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 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 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 ai 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 {ai},
i =0 to n, i.e.
f(x) = 1: Integral -1 to 1 a0
+ a1x + ... + anxn dx = 0; a0 + (a2/3) +
(a4/5) + (a6/7) + .... = 0;
f(x) = x: Integral -1 to 1 a0x
+ a1x2 + ... + anxn+1 dx = 0; a1 + (a3/3) +
(a5/5) + (a7/7) + .... = 0;
... etc. ...
f(x) = xi (i <= n); Integral -1 to 1 a0xi
+ a1xi+1 +
... + anxi+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 [Mnxn]
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 concep