I see no reason why we should have less confidence in mathematical intuition, than in sense-perception.

-- Kurt Gödel

To grant necessary ontological status only to the non-negative integers, as does Kronecker and his thought-mates, resembles the attitude which prevailed among geometers from Euclid down to Gauss and Lobachevsky, which considered geometry as described by Euclid to be the only possible geometry. It was a word with no plural, like “universe”; that both these terms are now pluralizable (the latter in a conjectural, the former in a now very precise sense) represents a triumph of the human spirit, which now knocks tankards with the Invisible, in a toast to Him who made us all, the math along with the meat. The old geometers believed this because they fetishized the visible: the flat, drab world (well, pied with beauty, true, but drab compared with the full panoply of Riemann surfaces and Finsler space) whose mensuration is approximated so closely by Euclidean geometry -- at least, that stretch of turf that lies visibly close to hand – that the actual deviations that do exist cannot be detected by ordinary, plain-man means; and those who scoff at the invisible infinite, are very plain men indeed. And it was this servitude to the locally visible – which is in particular to say, to the

*contingent*– which caused the finest minds to fritter fruitlessly after a derivation of the Parallel Postulate from the rest of the Euclidean axioms, a fiasco that lasted literally for over two thousand years, from antiquity down to the nineteenth century. Indeed, it was not until we became familiar with the “invisible” worlds revealed to us by Lobachevsky, Riemann, Klein and Poincaré that we became fully clear on the status of Euclid’s axioms, and the distinction between axiomatics and model theory.
So, to argue concretely: If the number

*one*is real, then so is a half, for I give you half this pie. And if “one” is real then so must be the square root of two, as being the measure of the hypotenuse of the isosceles right triangle, by the inexorable evidence of the Pythagorean Theorem. (Likewise the square root of 5, 13, 17, etc., and thus their products.) And the square root of two turns out not to be a ratio of natural numbers. Now, Pythagorus himself shrank from this conclusion, and stigmatized the postulant entity as irrational; yet now we take them for granted. And if third roots or electricity are real, then so are the imaginary numbers required for their description; and if matter is real, and with it atoms, then so is quantum mechanics: which mean that compact operators on Hilbert Space are real: you cannot see them or touch them, but you can almost hear them, buzzing all around us… It is a slippery slope (we might almost say, a declivity whose slope is infinite) when we admit the reality of the visible world: it quickly (“quickly”, considered*sub specie aeternitatis*) drags in all the invisibles wíth it.
It is true, the atmosphere around those higher turrets is rather rarefied. Sometimes we become light-headed, and wonder if we are not after all just making some of this stuff up. Yet no sooner do we begin to doubt our senses – or rather, to put

*too much trust*in our senses, and too little in our carefully nurtured sense of the unseen – than Nature coughs up some concrete correspondence with our most arcane designs. That same Riemann hypothesis now seems to be in some strange harmony with the energy-levels of atoms. And as for connections on fibre bundles – meet the gauge fields of particle physics, your twin, separated at birth!
*

The question remains, whether mathematical entities are, so to speak,

*real in general*, or only real within a particular reality: much as a planet might pursue its course, in our universe but not another. Now, the examples of mathematical reality adduced thus far, have all been given a clean bill of health by our actual, particular universe. Hilbert space is as much a part of our daily reality as are porpoises – quite as vibrant, almost as much fun, and much less likely to go extinct. But the*Continuum Hypothesis*… ahh. That’s another matter. One would really like to have a better handle on that one. There are models for set theory in which it is true, and models in which it is false. This tends to make us acutely uncomfortable, since, unlike the logically equivalent but intuitively more ethereal Axiom of Choice, it’s the sort of thing where you feel there ought to be a plain fact of the matter. Nevertheless, its status may be ultimately no worse than that of the parallel postulate, which is quite placidly and understandably true in some geometries, false in others, and indeed true in our own universe at appropriately small scales (here I mean, of course, not what happens unobserved at infinity, but such local effects as the sum of the angles of a triangle), while false at others.[concluded here]

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