The relationship between mathematics and intuition
has been through a storm over the last century.
Mathematicians once embraced what they described as intuition; indeed, at some
level, all our mathematical notions are founded on some form of intuition
– what it means for a value to be a member of a set, or for a relation to
relate one value to another – and our choice of which mathematical
structures to define and explore rests on the intuitive familiarity of some of
the results that arise from those choices – natural counting, the
continuum of the number line and the smooth structures we are able to build from
it. On the other hand, mathematics draws out from these intuitions some results
against which our intuitions revolt – Bannach and Tarski dismembering a
sphere into seven distinct parts and reassembling them to make two spheres, each
occupying the full volume of the original, for example; but, even before this,
the idea that the infinite number of whole-number ratios, dense in the real
line, is no bigger than the subset of them between 0 and 1 or the set of whole
numbers, yet that the real numbers between 0 and 1 out-number
these countable
infinities; while, all the same, only countably many even
of the reals are exhibitable, all the rest being (in some sense)
mythical beasts, in whose existence we are forced to believe even though we can
never identify them. This lead to a formalist approach to mathematics, which
insisted on pedantic precision about the statement of axioms and all reasoning
from them, mistrusting our intuitions on account of their revolt against the
results that flow from that reasoning.
Closely related to this was a split between the intuitionists
, who
accept various axioms – that were accepted because their adherents find
them intuitive
, mainly because they had long been accepted without
question; but that had lead to the results alluded to above – and a new
school of thought, called constructivist
, which rejected these axioms,
most prominently the law of the excluded middle
(a.k.a. reductio ad
absurdum) and the axiom of (transfinite) choice. I object, against the
name of the former, traditional, school that different minds have different
intuitions; and, in particular, that our intuitions are not some mystical
insight from beyond the realm of reason but, in fact, always learned from our
experiences.
I do not find the law of the excluded middle intuitive – if you show that one of two parties to an argument is wrong, you have not necessarily shown their opponent to be right, however commonplace it may be for the one not proven wrong to claim victory. The nature of infinities, once we understand them, leads to startling results – such as the lack of distinction between the seemingly larger infinity of ratios and the manifestly minimal infinity of counting numbers, contrasted with the inescapable fact that the set of subsets of the whole numbers is bigger (in a natural sense) than this infinity. My intuition is thereby put on its guard against expecting (without detailed proof) what is true of finite things to be true also of infinite things – consequently, although I see the motivation for the axiom of choice (and its assorted equivalents), my intuition remains wary.
Although the intuition-disturbing results alluded to above do not disprove the axioms from which they flow – there is no actual inconsistency – they do encourage me to avoid the use of those axioms, where I can, in large part because a careful examination of how they lead to these results has undermined my acceptance of their modes of reasoning. The use of reductio allows us to prove the existence of mythical beasts that we can also prove can never be exhibited – almost all of the real numbers (and all of their non-measurable sub-sets, which constitute almost all of the subsets of the reals) are non-exhibitable. However happily one may accept the existence of things that we can never witness – a state of affairs that any physicist necessarily mistrusts, even though we accept quarks – I very much prefer to prove the existence of anything by telling you how to find or construct that thing – i.e., how to exhibit it. So I prefer constructive proof, wherever my faculties (crippled though they are by my intuitionist education) can produce one.
This is not merely a matter of æsthetic preference: it also has pedagogic value. If you wish to teach a pupil how to do something, showing them how to do so works better than proving that, were it not possible, there would be something fundamentally wrong with the whole subject you are teaching them. While you may have faith that your logical system is nicely-behaved, your student may yet be somewhat sceptical, especially if familiar with Gödel's results.
There is an art, that mathematicians must study, to the fine balance between formalism and intuition, preferring those formalisms that lead to intuition and rigour agreeing more of the time and, where these do diverge, having them do so in places that illuminate rather than confusing and, thereby, serve to educate the intuition to better lead where rigour can follow, rather than where the two conflict. In particular, I do believe that rigour always does follow intuition; hence that it is valuable to use formalisms whose consequences our intuition can learn to anticipate. This not only makes it easier for us, as mathematicians, to discover new truth: it also makes it easier for us to express what we already know in clearer terms and communicate it to others, most especially to each new generation of potential researchers.
In Conway's excellent book On Numbers and
Games
, in between its zeroth and first parts, he includes an appendix
(formally to the zeroth part) which he describes as in fact a cry for a
Mathematicians' Liberation Movement
– at the heart of which is this
bold assertion:
We believe that mathematics itself can be founded in an invariant way, which would be equivalent to, but would not involve, formalisation within some theory like ZF. No particular axiomatic theory like ZF would be needed, and indeed attempts to force arbitrary theories into a single formal strait-jacket will probably continue to produce unnecessarily cumbrous and inelegant contortions.
The seed of this proposal is that his description of the surreal numbers is all formalised in a way (that he can justly show could be formalised in terms of ZF) that is designed to make the treatment of his subject matter flow as naturally as possible – in short, to prime the reader's intuition to be comfortable where his leads, using rigour to show the way but chosing the terms of that rigour to make the showing flow naturally into intuition.
It is likewise my belief that unshackling mathematics from artificial chains, such as those of Zermelo-Fränkel set theory, can be done without sacrificing rigour – which will not only liberate mathematicians but indeed empower (mathematicians generally, and especially) educators to convey mathematics more fluently and enable each new generation to learn with greater clarity the truths that earlier generations struggled to comprehend, encumbered as they were with the formal baggage that was needed while we understood the nature of the subject less well. In particular, that unshackling will allow for mathematics to be expressed in terms more accessible to intuition and better designed to educate our intuitions to more closely match what the expression's associated rigour will actually sustain, once our intuition suggests it. Rigour, properly understood, empowers untuition; when (both are) Done Right, these two are not in opposition to one another.
The math with bad drawings
blog
includes a
post, titled Is algebra just a series of footnotes to the distributive
property?
, in whch the author explores related subject matter, around the
specific example of
understanding the
distributive property, notably declaring at one point: Intuition before
formalism.
Written by Eddy.