Edge: What Are Numbers, Really? - Stanislas Dehaene [page 5]
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At first sight, the instantaneous realization of this fact on a hospital bed seems incredible, too amazingly bright to be humanly possible. But in fact a minute of reflection suggests a simple way in which the Indian mathematician could have recognized this fact. Having worked for decades with numbers, Ramanujan evidently had memorized scores of facts, including the following list of cubes:

1x1x1 = 1

2x2x2 = 8

3x3x3 = 27

4x4x4 = 64

5x5x5 = 125

6x6x6 = 216

7x7x7 = 343

8x8x8 = 512

9x9x9 = 729

10x10x10 = 1000

11x11x11 = 1331

12x12x12 = 1728

Now if you look at this list you see that (a) 1728 is a cube; (b) 1728 is one unit off 1729, and 1 is also a cube; (c) 729 is also a cube; and (d) 1000 is also a cube. Hence, it is absolutely OBVIOUS to someone with Ramanujan's training that 1729 is the sum of two cubes in two different ways, naming 1728+1 and 1000+729. Finding out that it is the smallest such number is more tricky, but can be done by trial and error. Eventually, the magic of this anecdote totally dissolves when one learns that Ramanujan had written this computation in his notebooks as an adolescent , and hence did not compute this on the spur of the moment in his hospital bed: he already knew it!

Would it be farfetched to suggest that we could all match Ramanujan's feat with sufficient training? Perhaps that suggestion would seem less absurd if you consider that any high school student, even one that is not considered particularly bright, knows at least as much about mathematics as the most advanced mathematical scholars of the Middle Ages. We all start out in life with very similar brains, all endowed with an elementary number sense which has some innate structure, but also a degree of plasticity that allows it to be shaped by culture.

Back to the philosophy of mathematics, then. What are numbers, really? If we grant that we are all born with a rudimentary number sense that is engraved in the very architecture of our brains by evolution, then clearly numbers should be viewed as a construction of our brains. However, contrary to many social constructs such as art and religion, number and arithmetic are not arbitrary mental constructions. Rather, they are tightly adapted to the external world. Whence this adaptation? The puzzle about the adequacy of our mathematical constructions for the external world loses some of its mystery when one considers two facts.

  • First, the basic elements on which our mathematical constructions are based, such as numbers, sets, space, and so on, have been rooted in the architecture of our brains by a long evolutionary process. Evolution has incorporated in our minds/brains structures that are essential to survival and hence to veridical perception of the external world. At the scale we live in, number is essential because we live in a world made of movable, denumerable objects. Things might have been very different if we lived in a purely fluid world, or at an atomic scale and hence I concur with a few other mathematicians such as Henri Poincare, Max Delbruck, or Reuben Hersh in thinking that other life forms could have had mathematics very different from our own.
  • Second, our mathematics has seen another evolution, a much faster one: a cultural evolution. Mathematical objects have been generated at will in the minds of mathematicians of the past thirty centuries (this is what we call "pure mathematics"). But then they have been selected for their usefulness in solving real world problems, for instance in physics. Hence, many of our current mathematical tools are well adapted to the outside world, precisely because they were selected as a function of this fit.
Many mathematicians are Platonists. They think that the Universe is made of mathematical stuff, and that the job of mathematicians is merely to discover it. I strongly deny this point of view. This does not mean, however, that I am a "social constructivist", as Martin Gardner would like to call me. I agree with Gardner, and against many social constructivists, that mathematical constructions transcend specific human cultures. In my view, however, this is because all human cultures have the same brain architecture that "resonates" to the same mathematical tunes. The value of Pi, thank God, does not change with culture ! (cf. the Sokal affair). Furthermore, I am in no way denying that the external world provides a lot of structure, which gets incorporated into our mathematics. I only object to calling the structure of the Universe "mathematical ". We develop mathematical models of the world, but these are only models, and they are never fully adequate. Planets do not move in ellipses - elliptic trajectories are a good, but far from perfect approximation. Matter is not made of atoms, electrons, or quarks - all these are good models (indeed, very good ones), but ones that are bound to require revision some day. A lot of conceptual difficulties could be clarified if mathematicians and theoretical physicists paid more attention to the basic distinction between model and reality, a concept familiar to biologists.

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