THE NATURE AND ROLE OF INTUITION

IN MATHEMATICAL EPISTEMOLOGY

Paul Thompson

"MATHEMATICS is not a unique and rigorous structure, but a series of great intuitions carefully sifted, and organised, by the logic men are willing, and able, to apply at any time".

- Morris Kline (1)

INTRODUCTION

Philosophers of mathematics have, for thousands of years, repeatedly been engaged in debates over paradoxes and difficulties they have seen emerging from the midst of their strongest and most intuitive convictions. From the rise of non-Euclidean geometry, to present-day problems in the analytic theory of the continuum, and from Cantor's discovery of a transfinite hierarchy to the fall of Frege's system, mathematicians have also voiced their concern at how we blindly cash our naïve everyday intuitions in unfamiliar domains, and wildly extend our mathematics where intuition either has given out, or becomes prone to new and hitherto unforeseen pitfalls, or outright contradiction.

At the heart of these debates lies the task of isolating precisely what it is that our intuition provides us with, and deciding when we should be particularly circumspect about applying it. Nevertheless, those who seek an epistemologically satisfying account of the role of intuition in mathematics are often faced with an unappealing choice, between the smoky metaphysics of Brouwer, and the mystical affidavit of Gödel and the Platonists that we can intuitively discern the realm of mathematical truth. In the proposed thesis I hope to supply, as an alternative, the lineaments of a more plausible and naturalistic account of mathematical intuition.

The analysis combines a cognitive, psychological account of the great "intuitions" which are fundamental to conjecture and discovery in mathematics, with an epistemic account of what role the intuitiveness of mathematical propositions should play in their justification. I continue by examining the extent to which our intuitive conjectures are limited both by the nature of our sense-experience, and by our capacity for conceptualisation. This leads me to investigate whether we can, as Gödel hints, avoid mixing our pre-theoretic intuitions with our more refined, analytic and topological ones, and - more fundamentally - whether we can, in practice, discriminate reliable intuitions from processes known, in retrospect, to lead to false beliefs. Finally, I suggest how we can use visual and formal heuristics to cultivate our mathematical intuition, and in particular I discuss their interplay in familiarising us with acceptable proof-procedures in functional analysis, with consequences of the Generalised Continuum Hypothesis and its negation in Zermelo-Fraenkel set theory, and with non-standard systems of geometry.

Such a working familiarity with unorthodox systems, then, could in turn enable us to delimit and subsequently refine our geometrical and analytical intuitions, and the breadth of this new epistemic perspective could ultimately allow us to appeal to intuitive (or 'intrinsic') support in cases where intuition has traditionally been seen as out of its depth, as misleading, or invariably counterproductive.

1. GÖDEL'S HOPE : REFINED INTUITION

One qualm which is often expressed, in set-theoretic research, rests on the fact that the meaning (and therefore truth) of certain hypotheses, whose plausibility is being tested by means of relative consistency proofs, depends strongly upon the intuitive embedding theory used in the consistency proof. In particular, although we may be keen to ensure that a proof-theoretically weak standard or arbiter is used, in other words one which seems epistemologically easier to defend, this may generally be insufficient to decide strong postulates, such as the Generalised Continuum Hypothesis, or the Inaccessible Cardinal Axioms.

Accordingly, I suggest, we must either seek a way of gradually ramifying, or extending, the scope of what we call intuitively clear (within strong and substantive constraints, of course), or instead be resigned to the view that the bounds of intuition are, as a matter of fact, well-defined, and may not even be extensive enough to back up, say, second-order real analysis, let alone any stronger theory which may, in the future, be expedient in the formal characterisation of physics.

2. MENTAL SIGNALS, AND THE SEDUCTIVE 'FEELING OF FAMILIARITY'

The Gödelian brand of Platonism, in particular, takes its lead from the actual experience of doing mathematics, and Gödel accounts for the obviousness of the elementary set-theoretical axioms by positing a faculty of mathematical intuition, analogous to sense-perception in physics, so that, presumably, the axioms 'force themselves upon us' much as the assumption of 'medium-sized physical objects' forces itself upon us as an explanation of our physical experiences. We might suppose then, (as Gödel does indeed suppose) that the presence of a 'feeling of familiarity' with basic principles, a sense of their obvious correctness, signals the fact that our belief in them has been generated by an intuition of mathematical reality, (whether this be construed Platonistically, or by a more moderate form of realism). This state of 'at-homeness' or plausibility might therefore be used as an indicator, to identify the occurrence in us, of mathematical intuitions.

Even ignoring the tremendous attack (found in Wittgenstein's discursus on 'reading' in the Investigations) on using 'characteristic experiences' as a reliable criterion for establishing whether certain cognitive processes are going on or not (2), and even if we could rely on readily isolating such a faculty by introspection, it is not clear how much work it could do for us in appraising new principles decisively, or in detecting illegitimate reasoning.

For a start, even a feeling of familiarity in the case where (say) the axioms of set theory (or, better still, geometry) strike us as obvious, does not guarantee that our belief in those axioms is a direct consequence of a mental process in which we apprehend mathematical objects. Perhaps the natural feeling of self-evidence, and the ensuing dogma of apriority, results from the effortless exercise of those conceptual abilities we have acquired in our mathematical youth, in learning to talk about sets, points or lines. This enculturation process seduces us into a mode of reasoning which becomes second nature to us, despite the inevitable fact that our language, at any stage in its evolution, remains a 'slap-dash, poorly-tuned' categoriser, often glossing over latent counterintuitive features instead of exposing them all to view in an instant (cf. Frege's "Unrestricted Comprehension").

Moreover, even the term 'counterintuitive' has acquired an ambiguous role in our language use: when applied to a strange but true principle (which has passed into our mathematical practice after receiving overwhelming extrinsic justification) 'counterintuitive' can now mean anything on a continuum from 'intuitively false' to 'not intuitively true', depending on the strength of the conjecture we would have been predisposed to make against it, had we not seen, and been won over by, the proof. Indeed, to our surprise, we often find out, in times of paradox, how weak and defeasible our ordinary intuitions are - how their varying strength often peters out to neutrality.

The very idea that our intuitions should be both decisive and failsafe, derives historically from the maelstrom of senses which the term 'intuition' has acquired in a series of primitive epistemic theories. Some of these senses have been inherited from the large role introspection played in the indubitable bedrock of Cartesian-style philosophy, and some simply from the pervasiveness of out-moded theological convictions which seek to make certain modes of justification unassailable.

3. KITCHER'S "GESTALT"

Philip Kitcher (3) has remarked that to admire the intuitions of a conceptual manipulator - in particular of a great creative mathematician such as Euler, Riemann, or Ramanujan - is to recognise an ability to obtain an unusual and fruitful gestalt on a problem. Crucially, intuition of this sort is not something which in itself warrants belief, but it may well play an important heuristic role, and also serve as part of the warranting process. When Wittgenstein thinks up a particularly graphic metaphor, or when Descartes or Fermat notices the similarity between plane Euclidean geometry and part of algebra, they are each conjecturing or selecting a net of logical relations, which can be fruitfully isolated from one area, and mapped onto another in an extremely incisive way. This technique is what I shall call abstracting a paradigm or schema, and cashing the metaphor in a new domain. As we shall see, it is arguably one of the most powerful (and dangerous) heuristics for generating extensions of mathematics. And the conjecturing of those types of metaphors which can safely and profitably be cashed, lies at the heart of intuition's fundamental role in mathematics.

4. THE LADDER ANALOGY

Although the talented mathematician looks at a recalcitrant puzzle from a new point of view, 'intuiting' that a particular manoeuvre will help in the summation of a series, say, (or with the evaluation of an integral, or that a certain number-theoretic problem reduces to a result in the theory of functions), the secret of this type of success is not to be taken to be some special autonomous ability to discern features of mathematical reality; in particular, not an ability to gaze at mathematical objects and, independently of both the breadth of the problem-solver's memory and his powers of analogy and association, come up with the fruitful idea. Consequently, the intuitions of which mathematicians speak are exercised more in the solution of research problems than in the knowledge of axioms, and are not, crucially, those which Platonism requires. The role of intuition then - conceived of as a sort of reactional versatility, a source of conjecture or of a fruitful new gestalt on a problem - is, in sum, more like the ability to leap effortlessly between a large number of rungs on an infinite ladder, rather than the considerably more Herculean ability to anchor it.

5. SCHEMATIC INTUITION

It is a universal phenomenon of expert mental life that points occur in a problem-solving process, which may be undetectable to a novice, but where the specialist sees that one can 'turn on to a familiar road' - a road traversed so many times before that the novelty and challenge of the particular problem disappears.

This remarkable ability, I suggest, can be explained more readily if we suppose that for a skilled mathematician - as uncontroversially as for the chess grandmaster - the mental representation of their respective problems, or positions, are not copies of either the physical symbols or of pieces on a physical board. They are much more abstract structural descriptions of the meaningful relationships between groups of symbols or pieces. Through many years of experience, both experts have acquired automatic perceptual mechanisms which rapidly pick out frequently-occurring strategic patterns, or 'schemas', from the input.

I am indebted to Richard Skemp [8] for observing that similar schemas to those which enable our specialist to recognise several thousands of patterns, also integrate our existing knowledge, and act as a tool for future learning by making understanding possible (now to be construed along the lines of assimilating something to an appropriate schema). I suggest then that intuition's role lies in guiding our search for the appropriate schema; if this model is correct, then two morals can immediately be drawn:

i) Our intuition, which depends strongly on our cultural and scientific heritage, has largely been developed by others (and not in any perspicuous way):

Our ability to isolate and detach our concepts from the examples that give rise to them, and subsequently to attach them instead to language, enables us to bring past experiences usefully to bear on the present situation. But even more insidiously, past conceptual structures, painstakingly abstracted and slowly accumulated over successive generations, become available to us as well, and these quietly by-pass the individual's scrutiny as they become part of his self-built 'intuitive' conceptual system. Although a potential source of prejudice, the value of this type of latent ramification is highlighted by Newton's modest remark:

"If I have seen a little further than others, it is because I have stood on the shoulders of giants."

ii) 'Conjectural Intuition' can also be modelled.

When approaching unfamiliar territory, we often, as observed earlier, try to describe or frame the novel situation using metaphors based on relations perceived in a familiar domain, and by using our powers of association, and our ability to exploit the structural similarity, we go on to conjecture new features for consideration, often not noticed at the outset. The metaphor works, according to Max Black (4) by transferring the associated ideas and implications of the secondary to the primary system, and by selecting, emphasising and suppressing features of the primary in such a way that new slants on it are illuminated. Skemp, too, is optimistic (p.61):

"The process of mathematical generalisation is sophisticated because it involves reflecting on the general form of the method, while temporarily ignoring its content, and powerful because it makes possible conscious, controlled, and accurate accommodation of {new scenarios, to} existing schemas, not only in response to the demands for assimilation of new situations as they are encountered, but also ahead of these demands, seeking or creating new examples to fit the enlarged concept."

While I agree that the intuitive leap is the frequent forerunner of the deliberate generalisation, I feel that the intuitive selection of those schemas whose cashing generates pertinent conjectures, as new enigmas arise, is not generally a conscious process, and the ability to survey one's own inventory of schemas (which Skemp calls 'Reflective Intelligence') is not a faculty which is genuinely available to most creative thinkers. Some schemas are simply too insidious, too deep down to grasp.

In order to help us see this though, let us pause for a moment and consider an illustration (5).

6. THE RHAPSODE AND THE PAPYROLOGIST

Conjectures tend to emerge through a vast sieve of intuitive, and generally unconscious, schemas.

When composing Latin elegiacs, Homeric epic, or Aeolian lyric verse in the tortuous styles of Sappho or Anacreon, I was often amazed, as a Classics student, at the way in which classical audiences were regarded as able to perceive the finer nuances of literary genres whose structural demands, on the composer, were astounding. And yet the speed and fluency of the oral rhapsode can only suggest that this whole panoply of 'rules', (whose explicit form fills whole text books on lyric structure and metrical analysis), were second nature to the composer, and, in particular, were not applied in any piece-meal or conscious way at all. On the other hand, when the papyrologist tentatively reconstructs whole lines of verse (from tiny papyrus fragments), his conjectures are vetted by measuring them up against a conscious inventory of schemas, each one acting as an added constraint on how suitable his various hunches really are. But perhaps another classical rhapsode, inspecting the fragment, would conjecture his own extension of the line because the fragment fell into place with a feeling of metrical cohesion, like Wittgenstein's object fitting the contour of a sheath.

Accordingly, this unconscious form of intuition - which may well produce substantially the same conjectures as the more conscious, and explicit, method of the papyrologist - could therefore indicate how the mental 'bingo-machine' itself which generates the conjectures for many creative mathematicians, remains so inscrutable. The inscrutability accords well with the coherent and articulate accounts (6) that innovative thinkers, in every branch of creative activity, have given of their inner experiences. These suggest that the skeletal idea, or conjectured schema, 'appears unbidden' in consciousness, only later to be subjected to a series of more conscious processes of extension and transformation. Moreover, it would be satisfying, to the cognitive scientist at least, if this inscrutability could be explained without leaving the way open for less naturalistic, more fanciful accounts of intuition.

7. THE MOVE FROM THE CONTEXT OF DISCOVERY TO THE CONTEXT OF JUSTIFICATION

So far in discussing the nature and role of mathematical intuition, I have concentrated primarily on the context of discovery, offering a psychological account of how intuition could be conceived as a type of reactional versatility, which generates conjectures and fruitful new angles on intractable problems in mathematics - while the conjectures, in turn, later become subjected to the cut and thrust of a more rigorous logical analysis.

However, not just any explanation will do in giving an account of intuition, because it may well be that all our beliefs - even lucky guesses - have explanations, and beliefs which are merely true by accident lack the epistemic status necessary for them to be called knowledge, in that we cannot provide anything that might serve to justify the assertion of such a belief, nor any connection (causal or otherwise) which connects the belief with the fact that makes it true.

A satisfactory explanation of conjecture and intuitiveness in mathematics must therefore do more than explain the origin of a belief which falls into either of the two categories - it must also show that such beliefs are generally speaking trustworthy, and how we can expect to rely on them at all.

In the next four sections then, I hope to make good the deficit, in a sense, by supplementing my psychological and schematic account of how we make mathematical conjectures and find things intuitive in mathematics, with an epistemic account under which analogical thinking (that is, applying a familiar schema in a new context) can be thought of as a form of inductive inference. This I hope will serve to indicate both why testing for intuitiveness can ever be a reliable method for finding out whether something is true, and why conjectures, although fallible, can be regularly correlated with what we later find to be correct.

8. AN EXTERNALIST THEORY: "PRIMA FACIE SUPPORT"

Now, of course, the most violent objection to intuitiveness counting in the process of justification could be some type of internalist stipulation that a justification must always take the form of a convincing series of reasons available, or cognitively accessible, to the knower. In my example of the rhapsode and the papyrologist, I argued (against the originator of Schematic Intelligence, Richard Skemp) that not only are we very often ignorant of any explicit justification for our intuitive schemas, but the ability to survey one's own inventory of schemas is not a faculty genuinely available to creative thinkers, and so, what seems to be responsible for many of their intuitive beliefs, if our model is acceptable, should not be expected to be cognitively accessible to the knowers, in any case.

While the novice papyrologist, as well as the amateur mathematician, vets conjectures and seeks to justify them intrinsically by measuring them up against a conscious inventory of schemas (each one acting as an added constraint on how suitable his various hunches really are), an expert may well feel he has justified substantially the same conjectures as the explicitly methodical novice, sensing a cohesion with his intuitive beliefs. And, here, there is no cognitively accessible justification at all.

Even my straightforward perceptual belief that there is a tree outside my window is generated by a causal process of a type which is highly regarded by the epistemologist, but, since I know next to nothing about optics, retinas and brain function, I can produce no explicit reasons for my belief. On my admittedly externalist reliabilist account then, in order for a belief to acquire some form of intrinsic justification, it is enough that the causal process inculcating my beliefs merely be reliable in fact, and since, on a sympathetic reading (which does not invoke the future's retrospect for example) my intuitions generally do lead to true beliefs, this, together with the prevalence of similar convictions in others, suggests that the intuitiveness of a belief p at least lends prima facie support to the claim that p is true.

9. THE NOTION OF A CONTINUUM OF SUPPORT - THE WEAK END OF THE SPECTRUM OF JUSTIFICATION

In discussions where our epistemic standards are necessarily very high, such as in deciding which axioms are suitable as a basis for set theory or for different types of geometry, commentators such as Maddy have traditionally been rather modest and tentative about the importance of intrinsic, or intuitive, support for axioms, keen to amplify the repository of supports as soon as possible with other independent modes of confirmation. Without extrinsic appraisal as well, which includes the corroboration available from suitable theoretical supports and independently verifiable consequences, no intuitive belief can count as more than mere conjecture, and, in many cases, the set-theoretic methodology has more in common with the natural scientist's hypothesis formation and testing than with the caricature of the mathematician writing down a few obvious truths, and proceeding to draw logical consequences. Besides the intrinsic appraisal criteria of intuitive plausibility, simplicity, elegance and aesthetic appeal, among the rich inventory of independent sources of justification for fully-developed axiomatic systems will be 'lack of disconfirmation', breadth and explanatory power, and, ultimately, the intertheoretic connections and confluences which the newly-devised systems and theories generate. For instance, the eventual ability of set-theoretic methods to generate the analytic theory of the continuum, a consistent rendering of our confused intuitive beliefs about the relation of the line to its smallest parts, is one of its greatest achievements, and one of its strongest post facto supports.

Nevertheless, the conjectures (some would say 'intuitions') which are generated in the course of ordinary investigative mathematics - for instance, in using associative or analogical thinking as a heuristic in the course of devising a proof - are not underpinned by a similar army of supports, and yet it would be highly desirable to be able to judge their epistemic status.

In keeping with my psychological account, then, those conjectures reached by an informal and unstructured mode of association, without the use of analytical methods or deliberate calculation, I will call intuitive conjectures. The importance of an account which can lend prima facie justification, in varying degrees, to a belief generated by associative thinking or by what one might call an 'educated guess' in mathematics, should become obvious for two reasons. Firstly, all but a vanishingly small proportion of the time spent in creating a proof in mathematics is taken up in this type of strategic thinking (rather than in tying up the ends and reflecting on the whole edifice of the proof as a vehicle of persuasion). Secondly, during a substantial amount of this strategic thinking time, when we select schema after schema to bring to bear on the proof-stage we have arrived at so far, there is a strong temptation to say we are some way between

(a) having no evidence at all for our desired conclusion, and

(b) actually knowing it.

This sort of 'progress in justification' is undoubtedly a familiar feeling among mathematicians, and may well provide feedstock for a more careful analysis of the weaker end of the spectrum of justification. But before I provide an attempt at such an analysis, let me cite an example, by way of illustration, which suggests we can differentiate the educated guess from the lucky guess in associative thinking. We may thereby accord such an educated guess an epistemic status which is somewhat weaker than that of inductive or deductive inference, but which at least provides us with a source of justification which registers on the epistemic scale, and registers independently of those supports introduced by subsequent processes of testing and analysis of consequences.

10. THE DYNAMICS OF SUPPORT - AN ILLUSTRATION

Let us say, for example, that I am considering Vn, the vector space of polynomials of degree at most n over R, with a view to finding a basis for the dual space Vn*, of functionals operating on these polynomials. I believe that it is sufficient if I can devise n+1 functionals which are linearly independent, and for even this belief to be justified there are two requirements, on my bolstered externalist theory of justification:

(i) that my producing functionals in this way has been reliable in the past in producing a basis, that is, calling them a basis has not led to any errors or unacceptable consequences, when I have done it before, either myself or vicariously by being shown it;

(ii) that my epistemic perspective is sufficiently wide for me to be actually aware that doing this ought to be reliable as well, in other words, I have enough experience of inductive inference in general (and in particular of ones which are similar in various ways), to show that transferring the previous manoeuvre or schema to the present environment is relevant.

In this case my background beliefs about V* having the same dimension as V, when V is finite dimensional, provides this epistemic perspective; analogy with the geometrical decomposition of Rⁿ into subspaces provides it to a degree, and the justification would accordingly be weaker; guesswork or wishful thinking that n+1 functionals need only be linearly independent to do the required work is not a satisfactory epistemic perspective, and my beliefs, even if they seemed to be qualitatively the same as those in more enlightened subjects, would not ipso facto be justified. While I shall return to the idea of epistemic perspective shortly, let us for a moment assume that my knowledge that Vn* can only have n+1 dimensions is unquestionable, and provides the necessary epistemic perspective for my believing justifiably that if I find n+1 linearly independent functionals I will have arrived at something stronger, namely a basis.

Now, using a combination of memory and association, I begin to decide what the functionals might look like. I have a limited range of functionals that I am familiar with, represented by inner products, say, involving integrals of terms which are composed by simple operations such as conjugation, or exponentiation.

Even my schematic classification of certain concepts as functionals at all, furnishes me with a minimal but significant justification in hypothesising any one of them that I might select (so long as there is no prima facie reason to believe that it will be poor) as a suitable starting-point for constructing a basis. But, more importantly, my background beliefs about Fourier analysis, an entirely different domain, suggest that orthonormal series are quite readily constructed using integrals of simple products, so I conjecture, say, at this slightly higher level of justification, the functionals

{ psi i(f(x))}i=0 to n = { Integral _{-1 to 1} xⁱf(x) | i = 0 to n }

with a view to testing their linear independence. At this point I experience Intuition 1, that all n x n determinants of the form

(Note that this is sufficient; for contradiction, assume a non-trivial

0 = Sum _{i=0 to n} a_i psi _i

exists. Then, considering the effect of this functional on Vn's basis { 1,x,x²,...xⁿ }, we have (n+1) homogeneous linear equations, in (n+1) variables {a_i}, i =0 to n, i.e.

f(x) = 1: Integral -1 to 1 a₀ + a₁x + ... + a_nxⁿ dx = 0; a₀ + (a₂/3) + (a₄/5) + (a₆/7) + .... = 0;

f(x) = x: Integral -1 to 1 a₀x + a₁x² + ... + a_nxⁿ⁺¹ dx = 0; a₁ + (a₃/3) + (a₅/5) + (a₇/7) + .... = 0;

... etc. ...

f(x) = xⁱ (i <= n); Integral -1 to 1 a₀xⁱ + a₁xⁱ⁺¹ + ... + a_nxⁱ⁺ⁿ dx = 0; etc.

whose solution is trivial when all the above determinants are non-zero. Otherwise, considering the n x n arrays as the matrices of linear transformations on Rⁿ, we can see that there will be a non-trivial eigenspace of solutions). My 'progress in justifying' this conjecture goes through 3 stages:

(i) Association with weakly similar very general situations ('there are so many positive terms in it');

(ii) Structural analogy from a small sample, backed by successful reliance on similar extrapolations (I try for n=1,2,3...and generalise, framing an inductive hypothesis);

(iii) Formal induction (I try to perform a double induction over n, but fail because I lack the schematic resources to discern a relevant similarity between the highly complex algebraic forms for det [M_nxn] and det [M_(n+1)x(n+1) ]

and the other pair of determinants).

I would like to say, at this impasse, that I nevertheless have analogical evidence for my conjecture, which has not yet been susceptible to the standard types of confirmation for various contingent reasons, such as the symbolic unsurveyability of the determinant expression, and my schematic bias towards seeing only simple patterns in my perceptual input. This latter point, that there is a cognitive/perceptual bias that slants our data in favour of structurally rather simple patterns, is substantially one of Quine's points, when he argues (7) that simpler hypotheses stand better chance of confirmation. Nevertheless, here it is operating negatively, since at this level of conceptual agility, we are not yet sufficiently equipped to be able to implement suitably powerful independent confirmation procedures. Having said that though, this bias is not, in the long term, a serious barrier to mathematical knowledge, because the heuristic inventory of our intuition can not only be trained to recognise yet finer distinctions (as in the case of, say, training an ornithologist), but it is often versatile enough to provide a fruitful alternative gestalt on a problem. In the above case, during repeatedly unsuccessful attempts at the double induction over determinant size, I intuited that, perhaps, instead of placing n+1 constraints on the coefficients a_i to restrict their freedom, I should regard them as fixed, Sum _{i=0 to n} a_i psi _i = 0, say, and, in this case, deviously setting

f(x) = Sum _{i=0 to n} a_ixⁱ as a trick,

0 = Integral i=0 to n a_i psi_i = Integral _{-1 to 1} a_i xⁱf(x) dx = Integral _{-1 to 1} f(x)² dx,

forcing f(x) = 0, so all a_i = 0.

I could consequently justify my belief, with the full sanction of deductive inference, that the { psi_i }_{i=0 to n} formed a basis for the dual space. Nevertheless, all along I could reasonably be said to have already possessed evidence or support (in various minimal amounts) for my attitude to my conclusion, and it would be a dogmatic theory indeed which insisted on claiming that I had no evidence either way until I came up with the final strategem.

11. THE EPISTEMIC PERSPECTIVE

When we tackle Steiner's question then, "Is mere true conjecture knowledge?", we must centralise the epistemic perspective from which the conjectures were made, rather than study them propositionally, or in isolation. We must aim to partition (in some sensible, but not necessarily categorical way) our true conjectures into (i) lucky accidents, and (ii) suggestions of a type which have some history of reliability in similar situations, investing the latter, in an important sense, with some prima facie intrinsic justificatory support. With the new conjecture we naturally associate similar manoeuvres, so that factors that include their range and degree of similarity, together with experimental information about their general reliability, will determine the strength of the initial support for our new conjecture, prior to the cut and thrust of more methodical, rigorous analysis. If present at all, prima facie intrinsic justification is present only in some cases of true conjecture, and as we have often seen, an epistemic theory which aims to be honest about it runs the risk of either inventing conditions which are too severe, because of a mistaken analysis of knowledge in terms of necessary and sufficient conditions, or on the other hand it risks appearing to provide too much warrant at the outset, for what are often no more than fortuitous speculations. In order to reject a 'higher register of consciousness' account, where we can only justify our true conjectures or intuitive beliefs by actually producing a cognitively accessible series of reasons for their appearance, a faculty whose universality I argued against, I weakened my epistemic theory to an externalist one, a kind of reliabilism in which justified belief is belief resulting from a cognitive process which is reliable in fact.

Now I wish to strengthen it by claiming that the degree of intuitive support a mathematician may attach to a conjecture he is investigating will depend crucially on his own heuristic inventory and range of natural associations, together with all the distributional information he has acquired about their reliability in practice, in a whole host of similar situations. In other words, fallacies apart, experience is enough to tell us when the beliefs that we arrive at by associative thinking, are reliable in their individual contexts. Appeal to the metrical properties of R² , say, in analytical topology may suggest justified beliefs about other finite-dimensional Banach spaces, with Lipschitz-equivalent norms. Strictly, conjectures of this type are analogies, and yet they all share a strong component which is akin to inductive inference.

On the other hand, though, there are true conjectures which are analogical in form but may lack justification for us simply because of our limited epistemic perspective. For instance, when considering the annihilators of intersecting subspaces in finite dimensions, I may conjecture truly, but without justification, that for subspaces U and W,

(U intersection W) ^o is contained in or equal to (U ^o direct sum W ^o)

even in infinite dimensions, by analogy with hyperplanes of co-dimension 1 in Rⁿ. If this is arrived at by association from finite-dimensional geometry, where the result holds with equality by a much stronger dimensional argument, then the conjecture, for me, will have extrinsic (but have very little intrinsic) justification. If, on the other hand, such or similar transformations are not unprecedented on my part, and they have been reliable in the past - if, for example, I know a fair amount about other aspects of duality in infinite-dimensional space - then my analogical intuitive belief in this case possesses not only heuristic, but also evidential, value.

12. FINITE SCHEMAS BEING CASHED IN INFINITE DOMAINS : BROUWER'S WARNING

One of the dominant leitmotifs of 19th century analysis - dominated, as this was, by the Critical Movement of Cauchy and Weierstrass - had been a caution or reserve over the mathematical use of the infinite, except as a façon de parler in summing series or taking limits, where it really behaved as a convenient metaphor, or mode of abbreviation, for clumsier expressions only involving finite numbers. It is well-known, however, that when Cantor came on the scene, the German mathematician Leopold Kronecker, who had already 'constructively' re-written the theory of algebraic number fields, objected violently to Cantor's belief that, so long as logic was respected, statements about the 'Completed Infinite' were perfectly significant. Cantor had further urged that we should be fully prepared to use familiar words in altogether new contexts, or with reference to situations not previously envisaged. Kronecker, however, felt that Cantor was blindly cashing finite schemas in infinite domains, both by attributing a cardinal to any aggregate whatsoever, finite or infinite, and worse still, in his subsequent elaboration of transfinite arithmetic.

Of course, Euler had, many years earlier, summed Bernouilli's series Sum _{n=1 to} (1/n²), by applying rules originally designed for the finite, to the infinite. Conjectural Intuition led Euler to consider the familiar decomposition-schema, for an even polynomial in terms of its non-zero roots i :

0 = Sum _{i=1 to infinity} (-1)ⁱb_ix²ⁱ = b₀ (1-{x²/beta _i ²}).

By analogy with this finite decomposition, perhaps (sin x)/x, despite its infinity of roots at ±n , might also admit a dual representation:

1 - (x²/2.3) + (x⁴/2.3.4.5) - (x⁶/2.3.4.5.6.7) + ... = (1 - {x²/pi²})(1 - {x²/4 pi²})(1 - {x²/9 pi²})....

Whether a refined intuition of this product could be genuinely developed or not, what mattered was that simply looking at x² 's coefficient

(1/2.3) = (1/ pi²)+(1/4 pi ²)+(1/9 pi²)+....... meant that 1+(1/4)+(1/9)+..... = pi²/6.

Although the interim 'strain on the intuition' (at the time) was crucial to Euler's heuristic approach, this particular infinite detour had been analysed out of his subsequent proofs of the result, which appeared almost 10 years after its discovery.

More awkwardly though, Cantor was, by the turn of the century, busy generating a whole hierarchy of actual infinities, and shortly afterwards in 1904, Zermelo's licentious appeal to the 'selections of representative elements' from even uncountably-infinite families of sets (in his Well-Ordering Theorem), provoked a heated debate dominated by the French Analysts. The Paris School attacked the blind cashing of unwarranted finitary schemas in wider domains, criticising the unfettered development of the theory of selections and rejecting the tacit use of non-effective procedures in topology, measure theory, and functional analysis. When asked to give his opinion on Zermelo's proof, by Hilbert in December of the same year, Emile Borel, who rejected transfinite ordinals beyond those in Cantor's second number-class (8), wrote an article for Mathematische Annalen which stated that he was unimpressed. Zermelo, he said, had merely shown the equivalence of the problems of

i) Well-ordering a set M, (by expressing it as the union of an increasing chain of gamma-sets:

UNION of {M(gamma) well-ordered, and contained in or equal to M | a in M(gamma) & A={x|x in M & (x<a)} -> Ch(M-A)=a} ),

and

ii) Choosing a distinguished element from each non-empty subset of M.

In spite of the desire to remove unnecessary countability assumptions, which Borel regarded as one of the causes célèbres of mathematics since the Cantor-Kronecker feud, he insisted (as Brouwer would reiterate many years later), that Effective-Choice Principles, which had historically been isolated from the mathematics of finite (or, at worst, denumerable) sets (9), were not devised as a tool for dealing with the infinite as Cantor's contemporary followers hoped to conceive of it. Borel ostracised the practice of making uncountably-many arbitrary selections from sets, saying that it was 'outside mathematics', while his even more ascetic colleague, Baire, pointed out that on the one hand the continuum (namely the set of all sequences of natural numbers), and, on the other hand, the well-ordered sets, were two things which for him were only defined in potentiality, an incompleteness which made it unlikely that either of the two potentialities could ever be reduced to the other by familiar comparison techniques such as the Schröder-Bernstein Theorem.

While Borel later replaced his 'denumerability criterion for effective selections' with one based on effective enumerability (10), (paving the way for investigations into related notions of 'effectivity' for N, by the 1930's Recursion Theorists), Brouwer, for one, issued a further, forceful caveat concerning how far we should project, or stretch, our basal intuition - or even be gullible as to its generality.

"The common uncircumspect belief in the applicability of traditional logic to mathematics was caused historically," says Brouwer, "by the fact that, firstly, classical logic was abstracted from the mathematics of subsets of a definite finite set, that, secondly, an a priori existence independent of mathematics was ascribed to this logic,and finally, on the basis of this suppositious apriority, it was unjustifiably applied to the mathematics of infinite sets". (11)

The heroic course on which Brouwer embarked, in order to avoid unwarranted ramification of intuitive procedures valid only in more limited domains, was to elaborate constructively (by a Kantian 'successive synthesis') on our rudimentary awareness of mental states. This consisted of the observation of our 'life-moments falling apart', and succeeding one another in time, thereby familiarising us with the unmistakable schema of a linear series. If it is possible then, according to Brouwer's 'smoky metaphysics', to experience distinct acts of awareness, then the natural numbers are not only implicit in the stream of our consciousness, but their arithmetic, isolated by more intellectual mental processes, may also be conceived as growing directly out of primary awareness.

Classical set-theorists however, (such as Fraenkel, Bar-Hillel, and Levy (12)) seem more closely to represent current mathematical practice, which has grown impatient with the intuitionist's neoteric and unwieldy account of the continuum - conceived not as a classical Banach space, but as a spectrum of ever-emerging points only potentially realisable, by repeatedly applying Heyting's Cauchy-criterion to infinitely proceeding sequences, whose individual continuation is itself governed by spread-laws. Modern analysts therefore prefer to regard the existence of 'an infinite stage' (13) as part of the intuitive picture, adding an existence axiom to provide us with a ready-made infinite set. Nevertheless, while it is universally agreed to be 'necessary for science' (14), the subsequent cashing of power-set, choice, and (post-1922:) replacement-schemas on the given infinite set, is often granted much more than a heuristic status, and is awarded the 'intuitive' medal as well (15). There is some sense though, in which the cumulative hierarchy is intuitive, and, as working mathematicians are keen to emphasise, it is not as though we are just veiling the bare pragmatics of using these principles with an aura of epistemic respectability.

But familiarity with our arrows alone would not guarantee us success, as archers, against an ever-increasing range of targets; and, as the Hausdorff Paradox will show (section 15), while some of our schemas may well be very familiar, the combination of both schema and context of application may not be backed per se by any obvious aspect of our conceptual life. We should not then be unduly surprised when both transfinite set theory and functional analysis, though they rest on a very naturally conceived mathematics of the infinite, soon begin to display to us whole new brands of paradox.

13. ARE OUR INTUITIVE CONJECTURES LIMITED, OR BOUNDED ABOVE, EITHER BY THE NATURE OF OUR SENSE-EXPERIENCE, OR BY OUR CAPACITY FOR CONCEPTUALISATION?

Although both our mathematical heritage, and the style of our educational development within it, place strong constraints on what we tend to call 'intuitive', this is more of a social accident than a contingency which prevails at the cognitive level, where far more formidable constraints may be found.

The belief that our intuitions are bounded above by the nature of our sense-experience together with the interpretative skills we can apply to it, is a view I shall call the 'Cut-Off Theory'. This is perhaps corroborated in the extent to which non-standard systems proliferate in physics, geometry and set theory, once we have got beyond a certain level of proof-theoretic power. The opposing view, I shall call that of 'Ramified Intuition', referring to how one might somehow be able to carve a path through the different formalisms generated at the crucial stage, and select only those which are best corroborated not just by their extrinsic applicability, but also by their compatibility with other intuitive theories, supported by cautious "Gedankenexperimenten".

Gödel's feeling is that our intuition can be suitably extended to a familiarity with very strongly axiomatised domains, such as extensions of ZF, or calculus on smooth space-time manifolds, thereby providing us with backgrounds for either accepting or rejecting hypotheses independently of our pre-theoretic prejudices or preconceptions about them. And that might help to extricate us from a situation in which 'the dial is tied to the hands of the clock'. (16)

Perhaps the reaction of the cut-off theorists would be to say that the multiplicity of ways in, the versatility of outlooks provided by our intuitive schemas so often at lower levels, is simply unavailable at higher levels, to produce enough intuitive corroboration as is needed, on as many independent fronts as possible.

In response though, we may point to the fact that, in appraising, say, the Church-Kleene equivalence theorems for -definability, Turing-computability, and general recursiveness (in the Gödel and Herbrand sense), with regard to their claims to be collectively demarcating the limits of 'intuitive computability', it is a feature of this particular problem that it is susceptible to a diversity of equally restrictive intuitive re-characterisations, whose unexpected confluence gives each of them a strong intuitive recommendation.

Moreover, this confluence turns out to be a surprisingly valuable asset in appraising our rather more recondite extensions of our intuitive concepts:

"It turned out that weak compactness has many diverse characterisations, which is good evidence for the naturalness, and efficacy, of the concept." (Kanamori and Magidor, (1978), p.113).(17).

In spite of this, among those who remained sceptical about developing our powers of intuitive appraisal was Hermann Weyl, who often spoke rather sardonically about Gödel's optimism:

"Gödel, with his basic trust in transcendental logic, likes to think that our logical optics is only slightly out of focus, and hopes that after some minor correction of it, we shall see sharp, and then everyone will agree that we are right. But he who does not share such a trust will be disturbed by the high degree of arbitrariness in a system like Z (Zermelo's), or even in Hilbert's system ... No Hilbert will be able to assure us of consistency forever; we must be content if a simple axiomatic system of mathematics has met the test of our mathematical experiences so far. It will be early enough to change the foundations when, at a later stage, discrepancies appear." (Philosophy of Mathematics & Natural Science).

14. CATCHING STRONG POSTULATES IN A BROADER INTUITIVE NET

There are several types of cut-off arguments which seem devastating against any ramifying plan such as that advocated by Gödel. By way of illustration, one of Gödel's original arguments in favour of the unsolvability of the Generalised Continuum Problem (on the basis of our usual intuitive axioms), was based upon certain facts, which were unknown to Cantor, which seemed to indicate intuitively that the continuum hypothesis will ultimately turn out to be wrong (while, on the other hand, we know that its disproof is demonstrably impossible, on the basis of the axioms being used today).

In the ensuing debate there arose a search for dramatic or outrageous consequences of either CH or its negation when appended to ZF, which did so much violence to our intuition that the case for accepting either one, or the other, of the two conjectures, was undermined. (18). Unfortunately though, in many cases the suggested thought-experiments were weak and parochial, in that either the prejudices being appealed to were hopelessly naive (e.g. Freiling (1986), (19)), or the investigations are carried out on a 'vanishingly small subsystem' (Hausdorff, (1914)) of the principle's potential range of application.

This latter problem was exacerbated because our cognitive grasp of the (2^(2^aleph₀)) sets of reals has inevitably been mediated, and more finely developed by forming intuitive schemas on the (2^aleph₀) well-behaved sets of the Borel hierarchy. Accordingly, the Cantor-Bendixson theorem (that CH holds for closed sets of reals) was readily generalised to PI⁰₂ (20) and all Borel sets (21) due to the schematic simplicity of the projective operations involved in generating them, and, as it were, by pulling the weighty hypothesis a 'vanishingly small distance further' along the new schematic rails. (22).

Even our schematic means of definition (in creating an apparently substantial hierarchy by recursion of our intuitive operations over the countable ordinals), guarantees that we have insidiously conferred an unwanted simplicity on what point-sets we are equipped with, to act as feedstock for ramifying our intuition.

Moreover, even Lusin's drastic extension (1925) (23) of the Hausdorff result to Souslin's analytic sets SIGMA¹₁ (24), still incorporated the 'perfect subset property' as the vital 'hidden variable'. This crucially makes all the point-sets we tend to consider 'structurally too special to act as a guide to the continuum hypothesis overall.' (25).

It is this in-built cognitive tendency, which hampers our attempts to ramify our intuition: we extend our mathematics into strongly-axiomatised domains, where new principles have a much freer rein than before, so that the potential domain of their application outstrips what we can readily specify using our old schemas, even suitably bolstered by using transfinite induction, or recursion, as ramifiers. Consequently, any familiarity we pretend to develop with these domains will be largely mediated by schemas developed on the subsystem, which we must therefore guard ourselves against cashing - as far as is consciously possible - in the surrounding global domain.

Moreover, it might be thought that this 'inability to escape' - from intuiting formally simple subsystems of those domains into which we extend our mathematics - guarantees that the progress of ramifying our intuition will inevitably be jejeune, and - in both senses - skeletal. (26).

But before such a qualm could seriously challenge our current styles of intuitive thinking in higher mathematics, and induce us somehow to compensate for our affective responses, or de-bias the contents of associative memory, we would need further examples of cases where there was some compelling reason to believe that we were effectively 'inducting from a biased sample', in so exercising our intuition.

15. THE GREATER DISTANCE BETWEEN ARCHER & TARGET: SORITES SITUATIONS & THE HAUSDORFF PARADOX

Let us consider, by way of illustration, the significance mathematicians have often attached to the Tarski-Banach theorem. (This asserts what has otherwise become known as the "Hausdorff paradox", namely the deduction that any sphere in R³ of unit radius, may be partitioned into a finite number of (non-measurable) pieces which can be rotated in R³ to form a partition of two disjoint spheres of unit radius).

Many people find this result implausible, although it is a consequence of the Axiom of Choice (when appended to RA² - second-order real analysis) - all of whose axioms are allegedly intuitively plausible. But the actual significance of this result should not be overstated. It even seems rather benign and natural, once we examine the processes of extension and transformation which lead our chain of reasoning away from what can be readily accommodated by our intuitive schemas. Those who are more anxious, however, might claim that the iteration of axioms one-on-one has led to a Sorites situation, generating a theorem eventually, or ultimately, which is not merely not intuitively true, but intuitively false. This could have happened in one of two ways, depending on whether our formal apparatus, the axiom-system involved, is poor at playing intuition's role (Brouwer's qualm) or whether that role is hardly worth representing anyway (Frege's qualm). In the first case perhaps the axioms, supposedly backed by our intuitions, fail to represent them precisely, and, since they are very slightly incorrect, the subsequent process of inference has merely built up the error. This explanation at least has the recommendation that it accounts for different degrees of implausibility. Alternatively, it may simply be that the notions involved were inherently incoherent, and it required the building of an edifice substantial enough to fall down itself, before the latent inconsistencies had worked themselves to the surface.

16. BEYOND INTUITION: WHAT IF THE TARGET VANISHES?

While these Sorites situations are pernicious, their progressive insinuation into the epistemologically-safer subdomains of mathematics, can, in practice, be partially held back by a revisionist struggle such as that advocated by Hermann Weyl (section 13). This consists of successively

i) updating, altering and refining our naive intuitions (to diminish what I earlier called Frege's qualm),

and subsequently

ii) decreasing the shortfall between our formal systems and the intuitions of the day, which they claim to represent (reducing Brouwer's qualm).

The Sorites situations therefore have by no means the same power to block the extension and revision of our intuitions, as the arguments that our intuition will inevitably give out altogether at a certain stage.

Here the conclusions will not be intuitively false, but simply not intuitively true, and the candidates for appraisal will behave like targets which are no longer just very far from the archer, but no longer even visible at all.

17. THE ANALYSTS DISTANCE THEMSELVES FROM GEOMETRICAL INTUITION: ITS ROLE IN EXTENSION PROBLEMS

The 19th century belief that our geometrical prejudices should be isolated and withdrawn from the formal presentation of proofs in analysis, led to the idea that our basic intuitions were too weak to have any decisive role to play in the subsequent development of mathematics. This, however, often meant that we had now begun to notice when inappropriate schemas were being used, or that we had become impatient on noticing that their unquestionable success at the conjectural stage - the context of discovery - was not mirrored or redoubled later on when it came to finding a sufficiently acceptable justification.

Imagining perhaps, that he was banishing deceptive intuition forever from analysis, Cauchy merely succeeded in driving it down to a far deeper level where it could continue its subtle mischief unabated. Before recognising 'uniform convergence' (in the Stokes-Siedel sense), Cauchy for a time (27) believed that the sum of any convergent series of continuous functions was not only continuous, but could be integrated term-wise, while he also followed Gauss into the minefield of casually interchanging limits in double-limit processes. Weierstrass, however, whose terminology brought an unprecedented rigour to proofs involving notions of continuity, supported his call for a strictly logical approach to mathematics by constructing disconcerting counterexamples to plausible and widely-held notions. The prime instance of this was the case of the continuous but nowhere-differentiable function

Sum _{n=1 to infinity} a_n cos (bⁿx), 0<a<1, ab >1+(3/2);

whose surprise presentation in a paper read to the Berlin Academy of Sciences in 1872, challenged the loose way in which geometical and other intuitive ideas were used in proofs. A further bolt from the blue came in 1890 when Giuseppe Peano (whose later logical work represents a permanent landmark in the modern foundations of mathematics), discovered a continuous curve that passes through every point of a square. This, too, challenged the over-reliance on 'spatial intuition' as a source of comfort in handling the developing functional calculus. Nevertheless, the geometry of our youth, now seen behaving like a maladjusted 'enfant terrible', had insinuated itself so deeply into our schematic grasp of functions on the real line, that it now seemed ineradicable, and maybe even indispensable. Our learning of the calculus was made considerably easier by the use of such pedagogic devices as graphs and trigonometry, and although, strictly speaking, no analytic geometry is needed for either calculus or the theory of analytic functions, we find it advantageous, and in practice necessary, to continue to use geometric interpretations (just as keeping certain familiar interpretations in mind facilitates our handling of the undefined terms in any axiomatic system).

Mathematicians such as Hermite, who considered the class of functions to be co-extensive with the functions pictured by their geometric intuition, treated the examples of Weierstrass and Peano as pathological cases, quite outside the field of orthodox mathematics. But the real significance of the varieties of abnormal behaviour that continuous functions can show, is that it taught mathematicians caution about cashing naive intuitive schemas, derived from the limited world of basic geometric experience. In particular, the ardent 'cut-off' theorist can point to a whole plethora of theorems in analysis where our naive pre-theoretic intuitions about 'geometrical' sets (say) in Euclidean space, retain their heuristic status, but their 'allegorical' quality debars them from carrying any weight in a proof. That is to say, the conceptual relations in the theorem can often be translated into an equivalent intuitive form - a form, that is, which is felt to be plausible because our schemas readily accommodate it, but which ultimately requires independent justification. Having said that though, there are still those who claim that results such as the earlier Hausdorff paradox, with its labyrinthine reassembly of non-measurable point-sets, defies intuitive characterisation and that ultimately even our refined intuition will be neutral as to the status of certain extensions, maybe even using familiar Choice Principles in strange new contexts. Henri Lebesgue hints at this view when he says that it would be no surprise if our intuitions were represented mathematically by measurable sets.

Some hypotheses then, seem hopelessly out of reach. We must therefore be content to admit that our finer intuitions about the universe of point-sets - or about any other transfinite constructions we tend to employ - may not ultimately be sufficiently far-reaching to produce clear and unambiguous answers for either the continuum hypothesis, or any other pertinent questions or enigmas about the analytic theory of the continuum.

The only alternative is to seek a whole new brand of theoretical intuition which goes much further in heuristic strength than our pre-theoretic prejudices.

18. GÖDEL'S WEDGE

Gödel (28) explains our surprise at the emergence of paradoxes such as Peano's construction of space-filling curves, or Weierstrass's discovery of continuous but nowhere-differentiable functions, by accusing us of carelessly mixing our pre-theoretic intuitions, with our more refined, analytic and topological ones. Such a clash, between familiar geometry, say, and the set-theoretic reduction of point-sets, will undoubtedly arise at some stage, in considering not just the Hausdorff result, but also similar chimaeras such as Cantor's ternary null set. In other words,

"the [paradoxical] appearance ... can be explained by a lack of agreement between our intuitive geometrical concepts and the set-theoretical ones occurring in the theorem". (29).

Accordingly we must drive a wedge between our pre-formal and formal intuitions, in the hope of 'separating out errors coming from using the pre-theoretical intuition'. (Wang, (1974), p.549). This however, is easier said than done, and although Gödel indicates the need for vigilance, and provides a further valuable perspective on both Brouwer's qualm (section 16) and the Sorites situations which provoke it, commentators (29) have repeatedly been bemused at how Gödel proposes to apply the wedge in practice. The suggested exercise in discrimination seems notoriously difficult to carry out, especially when it is tempting to say that the 'refined' intuitions of one generation - far from being a once-and-for-all clarification of our logical optics - have historically either turned out to be fallacies, or at best become the naivest intuitions of the next (cf. section 5(i)).

19. FALLACIES OF INTUITION

Those who are eager to argue how futile it is to try and demarcate or even seek out an epistemologically safe subsystem of 'pure intuitive propositions' (which could be used as the basis for an unproblematic branch of mathematics), also tend to emphasise how often we fail to discriminate reliable intuitions from processes known, post facto, to lead to false beliefs. For instance, in his attack on various popular accounts of intuition, insofar as they claim that intuition provides us with an incorrigible a priori knowledge of mathematics, Philip Kitcher (30) cites several episodes from the history of mathematics when mathematicians have hailed something as intuitively self-evident - giving it much the same status as we give to the Zermelo-Fraenkel axioms of set theory - even though these 'Conjectures of the Day' have, subsequently turned out to be false. The most familiar example, perhaps, is that of Frege (or even of Dedekind or Cantor), each of whom advanced a 'Universal Comprehension' principle, taking any property to determine a set. But this is by no means the only case of its kind. Shortly before this, the great Gauss and Cauchy went astray (section 14), by surrendering themselves to the guidance of intuition, and earlier still many mathematicians of the 18th century believed in the self-evidence of the 'law of continuity', (which states that what holds up to the limit, also holds at the limit). This also turns out to be a natural fallacy.

These disconcerting cases show that we cannot always apply 'Gödel's wedge' and discriminate reliable (or even a priori) intuitions from processes known (in retrospect) to lead to false beliefs. They do not, however, in themselves, impugn our ability to accrue mathematical knowledge, any more than the existence of sensory illusions, or 'deceptions of the senses' (to borrow Gödel's analogy) create insuperable obstacles to our knowledge of physics.

But where our discriminatory shortcomings do matter, though, is in cases where experience suggests that the intuitive belief we have formed is misguided, and this provides a stumbling-block for the thesis that our intuitions occupy the position of being a privileged warrant, by their very nature, for our beliefs, and somehow continue to justify them, whatever recalcitrant experience we come up against.

Similarly, the set-theoretical paradoxes threaten not so much the possibility of mathematical knowledge, as they now threaten either an a priori, or any other unduly perspicuous account of its nature.

These fallacies of intuition then, have gained a significance in the contemporary epistemology of mathematics, which, as Georg Kreisel suggests (Informal Rigour and Completeness Proofs) has been somewhat overplayed. This, no doubt, results from our memory bias which makes us, for the most part, recall surprises, memorable cases in which strong initial impressions were later disconfirmed, and ultimately it also leads to an overestimate of the dangers of intuitive thinking.

Favourite examples of intuition going astray are often cases of over-simplifications, of applying schemas too generously where their domain of application has to be more finally demarcated. This happened when the Weierstrass M-test undermined the epistemic status of most proofs with casual interchanges of limits in double-limit or integral-summation processes, and, similarly, Zermelo's separation axiom was designed to allow limited comprehension on previously-constructed sets. Some intuitive beliefs have in fact been falsified by the progress of science - for example, the belief that at any given moment, a physical object is in a certain location and moving at a certain speed (pre-Heisenberg), or the pre-relativistic belief that time doesn't slow down when you travel at ten miles an hour.

But the feeling is, that these examples only replace one form of intuitive justification with a finer one, so that in scientific contexts intuitive beliefs must be tested like any other hypothesis - they are equally defeasible, can be outweighed by theoretical evidence, and, like any other hypothesis, they can be overthrown.

In the words of Imre Lakatos (31):

"Why not honestly admit mathematical fallibility, and try to defend the dignity of fallible knowledge from cynical scepticism, rather than delude ourselves that we can invisibly mend the latest tear in the fabric of our 'ultimate' intuitions?"

20. EMPIRICALLY BOLSTERED INTUITION: THOUGHT-EXPERIMENTS AS RAMIFIERS FOR RIEMANNIAN GEOMETRY

It is worth remarking perhaps, that there are, in current usage, already two different ways (32) of conceiving of the extension of the term 'counterintuitive', depending on whether or not our informal, natural preconceptions are allowed to be extended or modified to accommodate non-standard systems, which may be corroborated by empirical science, and which we can - to a point - familiarise ourselves with, by means of simple thought-experiments.

When, in the 19th century, the German mathematician Riemann, and independently Helmholtz, developed another type of geometry (33) which in effect corresponded to Sacchieri's obtuse angle hypothesis (a postulate which Sacchieri himself regarded as ridiculous), Riemann developed his conception not by means of the postulational approach, but by generalising and extending an intuitive notion of 'curvature', originally developed by Gauss, to geodesics. Geodesics are those paths which lie on a general surface of co-dimension 1, enabling us to speak of the curvature of 3-dimensional regions of space. Riemannian, Lobachewskian, and Euclidean geometries therefore envisage a space all regions of which are alike in having positive, negative, and zero 'curvature', respectively.

To 'restricted' intuition, this way of speaking sounds paradoxical, and, at first sight, to the layman, it is as if he were being asked to visualise 3-space as somehow bent and twisted, perhaps in the 4th dimension - whatever that means. But when Rom Harré and Karl Popper discuss conceptualisability as a constraint on theory-formation in physics, they still mean to allow intuition to be bolstered by conceptual heuristics, so that actual feats of the imagination become as unnecessary as they are impossible. Perhaps this will become clearer with an example.

There seems to be no loss of generality, if we use, as our intuitive heuristic here, the case of a blind map-maker imprisoned within a surface (34), so that he can never move above or below it, and so that a 'straight line' from his point of view, could, for practical purposes, be identified with the shortest distance, for him, between two points. But then the geometry of the surface (i.e. what the angle-sums of triangles composed of 3 geodesics will be, and so forth) will be determined by the curvature of the surface, so that any two regions that are similar in curvature, will be similar in geometry. Consequently, although the mathematical definition of curvature is not simple, bolstered intuition can indeed visualise it, in some sense, by creative analogy with ordinary surfaces, and there is nothing inherently paradoxical about it.

21. THE A PRIORI GEOMETERS

The trouble is though, advocates of the so-called 'a priori' interpretation of geometry regard Riemannian geometry as false, no matter how far it is corroborated as the underlying geometry of our universe by astrophysics, or relativistic empirical science. Confusion about how the apriorists can make any headway here at all, arises because people often get the idea that if the postulates of some non-Euclidean geometry are true, then the postulates of Euclidean geometry cannot be true at all, due to some sort of mutual incompatibility. This, however, is a mistake. People who have this idea do not fully realise that geometrical axioms are capable of truth or falsity only when interpreted in some specific way, so that a pure uninterpreted set of axioms is, in itself, neither true nor false. It is therefore misleading to say that Riemannian postulates are 'true empirically' when in fact they are true under some interpretations and false under others. Just as many structuralists have been inspired by Benacerraf's attack on the set-theoretic reduction of N, (fleshed out, as it is, with arbitrary features that go beyond the role of the counting numbers in language), the apriorists have similarly been led to say that 'straight lines' (or 'real numbers', for that matter) are only determined insofar as they are whatever obeys their 'normal-usage' axioms, namely the Euclidean postulates, or in the case of the reals, RA². Consequently, any revolutionary interpretation which tries to tie these terms by fiat to empirical outcomes concerning, say, geodesics, will merely show that the sides of empirical figures, for the apriorists, are not really 'straight lines'. Furthermore, it would be just as perverse and counterintuitive - to them - to extend our notion of straight lines to geodesics, as it would be to put forward, first of all, a neoteric account of the continuum, not as a classical Banach space but couched in terms of Heyting's 'Cauchy-criterion' on infinitely proceeding sequences, and then to proceed to endorse it as supplying the currently most natural analysis of our intuitive concepts.

Unfortunately, though, the a priori interpretation not only seems prescriptive in itself, if it stands rigidly by Euclidean geometry, but it may well be just as arbitrary in deciding questions about proposed extensions of our terminology into radically different new domains, such as quantum mechanics or the calculus of manifolds. Worse still, the apriorist's view is not merely a parochial stand-point, but it also fails to allow for the evolution of our concepts, and the subsequent modifications and realignments of our intuitive schemas. These realignments presumably occur when useful new concepts are introduced into the domain of discourse, and as new flaws, in old ones, come to light. Even if there is a considerable reticence - or even just a natural time-lag - before the new schemas are more generally incorporated into people's conceptual systems, the diaspora is irrepressible, and such a development inevitably takes place (cf. Newton's schemas in section 5(i)).

The empirical scientist who is also an enlightened 'use-theorist', would therefore attack Euclidean geometry as an unassailable arbiter of intuition, since he will probably regard any tentative axiomatisations from our intuition not simply as distortions of our ideas of our intuitive conceptual relations (like Brouwer's view of the Heyting calculus), but as heavily idealised versions, of much less well-behaved patterns of terminological usage. 'It is fair enough', the Wittgensteinian physicist concedes, 'if the Euclidean calculi and other logical systems we generate (which are rather similar to language-games with well-defined fixed rules) are treated as the great mathematician Frank Ramsey suggests' - in other words strictly normatively, or for purposes of comparison with a proliferation of non-standard theories - 'but it is all too tempting to regard them as in some way prescriptive of the future course of our intuition. That would be to behave as if these axiomatisations were better, rather than worse, than the informal language the physicist develops - the latter being precisely the domain in which his intuitions roam up and down'.

To this charge though, the reticent apriorists can always say - for a time - that the modern empirical scientist, in his eagerness to support many new realignments of our intuitive schemas comparatively soon after they are conjectured, is simply playing a different language-game, and furthermore, a dangerous one: in the short term, there is no regulative ideal, except perhaps patent contradiction, to prevent ramified intuition from going too far; whereas in the long term, 'the bold bridgeheads seized by intuition must be secured, by thorough scouring for hostile bands that might surround ... and destroy them'. (35).

In the meantime then, in discussing our primitive inklings of plausibility and the epistemological status of their currently-sanctioned extensions, what is at stake is not a simple empirical question of truth or falsity, nor is the issue one of analysing the semantics of ordinary usage: this is the kind of case where our problem is to decide which are the deeper of many conflicting tendencies, all present in our usage of the terms involved, and each of which enjoys its own ephemeral rise and fall in the conceptual evolution of our particular culture.

22. VISUAL HEURISTICS AND THEIR LIMITATIONS

When Frege speaks (36) of the 'truths of Euclidean geometry' as governing all that is spatially intuitable, it looks as though, at last, we may have found a domain in which our intuitions are constrained and held within strict and well-defined bounds.

"Conceptual thought", he says, "can after a fashion shake off the [Euclidean] yoke, when it assumes, say, a space... of positive curvature. To study such conceptions is not useless, by any means; but it leaves the ground of intuition entirely behind".

But when Ewing (1938), and Strawson (1966) go on to endorse this essentially Kantian line, claiming that 'phenomenal geometry' (i.e. what we can spatially intuit) is necessarily Euclidean, James Hopkins, in his famous 1970 article Visual Geometry (37) insists that (p.27) 'the geometry of imperfect sight in an unobvious world will be indeterminate'. Since our mental images are too crude to determine the curvature of our space, they will be neutral on the 5th postulate, since they effectively 'paint over a delicate design with a thick brush'.

Moreover, even if our mechanical and optical experiences allowed us to derive an absolutely exact metric for the spatial region intuitively accessible to us, we could not go on to apply the metric globally: a region homeomorphic to a connected subset of Euclidean 3-space can belong to many different topological spaces,as observed by Clifford in 1873. As well as the infinite space R³ itself, there are ten families of compact spaces which are also locally-isometric to R ³. (38). Intuition's default in furnishing criteria for choosing between these globally homeomorphic but locally-isometric space-forms, mirrors its uncertainty seen in the Euler-Cantor cases of section 12, over applying schemas derived from the finite to the infinite.

Nevertheless, the feeling is, perhaps, that the canvas of the imagination, is just one asset among many, to be found in the heuristic inventory of our intuition, and within each domain, be it visual or formal, the individual schemas behave like instruments with their own peculiar limitations.

The mainstay of intuitive geometry is no longer the pictorial imagination - though it often provides a mental starting-point or stimulus - but the set of conceptual relations, and their attendant schemas, determined by the definitions. Accordingly, not all our schemas are visual, and as we saw earlier (section 17) our formal schemas readily supersede our basic geometric ones in clarifying the subject-matter of real and complex analysis. As we saw there,

"a space is nothing but the verbal substantialisation (la substantialisation verbale) of mutually-compatible spatial relations. To say that a figure cannot enter into a space is tantamount to saying that it constitutes a system of relations which is incompatible with a more general system, embellished with the name of 'space' (décoré du nom d'espace)". (39).

A realisation of how we tend to amplify, formally, our impoverished visual intuitions, and of how we proceed to adopt the formal schemas as less unwieldy surrogates for the visual ones, led Ernest Mach to go beyond the classical empiricist posture and acknowledge the conjectural aspect of our intuition in autonomously generating concepts:

"The same economic impulse that prompts our children to retain only the typical features in their concepts and drawings, leads us also to the schematisation and conceptual idealisation of the images derived from our experience". (40).

This process of idealisation has so divorced geometry from sense-experience that, although we can induce the formation of new intuitive schemas by reasoning with the infinite, isotropic, homogeneous, (epsilon/delta)-continuous space of geometry, it would be something of a surprise if, at the end of all that, we were still able to visualise it by a Herculean stretch of the imagination. And there is no need for us to suggest, rashly, that perhaps our mental images are, strictly, if undetectably, non-Euclidean (41). Ultimately the 'change in visual congruence' which Reichenbach argues for in claiming we can become accustomed to visualise non-Euclidean geometry, is an unnecessary feat in actually familiarising ourselves with it. Even our complex formal presentations of geometry can similarly acquire a more or less intuitive status - induced by the enculturation process described in section 20 - so long as we are convincingly shown, along the way, that the new derivations and corresponding schemas, naturally arise form our attempts at intellectually dissecting and recomposing the idea of space we have always been familiar with.

Our intuitions then, of 'general geometry' (42) - which far from being the negation of Euclidean geometry, includes it as a special case - are derived from our familiarising ourselves with all our currently thought-of systems of forms in spaces of arbitrary dimension. Speaking of this newly-secured intuitive territory in his Development of Mathematics, Bell says (p.464):

"The revised definitions of curves and surfaces exposed much which was unsuspected, but perhaps partly implicit, in the intuitive concepts abstracted originally from sensory experience. In short, a deeper intuition broadened and deepened a shallower".

23. CONCLUSION: NURTURING OUR INTUITION

I opened this discussion with a plea to those embarking on any historical enquiry, to guard themselves against the fallacies and errors of the past. Of course, this is more easily said than done, in that we are largely the inheritors of conceptual systems peculiar to our scientific heritage, and even more constrained by the idioms peculiar to the present stage of its development.

But what is clear, though, is that while the patterns we are trained to recognise are codified as schemas, the schemas we are most keen to apply are occasionally poorly-tuned, not suitable for the context, or totally in default when we project them into new situations. They may be indispensable as a heuristic, but the fact that they are so familiar often seduces us into the jaws of paradox. The trouble seems to lie chiefly in the traditional assumption that what is intuitive, is, by nature, something absolute, unchanging with time and place, and therefore capable of being identified once the genius with the eye sharp enough to perceive and characterise it comes on the human scene.

Any plan as such as Gödel's, though, where a slight readjustment of our logical optics will bring large branches of mathematics into focus, seems to ignore the perennial rise and fall of individual movements in mathematics, with their own innovative axiomatic systems. While these are often heralded as self-evident, they are invariably superseded by the next in a seemingly interminable series of stronger or more general new conjectures - conjectures which in turn become subjected to the cut-and-thrust of logical analysis.

Noticing this incessant 'ebb and flow of the outgoing tide', while more and more intuitive territory comes into view, Morris Kline pertinently observes:

"Intuition throws caution to the winds, while logic teaches restraint." (43).

Spectators of this two-way interplay will perhaps be led to try and refine their intuitive abilities 'before the tide turns', modify them where weaknesses are found and constantly realign them into an increasingly cohesive structure. If intuition in mathematics is properly characterised as a living growing element of our intellect, an intellectual versatility with our present concepts about abstract structures and the relations between these structures, we must recognise that, as such, its content is variable and subject to cultural forces in much the same way as any other cultural element. Even the symbols designed for the expression and development of mathematics have variable meanings, often representing different things in the 19th and 20th centuries, by virtue of the underlying evolution of mathematical thought.

It must therefore remain an important strategy to aim to develop an increasingly versatile and expressive medium for the representation of familiar ideas.

Thankfully, our natural language is a subtle and highly structured vehicle of expression, and it exerts strong pressure on us to clarify our thinking, due to the necessity, during communciation, of linking our ideas with words that satisfactorily represent them. But it is largely by the use of symbols - words being a special case - that we achieve voluntary control over our thoughts. The process of inventing and employing artificial languages, which play a small role in our overall semantic vocabulary, often highlights the schematic relations between our concepts and ideas.

The auditory symbols of an orchestral score, for example, can only strike us as a masterpiece of representational economy. In this case, 'unacceptable' or 'uncongenial' harmonic progressions can be discerned and isolated purely formally, just like valid and invalid proofs in Frege's Begriffschrift, can be picked out by the eye, or unsuitable isomers can be identified visually in the graphical display of chemical reactions.

In these cases, visual imagery not only guides the formation of our schemas, but also enables us to spot strategic groups of symbols, thereby revealing their individual relations. Einstein remarked to the French Analyst Hadamard, in an oft-quoted letter, that for him, visual imagery was the type of representation most favourable to the integration of ideas (44), while more formal, algebraic operations were secondary, making possible more socialised communicative thinking.

Skemp [8] distinguishes visual from formal symbolism in that visual heuristics reveal global structure, and present data all at once, rather than in sequence, while algebraic language is more conducive to indicating details, being by nature more analytic, and abstracting properties from data independently of their spatial configuration.

Nevertheless, while we invariably do resort to graphs or diagrams to familiarise ourselves, as working mathematicians, with increasingly abstract material, it seems that the ability to reason formally, which requires the explicit formulation of ideas, together with the ability to show ideas to be logically derivable from other and more generally accepted ideas, are great assets in broadening the scope and range of the schemas which become second nature to us, and are instrumental in extending the familiar territory of our intuition.

To sum up then, the approach presented here is based on the following general notions about intuition:

First, that during all but a vanishingly small proportion of the time spent in investigative mathematics, we seem to be somewhere between having no evidence at all for our conclusions, and actually knowing them. Second, that during this time, intuition often comes to the forefront, both as a source of conjecture, and of epistemic support. Third, that our intuitive judgments in these situations are often biased, but in a predictable manner.

Hence, the problem is not whether to accept intuitive judgments or to reject them, but rather one of how they can be de-biased, developed, and refined.

Ultimately though, any satisfactory analysis of the role of intuition in mathematics should recognise it as a versatility in measuring up new situations, or even conjecturing them, using a rich repository of recurrent and strategically-important schemas or conceptual structures, painstakingly abstracted from sensory experience by the intellect, constrained by the languages available to us at the time, and influenced by the accumulated resources of our cultural and scientific heritage: what intuition does not do is constitute an insight gained by Reason, through some remarkable clairvoyant power - an insight, which, for Ramanujan and Gödel, seemingly paved the way towards a crystal-clear apocalyptic vision of mathematics, or, for Descartes, paved the way into the ultimate structure of the human mind.

...........................................................................................................

Paul Thompson,

University College,

OXFORD UNIVERSITY, U.K.

(March, 1993)

[Current address: Laboratory of Neuro-Imaging, Dept. of Neurology,

Center for the Health Sciences, 4238 Reed Neurology, Westwood,

Los Angeles CA 90095-1769, U.S.A.

E-mail: thompson@loni.ucla.edu ]