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1962 Oxford University DPhil Thesis

Knowing and Understanding

Relations between meaning and truth,
meaning and necessary truth,
meaning and synthetic necessary truth

Aaron Sloman
(Now at University of Birmingham)
Contact: a.sloman[AT]

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A carbon copy of the typed thesis was deposited in the Bodleian Library, Oxford University in May 1962. In 2007 the library started scanning old theses and this was the first one done. Although the text is easily human-readable, available OCR technology could not cope with it, so it was eventually re-typed manually, producing a much less bulky searchable version now available here. Luc Beaudoin and Carol Woodworth provided invaluable help with proof-reading. Further details and links to the (very bulky) original scanned files can be found below. The thesis is available in the following formats:

Formats: Full transcribed thesis -- 2018
Re-typed version: searchable HTML (669KB) and PDF (2.1MB)
Plain text embedded in html, with diagrams added on pages 287, 288, 307:
Searchable PDF version, preserving original page numbering, line breaks and underlining (used for emphasis in 1962 typescripts):

Scanned Original
The original, much more bulky (74.1MB), scanned-in version with no OCR processing is also available, based on files provided by the Bodleian library, Oxford, in 2007:

Versions in Oxford Library:

A list of contents of the thesis, with page numbers, is available in a separate text file:

Note regarding Computing, Artificial Intelligence, and Morons!
When working on this thesis between about 1959 and 1962 I knew little or nothing about computers (except that they were beginning to be used in business and engineering) and nothing about AI. I did not think of attempting to implement the ideas in a working computer model until after I met Max Clowes at Sussex University in 1969.
However the thesis occasionally refers to what "a moron" might do, a commonplace of philosophical thinking at the time. In retrospect, I suspect that most of the references to a moron in the thesis can be interpreted as references to a computer-based, working, but not very sophisticated, AI system.

Further information is provided below. Please report any errors to a.sloman[AT]


The new version is not guaranteed to be error free. Corrections will be made when transcription errors are reported.

A detailed table of contents of the thesis, including chapter abstracts and links to the original scanned PDF files is below.


A. Sloman, (1962). Knowing and understanding.
DPhil. thesis University of Oxford.

The aim of the thesis is to show that there are some synthetic necessary truths, and that synthetic apriori knowledge is possible. This is really a pretext for an investigation into the general connection between meaning and truth, or between understanding and knowing, which, as pointed out in the preface, is really the first stage in a more general enquiry concerning meaning. (Not all kinds of meaning are concerned with truth.) After the preliminaries (chapter one), in which the problem is stated and some methodological remarks made, the investigation proceeds in two stages. First there is a detailed inquiry into the manner in which the meanings or functions of words occurring in a statement help to determine the conditions in which that statement would be true (or false). This prepares the way for the second stage, which is an inquiry concerning the connection between meaning and necessary truth (between understanding and knowing apriori). The first stage occupies Part Two of the thesis, the second stage Part Three. In all this, only a restricted class of statements is discussed, namely those which contain nothing but logical words and descriptive words, such as "Not all round tables are scarlet" and "Every three-sided figure is three-angled". (The reasons for not discussing proper names and other singular definite referring expressions are given in Appendix I.)


 Meaning (Philosophy), Vagueness, Truth, Immanuel Kant, Gottlob Frege,
 Synthetic necessary truth, Synthetic apriori knowledge, Logic, Geometry,
 Arithmetic, Functions vs rogators.

Thesis contents list (in separate file)

Information about the thesis on Oxford University Web site:

Oxford University Research Archive (ORA)
Bodleian Libraries, University of Oxford

    Digital Origin:  Digitized other analog
                         Type of Award:      DPhil
                         Level of Award:     Doctoral
                         Awarding Institution:   University of Oxford

  About The Authors
     A. Sloman           Search for more by this author on ORA site
    institution          University of Oxford
    faculty              Faculty of Literae Humaniores
    oxford College       St Antony's College
    (Balliol College 1957-60, St Antony's 1960-62)

Current institution University of Birmingham Contributors Mr D.F. Pears Role Supervisor Bibliographic Details Issue Date: 1962 Identifiers Urn: uuid:cda7c325-e49f-485a-aa1d-7ea8ae692877 Item Description Type: thesis; Language: en Keywords (expanded below): Meaning (Philosophy) truth Relationships Member of collection : ora:thesis Rights Copyright Holder: Aaron Sloman Terms of Use: Click here for ORA Terms of Use

This thesis is part of the following Oxford collections:
Oxford University Research Archive (including theses).

Citable link to the Oxford page:

Contents of this page


With grateful thanks to Oxford University Library Services
Especially Sally Rumsey (ORA Service & Development Manager)

Scanned image files were made available at: which links to:
This was the first Oxford DPhil Thesis to be scanned for the archive.

The scanned image PDF files were downloaded from Oxford to Birmingham in 2007:
Suitable OCR technology was not available, so the PDF files contained only images of the pages.

Humanly transcribed versions of the files were added in April/May 2016, thanks to the efforts of Luc Beaudoin, who identified Hi-Tech Transcription Services as suitable for the task, then helped with proof-reading and getting the files to work in Libreoffice. (I am very grateful to Luc and Carol Woodworth for help with proofreading.).

Note on angle trisection (using neusis) and non-Euclidean geometries
(Added: 20 May 2016)

The thesis repeatedly uses the impossibility of trisecting an arbitrary angle in Euclidean geometry as an example. The proof of impossibility is quite complex and was unknown to the ancient Greeks. I don't recall whether I had ever encountered a proof of the impossibility before or during writing the thesis, though I knew it had been proved impossible.

Many years later I learnt that it is possible to trisect an arbitrary angle using origami geometry, and, perhaps more interestingly, it is possible using the "neusis" construction, which is a simple extension to Euclidean geometry that allows a straight edge to be marked in two places and moved around subject to constraints. The neusis construction was known to Archimedes, along with its support for trisection of an arbitrary angle. For more details and a demonstration of its use see:

In view of all this, all references to the impossibility of trisection should be interpreted as referring to impossibility using only the constructions specified in Euclid's Elements.

We cannot know what Kant would have said if he had learnt about the possibility of extending Euclidean geometry with the neusis construction and the demonstration that it can be used to trisect an arbitrary angle: he might have used those two possible extensions of what was previously known as examples of synthetic apriori (non-empirical) knowledge of necessary truths about Euclidean geometry.

Likewise we cannot know what he would have said if he had learnt about non-Euclidean geometries in which the parallel axiom is false. The axiom specifies that if P is a point in a plane S and L is an infinite straight line also in plane S, then there is exactly one infinite line through P in S that is parallel to L, i.e. it never intersects L. For some surfaces the axiom is false because there are NO pairs of parallel lines in the surface, e.g. the surface of a sphere, or an ellipsoid, or an egg. In other cases the axiom is false because infinitely many straight lines through a point P can be parallel to a line L (using a non-trivial mathematical generalisation of "parallel" to include lines on curved surfaces). Einstein's theoretical work, along with Eddington's astronomical observations, led to the rejection of Kant's belief that our space is necessarily Euclidean, though mathematicians had investigated non-Euclidean geometries before that.

Postulating a space with no parallel lines requires a modification of axioms of "betweenness", as explained in

These cases are not easy to imagine but have been investigated in depth by mathematicians. (Search for "elliptical geometries" -- no parallels, and "hyperbolic geometries" -- infinitely many lines parallel to L through P in S.)

When I was a student many philosophers mistakenly regarded the discovery that physical space is non-Euclidean as a refutation of Kant's philosophy of mathematics. But as far as I know Kant never claimed that mathematicians are infallible, and if he did that was simply an error, within a larger, largely correct, theory. Even great mathematicians can make mistakes either by failing to notice certain possibilities or by doing erroneous reasoning, as Imre Lakatos demonstrated in his PhD thesis, later published as Proofs and Refutations (Cambridge University Press, 1976). (While working on this thesis I heard Lakatos lecture, and referred to his work in Chapter 7, section 7.D.10.)

If Kant had learnt about all this, then instead of claiming that Euclidean geometry contains only synthetic necessary truths, he could have pointed out that the subset of Euclidean geometry without the parallel axiom can be consistently extended in three different ways, producing three different kinds of geometry: a far more interesting synthetic necessary truth!
For more on non-Euclidean geometries see (for example):

Summary: There is a type of synthetic necessary truth that, as far as I know, Kant did not notice and has not been discussed in relation to his philosophy of mathematics: namely such truths as that particular mathematical domains can be extended in particular ways with certain necessary consequences.

I have used the example of adding the neusis construction to Euclidean geometry, and the example of adding the parallel axiom, or one of two main alternatives, to Euclidean geometry minus the parallel axiom. (I did not know about the neusis construction when writing the thesis, unfortunately.) The ability to explore and learn about different geometries is a deeper feature of mathematics than the possibility of discovering truths about any particular geometry. I suspect Kant would have accepted this as illustrating the importance of synthetic necessary truths and synthetic non-empirical knowledge, in mathematics.

There is a discussion of the relevance of non-Euclidean geometries to Kant in

Shabel, Lisa, "Kant's Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.),
The article cites books and papers published in the decades after this thesis (which was never published as a whole until now).

Contents of this page

Note on "possible world semantics" for modal concepts
(Added: 24 Jun 2016)

Ideas about possible world semantics for modal concepts were developed in the 1960s by Saul Kripke, David Lewis and others as explained in
Menzel, Christopher, "Possible Worlds", The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.)
Occasional allusions to the notion of possible world occur in this thesis, written in 1962, e.g. in Section 2.C.6.
"We cannot perceive the set of all possible worlds, we can perceive only the actual one"
and in Section 7.B.6.
"But our definition of 'necessary truth' was restricted in such a way that we need not take account of all these complexities, for it is concerned only with classes of objects possessing properties which actually do exist in our world. We therefore have no need to talk about all possible worlds, since we can limit ourselves to talking about all possible states or configurations of this world, where 'this world' describes a world in which the same observable properties and relations exist as exist in our world. (It should be recalled that the existence of universals need not involve actual existence of instances. See section 2.D.) Thus, since we are talking only about states of this world, we need not consider worlds without space and time, or five-dimensional worlds."
So, in this thesis, the notion of the set of possible worlds was explicitly rejected as a basis for explaining how propositions can have modal properties of the sort assumed by Kant, i.e. being necessarily true or false or contingently true or false. It seems that an approach to modality similar to one used in this thesis was independently developed later by Barbara Vetter, referenced below, though I have not yet studied her work in detail.

Some of the later developments based on or related to ideas in this thesis are described in the historical notes below, followed by the list of related/derived publications.

History of this version of the thesis

In 2007, the "original" PDF files (containing bulky scanned images of thesis pages) scanned were provided by the Oxford library, as outlined above. The derived PDF and HTML files with machine-searchable text are much smaller.

The first three files were transcribed using an OCR package: tesseract. The results required a huge amount of tedious post-editing. That produced .TXT files from which the first three new PDF files were derived.

In 2016, thanks to help from Luc Beaudoin, who found a suitable company to do the work (HiTech), the remaining chapters (Chapter 2 on) and appendices, etc., were re-typed by humans, producing MS Word files. These required some post editing but most of the typing seems to have been accurate. Please report any remaining flaws.

For post-editing, the word files were converted to ODT format, using Libreoffice. After checking and fixing, three derived files were produced from each file using Libreoffice: .DOCX, .PDF, and .TXT. Eventually, the derived PDF and TXT files were concatenated to produce the versions of the full thesis now available. There may be remaining transcription errors.

(Including chapter abstracts.)

Thesis contents list (in separate file)

Abstracts of most portions of the thesis are provided below, to help readers decide whether they wish to look at some or all of the thesis.

The original page breaks and most line breaks have been preserved in the new PDF files, but the detailed format has changed. Despite proof reading, there may still be transcription errors in the new files.
Please report any errors to: a.sloman[AT]


        Front Matter Part Pages
          Title and Abstract      Pages i-ix
          Preface + Table of Contents      Pages i-v


Part or Chapter Title Start Page
Chapter one: Introduction . . 1
Extract from final section of Chapter 1
I shall try, making use of the assumptions and methods described in the previous section, to describe the general connection between the meanings of certain sorts of statements and the conditions in which they are true, and then show how it is possible for a proposition to be true solely in virtue of what it means, that is, to be analytic. The question will then be raised whether the class of analytic truths includes all necessary truths, and the negative answer will be illustrated by the description of examples of necessary truths which are synthetic. I hope that in the course of all this it will become clear why other philosophers have reached different conclusions, the most important reason being, I think, that they have used much looser (and fluctuating) criteria for identity of meanings and propositions than I use. Failing to make fine discriminations, they fail to notice interesting relationships. (See section 2.C.)

It is hoped that there will be something of interest in the general picture that will be painted, even if the details are neither new nor very interesting in themselves.

..... 1.A. The problems . . 1
..... 1.B. Methodological remarks . . 5
..... 1.C. The programme . 13
Chapter two: Propositions and meanings . 18
Before we can explain how the analytic-synthetic distinction and the necessary-contingent distinction are to be applied, and discuss the question whether they divide things up differently or not, we must be sure we know what sorts of things they are meant to distinguish. This applies also to the true-false distinction. Sometimes it is not clear whether philosophers think these distinctions apply to sentences or to statements or to ways of knowing, or something else, (c.f. section 6.A) and this leads them into ambiguity and confusion. I shall apply the distinctions to statements or propositions, which are expressed by sentences. When I talk about statements, I am talking about sentences together with the meanings they are understood or intended to have. When I talk about propositions, I shall be talking about the meanings of sentences (as understood by some person or group of persons). I shall often use the words "statement" and "proposition" interchangeably, as the difference between them is important only in contexts in which we are concerned about the actual form of words used to express a proposition. But this leaves unanswered the question: what is the meaning of a sentence, or the meaning which it is taken by some person or persons to have? The only way to answer this question is to describe the ways in which words and sentences can be used with meanings or understood with meanings, and to say clearly how to tell whether two words or sentences are used or understood with the same meaning or not. That is to say, we must describe criteria for identity of meanings and propositions.
..... 2.A. Criteria of identity . 18
..... 2.B. General facts about language . 24
..... 2.C. Universals and strict criteria . 38
..... 2.D. The independence of universals . 50
Chapter three: Semantic rules . 63
Chapter two contained an argument to show that in order to avoid begging questions we must look for the sharpest possible criteria for identity of meanings, and it was suggested that only by taking note of the universals (i.e. observable properties and relations) to which words are intended to refer could we find sufficiently sharp criteria. ... Section 2.D contained arguments to show that talk about universals can explain since their existence is a fact about the world, independent of the existence of instances or of our use of language. In this chapter an attempt will be made to show in more detail how properties may be used to give descriptive words their meanings, and how we may compare and distinguish meanings by examining the ways in which words refer to properties. This will provide many interesting examples to which the analytic-synthetic distinction may be applied later on. .... First of all the simplest type of correlation between words and properties will be discussed, and then it will be shown how more complicated correlations are possible, firstly by means of logical syntheses of concepts and secondly by means of non-logical syntheses..... Although the discussion is restricted to words which refer to properties, nevertheless similar remarks could be made about words referring to observable relations.
..... Introduction . 63
..... 3.A. F-words . 64
..... 3.B. Logical syntheses . 70
..... 3.C. How properties explain . 83
..... 3.D. Non-logical syntheses . 93
..... 3.E. Concluding remarks and qualifications 102
Chapter four: Semantic rules and living languages 107
In Chapter Three an attempt was made to describe various ways in which descriptive words may be correlated with universals by semantic rules. It was pointed out in section 3.E that our ordinary use of words is much more complex than the uses described in that chapter, and the purpose of this chapter is to describe some of those complexities. There are many respects in which the description of semantic correlations and logical and non-logical syntheses of meanings provided an oversimplified model. For example, it took no account of descriptive words which refer to tendencies or dispositions or unobservable properties or theoretical notions of the sciences, or those words, such as "angry", "hopes", "intends" which may be used to talk about conscious beings. However, even if we leave out these complicated concepts, and concern ourselves only with words which are correlated with observable properties in something like the manner described in the previous chapter, we shall find complications which have not been accounted for, though very briefly mentioned near the end.
..... 4.A. Indefiniteness 107
..... 4.B. Ordinary language works 117
..... 4.C. Purely verbal rules 125
Chapter five: Logical form and logical truth 129
We are now ready to set out upon the last lap of Part Two, in which our main aim has been to explain how certain kinds of words and sentences can have the meanings they do have, and how their having these meanings helps to determine the conditions in which propositions which they are used to express are true. This explanation serves two important purposes. First of all, it provides an answer to the question: what sorts of things are propositions, the entities to which the analytic-synthetic and necessary-contingent distinctions are to be applied? (Cf. 2.A.1.) Secondly it helps to display the general connection between truth and meaning, between knowing and understanding, at least in a certain class of cases. This prepares the way for the discussion of some more restricted kinds of connection, in Part Three. (Part of that discussion will be anticipated in the present chapter.)
..... Introduction 129
..... 5.A. Logic and syntax 130
..... 5.B. Logical techniques 144
..... 5.C. Logical Truth 166
..... 5.D. Some generalisations 176
..... 5.E. Conclusions and qualifications 181
Chapter six: Analytic propositions 194
The main stream in Part Three will be a continuation of the attempt to describe the various factors which can determine or help to determine the truth-value of a proposition. This will provide illustrations for my explanation of the meanings of "analytic", "necessary", "possible", and related words, which will proceed at the same time. It will be shown that there are several different ways in which a proposition may be necessarily true, corresponding to a number of different ways in which its truth-value may be discovered. In particular, it will be argued that, in the sense of "analytic" which is to be defined in this chapter, not all necessary truths are analytic. This is because there are some properties which are necessarily connected, although they can be completely identified independently of each other. Hence their necessary connection is not an identifying relation or a logical consequence of an identifying relation.
..... 6.A. Introduction 194
..... 6.B. Some unsatisfactory accounts of the distinction 199
..... 6.C. Identifying relations between meanings 217
..... 6.D. Indefiniteness of meaning 229
..... 6.E. Knowledge of analytic truth 236
..... 6.F. Concluding remarks 249
Chapter seven: Kinds of necessary truth 260
The features of an analytic proposition in virtue of which it is true ensure that it would be true in all possible states of affairs, so we can say that it could not possibly be false, that it must be true, that it is necessarily true, and so on. All these truth-guaranteeing features are topic-neutral and can be described in purely logical terms, such as that the proposition is made up of certain logical words in a certain order, with non-logical words whose meanings stand in certain identifying relations. This chapter will be concerned with the question whether there is any other way in which a proposition can be necessarily true. In order to give this question a clear sense I must explain what is meant by "necessary", that is, give an account of the way in which the necessary-contingent distinction is to be applied. I shall start off by talking about the meaning of "possible". The next section will attempt to explain the meaning of "necessary". The rest of the chapter will be concerned to describe and distinguish kinds of necessary truths, and ways in which a proposition may be known to be true independently of observation of contingent facts.
..... Introduction 260
..... 7.A. Possibility 261
..... 7.B. Necessity 272
..... 7.C. Synthetic necessary connections 283
..... 7.D. Informal proofs 294
..... 7.E. Additional remarks 319
Chapter eight: Concluding summary 329
This chapter concludes my answer to the main question raised in section 1.1 Many subsidiary questions have been raised which could not be answered in the limited space available - some of these are dealt with briefly in the appendices. I claim to have shown that Kant was justified in describing some kind of knowledge as both synthetic and a priori,1 and, which is perhaps more important, to have revealed some relations between very general concepts, such as "property", "meaning", "truth", "proof", "possibility" and "necessity".
(Contents below)
..... Bibliography 389


.....p 335 .. Appendix I. Singular referring expressions

.....p 340 .. Appendix II. Confusions of formal logicians
This appendix presents arguments against the view that a natural language must include a formal system, and that logic is just a matter of syntax. One of the key points, also made by Frege, is that semantics cannot emerge from syntax alone: we also need to take account of the functions of the symbols used, not just their form.

.....p 357 .. Appendix III. Implicit knowledge
This appendix gives examples of several kinds of implicit knowledge, including allowing for the deployment of implicit knowledge to be unreliable sometimes (Compare Chomsky's Competence/Performance distinction, 1965). The ability to do logic and mathematics, as well as many other kinds of things, depends on the use of implicit knowledge, which can be very difficult to make explicit. (At that point I knew nothing about the young science of AI which was beginning to provide new techniques for articulating implicit knowledge.)

.....p 372 .. Appendix IV. Philosophical analysis
The ideas about implicit knowledge in Appendix III are used in Appendix IV to explain some of the puzzling features of the activity of conceptual analysis (disagreeing with R.M. Hare's explanation). This leads to further discussion of the nature of philosophical analysis and the claim that it cannot be concerned merely with properties of concepts: it must also be concerned with the world those concepts are used to describe, which may support different sets of concepts.

Note added May 2016
This theme was taken up again many years later in my paper distinguishing logical topography from logical geography in

.....p 381 .. Appendix V. Further examples

.....p 386 .. Appendix VI. Apriori knowledge

Contents of this page

(Draft) Historical note
(Added 7 May 2016 -- Updated 23 June 2016)

[This note was a result of re-reading parts of my thesis as part of the process of checking the transcription from image files to text files. The thoughts and recollections recorded here may be extended or revised later.]

My first degree was in mathematics and physics (Cape Town, 1957) after which I went to Oxford planning to become a research mathematician. In Oxford I became friendly with several philosophy graduate students and attended their seminars and some philosophy lectures. I soon realised that the philosophers I encountered had a view of the nature of mathematics that was deeply mistaken, and did not fit my experience of doing mathematics, including discovering and proving, or disproving, conjectures. In particular, it seemed to be commonly thought that the rejection of Euclid's parallel postulate on the basis of work by Einstein and Eddington demonstrated that Euclidean geometry was empirical.

However this ignored the fact that a great deal of Euclidean geometry is also common to its alternatives. And work of Imre Lakatos (mentioned briefly in the thesis (Chapter 7), later published as Proofs and Refutations) showed that it was important to distinguish the non-empirical characteristics of mathematical discoveries from a claim that mathematicians are infallible.

After a year or two registered as a mathematics student (details forgotten) I switched from Mathematics to Logic (supervised for a while by Hao Wang) and then later switched to Philosophy and became a philosophy research student. David Pears was named my supervisor, though I continued to attend lectures by Hao Wang, and also by Michael Dummett, John Lemmon, and others. My "moral tutor" at Balliol College was Richard Hare, who helped me to broaden my (miniscule) philosophical education, including introducing me to his version of Kantian meta-ethics.

For one term my college arranged for me to be supervised by Michael Dummett, but that was a time when all his energies were spent on trying to help refugees who were being obstructed by immigration authorities, and phone calls continually disrupted our meetings. Insofar as we did communicate I tended to disagree with his anti-realism. (A few years later he decided that the political situation was beyond repair and re-focused on academic work.) I have never been able to understand how highly intelligent people can take religion seriously: and he was an example. But I did not challenge him on that.

(I did challenge Hare, whose response was that for people to believe in God it was not necessary for them to believe that God exists: for him that seemed to be a kind of moral stance rather than a factual belief.)

During my five years in Oxford, mostly after the second year, I attended lectures and seminars by Gilbert Ryle, John Austin, Peter Strawson, William Kneale, Martha Kneale, Friedrich Waismann, Michael Hinton, Anthony Quinton, Geoffrey Warnock, Mary Warnock, Paul Grice, Elizabeth Anscombe, Philippa Foot, Sybil Wolfram, Stuart Hampshire, among others, as well as lectures by visiting philosophers, including Karl Popper, Carl Hempel, Hilary Putnam, Georg Kreisel, John Mackie, and John Wisdom, some of whom kindly gave me some of their time. I met A.J.Ayer after he moved from London to Oxford around 1960, and I believe it was his presence on the interview panel that persuaded St.Antony's College to give me one of the two two-year Senior Scholarships in Philosophy that they made available in 1960. That award allowed me to complete my D.Phil. (The other Scholarship was awarded to another philosopher, Bob Stoothoff. As far as I know, St.Antony's never again appointed philosophers, other than social and political philosophers, after Bob and I left, in 1962.)

This list may be extended, as memories return.

I did not find that the quality of what I learnt from well known philosophers was always correlated with their reputation.

Note on Hempel's critique of Kant (Added: 12 Aug 2019+12 Mar 2020)
Carl Hempel's extremely clear, and superficially devastating critique of Kant's view of mathematics is summarised in his paper: Geometry and Empirical Science American Mathematical Monthly 52, 1945, reprinted in several places and made available online by Andrew Chrucky, Feb. 7, 2001 here:

The paper is referenced in the thesis in section 7.C.7 (note), where I quote Hempel as saying: "The fact that these different types of geometry have been developed in modern mathematics shows clearly that mathematics cannot be said to assert the truth of any particular set of geometrical postulates; all that pure mathematics is interested in, and all that it can establish, is the deductive consequences ....." -- a view that now seems to be very widespread, along with the assumption that mathematics can start with many different formal (logical) specifications of mathematical structures or spaces, and investigate their consequences, e.g. what new (logical) statements can be derived from particular specifications. This view seems to be shared by many mathematicians and philosophers of mathematics. However, it does not describe what I learnt to do when studying geometry at school and I don't think it describes what the great ancient mathematicians were doing before, during and after Euclid's work.

Hempel visited Oxford while I was working on this thesis and I attended a lecture he gave. My college tutor R.M. Hare arranged a meeting where I told Hempel what I was trying to do, and although I have no recollection of details, I am sure I failed to convince him that his criticism of Kant was misplaced, like many others. I thought it misplaced for two reasons, first the fact that Kant was not claiming that there is some infallible mode of mathematical discovery, and second because the modes of discovery when exploring Euclidean space can also be used, with differences of detail, in exploring non-Euclidean spaces, e.g. the surface of a sphere or torus.

So just as Euclid's work is not refuted by the fact that there exist non-Euclidean surfaces such as the surface of a sphere, Kant's work is not refuted by pointing out that some, or all, regions of the space we inhabit are not Euclidean. This is in part related to Kant's claim that the concepts used in mathematics are not empirical (derived from experience) and neither are the truths discovered empirical, though in both cases experience may be an important stimulus -- including cases where mathematicians need to use external objects, such as diagrams in sand, or on pieces of slate, or paper, etc. to help their thinking. I am not sure I made that point clearly enough in the thesis itself.

Around that time Imre Lakatos, referenced on page 309 of the thesis (section 7.D.10), had begun to give talks based on his PhD research --- later published as Proofs and Refutations (1976) into errors made by Euler and other mathematicians in relation to Euler's theorem about polyhedra, discussing in great detail some of the consequences of the fact that even great mathematicians can make various kinds of mistake.

I don't think there is anything in Kant's view of the nature of mathematical discovery --- as involving truths that are synthetic (not-analytic), necessary (not contingent) and non-empirical (a priori) --- that implies that mathematicians are infallible.

Everyone who studies mathematics will know from personal experience that mathematicians can make mistakes. Kant may have mistakenly believed that physical space is necessarily Euclidean (though I suspect he was aware of non-Euclidean surfaces such as the surface of a sphere or a torus -- or cup with handle). But Kant was not mistaken insofar as he claimed that Euclidean space is something we can identify and study, making surprising discoveries and finding proofs of unobvious features derived from more obvious features, where the proofs do not all start from definitions and derive theorems using only logically valid inferences. Moreover, if he had known about extensions to Euclidean geometry, e.g. the neusis construction that makes it easy to trisect an arbitrary angle, although that is impossible using only Euclid's operators, I suspect he would also have used those extensions as examples.

I think Hempel's attempt to demolish Kant's claims about mathematics ignores these points. For similar reasons, I thought all the philosophers who believed Kant's philosophy of mathematics had been demolished by Einstein and Eddington had misunderstood what Kant had discovered, which I thought was correct.

Defending this claim requires a demonstration that there is a mode of mathematical discovery that has the features claimed by Kant (and my thesis). After learning about AI, about seven years later, I thought, for several years, that such a demonstration, defending Kant, could be given by using AI techniques to build a working model of Kantian mathematical discovery, running on computers. But I now suspect it requires a more general kind of computer that can combine discrete operations with continuous ones, as mathematicians working in geometry frequently do. Perhaps such a computer can be made from sub-neural chemical components, since chemical processes combine discrete changes (e.g. formation and removal of chemical bonds) and continuous changes (e.g. when molecules twist, move together or apart, etc.). This now seems to me to be closely related to the points Schrödinger made in 1944 in his little book What is life? about the importance of quantum theoretic explanations of high stability of some molecular bonds that can also be altered rapidly and with low energy catalytic influences. This explains the combination of long term stability required for a reliable inheritable mechanism and complex and varied developmental processes controlled by the genetic structures.

I suspect this is related to Alan Turing's claim in his PhD thesis, published 1938, that humans are capable of both mathematical intuition and mathematical ingenuity, but computers (e.g. Turing machines) are capable only of mathematical ingenuity. For more on this see the following (all "work-in-progress"):
  The Meta-Configured Genome

I owe a great deal to many philosophy graduate students in Oxford between about 1958 and 1962, who kindly spent time arguing with and educating me, several of whom later achieved great distinction, including Ian Hacking (visiting from Cambridge) whom I met first at a Philosophy graduate conference, and John and Dagmar Searle, whom I met on a skiing holiday arranged for Oxford students, and in seminars. They even kindly gave me a tutorial, criticising a paper I had written on Free Will.

Around the time of my official change to philosophy, I read enough of Kant's Critique of Pure Reason (in the translation by Kemp Smith) to be convinced (on the basis of my personal experience of doing mathematics, e.g. finding proofs) that he understood better than most contemporary philosophers what mathematical discovery was, and how it provided knowledge that was different from analytical truths, which could be established purely on the basis of logic and definitions, and empirical truths, that could only be discovered on the basis of observation and experiment and were liable to refutation only by some newly observed phenomenon. (As pointed out by Lakatos -- mathematical arguments are also liable to refutation, by finding flaws in the concepts, the arguments, etc., without making any new empirical discoveries. That difference requires more discussion.)

So I set out (around 1959, I think) to explain why Kant was right to describe mathematical knowledge as synthetic, not analytic, non-empirical, and non-contingent (i.e. mathematical truths are necessary truths, and mathematical falsehoods are necessarily false, e.g. "There is a largest prime number").

(Several years later, after learning about programming and Artificial Intelligence, mainly from Max Clowes[*], after he arrived at Sussex University in 1969, I came to the conclusion that Kant had already understood some of the deep ideas and would have been an enthusiastic user of AI to advance philosophy if he had had the opportunity.)

The thesis was finally submitted in 1962, and accepted. The examiners were Elizabeth Anscombe and Geoffrey Warnock. In those days Oxford University was too arrogant to involve external examiners. Theses were either accepted or rejected: only later did the practice of acceptance subject to revision evolve. I am sure my thesis would have benefitted from that!

The defense of Kant was built on Frege's distinction between Sense (Sinn) and Reference (Bedeutung) but showed how those concepts needed to be refined and used with great care. This led to the notion of a "rogator" a function that includes as one of its arguments the state of the world, or the state of a portion of the world, because an expression with a Sense can have a reference that is not fully determined by the Sense, only by the combination Sense-plus-how-the-world-is, e.g. the reference of "The oldest female member of the UK parliament on 1st December 2040". In such a case the world even determines whether there actually is an individual referred to. E.g. the UK may no longer exist, or parliament may have been replaced by some other institution, or there may be no female members, or the two oldest female members may have been born at the same time on the same day.

While writing the thesis, although I knew about Turing machines (thanks to having had Hao Wang as supervisor before I switched from logic to philosophy) I had had no experience with computers or programming, and I did not realise how important programming ideas and developments in Artificial Intelligence were going to be for philosophy (unfortunately still largely ignored by teachers of philosophy, except for those who believe or discuss greatly exaggerated claims about what machines will soon be able to do).

Without realising it, I was anticipating developments in AI by systematically interpreting Frege's notion of Sense in terms of semantic procedures, e.g. procedures for identifying referents or for establishing truth values.

However, since those procedures typically required interrogating some part of the world, they were not like Frege's functions, which simply, and unconditionally, associate each argument (input) with a value (output). In contrast, the semantic procedures discussed in the thesis associate an argument and some portion of the world with the value. The value of "the tallest person in my office" at a particular time might be a particular individual, but if the situation in the office had been different it could have been a different individual.

So unlike Frege's functions (modelled on functions in mathematics) these semantic procedures have to "ask the world some questions" in order to determine a value, and I therefore called them "rogators" (from the Latin for "ask"). These rogators produce results that depend on contents of limited portions of the world, which can change from time to time, and which might have been different if something had happened earlier to change things.

This idea is very different from the idea of "possible world semantics" which became popular later. It's more like "possible world-fragment semantics", where the relevant fragment of the world is determined by the semantic content for which a referent, or truth value is in question. For example, the question whether a particular ball is in contact with a particular vase at a particular time, and if so which parts of the surfaces of the ball and the vase are in contact, depend only on the contents of a tiny fragment of the universe at the time in question. Such a fragment is often inspectable by perceptual mechanisms, in some cases enhanced by measuring devices or magnifying glasses, or other domestic or laboratory apparatus. There are more complex cases where far more sophisticated apparatus and procedures are required, e.g. answering the question which of two fields has a larger area, or whether X can see Y, or what X sees about Y.

The main point is that the contents of the whole universe cannot be invoked in an explanatory role, or as part of the semantic analysis, since the whole universe is typically inaccessible and people have been using concepts of truth and falsity and modal concepts for centuries without any reference to the whole universe, or to possible worlds that are alternatives to the whole universe.

(It seems that Barbara Vetter has developed some closely related ideas independently: e.g. Barbara Vetter, 2013, 'Can' without possible worlds: semantics for anti-Humeans, Imprint Philosophers 13, 16, Aug, 2013.)

This defence of Kant required a theory of compositional semantics that allowed the semantic content of complex linguistic structures to depend not only on the semantic contents of parts and the syntactical relationships used, but also on relevant parts of the world, the parts to which the associated procedures had to be applied in order to determine referents and truth values. These ideas are closely relevant to requirements for design and implementation of language-using human-like machines. (In the thesis, this idea was loosely acknowledged, in the philosophical fashion of the time, by talking about what a "moron" might be able to do. In an up to date version of the thesis all those references would be replaced by comments regarding computers running programs, although many of the requirements for such machines discussed in this thesis have not yet been met by AI systems, for various reasons, including the difficulty of the task.)

Another unwitting anticipation of a theme in AI is the emphasis on what would now be called meta-cognitive abilities: the ability not only to apply procedures, but also to reflect on the process of application and in some cases discover that the result of applying the procedure can be determined without applying it. In other cases two processes need to be run in parallel: one applying some cognitive ability and the other inspecting features of the process.

This generalises to abilities to reflect on aspects of perception, learning, and reasoning and notice structural relationships that could go unnoticed in machines or organisms without the required meta-cognitive architecture. There has been work on meta-cognition in AI (including workshops and publications), but I think there is not yet a well understood and implemented specification for a meta-cognitive architecture capable of making the kinds of mathematical discovery that Kant drew attention to, for example, the ability to discover not merely what the result of applying a certain procedure is but also that that procedure cannot produce any other result no matter what portion of the world it is applied to. A Kantian example in the thesis is that no three planar surfaces can bound a finite region of space.

In some cases the discovered impossibility arises out of aspects of the logical/mathematical structure of the procedure. In other cases it is because of the structure of the portion or aspect of the world to which it is applied. Related meta-cognitive processes could lead to discoveries about what alternative results a procedure could produce if some feature of a situation were varied. E.g. "A is outside B" happens to be true, but if might be false if the location of A or B or both were changed. However, if A is incompressible and is much larger than B then it can be concluded that if neither A nor B changes shape or size then no rearrangement could make "A is outside B" false.

From this viewpoint discoveries in logic are a special case of a broader class of non-empirical, mathematical, discoveries about possible values for various procedures applied in specified situations. The thesis explored a subset of examples but later work covers a much broader variety, which turned out to be related to a generalisation of J.J.Gibson's theory of perception of affordances. See:

Unfortunately research in AI has recently become dominated by the assumption that intelligent agents constantly seek and make use of empirically based statistical regularities: this approach cannot shed light on discovery of non-statistical, structure-based regularities. The AI work that does attempt to model non-empirical (e.g. logical, mathematical) reasoning mostly assumes that that can be done by building machines that are presented with sets of axioms and rules of inference and abilities to determine which formulae are or are not derivable from the axioms by the rules. This ignores the possibility that there might be a prior form of discovery leading to the axioms and rules. This thesis suggests that that could be based on abilities to examine procedures and work out their constraints and powers, meta-cognitive abilities that go far beyond abilities to simply follow pre-specified rules and constraints. The vast majority of computers that can do the latter cannot (yet) do the former, which is not generally recognized as one of the aims of AI. In other words, AI is diminished by not yet being sufficiently Kantian.

The thesis is, in part, an attempt to spell out some of the requirements for satisfactory answers to Kant's questions. There is still work to be done to meet those requirements.

A decade after finishing the thesis I started trying to use ideas from programming and AI to demonstrate why Kant was right about mathematics, but the project turned out to be extremely difficult because of the mis-match between the procedural operations provided by computers and the sorts of procedures required for biological perceptual and reasoning mechanisms. Part of the problem that I think is still unsolved is to specify what the requirements were that were met by information-processing products of biological evolution.

My 1978 book The Computer Revolution in Philosophy (now freely available in a slightly revised online edition) was an attempt to spell out some of the unobvious requirements for the task. Unlike some of the leading AI researchers, the book did not proclaim that the problems would soon be solved: it said the opposite, namely the problems are very hard (e.g. the problems of modelling natural vision systems) and would not be solved by the end of the 20th century. That may be true of the 21st century also!

Work on spelling out requirements in terms of the problem of understanding and modelling achievements of biological evolution is now a key part of the Turing-inspired Meta-Morphogenesis project, attempting to understand the diversity of biological information processing mechanisms and how they evolved. It seems that evolution itself "discovered" and made productive use of many important mathematical aspects of the world, long before there were any animals that were able to extend those mathematical discoveries.

Derived publications

The following papers and book chapters present or develop ideas related to various portions of this thesis.
(Not a complete list.)
Contents of this page


This web page was originally loosely based on the corresponding ORA web page, from which the originally scanned chapters were downloaded. These pdf files contained only images of the pages, with no machine readable textual information. From time to time, this web site was expanded and/or reorganized, first in order to provide more information about the thesis, and, later on to provide transcribed versions of the PDF files. All chapters of the thesis have now been transcribed, along with bibliography and appendices. So the text is searchable and can be copied and pasted, e.g. into notes.

21 Jul 2016 Added separate simple contents list

10 July 2016 Contents of this page re-ordered.

23 June 2016
Chapter 3 had previously been formatted differently from the other chapters, using single-spacing. It has now been altered to use 1.5 spacing, like the remaining chapters.

8 Jun 2016
New versions of Chapter 5, proof-read and corrected by Carol Woodworth, and Chapter 2 proof-read and corrected by Luc Beaudoin. I am very grateful for this help. Remaining errors and style idiosyncracies are my responsibility. A temporary full-length PDF version of the whole thesis with corrections so far included is available on request.

8 Jun 2016
Updated and slightly reorganised.

9 May 2016
Added links to subsequent related publications, and historical note.

April-May 2016
I previously wrote: "If anyone is able to automate the conversion to text of the remaining chapters I shall be very grateful!". In April 2016, the conversion was done by manual typing, by an agency in India. Links to the newly provided PDF files are included for Chapters 3 to 8, Appendices and bibliography, alongside links to the original scanned images (also PDF).
There are still likely to be transcription errors. Please email me if you find any.

At a later stage, a single document combining all the new text will be available, possibly in PDF and HTML formats.

Updated 7 May 2016:
Found and fixed some problems in derived .TXT files.
Also re-formatted this page, to make things clearer, I hope.

Updated 5 May 2016:
Added Chapter 8, Appendices, and Bibliography.
Added diagrams to Chapter 7 and corrected more typos.
Updated 3 May 2016:
More corrections in transcribed chapters, namely
         Chapters 2, 3, 4, 5 and 6

Updated 28 Apr 2016:
I had previously produced text versions of the Abstract, Preface, Table of contents and Chapter 1, from which searchable PDF had been derived. Now (April 2016) thanks to much help from a former PhD student, Luc Beaudoin (, the chapters that had not previously been transcribed were re-typed by Hitech because OCR technology is not yet able to cope with the very fuzzy scanned carbon copies of the original thesis. Luc Beaudoin also helped with subsequent proof-reading, and a host of minor problems (still in progress).

Updated 13 Dec 2018
After a lot of proof reading and correcting, a (hopefully) final version of the transcribed thesis was produced, in PDF and HTML formats, including diagrams. So all the separate chapters have been withdrawn. The Oxford library version was also updated, incorporating the searchable version, and replacing the separate scanned chapters with a single PDF file (74.1MB), also available above. For For information on the Oxford site see:

Older Updates
Feb 2014:
The abstract, preface, table of contents and chapter 1 were translated by the tesseract OCR package, with post editing, into plain text files, whose contents are now searchable.

For more convenient printing, the text files were then converted to searchable PDF.

Updated 11 Feb 2014:
Split part 2 (preface, contents and chapter 1) into 2a (preface+contents) and 2b (Ch 1) (above).
Updated 10 Feb 2014:
Used OCR (tesseract) to create searchable, selectable plain text versions
of the Abstract, preface, table of contents and Chapter 1 (above).
Updated 9 Feb 2014:
Fixed link to Oxford Research Archive. (They change from time to time, unfortunately.)
Updated 8 Jan 2008:
Added more information about the contents of Appendices III and IV,
including links to some of my more recent work on those topics.