On “Rounding Out”

The shortest and best way between two truths of the real domain  often passes through the imaginary one.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 123

We have noticed Quine’s grudging acceptance of the irrationals given his unquestioning acceptance of the rationals, a process we alluded to as “rounding out”.

But it is really not so much rounding out as filling out -- or rather, filling in: filling in the gaps between the rationals.  And the result is, unfortunately, not so rational as the rationals themselves.  The rationals -- that is, fractions -- are forced upon you by Nature already in nursery school:  How shall we divide these two cupcakes among the three children? (Answer:  Each gets two-thirds.)  But the Reals are (we admit this, despite our Realism) a bit unreal.  Full of all manner of set-theoretic paradox.  Inscrutable.  You can still work with them in practical terms, because the rationals, which are well understood, are, though no more numerous than the integers, dense in R, providing a sort of well-defined ladder or footbridge along which we may proceed.

Here in any event  is the testimony of a first-rate mathematician,  to the effect that the transition to the full reals  is essentially a forced move:

We shall show how to construct a complete ordered field  from a simple chain [Think:  the natural numbers].  This … proves that any contradiction inherent in the postulates for a complete ordered field -- that is, the real number system -- is latent in the postulates for a simple chain, which is a far less complicated structure  whose consistency is almost guaranteed by our intuition.

Note that we do not discuss the existence of the simple chain.  In spite of its intuitive simplicity, a simple chain carries within itself  the germs of all the difficulties in logic and mathematics;  we are obliged to take its existence as axiomatic.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  112

Such is the very axiom -- we need but two -- which we seized upon to begin this whole series of essays.   We quoted it in the form made famous by the Nominalist (and our otherwise-foe) Kronecker, which invokes the deity -- possibly casually or ironically, but perhaps more pertinantly than he knew:  for whence this “intuition” of which we are so sure?

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The next step beyond the reals were named the “imaginary” numbers, and the name stuck.  Adjoin these, and you generate the Complex Plane.   And, unforced and arbitrary as this move might initially feel, it is on the Complex Plane  that you at last truly get a notion of a Natural Setting.  Everything just suddenly… works, and works better than you ever thought it could.  You have, all of a sudden, a Circle of Convergence -- sounds like something out of Lord of the Rings, and it is just as good.  Differentiable functions turn out to be perfectly smooth, and with a natural notion of their own domain.  (Try to define them on too small a region, and they will propagate themselves by analytic extension till they are nice and fat.)

On the complex plane, things are rounder.  (Note:  Round is good.) In R, a ‘ball’ is a line-segment, and a ‘sphere’ (the surface of a ball) is two points.  In C, they’re a disk and a circle respectively.  And you can round out or rather round off the complex plane yet further, by adjoining a single ‘point at infinity’, which is the limit of any ray pointing in any direction.  The plump, rotund result:  the Riemann Sphere.  This is homeomorphic to the surface of a penguin,  the world’s most perfect shape.

As Penrose puts it:

It is as though Nature had herself entrusted to these numbers  the operation of her universe.

(Again, note the theistic language which, all unbidden, surges forth at such a time, from even the driest of nibs.   It is a very early and natural theology, such as is depicted in that fine chapter of The Wind in the Willows, "The Piper at the Gates of Dawn".)

Another indication of the greater naturalness of the complex plane as a nursery for functions:  A real function may be C-infinity (infinitely differentiable) at a point, yet somehow “off” at this point, a fact revealed by the fact that its complex analogue is not there analytic.   Thus, as one writer put it, (complex) analytic functions are “smoother” than real functions.

[Example:  exp(-1/x), for x > 0; 0 at x = 0.  That last point is artificially “tacked on”, and in the complex picture, it shows.]

 

This Complex Plane  is a real find; it is not just a waystation to something better yet.  (David Berlinski calls complex numbers "instruments that providence had provided for the recovery of lost symmetries," a neatly postlapsarian formulation.)  There is very little beyond this, by way of fields suitable for the calculus -- certainly nothing that approaches the leap that the complex numbers represented beyond the reals.   There are the quaternions, which have their points, but are a very poor cousin indeed: the theory is poorer, not richer, for the extra generating elements, since the field of quaternions offers no analogue of holomorphic functions. ( “Quaternions have more or less dropped by the wayside.” -- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 325)

Then there are the octonions, for which no-one has ever found much of a use.  And there’s an end to it.

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Other mathematical instances of “rounding out”:

*  The adjunction of zero to the natural numbers, and of the empty-set to the world of sets.  Both function exactly like their less spectral congeners.

And a nice aesthetic note -- both are represented by a round symbol: respectively, a goose egg, and a goose egg barre sinistre.

* There are various elaborate ways of constructing things out of other things, like a Stone-Cech compactification.  But in “taking the power set”, we just stand back and let it happen.  Again and again.  Yielding the “beth numbers”, and more infinities than most folks know what to do with.

* The mathematics of string theory adds extra tiny “compactified” spatial dimensions to the three of everyday experience; in these, you just go round and round.  But this isn’t rounding-out, really, since the large spatial dimensions may themselves be compact, in which any sufficiently long journey circles back on itself.  (“Compact” doesn’t mean “tiny”;  it’s a topological, not a metrical notion.)  Space could even be flat, yet finite -- thus having the topology of a three-torus.

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Footnote:

It is well-known that it took mankind a long time to recognize zero as itself a number.  Less well known is that “not until modern times was unity considered a number” (D.E. Smith, History of Mathematics, vol. II, p. 26.)  Or that the negative numbers were long qualified as "false".

Compare the uncertainty over whether white qualifies as a “color”.  (And if it does, what about black, or grey?)

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So where is Minimalism in all this?  Are we just tacking on turrets and wing-additions to some increasingly sprawling McMansion?

Not a bit of it.  The operative word here really is round.  For, round things are minimal surfaces -- indeed, the very simplest class of these -- in the sense of using-up a minimal area to enclose a prescribed volume.   Our purpose is, indeed, to group like with like, and to enclose them in some stable structure.  This is no multiplication of entities for their own sake -- the itchy-clutching witchfingers of insensately proliferating fractals, which is the very architecture of the dungeons of Hell.   In rounding out, the mathematician is seeking a coherent minimal structure to regiment and account for what he has hitherto seen:  one which, upon acquaintance, may become more intuitive than the partial structures initially encountered.  (The “upon acquaintance” part may of course require a bunch of Ph.D.’s and several hundred years.)

            And the things which we have seen, and which need explanation -- or at least for agencement into some larger and more natural whole -- do keep arising.  Connections are detected among them which cry out for elucidation.  So we ascend to a yet loftier bird’s-eye -- eagle-eye -- phoenix-eye view.  To arrive, it may be, eventually at Topos Theory, or the Lord of Hosts.

(For the latter, though note:  that ladder reaches only so high.  We quote the saint:

Remaneret igitur humanum genus, si sola rationis via ad Deum cognoscendum pateret, in maximis ignorantiae tenebris.
-- Thomas Aquinas,  Contra Gentiles, lib. 1 cap. 4 n. 4 )

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The examples we gave were mathematical, merely for clarity.  But the principle of Rounding Out  applies to any field with structure.

These vague words ‘capable’ and ‘normal’  allow the grammarian scope for shaping his task to suit his convenience.  Seeking simplicity, he will round out and round off.

-- Quine, “Reply to Harmon”, in The Philosophy of W.V.O. Quine (1986)

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Footnote re the irrationals:

Dedekind sttressed the distinction of category  between cut and number  in 1888; against the view of his friend Heinrich Weber  that “the irrational number is nothing other than the cut itself”, he explained that “as I prefer it, to creat something New distinct from the cut, to which the cut corresponds.  We have the right to grant ourselves such power of creation”,  and cuts corresponding to both rational and irrational numbers were examples.

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 87

A seemingly slight, even pedantic distinction;  but like many another such, it might have its point.   Cf. my astonished delight in junior high-school, upon meeting the distinction between  x (the thing itself) and ‘x’ (the name of x) -- already adequately foreshadowed in Alice in Wonderland, but encountered now in a new context.  Likewise the difference between  x and {x} (the singleton-set of x).

In the case of an algebraic number like √2, a simple number staring you in the face out of a hypotenuse  versus the infinite train of rational pilgrims (never quite arriving at their destination) of a Dedekind cut,  one is reminded of the variety of definitions of something so familiar as a tangent:  the slope of a curve (at a point); the closest linear approximation to the curve (at that point); versus the distressing definition in Loomis & Sternberg as an infinite equivalence-class of curves (through that point).