metric in the early 1890's, he did not develop it further as an empirical theory, using it instead as an epistemological model. This suspicion of taste and smell is, I might add, as classical a philosophical prejudice as it is a false one, as Proust so well knew. The assimilation of qualities and spatial locations is not intended to be metaphorical. The entire point of Helmholtz's appendix is to reflate the claim that the axioms of Euclidean geometry, since they are statements about the conditions of experience, are not empirical judgments, and thus cannot be subject to experimental verification. As such a claim would entail that non-Euclidean geometry be unimaginable, Helmholtz argues that: First, he has no better criterion for the imaginability of an unknown object than that we be able to specify in advance all possible sense-impressions such an object would evoke under all possible conditions of observation.21 Second, he points out that Beltrami had already specified a way of mapping pseudo-spherical space onto portions of Euclidean space, thus that "this demand can be met for objects in spherical and pseudo-spherical spaces".22 The question as to which geometry is the geometry of our experience can be given empirical content, and it is therefore also subject to verification. This criterion of conceivability is tightly coupled to the will of the Helmholtzian observer. We learn at an early age to distinguish sensations that result from our acts of will from those which simply obtrude on us, and this means that we come in short order to the realisation that we are able to bring about the presence of various "presentabilia" by transporting our bodies and members to different locations. Even when we are not in their presence (or they not in ours) we can bring it about that they become present to us again through acts of will: Nennen wir die ganze Gruppe von Empfindungsaggregaten, welche während der besprochenen Zeitperiode durch, eine gewisse bestimmte und begrenzte Gruppe von Willensimpulsen herbeizuführen sind, die zeitweiligen Präsentabilien, dagegen präsent dasjenige Empfindungsaggregat aus dieser Gruppe, was gerade zur Perzeption kommt: so ist unser Beobachter zur Zeit an einen gewissen Kreis von Präsentabilien gebunden, aus dem er aber jedes Einzelne in jedem ihm beliebigen Augenblick durch Ausführung der betreffenden Bewegung präsent machen kann. Dadurch erscheint ihm jedes Einzelne aus dieser Gruppe der Präsentabilien als bestehend in jedem Augenblick dieser Zeitperiode.

21 22

Helmholtz. "Die Thatsachen in der Wahrnehmung", p. 230. H. von Helmholtz. "Über den Ursprung und Sinn der geometrischen Sätze: Antwort gegen Herrn Professor Land". In H. von Helmholtz. Wissenschaftliche Abhandlungen, vol. 2. Leipzig: Barth. 1883, p. 644.

2.3 "The Application of the Axioms to the Physical World"

33

Let us call the entire group of sensation-aggregates that may be brought about through a certain definite and finite group of will-impulses the temporary presentabilia, and consequently present, that sensation-aggregate from this group which is just coming to be perceived: so our observer is tied at the instant to a definite set of presentabilia, each of which he may make present by carrying out the appropriate motion in each and every moment he chooses. Thus each one of this group of presentabilia appears to him as obtaining in each moment of this period of time.23 The set of presentabilia corresponds to a physical location, fixed by our awareness that we have not in the immediate past willed any motions, and the sensation-aggregate present to us is the one on which we have just fixed our attention. The fact that we can choose at will which sensation-aggregate among the current presentabilia is present, means that we have the impression that all are present during this period of time. So Helmholtz's definition of imaginability appeals to just this capacity. We can imagine something if we can imagine all possible effects that it would have on our sense-organs from all possible perspectives, i.e. from all possible sets of presentabilia into whose immediate presence we can will ourselves. Helmholtz's argument against the transcendental nature of the axioms of Euclidean geometry depends on this ability of the observer to move about, and to be conscious of doing so. If neither I nor any objects in the room in which I find myself are in motion, then there is no way for me to establish physical congruence between the objects in my visual or tactile field. If I am aware that I am moving, I will notice the curvature of space in the fact that those objects which I thought to be at rest appear to expand and contract. And if I can manipulate the objects directly, I will note that perceived congruence (between, say, the visual impressions of two rulers) need not agree with physical congruence (the successful juxtaposition of the rulers). It would be possible to establish a deductive geometry based on generalisations of such actual measurements, a geometry whose statements always referred to the relationship between bodies being moved in space and in time. Such a geometry would be a physical geometry, not a pure intuitive geometry, whatever the latter may be. The only geometry which matters as far as our experience is concerned is the physical one. Obviously a geometry whose axioms are established empirically need not fear the question of why it applies to experience. It is instead intuitive geometry which needs to justify its existence. Helmholtz had always taken the view that our intuitions of spatial relationships were learned and not native to human consciousness. While he admitted the latter as a possible hypothesis, he found it intuitively objectionable, and the unexpected developments in non-Euclidean geometry offered him a forceful new argument in support of his view. But it was not much more than that, 23

Helmholtz. "Die Thatsachen in der Wahrnehmung", p. 226.

34

2. Helmholtz's Perceptual Manifold

above all because Darwinian theory had offered up a plausible explanation for the assumption of that "pre-established harmony" between mind and world which Helmholtz had so regularly lampooned. 24 In these later works, we see Helmholtz adopting a far more conservative position. The axioms of Euclidean geometry would appear to be transcendental if our minds had a peculiar twofold relationship to the objects of experience, but the question of whether or not there is such a relationship could never be answered empirically. The relationship would b e of the following sort: the causes responsible for the appearance of particular objects at particular locations would act (1) mediately on our consciousness, in other words temporally and in conjunction with sensible qualities, and (2) immediately as well, such that direct measurement (a process depending on both time and distinguishable qualities) was unnecessary. Helmholtz defines his terms in such a way that his argument need be interpreted neither realistically nor idealistically, although it is clear that he prefers the former approach. Either way, geometric propositions would have the peculiar characteristic of expressing time-independent regularities between all possible experiences. They would have that property because they described correlations between elements of our most fundamental perceptions, and they would thus appear to be binding on these concrete and temporal experiences. But their necessity would not draw on a deeper source than their independence of physical measurement, that is, from bodies moving in time and space. And those sorts of truths might well be found to hold for other Qualitätskreise of our most basic experiences, those of colours, acoustic tones, etc. Helmholtz m a k e s one fundamental assumption in this argument: that our representations come to be according to some sort of fixed laws, and that from a diversity in these representations we may infer a diversity in the conditions giving rise to them. 25 The causes of the perceptions in consciousness he terms moments, and he divides them along these lines: Nun finden wir als Tatsache des Bewußtseins, daß wir Objekte wahrzunehmen glauben, die sich an bestimmten Orten im Raum befinden. Daß ein Objekt an einem bestimmten besonderen Orte erscheint und nicht an einem anderen, wird abhängen müssen von der Art der realen Bedingungen, welche die Vorstellung hervorrufen. Wir müssen schließen, daß andere reale Bedingungen hätten vorhanden sein müssen, um zu bewirken, daß die Wahrnehmung eines anderen Orts des gleichen Objekts eintrete. Es müssen also in dem Realen irgend welche Verhältnisse oder Komplexe von Verhältnissen bestehen, welche bestimmen, an welchem Ort im Räume uns ein Objekt erscheint. Ich will diese, um sie kurz zu bezeichnen, topogene Momente nennen. Von ihrer Natur wissen wir nichts, wir wissen nur, daß das Zustan24

25

Cf. G.C. Hatfield. Mind and Space from Kant to Helmholtz: The Development of the Empiristic Theory of Spatial Vision. Madison: University of Wisconsin (Diss.). 1979. Chapter 5: "Helmholtz's Empirism: The Grand Synthesis". Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt", p. 401.

2.3 "The Application of the Axioms to the Physical World"

35

dekommen räumlich verschiedener Wahrnehmungen eine Verschiedenheit der topogenen Momente voraussetzt. Daneben muß es im Gebiet des Realen andere Ursachen geben, welche bewirken, daß wir zu verschiedener Zeit am gleichen Ort verschiedene stoffliche Dinge von verschiedenen Eigenschaften wahrzunehmen glauben. Ich will mir erlauben diese mit dem Namen der hylogenen Momente zu bezeichnen. Now we find as a fact of consciousness that we believe ourselves to perceive objects which find themselves at particular locations in space. That an object appears at a particular definite location and not at another will have to depend on the kind of real conditions bringing about the representation. We must conclude that other real conditions would have to have been at hand in order to bring about the perception of the same object at a different location. There must therefore be in the Real some sort of relationship, or complex of relationships, which determine the location in space at which an object appears to us. I want to call these, in order to refer to them economically, topogeneous moments. We know nothing of their nature; we know only that the coming-to-be [das Zustandekommen] of spatially distinct perceptions requires a distinction of their topogeneous moments. On the other hand there must be other causes in the realm of the real which bring about that we believe ourselves to perceive distinct material things at the same location at different times. I will allow myself to designate these with the name hylogeneous moments }b Topogeneous moments are responsible for an object's appearing at a given location, and not at another. The definition does not say how we identify an object, nor does it make any mention of time. In fact, it expresses a necessary connection between identity and location: if two things are identical with one another then they are, at any given time, at the same location. Hylogeneous moments account for the presence of qualitatively different things at the same place at different times. Here we assume that we do have such a criterion for identity—identical qualities, excluding spatial ones—and that two things not identical with each other must, if they are at the same place, be there at different times. This all sounds more complicated than it is, however. From the example that Helmholtz gives us of "something that must have a ground"—Red at this location now—and the lengthier, less abstract account in "The Facts in Perception", we know that he thinks of primitive perceptual judgments as points in a manifold of possible sensations. Space has a special role to play here, since spatial determinations show up in conjunction with all external Qualitätskreise. I can correlate a spatial location with tactile qualities as well as with visual ones—space is the common sense. Thus Kant, at the opening of the "Transcendental Aesthetic", states that space "is not an empirical concept that can be derived from external experience", because only by means of space can we repre26

Helmholtz, "Die Anwendbarkeit der Axiome auf die physische Welt", pp. 402-403.

36

2. Helmholtz's Perceptual Manifold

sent objects to ourselves as being outside of us, individuated and alongside one another. Space grounds all of the last three aspects of experience for Helmholtz as well; however he denies the conclusion that spatial properties are consequently prior to empirical experience. In essence he rejects the second part of Kant's contention that, "one can never make oneself a representation that there is no space, while one can very well think to oneself that no objects are to be encountered in it".27 To think of a space empty of qualia is, on Helmholtz's account, possible only if we prescind from the qualitative part of the internal manifold. If we do so, we prescind also from the possibility of measurement. For the fact that every Qualitätskreis is associated with the spatial manifold, but not conversely, it does not follow that the latter can be thought without reference to the former. I cannot, so to speak, imagine a meter without imagining a meter-stick. Qualia are as much an essential part of the perceptual manifold as are spatial determinations: Nämlich die im Räume vorhandenen Objekte erscheinen uns mit den Qualitäten unserer Empfindungen bekleidet. Sie erscheinen uns rot oder grün, kalt oder warm, riechen oder schmecken u.s.w., während diese Empfindungsqualitäten doch nur unserem Nervensystem angehören und gar nicht in den äußeren Raum hinausreichen. Selbst, wenn wir dies wissen, hört der Schein nicht auf, weil dieser Schein in der Tat die ursprüngliche Wahrheit ist; es sind eben die Empfindungen, die sich zuerst in räumlicher Ordnung uns darbieten. Namely the objects present in space appear to us vested with the qualities of our sensations. They appear to us red or green, cold or warm, they taste or smell etc., while in fact these sensations belong only to our nervous system and do not reach out into the external space. Even when we know this, the illusion does not cease, because this illusion is the original truth; it is indeed the sensations which present themselves from the beginning in a spatial order.28

The real correlates of such qualia, the hylogeneous moments, are unknown, although we have every expectation that these qualities will, at some level of analysis be "resolved into mechanics",29 i.e. into regular motions of particles in space. In actual perceptions, the effects of the hylogeneous moments are always conjoined with spatial determinations. So the elementary perceptions "white at [2, 3, 4]", "hard at [2, 3, 4]" may combine to form the perception that something white and hard is situated at location [2, 3, 4], Since the species of sensation are finite, we may associate with each point in space a finite vector of qualities, so that the state of each such point at a given time is represented 27 28 29

Kant. Critique of Pure Reason, B38. Helmholtz. "Die Thatsachen in der Wahrnehmung", pp. 228-229. Η. von Helmholtz. "Über das Ziel und die Fortschritte der Naturwissenschaft". In H. von Helmholtz. Vorträge und Reden, vol. 1, p. 379.

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as[x,y,z][A,B,...,Z], where the first three coordinates are spatial, and the following ones are coordinates of various property-manifolds. The causes of the first three coordinates in a given experience are its topogeneous moments, and those of the rest its hylogeneous moments. The properties Helmholtz ascribes to these moments follow directly from this characterisation, if we adopt the simple definition that two entities are identical if they consist of the same vector (or vectors) of qualia. Consider first the hylogeneous moments. They account for the various material things with diverse qualities appearing at the same place at different times, that is, for the fact that we, when confining our attention to the location [x,y,z] observe changes in some of the qualia [Α,.,.,Ζ], Of course "to see another material thing" need mean nothing more than to see another set of qualities at that location: we never assumed a thing which was the bearer of these predicates except in so far as we considered the spatial location itself to be such a bearer. The matter is slightly more complicated when we come to the topogeneous moments. They account for the fact that a given object appears where it does: "other real conditions would have to have been at hand in order to bring about the perception of the same object at a different location".30 In the definition of hylogeneous moments Helmholtz does not speak of objects, but of material things [stoffliche Dinge] where the pairing of hyle and Stoff is deliberate. For it is the changes in the hylogeneous moments associated with a topogeneous moment—the causes of changes in the qualities appearing at a given spatial location—that allow us to mark the passage of time. If we associate with the concept "object" the entity responsible for a set of elementary perceptions, then it is clear that "object" contains "spatially situated" as part of its definition, that a determinate object is always spatially situated. We suppose the real object corresponding to [x,y,z][A,B,...,Z] to be the entity responsible for all the elementary perceptions making up this complex perception. However, that there is one such entity is an unjustified inference, a consequence only of the perceptions being unified by their simultaneous situation at a single location in space. The topogeneous moments are indeed responsible for our perception of this object's being at this location, for anything located somewhere else must need be a different object. But if the unique spatial location is part of the definition of the object, then is the definition not trivial? It says that the topogeneous moments are responsible for an object's appearing where it does, and I have just said that the concept "object" includes spatial situation in its definition. Surely the fact that we see one and the same object in motion overturns it on the first stroke? If an object's definition includes its location, then by changing the location we change the definition of the thing. In answer to the first objection we should recall that the 30

Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt", p. 402.

38

2. Helmholtz's Perceptual Manifold

relation of the spatial coordinates [x,y,z] to the coordinates of the propertymanifolds[v4,ß,...,Z] is asymmetric: the other coordinates are unified in their all being associated with the single spatial location—according to Helmholtz, "white and hard" is a concept, not an elementary perception like "white at [2,3,4]" or "hard at [2, 3,4]". So to say that the spatial location is an essential constituent of the object, that real conditions would have to have been at hand for the same object to appear at a different place is strained, but not empty. Helmholtz's definition refers to a single moment in time—the instant in which the observer directs her attention to a group of elementary representations—and it expresses the following thought: groups of sensibilia which obtain simultaneously at different places are not identical, that is they are not experienced as such. To say that the same object could have appeared at a different place at this moment means: an object qualitatively identical, indistinguishable with respect to the consequences of its hylogeneous moments, but at another location. The value of the definition becomes clear when we answer the second question, and turn our attention to descriptions of moving objects, for the perception of a single object moving through space can be arrived at only by means of a regular connection between groups of the topogeneous and hylogeneous moments. I will illustrate this connection by means of a reduced property/space manifold.

1

2

h

Β

R

h

G

Β

ti

Β

Β

Each row represents two locations 1 and 2 in a finite one-dimensional space. B, R, and G are three qualities from a single property-manifold, which we will take, for simplicity's sake, to consist only of these three qualities. Each pair of cells in a row is associated with a time tn. We consider initially only the first two—t, and t2. Giving the property-manifold its own axis yields three temporal

2.3 "The Application of the Axioms to the Physical World"

39

cuts of our time/matter/space manifold. I will also assume in this example that there must be in each column exactly one filled square: each spatial point must have exactly one of R, B, or G. Helmholtz did not hold this position, for he believed that the manifold of colours as we experience them could be rendered as superimpositions of three primitive colours. However, this simplification does not affect the example materially, since indeed a "completed picture" of the points 1 and 2 would involve many more quality dimensions, each of which would require a separate coordinate system.

t,

R

1

1

2

R G

Β

Β

2

q

1

2

R

B

i

We consider the diagrams for t, and t2 to represent the movement of an object from location 1 to location 2. How are we to describe this in terms of topogeneous and hylogeneous moments? A physical process, according to Helmholtz, is the coming-to-be of certain hylogeneous moments and their consequences (that is, the perceptions they cause) and the procession [.Ablauf] of these moments in conjunction with groups of distinct topogeneous moments. In my greatly simplified case, we will consider the one consequence of some hylogeneous moments (the sensation B) conjoined in succession to the group of two distinct topogeneous moments (the locations 1 and 2). At t, the topogeneous moments associated with 1 are conjoined to some hylogeneous moments giving rise to the sensation of B, and together they give us the perception "B at 1" or B(l), for short. At t2 the hylogeneous moments associated with Β are conjoined to the topogeneous moments of 2, and yield the perception B(2). But these conjunctions alone need not give rise to the perception that a single object has moved from 1 to 2. That perception will only result if the perceptions at 2 and 1, at times t, and t2 respectively, are correlated with this process in such a way that a single quality appears to be displaced, i.e. that Β was not perceptible at 2 at t2, and is no longer perceptible at 1 at time t2. If we had t'2 instead of t2, then there would be a perception of something stretching perhaps, but not of the displacement of a single thing. Indeed, if the perceptual manifold were to change randomly, like "snow" on a television screen, we would not be able to form coherent images of any stable objects, let alone see them move in a regular manner.

40

2. Helmholtz's Perceptual Manifold

"We could be living in a world in which every atom was different from every other, where there was nothing at rest. There would be not the slightest regularity to be found, and our thought-activity would be stilled".31 The perception of a displacement of a B[lue] object from one location to another is thus the consequence of an intricate interaction of topogeneous and hylogeneous moments. High-level, conscious perception is only ever of regularities among the ever-changing associations of the latter. The movement of a Β object appears as the sequence: t,\ B(l), R(2); and t2: G(l), B(2); so the hylogeneous moments associated with the sensation Β proceed from tt to t2 in conjunction with the group of topogeneous moments (1, 2), and, simultaneously other hylogeneous moments come to be in conjunction, from t, to t2, with the group (2, 1). Without such regularities in the effects of the two species of moments, there will be no stable objects to observe. At the same time, the supposition that distinct effects imply distinct causes entails that whenever two sets of regular behaviours differ, there is a difference in the constellations of topogeneous and hylogeneous moments which gave rise to them. So a systematic difference in the behaviours of the physical objects used as measuring instruments would imply systematic differences among the real processes standing behind them, that is, among the causes giving rise to these representations. Now that we have an understanding of Helmholtz's definitions in a simple case, we can perhaps make better sense of his account of physically equivalent processes and their relation to on the one hand physical, and on the other hand pure intuitive geometry: Wenn wir beobachten, daß verschiedenartige physikalische Prozesse in kongruenten Räumen während gleicher Zeitperioden verlaufen können, so heißt dies, daß im Gebiete des Realen gleiche Aggregate und Folgen gewisser hylogener Momente zu Stande kommen und ablaufen können in Verbindung mit gewissen bestimmten Gruppen verschiedener topogener Momente, solcher nämlich, die uns die Wahrnehmung physisch gleichwertiger Raumteile geben. Und wenn uns dann die Erfahrung belehrt, daß jede Verbindung oder jede Folge hylogener Momente, die in Verbindung mit der einen Gruppe topogener Momente bestehen oder ablaufen kann, auch mit jeder physikalisch äquivalenten Gruppe anderer topogener Momente möglich ist, so ist dies jedenfalls ein Satz, der einen realen Inhalt hat, und die topogenen Momente beeinflussen also unzweifelhaft den Ablauf realer Prozesse.32 When we observe that divers physical processes can proceed in congruent spaces in equal periods of time, this means that, in the domain of the Real, equal aggregates and consequences of specific hylogeneous moments can come to be and unfold in conjunction with specific definite groups of different topogeneous moments—namely those which give us the perceptions of physically equivalent parts of space. And if experience then teaches us that every conjunction or every conse31 32

Helmholtz. "Die Thatsachen in der Wahrnehmung", p. 243. Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt", p. 403.

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41

quence of hylogeneous moments that can exist or unfold with that one group of topogeneous moments is also possible in conjunction with every physically equivalent group of other topogeneous moments—well this is in any case a proposition that has a real content, and the topogeneous moments thus doubtless influence the unfolding (Ablauf) of real processes. "Physically equivalent" spatial magnitudes are ones in which the same processes may "exist and unfold" [bestehen und ablaufen] under the same conditions and in equal periods of time. The "most commonly used process for determining physically equivalent magnitudes," he goes on, "is the transport of rigid bodies, such as compasses and rulers, from one location to another". This leaves open the possibility that a "pure intuition" of these magnitudes might also be possible. But however the groups of equivalent magnitudes are to be determined, we still have to establish that they represent physically equivalent ones, if they are to do any work for us. That is, neither the geometry arrived at with ruler and compass, nor the one intuited directly must be the one that best meets the needs of physics. But a purely intuitive geometry can only be applied to the physical world if physical operations are used to connect its propositions to changing phenomena, and so it is at a distinct disadvantage. Now on Helmholtz's account, the behaviours of rulers and compasses are also represented as varying aggregates of primitive sensations, which imply changing connections between groups of hylogeneous and topogeneous moments. A statement about a rotation to the effect that, say,

C

"The rod A'Β may be rotated about Β in such a way that, without the rod AC moving, A' is made to coincide with C" will be a statement with real content,

42

2. Helmholtz's Perceptual Manifold

since, as we have seen, the very identity of the bodies AC and AB as well as, of course, their rigidity, depends on regularities in our perceptions—regularities in no way implied in the thin fabric of Helmholtz's definitions. But this proposition would appear, from a pure intuitive perspective, as necessarily true, and, furthermore, as being quite independent of any facts concerning the actual physical behaviour of a rotating rod. The argument as a whole thus has the form of a modus tollens. The same causes imply the same effects. Both idealists and realists must assume that appearances have some causal basis. In a Euclidean world, objects behave differently than they do in a pseudo-spherical one. Since different behaviours of physical objects correspond to different event-sequences in the perceptual manifold, they imply different connections between the topogeneous and hylogeneous moments, the real causes giving rise to the event-sequences. And since a world in which measuring instruments exhibited pseudo-spherical behaviours can be described by means of such sequences, which would differ from those traced out by such objects in a non-Euclidean one, such a world is conceivable and distinct. Therefore this possibility is real, in its implying a real difference among causes.

2.4 Pre-established Harmony Helmholtz now proposes a shift in perspective that allows us to see how a particular kind of consciousness might come to experience Euclidean geometry as a priori. He suggests, as I mentioned before, that our consciousness might be related to the topogeneous moments in a two-fold manner: Man könnte ζ. Β. annehmen, daß eine Anschauung von der Gleichheit zweier Raumgrößen ohne physische Messung unmittelbar durch die Einwirkung der topogenen Momente auf unser Bewußtsein hervorgebracht werde, daß also gewisse Aggregate topogener Momente auch in Bezug auf eine psychische, unmittelbar wahrnehmbare Wirkung äquivalent seien. One could for example assume that an intuition of the equality of two spatial magnitudes could be brought about directly by means of the action of the topogeneous moments on our consciousness, thus that determinate aggregates of topogeneous moments would be equivalent also with respect to a direct perceptual effect.33

Physical measurement, as we have seen, always assumes regularities in our perceptions which manifest themselves as the appearance of stable, subsisting objects, which in turn may be moved, i.e. put in association with groups of spatial points (the effects of the topogeneous moments). But the spatial points never 33

Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt", p. 404.

2.4 Pre-established Harmony

43

appear divested of sensible qualities, just as the sensible qualities are always experienced as being located somewhere in space. If on the other hand the topogeneous moments could act on us directly, without our needing to measure their relationships, and, furthermore, they did so in such a way that their effects on our consciousness mirrored that of the metric of Euclidean space, then we would intuit a pure geometry: Nehmen wir an, daß die Intensität jener psychischen Wirkung, deren Gleichheit als Gleichheit der Entfernung zweier Punkte im Vorstellen erscheint, in derselben Weise von irgend welchen drei Funktionen der topogenen Momente jedes Punktes abhängt, wie die Entfernung im Euklidischen Räume von den drei Koordinaten eines jeden, so müßte das System der reinen Geometrie eines solchen Bewußtseins die Axiome des Euklid erfüllen, wie auch übrigens die topogenen Momente der realen Welt und ihre physische Äquivalenz sich verhielten. Let us assume that the intensity of the psychical effect whose equality appears in Representation as the equality of the distance between two points should depend in the same way on some three functions of the topogeneous moments of every point, as the distance in Euclidean space depends on the three coordinates of each point—then the system of pure geometry of such a consciousness would have to fulfil Euclid's axioms, no matter how the topogeneous moments of the real world and their physical equivalence were to behave.34

The physical equivalences referred to in the last sentence are the metric of space as determined by physical geometry, by measurement with rigid bodies in time. If both that metric and the metric given by the direct effects of the topogeneous moments on our consciousness were to agree with the Helmholtz/Riemann analytic description of the metric of Euclidean space, then we would have a pre-established harmony between physical and psychological space, a psychophysical parallelism of the sort Helmholtz denied. The kind of intuitive geometry which is here being described is a static, atemporal one. Geometric statements are always statements about "regular [Gesetzmäßige] connections between topogeneous moments",35 and in the scenario Helmholtz envisages, we have direct access to such connections, although of course never to the topogeneous moments themselves. It is as if I considered the space of my perceptions at an instant, abstracting from the other properties distributed within it, and thus also from the possibility of utilising them as a means of comparison. This form of imagination cannot peel this space off from that of my basic experiences entirely: I must be able to identify the direct and indirect effects of the same topogeneous moments with each other, otherwise the geometry based on pure intuitions of space would have no empirical referent, and I would literally be unable to apply it physically. There is here no question of 34 35

Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt", p. 404. Helmholtz. "Die Anwendbarkeit der Axiome auf die physische Welt", p. 403.

44

2. Helmholtz's Perceptual Manifold

the one or the other metric being "correct", that is given by the actual relations of the topogeneous moments to one another outside of perception, for we have no direct knowledge of them. So the two-fold relation Helmholtz describes, though it lends itself to a straightforward realistic interpretation, is, as he maintained at the outset, neutral on the question of realism or idealism. The agreement or disagreement between the two geometries could play out entirely within an idealistic and solipsistic consciousness, without any assumption of an independent reality. The one geometry would be an atemporal one, abstracted from the elementary perceptions that make up our experiences, the other a temporal one, derived from observed regularities among these elementary perceptions. Helmholtz does not believe that this is the situation in which we find ourselves, but he also cannot dismiss it as a logical possibility. Nor does he adopt the position of later interpreters (e.g. Schlick or Reichenbach) who maintained that only the geometry of experience, of measurement, was meaningful. Indeed this latter interpretation results from the coincidence of two chains of influence originating in this article: the one remaining within the physical and mathematical problems discussed here, the other resulting from Wittgenstein's use of similar ideas in the Tractatus. For Wittgenstein proposes that the structure of a logical space, in that it is the space of possible experience, is isomorphic to the deep structure of language. In consequence, it is also the determinant of logical truth. Such a view entails automatically that all propositions that are empirically meaningful express correlations between contingent events occurring in the space. Now, as I have said above, I do not mean to suggest that Wittgenstein directly adapted the arguments advanced in Helmholtz's paper—I don't even know whether he read it. But the forms of argument advanced this paper, which become a commonplace in the time after its publication, recur in a striking fashion in Wittgenstein's book. All the following notions are found in the Tractatus in their logical guise, although with the additional assumption that we must be in possession of a sign-language that mirrors the combinatory possibilities of the elements of the "logical space": (1) The basic data of our experience consist of instantaneous judgments about the existence of primitive complexes or events organised in a manifold. (2) The cognitive operations that combine these primitive data into higher-level ones count as thought in just the same sense as our conscious thinking does. (3) The elements of the elementary facts are not objects in the colloquial sense. They have no existence except in the context such primitive experiences.

2.4 Pre-established Harmony

45

(4) Conscious perception of objects and motions of objects result from constellations of these elementary facts operating in regular connections in time. (5) There is an intuitive a priori set of statements describing fixed properties of the primitive manifolds. These statements are such as define the most general topological properties of the space in question. It is to such propositions that Wittgenstein compares the fundamental propositions of logic. I cannot hope to explain why Wittgenstein came to hold such similar views simply by referring to their presence in this one text of Helmholtz's, nor indeed in any the writings of any given author in the tradition that grew out of Helmholtz's work. The logical reasons for adopting a model of this sort will be outlined in the following chapters. Still, I think it beyond doubt that a model of this sort is what lurks behind the often cryptic pronouncements of Wittgenstein's book. Indeed, it is not to Helmholtz but to his student, Heinrich Hertz, that Wittgenstein directs the Tractatus's reader. But this should not dissuade us f r o m seeing in Helmholtz's epistemological work the deep background of Wittgenstein's notion of a logical space, and this for several reasons. First, the conception of scientific theories held to by Hertz, as well as later figures such as Boltzmann and Poincare, is fundamentally conditioned by Helmholtz's epistemology. C o m m o n to all of these scientists is the conviction that (1) sensory experience is structured in perceptual manifolds, and (2) scientific theories must take the form of manifold-based models. Boltzmann held a still stronger view, claiming that each such model "must in its essence be atomistic, that is, an instruction to conceive of the temporal changes of a very large number of things arrayed in a manifold of three dimensions according to certain rules". 36 From these two premisses, one arrives quite naturally at the conclusion that the business of scientific theories is to construct models in an extended manifold (for Boltzmann, a discrete manifold) of the observed changes of appearances in the manifolds of our direct experience. This view of the relation between theory and experience is so widely held by German-speaking philosophers of science at the turn of the century, that it will most likely remain impossible to determine where exactly Wittgenstein was exposed to it. Instead of detailing all of its various manifestations, it suffices, in my opinion to have some familiarity with Helmholtz's early and influential account. But there is a second reason to read Wittgenstein against the background of Helmholtz, namely the direct connection between Helmholtz and Hertz on the

36

L. Boltzmann. "Die Unentbehrlichkeit der Atomistik". In L. Boltzmann. Populäre Schriften. Leipzig: Barth. 1905, p. 156. Cf. also p. 145, footnote 1, where Boltzmann claims that a visual impression is more likely a "mosaic" than a continuous surface, because the number of elements of the retina is finite.

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one hand, and Hertz and Wittgenstein on the other.37 Both of these physicists held to Kantian views of the nature of scientific theory, indeed the process of "construction" in manifolds outlined in the preceding paragraph represents in my view an extension of the procedure advocated by Kant, in his Metaphysical Foundations of Natural Science. Kant argues there that if we are to have a unified mathematical theory of nature, we must first develop what he called a "metaphysics of corporeal nature". In such a metaphysics, the pure concept of matter is determined with regard to the pure concepts of the understanding by schematising it onto the manifolds of space and time. Such a fundamental scientific metaphysics then provides a framework within which the empirical sciences may be progressively developed by adding empirical postulates.38 These theoretical models will all ultimately connect to the field of immediate experience of sensations in time and space, which is made up of "intensive" (sensory) and "extensive" (spatio-temporal) magnitudes. The approach of neo-Kantian scientific epistemologists such as Helmholtz and his successors is essentially the same. But after Helmholtz, Kant's picture is extended in two fundamental regards: the concept of a manifold is enlarged to include multiple dimensions and, as in Boltzmann and Riemann, "discrete" manifolds; and the spatial character of experience is now seen as encompassing the intensive magnitudes of sensations as well, for sensations are taken to be organised in manifolds. That is to say, in Kant's philosophy, the aim of science is to map sensory experience which is fundamentally not spatial in character onto the pure manifolds of time and space. After Helmholtz, the sense-data of experience are already "spatially" structured, but in the extended sense "spatially" that I have described above. Thus the relation between phenomenal data and the manifolds of scientific theories is a mapping relation, albeit one which will be of great complexity (and whose specification will of course require empirical research).

37

38

Susan Sterrett has recently pointed out similarities between the theory of the Tractatus and model-theoretical approaches to scientific theory that were widely disseminated among the engineering community. I agree with her that there are a plethora of possible modeltheoretical predecessors (all of a more or less Langragian stripe) to Wittgenstein's picturetheory, and I do not see that we have to choose just one. I do find a quite specific neoKantian argument in Helmholtz, Hertz and their German-speaking successors that cannot come from the engineering side of things. The question concerning Wittgenstein's scientific sources still remains to be explored adequately, and I would not rule out any one in advance. Cf. S. G. Sterrett. "Physical Pictures: Engineering Models circa 1914 and in Wittgenstein's »Tractatus«". In History of Philosophy of Science: NewTrends and Perspectives. Vienna Circle Institute Yearbook 2001. vol. 9. ed. M. Heidelberger and F. Stadler. Dordrecht: Kluwer. 2002. pp. 121-136. For a detailed discussion of Hertz's adaptation of this Kantian paradigm, see D.J. Hyder. "Kantian Metaphysics and Hertzian Mechanics". In Vienna Circle Institute Yearbook 2002. ed. F. Stadler. Dordrecht: Kluwer. 2002. forthcoming.

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As I will argue in the concluding chapter of this book, the absolute division in Hertz's and Helmholtz's philosophies of science between a priori "spatial" structures, and their a posteriori contents is replicated in Wittgenstein's analysis of language and cognition. Hertz divides his mechanics into two books, the first one geometric, being a purely mathematical development of the properties of systems considered only as they appear in the intuition of the subject; the second an empirical kinematic portion, in which the range of "thinkable" systems described in the first book is reduced, by means of definitions and Hertz's "Fundamental Law", to those systems that are physically possible. The fundamental law is, on Hertz's view, the only truly empirical proposition in his book. It is empirical precisely in that it restricts the scope of possible appearances in the multi-dimensional space described in the first half ("Book 1"). In the Tractatus, the relation between geometry, a priori space, and the empirical propositions of science is transferred to language, yielding the triad: logic, the space of possible elementary propositions, and the empirical propositions which constrain the possible occurrences within this space. The a priori structure of experience thus sets the bounds on everything that can be meaningfully said. "Mechanics", says Wittgenstein in 6.343, "is an attempt to construct all true propositions that we need for the world-description according to one plan". Logic, he might just as well have added, is an attempt to construct all significant sentences according to one plan. That is the "relative position" of logic and mechanics alluded to in the Tractatus. Before we can treat these topics adequately, we need to get a better understanding of these logical doctrines, and I therefore turn in the following chapters to a discussion of Russell's, Frege's and Wittgenstein's work on the theory of prepositional judgment. As I explained in the Introduction, we should not assume that Wittgenstein adopted the concept of a logical space because of a prior commitment to the "scientific epistemologies" of Helmholtz, Boltzmann and Hertz. It may well be that he had such a commitment, but there are also clear logical problems that compelled him in this direction. Indeed there is no manuscript material prior to the introduction of the term "logical place" in the Notebooks dealing with the philosophy of science. Such material appears much later, and seems to represent a later interpretation of the earlier, logical results. Thus I attempt in the following chapters to follow the chronology of Wittgenstein's own work. Of course this approach should not rule out philosophical analyses which take account of both of these aspects simultaneously. Just to take just one example, it is clear that the notions of schematisation and construction in Kant are connected to that of the determination of a concept. Thus in Kant's philosophy as well, natural science unpacks the meaning of our everyday assertions maximally. Because it gives the most precise schematisation possible of our concepts onto the manifolds of space and time, it is also, in a sense, demanded by our only partially determinate, everyday intentions. It is the

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2. Helmholtz's Perceptual Manifold

most complete specification of our intentions, so to speak. And from this point of view, Kant's philosophy of science rests on the very arguments concerning intentionality and determinate judgment that Frege employed in his logical work. Nevertheless, in order to pursue such connections, one would have already to have made the case I am making in this book. The best way to make it at present is, in my view, to keep these two strands in Wittgenstein's thought distinct, in order that their interrelation in the text of the Tractatus may be as clearly delineated as possible.

3. Russell's Theory of Judgment In the preceding chapter, we have encountered what I take to be one half of the background to Wittgenstein's theory of logical space. This is a theory of perception that asserts both that immediate sensibilia are organised in manifolds, and that the relation of these sensory manifolds to physical reality is a mapping or Abbildung onto their physical causes. In Chapter 6, we will return to this model and its role in Heinrich Hertz's picture-theory of science. But before we are in a position to discuss these themes in connection with the Tractatus, we need to get a clear view of Wittgenstein's reasons for introducing manifolds and mappings into his theory of logic. This means looking not to antecedents within German-language "scientific epistemology", but to the work of Russell and Frege. This treatment will occupy us in this chapter, as well as in Chapters 4 and 5. My concern in the pages immediately following is the origins of Wittgenstein's theory of propositional judgment. As I indicated in the Introduction, the single most important source for our understanding of Wittgenstein's theory is Bertrand Russell's manuscript of 1913, which was published posthumously under the title Theory of Knowledge. On his own account, Russell abandoned this work under the weight of Wittgenstein's criticisms; but for this very reason its study gives useful insight into the problems occupying Wittgenstein when he began working independently on what he called the theory of the proposition. The title of Russell's manuscript is somewhat misleading, for its centrepiece is a theory of propositional judgment. Although many of the planned sections of the book were indeed to concern classical epistemological problems, for instance the relation of scientific knowledge concerning the external world to our immediate knowledge of sense-data, those portions that Russell did complete deal mainly with the metaphysical analysis of propositions. On the basis of an ontology of relations and individuals, Russell develops a non-mental account of judgment in which propositions and Fregean senses are reduced to the same ontological plane as the objects, relations and facts such judgments concern. Far from being a freestanding epistemological treatise, Russell's Theory of Knowledge was intended to mesh with Whitehead's and his Principia Mathematica by providing a metaphysical analysis of the basic "indefinables" employed in the earlier work, such as the terms proposition and function, the predicates true and false, and the basic logical relation of implication.

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In characterising the relations between Principia and Theory of Knowledge in this manner, I am perhaps making too sharp a division, for the theory of propositional judgment presented in the later book has a long history in Russell's thought, one running parallel to the development of Principia itself. Russell, like Frege, developed his theory of judgment in conjunction with his formal research. Neither philosopher viewed his logical work as concerned with pure, uninterpreted symbolisms, rather each sought to justify his logic by means of a theory of propositional judgment that specified the cognitive elements whose workings the symbolism reflected. Thus in the writings of both men, the logical and the epistemological theories are intertwined. For example, the fundamental insight that led Russell to his judgment-theory was arrived at on the same day he first articulated the theory of descriptions'—indeed I will be arguing that Russell's theory is in essence an application of the theory of descriptions to propositional contents—and he gives an early version of the judgment-theory in the Introduction to Principia. In this chapter and the one following, I will show how Russell's difficulties in developing this theory of judgment led Wittgenstein to develop solutions that reappeared as fundamental doctrines of the Tractatus. This is hardly a surprising development, for Wittgenstein, having been enlisted by Russell in 1912 as a collaborator on the judgment-theory, eventually broke with him. He proceeded to develop his own alternative theory, which forms a central part, if not the core, of the Tractatus. Many of the basic doctrines of that theory are by now well known. In the Tractatus, Wittgenstein holds: (1) That an elementary proposition is a fact which, when it is true, stands in a depictive or mapping relation to a corresponding fact. (2) That groups of elementary propositions share a common form, as do the groups of corresponding facts. (3) That this form is determined by the objects making up the facts on the one hand, and the names making up the propositions on the other. (4) That the objects and names are consequently also organised in groups, such that all possible combinations of names on the one hand, and objects on the other, form isomorphic groups—the groups of propositions and facts of (2).

Cf. B. Russell. "On Fundamentals". In B. Russell. Foundations of Logic. 1903-1905. The Collected Papers of Bertrand Russell, vol. 4. ed. A. Urquhart. London: Routledge. 1994, pp. 359-413. In this remarkable unpublished work, Russell develops his theory of incomplete symbols for descriptions, classes and propositions in parallel. See above all the passages beginning on p. 368 for the developments of the theory of non-denoting signs, which deals with classes, names and propositions.

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(5) That the objects, names and their group-membership are invariant: if a name belongs to a given group and has a given denotation, then both denotation and group-membership are fixed.2 Taken together, these five planks of the Tractatus theory of propositional judgment amount to the claim that our language and the world it concerns share a spatial structure. Language and world form a pair of isomorphic manifolds, where the so-called "pictorial relation" consists in a mapping between the elements of the two manifolds. The word for "picturing" and "mapping" in the Tractatus is the same in German: the relation between propositions and facts is one of Abbildung, so that an elementary proposition is "mapped" onto a unique location in the represented space (Wittgenstein occasionally calls this location a "logical place"). It does not seem to me controversial that the Tractatus advances such a theory of language. What is missing from the literature, however, is a unified account of Wittgenstein's reasons for developing just these views, of why he felt justified in postulating such a logical space. On my account, most of these theses can be shown to be explicit correctives to Russell's judgment-theory, so that even when Wittgenstein draws on Frege's work for his solutions to these problems, the result is still best understood in the light of Russell's project. Many of the remarks making up his 1913 "Notes on Logic",3 and the 1914 "Notes Dictated to Moore",4 as well as numerous entries in the Notebooks 1914-1916 are still directly concerned with this collaborative project. These remarks often reappear with slight modification in the Tractatus. Aside from discussing the aims and structures of Russell's and Wittgenstein's propositional theories, I will also show how the principal negative doctrine of Tractatus emerges in this story. For Wittgenstein not only requires that there be a logical space of elementary facts and propositions, but he also insists that we cannot say exactly what this space consists in, indeed that we shouldn't even say that it exists. As I suggested in the Introduction, there is a direct connection between the claim that we cannot state the conditions of significance placed on elementary judgments and the later regress argument concerning rule-following. In both cases, Wittgenstein argues that the distinction between significant language (language that can be right or wrong) and nonsensical utterances cannot derive from a prior understanding of grammatical rules, so long 2

3

4

My interpretation of the Tractatus's propositional theory is probably closest to that of the Hintikkas. In this book, I do not so much assume the interpretational accuracy of 1-5, as try to provide genetic arguments in favour of them. On the status of the different kinds of objects in the Tractatus, see J. Hintikka and M. Hintikka. Investigating Wittgenstein. Blackwell: Oxford. 1986, above all Chapter 2, "The Categorial Status of the Objects of Wittgenstein's »Tractatus«", pp. 30-85. In L. Wittgenstein. Notebooks 1914-1916. 2nd edition, ed. G.H. von Wright and G.E.M. Anscombe. Oxford: Blackwell. 1979, pp. 93-107. In Wittgenstein. Notebooks, pp. 108-119.

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as these are themselves construed as linguistic expressions. This claim is present already in one of the earliest sources we have, namely a letter from Wittgenstein to Russell in the summer of 1913, in which Wittgenstein identifies what he takes to be the core problem with Russell's theory: ... I can now express my objection to your theory of judgment exactly: I believe it is obvious that, from the proposition "A judges that (say) α is in a relation R to b ", if correctly analyzed, the proposition "aRb.v.~aRb" must follow directly without the use of any other premiss. This condition is not fulfilled by your theory.5 This letter has been the subject of sporadic commentary over the years; however, only since the discovery of the Theory of Knowledge manuscript has its significance for Wittgenstein's and Russell's work in the period in question become evident. For in criticising Russell's theory of judgment, Wittgenstein was also criticising his own efforts in 1912 to develop a similar theory. I have already indicated why this criticism was unsettling to Russell. He wanted his theory of judgment to reduce intentions to the plane of facts. If there was to be a distinction between meaningful and meaningless language (or between meaningful and meaningless thought), this distinction would have to be reflected in the judgment-relation at this factual level. If the theory of judgment was unable to secure the significance of elementary judgments here, then no appeals to linguistic or mental resources would avail. In order to see what specific demands Wittgenstein was making in this letter, we need to look at this project in some detail. Russell's propositional theory sought to eliminate senses or intentions by replacing them with what Russell called "judgment-complexes". Following a strategy of "contextual elimination" on the pattern of the theory of descriptions, Russell reasoned roughly as follows. Suppose one regards a proposition as the name of a fact, so that the meaning of the proposition is the fact that it represents, just as the meaning of a name is its bearer. What then does the proposition mean when it is false? A standard response, for instance that of Frege, is that the meaning of a proposition is an intentional entity that can continue to subsist even when the intended fact does not. Whether or not the proposition is true, its meaning is Frege's sense (or Meinong's object, or Husserl's intention), and not the fact. But Russell did not want to admit intentional entities, for he felt that any such solution led to idealism. Instead, he argued that propositions, like names of classes or definite descriptions, were pseudo-names or "incomplete symbols". He argued that a proposition is abstracted from a context which "completes" it, and this context is an intentional act of a subject, for instance an assertion or a judgment.

5

Wittgenstein. Notebooks, p. 122. The remainder of the letter concerns a luncheon-date.

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In contrast to the theories of his idealist opponents, Russell's intentional act does not involve a single entity, such as a proposition or its name. Such an act occurs when a judgment-relation holds between the judging subject and the components of the unique fact which would exist were the judgment true. These components appear as distinct relata of the relation, so that the intended content in no sense depends on the existence of the intended fact. The theory requires, as we shall see in greater detail in the following, that for each possible fact, there must be a second relational fact composed of the very same elements, which can subsist even when the fact does not. Russell calls this second fact a "judgment-complex". It describes the fact whose existence it asserts. Loosely put, Russell construed the proposition "aRb" as saying "There is a complex fact with a two-placed relational structure, and the first element of the structure is a, the second is b, and the relation involved is R". In this expression, all the elements of the possible fact aRb reappear, but they do so in dissociation. Thus, Russell reasoned, they can be related to one another and to a subject by means of a relation, and this relational complex will capture the intentional content of the original proposition. No single element of the judgment corresponds to the proposition or its sense. So Russell's theory replaces propositions which seem to name facts with d escriptions of these facts. But there is a second twist to it that must be emphasised if we are to understand its full scope, which is that Russell's is a theory of "direct reference". The elements that are involved in Russell's judgmentcomplexes are not the names of objects and relations, but the objects and relations themselves. This approach is indeed required by the anti-idealist and reductive thrust of the project, for there would be little point in eliminating propositional senses if one were to employ senses of names later in the reduction. According to Russell, when a subject forms a judgment concerning a possible fact, he stands in an unmediated relation to the very objects with which his judgment is concerned—objects with which he is acquainted (to which he is related by the relation of acquaintance). This is a bold move on Russell's part. The advantages are obvious enough: on this theory, it is not only the meanings of spoken or written expressions that depend on thoughts, but thoughts themselves are brought down to the same ontological plane as the facts they concern. One desirable result is that logic and mathematics can be interpreted as objectively valid theories, differing from the natural sciences in the scope and generality of their propositions, but not in their semantic domains. For every true judgment consists in a direct and determinate relation between a thinking subject and an objective state of affairs. So a judgment about a relation between judgments, for instance the judgment that ρ z> q, is therefore also a purely objective statement. It asserts that there is a particular relation between two judgment-complexes, namely those two from which the propositional contents ρ and q are abstracted. These judgment-complexes are in no sense mental, for they are

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just as much a part of the objective world as the facts with which they are concerned. The main disadvantage is that by reducing meaning to the plane of objects, Russell cuts off his access to all meta-logical resources. Russell cannot, for instance, appeal to a theory of symbolism, as Whitehead and he had done in Principia, to regulate the syntax of legitimate propositional judgments, because the relations of judgment and of acquaintance are ontologically prior to any name or proposition. Symbols have meanings only when they are correlated with judgments, and judgments are not concerned with symbols (symbols are not involved in acts of judgment) but with things. If some combinations of symbols count as significant propositions, whereas others do not, then this can be so only because they refer to significant judgments. A further consequence of Russell's approach is that it forces us to concede, as Wittgenstein came to argue, that the objects with which one's judgments are concerned always exist. This requirement alone virtually forces one to adopt an atomistic theory, if not a sense-datum theory.6 In order to get a sense of Russell's purpose in developing the judgment-theory, as well as the reasons for which the notion of a constraint on significance came to be such a problem for this theory, I will outline in the following sections the successive stages in Russell's thought that led to the formulation of the theory. With that analysis in place, I will return to Wittgenstein's objections, with particular attention to the letter from Wittgenstein to Russell that I cited above and to his later criticisms in the notes dictated to Moore and Russell in 1913-14. I will begin by looking at Russell's definition of logic in the Principles of Mathematics, in which he argued that logic (and thus mathematics) is singled out by its complete generality. Such a complete generality required, in turn, that the variables involved in the so-called "logical propositions" be completely general in their application. In consequence, each such proposition had to have what I shall call "significance constraints" which restricted the proposition in question to its proper domain of objects. Quite simply, in order to ensure absolute generality, every proposition had to be of the form: φχ only if χ is of type T. As we shall see, this approach was carried over to Principia Mathematica, with the important difference that the significance constraints were now posited as implicitly given by the functions appearing in the propositions. This did create problems for the theory of logic employed in Principia, but Whitehead and Russell were able to finesse these difficulties by arguing that such constraints, in those few cases where one had to make them explicit, could be interpreted as 6

Since Russell supposed that we had to be "acquainted" with the objects entering into judgments, it is very hard to see how I could make objective judgments concerning, for example, places I have never visited, unless I was in some sense acquainted with these places. On a sense-datum theory, one can evade this difficulty by arguing that I can describe these places as collections of sense-data that I would experience were I to go there.

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constraints concerning the use of symbols. But such constraints on significance also posed a challenge to the reductive programme of the judgment-theory, for they forced one into the position that Wittgenstein describes in his 1913 letter as "obvious". If functions are to implicitly constrain the range of their application, then this can only be because there is an implicit difference between those substitution instances of a function that correspond to a significant judgment, and those which do not. There must be, in other words, a difference between "Your hat is brown" and "Your brown is brown" at the level of the judgmentrelation, so that there is a significant judgment corresponding to the first, and there isn't one corresponding to the second. And this difference must be such as to ensure that a judgment-complex corresponding to "Your brown is brown" cannot in fact occur.

3.1 The Origins and Purpose of Russell's Judgment-theory 3.1.1 The universality of logic in the Principles of Mathematics As both Griffin and van Heijenoort have argued, Russell had from the outset conceived of logic as an absolutely general science, indeed he took this to be its defining attribute. Thus he defined pure mathematics in the Principles of Mathematics as "the class of all propositions of the form 'p implies q\ where ρ and q are propositions containing one or more variables, the same in the two propositions, and neither ρ nor q contains any constants except logical constants".7 Such propositions, which he termed "formal implications", had "the elegant property of being always true".8 Formal implications were therefore to be distinguished from other sorts of implications which, although they were true within a restricted domain, were nevertheless not universally valid. For Russell, as for Frege,9 the suggestion that logical propositions should have only restricted ranges would imply that logic was not a truly universal science. Conversely, the mark of a logico-mathematical proposition is its universal validity. So Russell's variables extended over all objects (or "terms") in the universe, and their range was unrestricted: even when one substituted proper-names for the propositional variables in a formal implication, the result had to be true if the implication was to qualify as a logical or mathematical truth. This led to some slight complications in the structure of such propositions, because one had to ensure that the range of these variables in a logical proposition was ex7 8 9

B. Russell. The Principles of Mathematics. 2nd ed. London: Allan & Unwin. 1937, p. 3. Russell. The Principles of Mathematics, p. 38. J. van Heijenoort. "Systeme et Metasysteme chez Russell". In Logic Colloquium '85. Proceedings of the Colloquium held in Orsay, France, July 1985. Amsterdam: North Holland. 1987, p. 113.

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plicitly restricted to the kinds of objects for which the proposition in question was meant to hold. Consider, for example, the expression p v ~ ρ from the point of view of the Principles of Mathematics .10 This prepositional function yields a proposition whatever values are substituted for ρ: ρ can be replaced by Socrates just as well as it can by Socrates is a man. But in the case of the first substitution, the disjunction is false, whereas in the second case it is true. Thus Russell prefixed pv~ ρ with a constraint equivalent to "p is a proposition" (the tautology ρ zd p) in order to transform it into a prepositional function that is true on all values o f p , i.e. p^>p.ZD.pv~ ρ Suppose now that we substitute Socrates" in for p. On Russell's definition of implication in the Principles of Mathematics, the antecedent constraint, Socrates zd Socrates is false, so that the implication as a whole remains true even when its argument is not a proposition. By prefixing logical propositions with appropriate constraints, we ensure their universal validity. In this early theory, propositions are clearly "entities" in the very sense that Russell was denying when he wrote Theory of Knowledge. It is because they are entities that there can be a confusion between objects and propositions as regards substitution. This confusion parallels that which we find in Frege, for whom propositions and names of objects are both proper-names. The notion of a constraint on significance is, in turn, introduced to ensure that variables take the appropriate significance ranges. When the wrong kind of substitution is made, the antecedent will be falsified, and thus the proposition will remain true even for objects falling outside its intended range of application.

3.1.2 Principia Mathematica and the theory of logical types By cipia types, tional 10

11

1910, when Whitehead and Russell published the first volume of PrinMathematica, this approach had been compromised by the theory of which rendered significance constraints on the arguments to a preposifunction useless. The bulk of the propositions of Principia Mathematica

Here I follow Griffin's analysis in N. Griffin. "Russell on the Nature of Logic". Synthese. 45 (1980), pp. 125-128. For Russell in this period, Socrates is not a symbol, but the philosopher himself, just as ρ is not a symbol, but a prepositional entity, which is neither mental nor linguistic.

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are what Whitehead and Russell called "systematically ambiguous'" 2 propositions. Their variables are unbound, and they function as schemata for indefinitely many universal propositions in each of which the variables are bound to a given type. Whitehead and Russell had to make this distinction, because they wanted to meet two demands simultaneously: on the one hand, they wished to derive propositions with the same absolute universality that Russell had ascribed to the propositions of pure logic and mathematics in his Principles of Mathematics; on the other hand, they had to ensure that the problematic substitutions which gave rise to the various paradoxes would be blocked by limiting the range of variables to the appropriate types and orders. Their solution was to propose that propositions with real variables did indeed have the complete generality demanded by Russell's earlier conception, but that they only did so by means of their "systematic ambiguity". The paradoxes forced one to regard truth-functions at the various levels of the propositional hierarchy as distinct functions, and this meant that an axiom such as p v ~ ρ had to be considered to have a distinct sense for each type of proposition that might be substituted for the variable p. Propositions with bound variables, on the other hand, contained an implicit restriction on these variables. These could take all and only those values on which the proposition in question would be "significant", which Russell and Whitehead defined as being "either true or false". This restriction could indeed only be implicit, because, as they explained, "... statements concerning the significance of a phrase containing "φζ" concern the symbol "φζ" ... . Significance is a property of signs".13 That is, if one seeks to constrain the arguments to a function (pi14 to a particular type by prefixing it with a function Cx, so that Cxzxpx will be a formal implication on the model of the Principles, then either: Cx is significant for all and only the arguments for which (px is—in which case the constraint would be redundant; or, Cx is false (thus significant) for an argument outside the type of (px's arguments, and this is absurd.

12

B

14

The term "typical ambiguity" is more common in the literature today.

A.N. Whitehead and B. Russell. Principia Mathematica. vol. 1. 2nd ed. Cambridge: Cambridge University Press. 1925-1927, p. 48. Whitehead and Russell use the circumflex to indicate that we are talking about the function (px, as opposed to the proposition φχ, which contains a real variable and asserts a (single) arbitrary value of the function. The latter is in turn distinguished from the proposition (χ).φκ, which asserts all the values of the function φχ with arguments of the appropriate type.

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Attaching a supplementary restriction on a real variable to ensure that it would take only significant arguments therefore forced one to interpret the entire proposition as one "dealing with the symbols rather than directly with the objects denoted by the symbols".15 Omitting the restriction, on the other hand, would require one to invoke the notion of systematic ambiguity, thus to withhold strict objective significance. The notion of a bound variable was intended to steer a course between these two unattractive alternatives: quantified propositions, whose variables were "bound" to a type, would indeed make definite statements about objects, since the functions involved would not be systematically ambiguous. And because the type-restriction was implicit, they would not be mere statements "dealing with the symbols" either. In short, only propositions with bound variables correspond to judgments. Propositional schemata containing real variables are symbolic expressions that refer to groups of distinct judgments by virtue of their ambiguity. They are not genuine assertions. This solution meant, however, that the functions entering into propositions with bound variables would somehow determine their own ranges of significance. A quantified proposition was to be interpreted as the assertion of all significant values (all instantiations which were either true or false propositions) of the function over which the quantifier ranged: The proposition "all men are mortal" is equivalent to '"jc is a man' implies 'x is mortal,' with all possible values of χ ". Here χ is not restricted to such values as are men, but may have any values with which "'χ is a man' implies 'x is mortal'" is significant, i.e. either true or false. Such a proposition is called a "formal implication". The advantage of this form is that the values which the variable may take are given by the function to which it is the argument: the values which the variable may take are all those with which the function is significant.16

This solution required that one explain what a function was, and how it "gave the values" that its variables might take. And this explanation was complicated by Principia Mathematical new account of propositions, that is to say by an early version of the judgment-theory. Furthermore, Whitehead and Russell were not able to complete their book without occasionally having to restrict the ranges of variables explicitly, by means of functions defining the type, that is to say the significance range, of a given variable. On the other hand, the problem of incorrect substitution we looked at above had now been removed by the judgment-theory. Socrates and Socrates is a man are not intersubstitutable on the new theory of propositions, because the latter is not an entity in its own right, but is instead an "incomplete symbol", like a description or the name of a class. But this gain had come at a cost, for Russell 15

16

Whitehead and Russell. Principia Mathematica. vol. 2, p. 34. The passage from which this phrase is taken is quoted at length on p. 73. Whitehead and Russell. Principia Mathematica. vol. 1, pp. 45-46.

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now had the problem of ensuring that the judgment-complexes from which such incomplete symbols were abstracted were all themselves significant. To see this, one need only consider the relation between propositional functions and their instances in Principia. Whitehead and Russell took the values of an elementary propositional function .Fav ~ Fa becomes a formal implication, with no implicit restrictions? What we need are functions, say T0 and T„ which pick out all and only those entities which are appropriate arguments for ει. The full conditional would then read, C:

T0(a).T\(F):z}\e^a,F)=>Fav~Fa

This means that "if a is an individual and F is a predicate, then if they stand in the ε,-relation, ' F a ' is a significant proposition". In adding these typerestrictions we have not disallowed the possibility of complexes such as ex(F,a). We have said only that they do not correspond to significant propositions. As we saw in Chapter 3, this is one of the difficulties that Wittgenstein pointed out in Russell's version of the judgment-theory. He objected that the very notion of a function, elementary or otherwise, which is both true of the members of one type and false of things outside it, is absurd. If Tax is such a 11

12 13

14

Cf. J. van Heijenoort. "Systöme et Metasysteme chez Russell". In Logic Colloquium '85. Proceedings of the Colloquium held in Orsoy, France, July 1985. Amsterdam: North Holland. 1987, p. 113. The references in this paragraph are taken from this short, but very informative paper. Russell. Principles of Mathematics, pp. 40-41. "Mais tous deux sont d'accord pour penser que la logique repose sur un univers unique et ne devrait pas s'abaisser ä considerer, successivement, de soi-disant univers de discours, univers desseches dont on peut changer a volonte. ... Une premiere cons6quence d'une telle conception, c'est que les quantificateurs liant des variables individuelles vont s'dtendre ä tous les objets, c'est-ä-dire ä tous les objets dans l'univers." J. van Heijenoort. "Systeme et Metasysteme chez Russell", p. 113. Cf. G. Frege. Begriffsschrift: Eine der arithmetischen nachgebildete Formalsprache des reinen Denkens. Halle: Nebert. 1879. Reprinted Hildesheim: Olms. 1969. §11, p. 19.

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function, and, say, ~ ToA is true—A does not belong to the type Ttx—then A is a significant argument for Tux. But then it ensues that A does belong to the type Τϋχ, and we have a contradiction. T0 and T{ must therefore be true of all arguments for which they are significant, from which it also follows that there are no arguments on which they both have truth-values. Suppose we try to use them to ensure the significance of an elementary copula. Rewriting C as a single implication, we get, C': T.iamne^n^.Fav

~ Fa

At first glance, it would appear as if the conditions under which the antecedent is true are more restricted than those under which e, alone is true. But this appearance is deceiving: if xTax15 is a type, and the functions T0x and £{(x,y) are sometimes both significant for some x, then there cannot be an χ for which εχχ,γ) is significant and not Ταχ, and similarly for T,y. If ε,(χ,χ) can take an argument that does not belong to xT0x, then either: (1) xT0x is not a type, or (2)

x£\(x,y) and xT0x never have common members, i.e. xT0x is not the type of χ in Ex(x,y).

Thus the supposition that type-restrictions could be usefully added to ε, (χ,y) leads to a contradiction. The contradiction can be avoided in three ways: (1) We can assume that the range of significance of χε^χ,γ) and xT0x coincide; however, that would make the introduction of T0 and T[ superfluous. (2) We can suppose that our functions T0 and Ti are defined at a higher order, and thus that they can range over both individuals and predicates without this implying that the latter belong to the same type. But as Griffin and Somerville argue, this would make the significance of elementary judgments depend on higher-order judgments, which leads to an unacceptable regress. (3) We could follow Russell and Whitehead's own prescription in Principia Mathematica, and interpret statements concerning types as statements about symbols as opposed to their referents. But this leads to a similar regress, for it implies that judgments depend on symbolic expressions for their significance, whereas the aim of the judgment-theory was to describe the object-level structures from which the meanings of symbols depend. Wittgenstein's reaction to this difficulty is given in the long letter of January 191316 above: the structure of the various elements in making up a proposition must, through their structure, guarantee its significance, since no supplementary 15

In Principia Mathematica, the expression**)» denotes the class determined by the function

16

Quoted on p. 78.

φχ, i.e. {χ : φκ\.

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riders will help. This account is obviously very close to Frege's implementation of type-restrictions at the symbolic level: we define the various types of signs not by differentiating among the universe of objects named by these signs, but by syntactic operations on sentences which we know already to have meaning. Functions that straddle the types cannot arise, and functions coextensive with the types, while perfectly possible, are not needed as hypotheses for limiting the range of the former, polytypic (and thus impossible) functions. From this point of view, a definition of a type is impossible, if by definition we understand the setting of limits to a concept. The types are maximal and discrete classes, and they cannot be delimited. As Wittgenstein put it to Russell: "We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other".17 As I have already suggested, it is quite possible that Wittgenstein abandoned his approach as a result of Frege's criticisms. In contrast to Russell's antiintentional theory, Frege's presupposes that the significant propositions whose analysis yields the function-names and proper-names of our logical language have a recognisable internal structure: if they did not, we could not very well parse them. A theory suggesting that the proposition, as it is judged, must dissociate its elements from one another is, on his view, absurd. For in dissociating these elements, we obliterate the information we need to distinguish meaningful and meaningless combinations of them. This problem is particularly critical when the proposition is a false one. In "Die Verneinung" Frege inveighs against a theory of negation in which "the negation of a thought is to be grasped as the dissolution of the thought into its components",18 and these remarks apply quite well to Russell's and Wittgenstein's "theory of symbolism". He rejects such a theory of negation with the argument that the persons judging (in this case, a jury) must recognise what they are denying, when they judge that something is not the case: Ist nun das Verneinen eines Gedankens als ein Auflösen des Gedankens in seine Bestandteile aufzufassen? Die Geschworenen können durch ihr veraeinendes Urteil an dem Bestände des in der ihnen Vorgelegten Frage ausgedruckten Gedankens nichts ändern. Der Gedanke ist wahr oder falsch ganz unabhängig davon, ob sie richtig oder unrichtig urteilen. Und wenn er falsch ist, ist er eben auch ein Gedanke. Wenn sich, nachdem die Geschworenen geurteilt, gar keine Gedanke [sie] vorfindet, sondern nur Gedankentrümmer, so ist derselbe Bestand schon vorher gewesen; ihnen ist in der scheinbaren Frage gar kein Gedanke, sondern ihnen sind nur Gedankentrümmer vorgelegt worden; sie haben gar nichts gehabt, was sie hätten beurteilen können.19 17 18

19

Wittgenstein. Notebooks, p. 98. G. Frege. "Die Verneinung" in G. Frege. Logische Untersuchungen. 3rd. edition, ed. G. Patzig. Göttingen: Vandenhoeck & Ruprecht. 1986, p. 59. Frege. "Die Verneinung", p. 59.

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4. The Breakdown of Wittgenstein's Copula-theory

So is the negation of a thought to be understood as the dissolution of a thought into its components? The jurors cannot through their judging change anything in the constitution of the thought expressed in the question set before them. The thought is true or false quite independently of their judging correctly or incorrectly. And if it is false, it is still a thought. If, after the jurors have judged, there is no more thought to be found, but only thought-rubble, then the same situation must have obtained beforehand; they were not proffered a thought, but only thought rubble; they had indeed nothing on which they could have passed judgment. This argument is indeed very close to those made by Wittgenstein in the "Notes on Logic",20 which were dictated in the year after Wittgenstein's meeting with Frege concerning his and Russell's theory. Here, Wittgenstein insists that "however, for instance, 'not-p' may be explained, the question of what is negated must have a meaning". Although Wittgenstein's and Russell's theory was not only a theory of negation, it was, as we have seen, most troublesome on just this point: What is it that we take to be true when we assert a proposition which is in fact false? And how can we be sure that what we are asserting is something that could be true, in other words that we are not simply asserting nonsense? Since the entity whose existence is asserted doesn't exist, we must have a clear notion of the conditions under which it might exist. For Russell and Wittgenstein in January 1913, meant " ' p ' is false", and thus in turn, "the fact described by the judgment 'that: p' does not obtain". Their theory faces precisely the difficulty to which both Frege and Wittgenstein later alluded: we have to know what would be the case if "p" were true, when we assert that "'p' is false". But if "'/?' is false" gets analysed into an expression in which the symbol 'p' is reduced to "thought rubble", then this demand will be impossible to meet. So by construing all the components of his propositions as simple names, Wittgenstein also obliterated the type distinctions among them, and with them the implicit structure of the proposition being analysed. In consequence, his theory of propositional content failed to secure a definite connection between content and possible fact—the very sort of connection that Wittgenstein later called the "internal connection" between proposition and state of affairs. If the list of objects, {F,a,(3x,y)ei(x,y)}, is supposed to render a proposition, we must know which of the various possible combinations of its elements is supposed to be its fact, or "corresponding complex", as Russell put it. Furthermore, we should be able to exclude a priori those combinations which are senseless, in that no fact could correspond to them. This means that when we judge that a proposition Fa is true, we must be 20

Cf. the set of quotations at the beginning of 3.2.

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certain that F and a could in fact form a complex. Whence does this certainty arise? Since we have only the symbol and the denotations of its components before us (assuming that we do not know whether Fa obtains), we cannot know this by looking to the fact in question. If we needed supplementary knowledge to the effect that F and a belonged to the appropriate categories of things, then this knowledge ought to be expressible. We would have to say, in essence, that "although a might not be an appropriate argument for F, in this case it is one". But no significant function can say this of any given pair (a, F): this knowledge cannot be said, as Wittgenstein later put it. Either we know already which objects can be possible argument for ε,(χ,_ν) or we don't, and, in the latter case, no further information will help us make the decision. Wittgenstein concluded in the Tractatus that we must already be in possession of a symbolism that allows us to settle such questions merely by inspecting the signs. The final impetus in this direction came from the decompositional theory's need to postulate a "logical form".

4.2 The ε-copula and the Propositional Form Treating predicates and relations as structurally distinct from individuals allows one to block invalid substitutions. But there is another problem at the heart of this theory that remains untouched by this solution. This concerns the existence of the class specified by the copula in Wittgenstein's theory. The other, more tentative solution referred to in the letter that I quoted at the opening of this chapter, is that the ε-copula itself must be done away with. Wittgenstein's initial proposal, namely that the role of this logical form be taken over by the predicate (or, in Russell's case, the relating relation) solves the wide directionproblem. But the status of these copulae and forms remains troubling. They are not simples, nor can they be functions, because they are components of elementary propositions. According to Principia Mathematica *62, such an £,copula is a relation. And we know that if it is to be an essential part of an analysed proposition, the relation in question must exist. According to *25, a relation exists whenever "there is at least one pair of terms between which it holds".21 In the case at hand, the relation ε, exists only if at least one fact of the form in question obtains. Russell's response to this difficulty, Pears suggests, was to take the high road: he argued that the logical form was a necessarily existing logical object. Wittgenstein took the Principia Mathematica definitions of the existence of a relation more seriously. As I stated above, Wittgenstein could only have arrived at this position after 1913; however, there can be little doubt that he did conclude that Russell's Platonist solution was unacceptable. 21

Whitehead and Russell. Principia Mathematica. vol. 1, p. 228.

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4. The Breakdown of Wittgenstein's Copula-theory

For it does indeed seem perfectly conceivable that, at any given time, no complex of a given form obtains. This difficulty was, I suspect, the final blow to the Russellian theory, and the one that moved Wittgenstein to postulate the propositional sign as an independent entity modelled on Frege's thought. For whether or not one attempts to solve the wide direction-problem by postulating that predicates and relations take on the role of the forms, one still has to explain what these forms are. If they are conceived as classes of facts, as they must be on the model of a descriptive decomposition, then this entails, as Pears points out, that they exist only contingently. From this it would follow that there was a contingent proposition on which others depended for their significance—worse still, that proposition would depend on itself for its significance.22 Lest this difficulty seem somewhat arcane, it is worth recalling both the logical function of these copulae, as well as the connection of this theory to views about meaning and intentions later criticised by Wittgenstein in the Investigations. Russell and Wittgenstein needed their forms because the core of their theory lay in the idea that we refer to facts indirectly, by means of formal properties. In short, they thought one could get around the problem of empty denotations by parsing facts into their formal and material aspects, and citing these individually. In doing so, one describes the intended fact as being a member of a given class. One way to do this is to imagine that the class consists of members sharing a given formal property. Certainly Russell was committed to this interpretation in 1913, if only by reason of the "no-class theory" of Principia Mathematica, according to which classes are to be replaced by the functions that define them. He could not admit that the classes required by the theory existed independently, and so he needed a form to stand in for them. Another possibility would be to argue that these classes are grounded in inherent cognitive or perceptual faculties of the intentional subject. Both of these approaches might explain our ability to project our intentions onto absent states of affairs by postulating that we can project formal properties. We know, for in22

Assume there were a fact Fa which, as matter of fact, always held, so that "Fa" was always true, and with it, " (3φ,χ)φχ". Fa can now serve as aproto-type for the class of propositions of the form φχ, and thus we can analyse propositions with that form into their constituents, one of which will be " (3φ,χ)φχ". Thus far the only difficulty is that the significance of all these others depends on the truth of "Fa". But since "Fa" is a proposition of the same form as its doubles, it ought to be susceptible to analysis in the same way. But then the truth of this proposition is a condition for the sense of its own propositional sign: as long as the fact Fa obtains, then (3