THE NATURE AND ROLE OF INTUITION

**IN MATHEMATICAL EPISTEMOLOGY**

** **

** **

**"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)**

** **

**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 naďve 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 V****n****, the vector space of polynomials of degree at most n over R, with a view to
finding a basis for the dual space V**

** **

** (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 R ^{n} 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 V**

** **

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}** x ^{i}f(x) |**

** **

**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}**

_{ }

**exists. Then, considering the effect
of this functional on Vn's basis { 1,x,x ^{2},...x^{n }}, 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 a _{0}
+ a_{1}x + ... + a_{n}x^{n} dx = 0; a_{0 } + (a_{2}/3) +
(a_{4}/5) + (a_{6}/7) + .... = 0;**

**f(x) = x: Integral -1 to 1 a _{0}x
+ a_{1}x^{2 }+ ... + a_{n}x^{n+1} dx = 0; a_{1 }+ (a_{3}/3) +
(a_{5}/5) + (a_{7}/7) + .... = 0;**

** **

**... etc. ...**

**f(x) = x ^{i }(i <= n); Integral -1 to 1 a_{0}x^{i
} + a_{1}x^{i+1} +
... + a_{n}x^{i+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 R^{n}, 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_{i}x^{i} as
a trick,**

**0 = Integral i=0 to n a _{i} psi_{i} = Integral _{ -1 to 1} a_{i }x^{i}f(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 R^{2} , 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 R^{n}. 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/n^{2}), 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)^{i}b_{i}x^{2i}
= b_{0} (1-{x^{2}/beta _{i }^{2}}).**

** **

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

** **

**1 - (x ^{2}/2.3) + (x^{4}/2.3.4.5)
- (x^{6}/2.3.4.5.6.7) + ... = (1 - {x^{2}/pi^{2}})(1 -
{x^{2}/4 pi^{2}})(1 - {x^{2}/9 pi^{2}})....**

** **

**Whether a refined intuition of this
product could be genuinely developed or not, what mattered was that simply
looking at x^{2 } 's coefficient**

** (1/2.3) = (1/ pi ^{2})+(1/4 pi ^{2})+(1/9 pi^{2})+....... meant that 1+(1/4)+(1/9)+..... = pi^{2}/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 célčbres 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^aleph _{0})) sets of reals has
inevitably been mediated, and more finely developed by forming intuitive
schemas on the (2^aleph_{0}) well-behaved sets of the Borel
hierarchy. Accordingly, the
Cantor-Bendixson theorem (that CH holds for closed sets of reals) was readily
generalised to PI^{0}_{2 }(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 SIGMA ^{1}_{1} (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 R ^{3} of unit radius, may be partitioned
into a finite number of (non-measurable) pieces which can be rotated in R^{3}
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 RA ^{2 }- 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 (b^{n}x), 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, RA^{2}.
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 R^{3 }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 ]*