Friday, August 12, 2011

Integers are our Friends

[Further reflections on a topic treated here, here, and here.]

It may be legitimately objected, that in beginning with a grubstake of mere integers, and proceeding stepwise to full mathematical Platonism, I am here sneaking past the goalposts via the fallacy of the sorites.  The classical example: We know that bald men and the hairy-headed equally exist, although we cannot specify, in that excruciating Gedankenexperiment in which each hair of the hirsute is plucked out (stop that!) one by one, at whích point precisely our unfortunate subject becomes glabrous.  Thus, suppose we agree to side with Kronecker and to grant ourselves the integers; and even grant, say, Arithmetic (that is, number theory using only elementary methods): Still, somewhere short of Topos Theory and Noncommutative Geometry – you’re not sure where, exactly, maybe you can’t even say specifically on which side of the divide Analytic Number Theory should fall, fair enough – somewhere this side that stuff, some right-thinking citizen needs to draw the line.  Noncommutative geometry – who ordered that?  You feel as though you’ve been sold a bill of goods.  Frchrssks, look at him:  that dude is bald.

Thus  the methodological objection.  There is also – especially these days, with our penchant for deconstructing and debunking – a psychological.  You may suggest that I have swallowed such a prodigious amount of abstract soup, merely because of some pre-existent hunger for it.  Now, I don’t believe that it was always pre-existent, in this particular case.  When I played cowboys and Indians, my little mind was on other things.  But, l’appetit vient en mangeant; and the integers were the appetizer.
            More concretely:  I have assumed less than may seem.  I have not so much as assumed any particular ontological status for the number “2”: I have merely taken Kronecker at his word, then attempted to refute, or at least to nuance, the second half of his epigram  (“… the rest is the work of Man”), the refutation being based simply upon the logical consequences of the first.  If, on the other hand, you were to begin by stubbornly denying that  one, two, and three (and I don’t mean “one, two, three, … infinity”, I mean: 1; 2; 3) formed an any more necessary part of the furniture of the universe, than Humpty Dumpty or Porky Pig, then I would be unable to convince you of anything by argument, having then no materials to work with.  We only got as far as I think we did, because of the perfectly enormous initial concession by the skeptic Kronecker.  You grant us the necessary reality of the natural numbers – their necessity bestowed, indeed, by the Necessary One – you have conceded a heck of a lot.  You have (it turns out) given away the ontological store.
            In fact, let us retrace our steps, and traverse some of the same terrain less hastily, and with less hunger for depth.  We have agreed to accept, as necessary, the natural numbers, and the simplest thing we can do is to count them – not worrying about primality, or odd-versus-even, or whether one number’s twice the size of another – not even necessarily ‘keeping track’: but just, ticking them off as they go by, like a bored doorman, waving the arriving spectators in to the stadium.  Now you will notice, in such a procedure, a tendency to nod off.  The numbers become dimmer and dimmer.  Has it been a thousand, or maybe twice that amount?  And should you ‘skip ahead’, and try to visualize, say, 17^(8371^545), you really can’t begin to imagine it.  The integers gradually wane, for all practical purposes, invisible.  Yet it is clear that these numbers are every bit as real, as the ones you noticed before you nodded off.  Furthermore, there are a whole heck of a lot more ‘invisible’ integers, than those that are (even with practice), visualizable:  to be precise, countably-infinitely-many more.  And if you are starting to stammer some objection about the possible non-necessity of numbers beyond Praxo (defined as the largest number that our species will ever actually need for anything; though  come to think of it  there is an interesting application of Praxo-plus-one …), then you are trumped, for we hold in our hands Dr. Kronecker’s get-out-of-nominalism-free card:  Every one of those dim distant integers  is as real as a rock, straight from the Maker’s quarry. By the time we are asked to swallow some new kind of quantity – say, a fraction, like “one-half” – we shall have swallowed a literal infinitude of whole ones.

[Update 10 IX 11] The title of this post, as well as this one,  is an example of the faux-naïf  -- a somewhat idiosyncratic concept which, like that of Minimalism (to which it bears some affinity) is slowly to be developed in the course of these posts.  In the meantime, let it remain a bit of a mystery.

Additionally, it has come to my attention that someone just found this post by searching on the words
            integers in our world
This is touching.  Though indeed, that search does not work very well, since the phrase in question did not appear in this post until this very moment.   To aid such sincere and innocent searchers in future, herewith some phrases for Google to match:
            =>  Integers at home and school
            =>  My favorite integers
            => The Campfire Book of Integers
            => Jonathan Livingston Integer
            => O Integer, my Integer !
            => Integers I have known


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