From Finitude to Infinity

[Good heavens, it’s Sunday, and there isn’t a stitch to post !  Well -- le mieux est l’ennemi du bien, so here’s a stub -- what Malkiel would call a “torso” -- to be worked on further  if I am spared.  Think of this as a construction site, which the curious stroller may peer at through a knothole in the fence, checking back in a week or so, to see how the thing is coming.]

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Thomas Nagel, The Last Word (1997), p. 71:

We draw this access to infinity  out of our distinctly finite ability to count, in virtue of its evident incompleteness.

Leave off, for the nonce, your incessant wrestling with the Riemann Hypothesis, and return to the simplicity of thought outlined in our parables of addition, one of the woodchuck, and one of Farmer John.

You awake in the night, in a cold sweat:  Might addition eventually break down?  For, aways down the number line -- somewhere we have never traveled, gladly beyond any experience -- there are these great big lumbering numbers -- the cube of googolplex, and whatnot.  Might they not eventually creak and crack beneath their own immensity?  Might not gigamegagoogolplex  simply refuse to have one digit more added to him, lest (like the glutton in Monty Python’s “The Meaning of Life”) he simply explode, splattering integers throughout the cosmos, in an arithmetical Big Bang ?

(Wittgenstein used to worry about that sort of thing, bless his heart.)

We do not, of course, believe anything of the sort;  though we’d be hard-pressed to explain just why we rule this out.  After all, some cosmologists, to account for certain perplexities in the red-shift, once put forward  in all seriousness  the hypothesis of “Tired Light”.  (Hey, if you had been stuck to the same old geodesic for thirteen billion years, wouldn’t you be tired too?  Or more likely, bored.)

Our metaphysical certainty  that finity is, so to speak, the same throughout, with no surprises down the line, is psychologically similar to various metaphysical assumptions in cosmology (uniformity in the large), but much more absolute.  The universe has, after all, so far proved lumpy at every scale, from the quark to galactic clusters and cosmic strings;  we can easily entertain the hypothesis that it might be gnarly all the way down.  Not so with the integers.  Our characterization of these as never ‘breaking down’  is a panoptical statement about their finished totality, thus about actual infinity.  Our faith in the well-behavedness of the finite  rests ultimately in our trust in the infinite.

Oh, and those photons ?  They do not get tired.  They whiz forever,
 tiny and trusting,

   communing with their Maker,

       in ways proportional to their understanding.

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Other cases of proceeding from the finite to the infinite (if only heuristically):

The main purpose of the study of operator theory is to discover, formulate, and prove  the proper generalizations, valid for all Hilbert spaces, of the powerful results known in the finite-dimensional case.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 74

It turns out that, for instance, every compact operator on a Hilbert space is the norm-limit of some sequence of finite-rank operators.  But properties do change in the process:  thus, a compact operator need not have any eigenvalues.

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A compact topological space is one in which every open cover has a finite subcover.  This turns out to be a very nice property indeed:  the classification theorem for compact surfaces (i.e., 2-manifolds) is one of the neatest and simplest and most satisfying and most startling in all of mathematics;  it completely settles that problem  once and for all.  All such shapes you can imagine  boil down to just:  a sphere; a sphere with handles; a sphere with cross-caps.  That’s it.

But as we leave compactness, we leave behind that finiteness -- and things get weird.

The classification theorem for compact surfaces  does not extend in any way to noncompact surfaces.  There is an incredible variety of noncompact surfaces.

-- Michael Henle, A Combinatorial Introduction to Topology (1979), p.  129

If, however, we retain compactness (and thus a kind of finiteness), but allow now surfaces with boundary, then everything settles back into place.  The choices are the same as before, only now you’re allowed to cut some holes.   As Henle puts it:

Every compact, connected surface with boundary  is equivalent to either a sphere or a connected sum of tori or a connected sum of projective planes, in any case with some (finite) number of disks removed.

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A platonist conception of the infinite as a completed actual totality  misrepresents, according to [the intuitionists], the very nature of the infinite, by illicitly assimilating its character to that of finite collections.

Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p.  34

Note that that critique is not McGinn’s own, but is that of the sloth-like finitist/constructivist tribe known to the préfecture de police as the “Intuitionists”;  nor are the preternaturally sophisticated working mathematicians of today  guilty of any such puerile reductionist assimilation.  If anything (for all I know), the smart money now views individual integers as (more or less tragic) restrictions of an original infinitude : much as we creatures here below, though fashioned by His transfinite hands, yet mostly muck about the malls and brothels, not that different, all told, from our lesser cousins  feathered or furry.

Something possibly a bit along those lines, in the supra-empyrean context of sheaf theory, is tantalizingly broached by Edward Frenkel  here:

The [finite] number 3 is a mere shadow of the [infinite] 3-dimensional space …

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For a merely morphological look at this word finitude, try this:

     http://worldofdrjustice.blogspot.com/2012/07/on-finitude-and-truthitude.html