ON THE NATURE OF MATHEMATICAL CONCEPTS: WHY AND
HOW DO MATHEMATICIANS JUMP TO CONCLUSIONS?
Notation: x for products: 2 x 3 =6, ^3 for cubes: 2^3 = 8, ^exponent: 2^11 = 2048.]
While engaged in the mathematical endeavor we simply jump, hardly ever asking "why" or "how". It is the only way we know of grappling with the mathematical problem that we are out to understand, to articulate as a question and to answer by a theorem or a whole theory. What drives our curiosity is a question for psychologists. Only after the jump has landed us on a viable branch the labor of proving the theorem or constructing a coherent theory can set in. The record of the end result, usually a presentation at a conference, a paper in a learned journal or a chapter in a book, is laid out in a sequence of rational deductions from clearly stated premisses and rarely conveys the process by which it has been arrived at.
The question why we have no other choice but to jump has received a remarkably precise answer through Gödel's Proof of Incompleteness in 1931 and Tarski's analysis of the concept of Truth in the thirties in Poland. Since then the development of a rigorous concept of an algorithm has led to a proliferation of so-called undecidability and inseparability results underscoring the limitations of the formal method.
The question how we jump has many aspects. First: what does the jumping consist of, what are we doing when we jump, what is going on in our minds when we are hunting down a mathematical phenomenon? And then: what is guiding us, how come we jump to CORRECT conclusions? Even if the guess was not quite correct, it usually was a good hunch that, properly adjusted, will open up new territory. Where do these hunches come from? Probably the simplest recorded answer to that question goes back to Plato and has spawned a school of thought in the Foundations of Mathematics that bears his name. It puts those hunches on a par with our spontaneous reactions to physical messages — "smell that? someone must be roasting a lamb in the next clearing", "there is a storm brewing in the South West, I can feel it in my bones". According to Plato's view mathematical objects exist eternally and immutably in a realm of ideas, an abstract reality accessible, if only dimly, to pure reasoning. That is how we discover them and their properties. By now, what with 2000 years of escalating experience with mathematics and painstaking critical analyses of its tenets, Platonism is no longer the accepted view in the Foundations. But, if nothing else, it is a wonderful allegory and an extremely useful working hypothesis.
To put it bluntly, while at work a mathematician is too busy concentrating on deciphering the hints he can gather from the trail he is following to stop and bother asking how the trail got here. It is enough for him to have a good hunch that the trail will lead to the goal.
The following is a slightly polished version of my spontaneous response to the assortment of EDGE-comments on Stanislas Dehaene's question "What Are Numbers, Really? A Cerebral Basis For Number Sense" and the subsequent discussion at The Reality Club. After a simple illustration of how we ponder, jump and then fill in the steps I address some general considerations raised on EDGE, which leads me to an exposition of the limitation phenomena.
Although keeping technicalities to a minimum, both conceptually and typographically, I am careful to be precise and correct. In our field the smallest inaccuracy can have disastrous consequences leading head on into contradictions.
1729 AN EXAMPLE OF MATHEMATICAL REASONINGStanislas Dehaene brings up the Ramanujan-G.H.Hardy anecdote concerning the number 1729. The idea of running through the cubes of all integers from 1 to 12 in order to arrive at Ramanujan's spontaneous recognition of 1729 as the smallest positive integer that can be written in two distinct ways as the sum of two integral cubes is inappropriate and obscures the workings of the naive mathematical mind. To be sure, a computer-mind could come up with that list at a wink. But what would induce it to pop it up when faced with the number 1729 if not prompted by some hunch? Here is a more likely account:
Confronted with 1729 you will recognize at a glance that:
i) 1729 = 1000 + (810-81) = 10^3 + 81 x (10-1) = 10^3 + 9^2 x 9 = 10^3 + 9^3 = (1 + 9)^3 + 9^3 = 1 + 3 x 9 + 3 x 9^2 + 9^3 + 9^3 = 1 + (3^3 + 3 x 3^2 x 9 + 3 x 3 x 9^2 + 9^3) = 1 + (3 + 9)^3 = 1^3 + 12^3 in view of the pattern ii) (a + b)^3 = a^3 + 3 x a^2 x b + 3 x a x b^2 + b^3.
Now all those 3's in the above expressions spring to attention, you fleetingly call up THE EQUATIONS
iii) (a + b)^3 + d^3 = a^3 + (c + d)^3 a^3 + (3 x a^2 x b + 3 x a x b^2 + b^3) + d^3 = a^3 + (c^3 + 3 x c^2 x d + 3 x c x d^2) + d^3
and JUMP to the conclusion that the choice of (1,9; 3,9) for a,b; c,d will give you the smallest positive integer that can be written as the sum the cubes of two integers (a+b) and d and also of a different pair a and (c+d). You have a well trained instinct. But, if called upon, it will be a simple matter to fill in that jump by a proof, the fixed coefficients 3 ruling out smaller choices for b,c,d, once the minimal possible value 1 is chosen for a.
ANALYSIS OF A TRAIN OF THOUGHTThe best way to understand the process encoded above in technical shorthand is via a metaphor, which should be spun out at leisure. Say you are driving into a strange town, and, for some reason or other, a building complex catches your attention. It does not just pop into your field of vision; at first glance you see it as a museum, a villa, a church or whatever. And then, depending on your particular interests and background, you may recognize its shape, size and purpose, muse over its style, venture a guess as to its vintage, and so forth.
Upon meeting 1729, your first reaction will probably be to break it up into the sum of 1000 and 729, because of our habit of counting in decimal notation. Stop for a moment to consider what would have been facing Ramanujan if Taxi cab companies were favoring binary notation! [11011000001 = 11011000000 + 1 = 11^3 x 100^3 + 1^3 = 101^3 x 10^3 + 11^3 x 11^3 = 1111101000 + 1011011001]. On the other hand, if you are one of those people obsessed with prime factorization you'll "see" the product 7 x 13 x 19 when somebody says "1729" to you while a before-Thompson-and-Feit but after-Burnside group theorist will say "Aha that is an interesting number, all groups of order 1729 are solvable" and anyone with engineering experience immediately thinks of the 1728 cubic inches contained in a cubic foot . But a historian of Mathematics will see 1729 as the year of Euler's friend and benefactress Catherine the Great's birth.
Next you decide, more or less deliberately, how to investigate the phenomenon. Do you drive to the nearest kiosk, buy a "Baedecker", search for that building and read through all you can find in there about it, before you make up your mind about what you want to know, in other words, assuming you have a kiosk full of lists handy in your own mind, do you run through all the integral cubes smaller than 1729? If so, why cubes?
If you have that kind of mind you probably would first run through the squares before getting to the cubes. The less methodical tourist, eager to enjoy rather than out to complete his (or her) knowledge, may choose to investigate in a haphazard way, spurred on by curiosity, guided by experience, using skills automatically while impulsively following hunches, prowling, sniffing, looking behind bushes, and then jump to rational conclusions.
Now return to Ramanujan and see how the first thing that springs to the naive eye beholding the number 729 is that adding 81 = 9^2 turns it into 810, whereupon 10 drops its disguise, shows one of its true natures as the sum of 1 and 9 and, lo and behold, all those powers of 3 start tumbling in. All the while you are aware of the pattern ii), just below the threshold of consciousness, exactly as a driver is aware of the traffic laws and of the coordinated efforts of his body and his jeep. That is how you find your way through the maze of mathematical possibilities to the "interesting" breakdown of 1729 into two distinct sums of integral cubes.
When you stop to ask yourself what is so great about that, something clicks in your mind: you are facing a positive integer with a certain property, you know that
iv) every collection of positive integers has a least member (in terms of its natural ordering).
That knowledge, always hovering below the threshold of consciousness, prompts the question whether 1729 might in fact be the LEAST positive integer expressible in distinct ways as the sum of two cubes. Having another look at the representation of 1729 as a sum of various powers of 3 as held in your mind's eye and exhibited in the third line of i) above, the more or less conscious awareness of ii) invites you to break up those sums of cubes according to the pattern iii) where you assume — "without loss of generality" — that a < d = a + b, and hence c < b. At this point the solution a = 1, b = c^2 = d and c = 3 surfaces by inspection as "obviously" yielding the minimal value for (a + b)^3 + d^3.
ABOUT MATHEMATICAL ACTIVITYI have gone through this simple illustrative example at such length in order to underscore a few of my pet contentions:
What we sorely need is a phenomenological study of mathematical practice. Polya and Lakatos had independently started out on that path, I do not know to what extent it has been followed up. Mathematicians are well aware of how they work, whether by themselves or in teams. But their goals are results that must be presented in a conclusive and "clean" form that makes them publicly accessible, at least within the profession, a form that necessarily obscures the path that led to them, just as the most beautiful tombstone will sum up a life but give no inkling of how it really has been lived, to use an observation by Claude Chevalley .
a) Much mathematical reasoning is done subconsciously, just as we automatically obey traffic rules and handle our cars, whether we know why and how they work or not. Symbolic notation is an "artificial aid" used to secure a hold like a piton, to survey a situation like a geological map and to encode general patterns for repeated application. But it is not mathematics. Mathematics can be done without symbols by a particularly "gifted" individual, like, e.g., Ramanujan. What that gift consists of is one of the questions raised in the EDGE piece. Obviously we are not all of us born with it. Nor do I believe that all people born as potential mathematicians become actual ones. Tenacity of motivation, an uncluttered and receptive mind, an unerring ability to concentrate the mind's focus on long intricate chains of reasoning and relational structures, the self discipline needed for snatching such a mind out of vicious circles, these are only a few characteristics that spring to mind. They can be cultivated. Experience will train the judgment to distinguish between blind alleys and sound trails and to divine hidden animal paths through the wilderness.
b) Free association plays an important role, an agility of mind that allows reasoning to jump ahead with a sure touch, after which comes the dogged toil of constructing proofs.
c) Conceptual visualization is an indispensable attendant to mathematical thinking. Formalization is only a tool and may encourage lazy thinking! Look at the freshmen who enroll in math because they assume they won't be expected to produce coherent arguments or to write grammatical text, that bureaucratic neatness in "plugging in numbers and turning the crank" will suffice to pass the course.
It is fascinating to browse through some of the essays on the Foundations of Mathematics by the topologist and logician L.E.J. Brouwer, the father of Intuitionism. You will find very few formulas in them, and yet they are rigorously reasoned, tightly and succinctly, more so than many formal texts. 
d) That practice, familiarity, experience and experimentation are important prerequisites for successful mathematical activity goes without saying. But less obvious and just as important is a tendency to "day dream", an ability to immerse oneself in contemplation oblivious of all surroundings, the way a very small child will abandon himself to his blocks. Anecdotes bearing witness to the enhancement of creative concentration by total relaxation abound, ranging from Archimedes' inspiration in a bath tub to Alfred Tarski's tales of theorems proved in a dental chair.