Existence and Use

 Now let us return yet again, with our crude wooden mallet, and hammer anew on that empirical numerical square peg  which we are attempting to pound into a (so we hold) round hole, as round as a cow – the Platonic realm -- or as it might be, to embed it smoothly into Euclidean n-space.

I am not saying that numbers are ideas.  Or if they are that, they are that in addition, where they keep such scruffy company as the Idea of a Unicorn and the Idea of a Ham Sandwich. They are, rather, facts as hard as baseballs, packing quite as much wallop. (Also, admittedly, we are interested in the Pitcher …)

In pointing out the “reality” of numbers, we’re really doing much what we do when demonstrating the reality of a hammer, without venturing so much as a murmur as to what either of these things are “in themselves”.  Philosophy pretty much gave up on the ding an sich a long time ago, and the new quantum perspectives makes even ordinary objects seem more remote than ever.  The reality of a hammer, as against that of my imaginary rabbit friend,  consists in its experiential predictability: I feel its weight, you do too, we both agree it’s heavier than a walnut; it looks a certain way from this angle, another way from that, and these match up in a natural way, both visually and tactually, and so forth.  (Whereas you, with your confounded skepticism, scare away my imaginary rabbit-friend, and thus can never experience his charming and comforting presence.) The fact that we can also state that the hammer is right there, as opposed to another place, is nice: it’s one more fact about the hammer, but it’s not the essence of being real. In the case of a quantum particle, we cannot state that it is precisely there, though it’s somewhere around here for sure; but it does other things predictably enough  that we count it as part of the furniture of the universe, and not just some fluke.  It is likewise difficult to say “where” the Big Bang was when it popped, nor “where” our universe is now. The latter questions might make sense and they might not; we might be comfortably embedded in a multiverse (itself endowed with a metric), just a stone’s throw from Cosmos 87; or we might simply be, with nothing else to be relative to and hence no “where”.  But we are here, wherever “here” is; we are real. 

            Now, let us hammer away at this metaphor of math as a useful tool, and thus real.  (The point feels Wittgenstinian, though in general I am little in sympathy with his approach to math.)

            So.  A rock isn’t for anything.  Hence, unpoliced by pragmatics, rocks shade off into pebbles and sand and dust and molecules at one end, into boulders and cliffs and mountains and tectonic plates (or the entire Moon) at the other, with no clear boundaries.  (Unlike the boundaries among mathematical objects – numbers but others as well – which are sharp as a guillotine.)  A hammer can’t shade very far without ceasing to serve its defining function – it needs to serve to hammer things, to earn its keep – and thus ceasing to be a true hammer.

            Mathematical objects have this tool-like usefulness, and in that respect are more like hammers than like rocks.  And this does tend rather to tether them to this world, thus to make them more familiar, more acceptable to the Chamber of Commerce: for these are practical men, and like to know that a thing has a use, and hasn’t just been dreamt up in some pooftah parlor-game on a rainy afternoon.  Integers are useful for counting, Riemann manifolds for relativistic physics, and so forth.  But: cavete, gentlemen of the C-of-C:  they are not defined by these (human) uses. A toaster is so defined, but Mathematica thrones though we perish.  These objects exist prior to and independent of any such simian or homosapiensan applications – rather like rocks, the bare unappreciated rocks, which exist, in their original and ultimate solidity, prior to their being hewn into ashlars.

            One thing I am not claiming  is any special inevitability of this or that application of mathematical objects;  on this special subject I am comfortable with the nominalist position (the “hocus-pocus” position, vice “God’s truth”).  It is all too familiar, that this or that feature of physics, or of anything else, may be modeled (approached) by this particular mathematical formalism or that. Indeed, to make it all painfully plain:  Take the simplest application of the simplest mathematical object: the use of the natural numbers for enumeration.  The association is psychologically so tight, that some people doubtless imagine that N is defined by that application, and was even invented for that applicaton (by Homo sapiens or perhaps Neanderthal  underpaid mathematicians), the way a toaster was invented to toast bread, and is defined by this end.  If bread no longer existed, the toaster would, in a sense, cease to be. 

            Not so the natural numbers.  Fie, lest ye imagine, that they require our little home-improvement projects for their existence.  And as a (particularly smarting)  proof, consider this:  They are not even needed for enumeration.  (Yet mark:  Though seemingly to their discredit, this rather redounds to their greater glory.)  For:

            One could quite well enumerate things, and even perform simple arithmetic, without the slightest notion of integers, or indeed of counting.  You do it thus:

            First sculpt a monkey, and set it next the temple. Then (using your duplicator-zapper, left behind by those visitors from the Companion of Sirius), duplicate the statue and add another one just like it.  Put the resulting statue-grouping a bit behind the (unitary) first.  Now duplicate that, and add a monkey. Put this a bit behind the last; and so on.

            Now, if you want to figure out (say) how many children there are in your family, have them line up next to the rows of monkeys, beginning with the one in front, and have each child hold hands with one monkey.  Keep walkng back along the ranks until each child is paired with exactly one monkey. 

            You can perform addition and subtraction (and, more laboriously, multiplication) by similar purely physical means, having no more arithmétic notion of what you’re up to than do the Pirana.  To be sure:  Various arithmetical relations are implicit in this system, for instance “greater-than” means you have to walk farther along, away from the temple, till  you find a match.  But these relations need never, in practice, become conceptually explicit.  Like the Pirana, you can get by with no names for numbers as such, certainly no systematic numbering.  First, operation of the system, being purely physical, can be performed without words or symbols.  But suppose you do want names for each individual sculpture-row (as for: “Meet you at the (117)”.) Simply color the nearest group (the unit group) yellow, the next one purple, the next one green, the next one (say) black-and-white stripes, and then the next one pink polka-dots on an argent field; mark the next one simply with a little white flag; the next, with a happy-face; and “so” forth. No rhyme and no reason; none needed.  It covers the (finite) maximum group of statuary you have thus far needed to resort to.  Should the sets ever fail you, simply zap-dupe the hindermost and add a monkey, dubbing the new addition however you like.

            The fable is not idle.  Some of the skepticism about the “reality” – that is, the transcendental uniformity – of mathematical objects, may stem, I suspect, from a more justified skepicism about the God’s-truth view of applications of mathematical objects.  Thus, “Tensors are real because the were precisely what Einstein required for his field equations.”  Morris Kline (among others) saw more clearly: it is possible (or, in his view, actually likely) that tensors are not precisely what Einstein needed, and that something will supercede them.  Tensors had already been discovered (or, as the fools say: invented), and  as it turned out, they served him well enough.  Einstein notoriously lacked the mathematical tools ready to hand, when he came up with his intuitive take on gravitation.  He wandered down the hall to the math department, or maybe met some guy in a bar, and the guy says, Yo, Al, try this, always worked for me; and Al says, Good enough.  Sort of like going next door to borrow a hammer only they don’t have one but they have a paperweight and it will do.

Let’s take it a step further:  We built up the integers above, concretely, Peano-fashion, via the successor-function.  But having reached that height, we may throw away the ladder.  Notice that the hidden isometry between the size of the integer and the number of monkeys in the statuary that represents it, is logically unnecessary.  So, to save raw materials, we replace each statuary group by a simple stele, surmounted by the arbitrary symbol that was chosen to represent the group.  Counting then will amount to reciting a memorized list, “… Tyrolean hat, bowling ball, chartreuse lozenge, woodchuck rampant, dot, ….”  This is essentially what we settled down to with: “January, February, March….” (originally the list was etymologically more numerical, before those designations were ousted in favor of various godlings and tyrants.)

This, by way of a reply to Quine, who states, in his reply to Charles Parsons (Hahn & Schilpp, ed., p.401):

I prefer to say, with Benacerraf, simply that there are no natural numbers, and there is no need of them, since whatever purposes we might have used them for  can be served by any progression.

 

True enough – witness our monkey-statues.  After all, anything an actual person will ever have to count is finite -- a stock of two googolplex counters should suffice nicely.  And if one is trying to answer the (probably ill-posed and unanswerable) question, “What is an integer, really?”, then the observation is perhaps telling.  We ourselves have no interest in the quiditas, the “inner threeness” of the number three. A monkey trio, or a tricorner hat, will do just fine. But the “purposes we might have used them for” are by no means as meager as that dismissive phrase might sound – the way one might use a radio as a paperweight, or a statuette as a hammer.  You can’t settle the Riemann Hypothesis with nothing but poker-chips.

The uses to which numbers can be put, and the results we shall obtain with them, inhere in the structure of the set of integers itself.  It is universal, it is absolute, it is culture-free.  Numbers are Necessary, however you choose concretely to characterize them.  We may never get our hands on more than the trunk of this elephant, plus a couple of legs, but the elephant’s there, in his mighty totality.