The Continuum: Mainstay or Menace? (erweitert)

The Continuum:  the original sin, from whose fecund loins
came all that is non-constructive in mathematics.

-- Anon.

Kronecker dismissed mathematical entities beyond the natural numbers as “Menschenwerk”.  An average practicing mathematician (who uses such entities all the time) may  agree with him to this extent:

(1)  Our intuitions about the natural numbers are clear and solid.   So long, indeed, as one deals only with some set of actual numbers (thus, a finite set), nothing especially surprising  or even all that interesting  turns up.  If we extend our horizon to the actual infinite of the set of all natural numbers, we meet some concepts that take getting used to (Hilbert's hotel):  but once we’ve done so, they seem natural enough.

(2) The rationals and negative integers  definitely, the algebraic numbers  probably, pretty much come along for the ride (that is, you can hardly exclude them once you’ve accepted N), and they still bring in no paradox – being, after all, of the same cardinality as the natural numbers themselves.  Though, a case could be made that these are not “entities” of the same standing as the integers, which in a sense we can hold in our hands (embodied in oranges, say), but rather abbreviations for operations on integers.  Thus, we cannot hold minus-two oranges in our hands; minus-two is not a thing, but a bookkeeping device. 

(3)  The real numbers, by contrast, are … a piece of work.  Maybe even Menschen-work, except that one could hardly imagine Menschen coming up with anything so intricate and even bizarre.  Their very cardinality baffles intuition  -- and the independence of the continuum hypothesis  shows that we are right to be baffled.  [Note:  The simple infinity of the integers already baffles *untutored* intuition;  but eventually you get the idea.  Click on the Label "Hilbert's Hotel" for further exemplification.  Whereas, the cardinality of the continuum is more like... Hilbert's Nightmare...] All sorts of queasy consequences arrive for simple quantification (cf. Quine re.  objectual vs. substitutional quantification).  The reals were invented (discovered?) for purposes of analysis, which in turn was developed largely for the sake of physics: but it now appears that physics (whether in its quantum cast, where Uncertainty provides a certain indissoluble granularity; or in the Wolframesque finite-automata approach) might not actually require, or afford, a continuum.

And yet standard mathematics speaks indeed ontologically of the reals, not merely pragmatically.  Thus for instance, Rudin’s standard text (Principles of Mathematical Analysis, 3^rd edn. 1976, p. 8):

We now state the existence theorem [emphasis in original] which is the core of this chapter.

Theorem. There exists an ordered field R which has the least-upper-bound property.

The author then mentions that the proof actually constructs the Reals out of the Rationals.  This is, of course, the most solid sort of proof of all – not one of those Cantorian diagonalization thingies that has you winding up assenting to the Infinite Woodchuck, without ever quite knowing how you got into such a fix.  It gives you an actual recipe for the construction of these extended numbers, as concrete and explicit as for baking a cake.  And yet… all kinds of things can be thus “constructed”, at will, including items which presumably are not part of the furniture of the universe, in the sense that angels actually sit on them.

~

A roaring vote of confidence in the continuum  is voiced by the noted mathematician René Thom:

“God created the integers and the rest is the work of man.”  This maxim spoken by the algebraist Kronecker  reveals more about his past as a banker who grew rich through monetary speculation  than about his philosophical insight.  There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity.

-- “’Modern’ Mathematics: An Educational and Philosophic Error?”, in American Scientist (1971), repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 74.

That is in-your-face Platonism, with which, quâ Realism, we have no quarrel.  But the psychological claim seems dubious:  Our intuition of the continuum is probably no more than a vague notion of a smear (and not very infinite at that, neither going out nor going down).   And as for the ontology … When we first meet the Real numbers mathematically (that was the very first thing we did in first-year calculus, with the opening chapter of Spivak’s text), we conceive them as the completion of the rationals.  And such they are indeed:  only, with respect to the metric provided by the absolute value.   With a p-adic valuation, you get a different completion of the rationals, the p-adic numbers.   Lastly, the surreal numbers augment the continuum in yet a different unexpected direction.  (I have less than no intuition about any of this.)

The physicist Schrödinger is less sure:

The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is really accessible to us.

-- Erwin Schrõdinger, “Causality and Wave Mechanics”, repr. in translation in: James R. Newman, ed. World of Mathematics (1956), p. 1059

And from an Intuitionist (close kin to a physicist):

This could be done  by seeing the continuum as something that is infinitely becoming, instead of already being.

-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 333

(Compare our old friend the actio/actum distinction.)
Might be fine for physics, doesn’t work for math.  ‘See’ it however you like; that uncompleted-account doesn’t jibe well with Cantor-style constructions.

~

One might say:  The continuum feels unproblematic enough, so long you take it for granted, as just some kind of smooth dense slippery thing, like mud.  Yet so soon as you pause to enquire more nearly, you are back in Saint Augustine’s predicament with regard to Time: “Quid est tempus? Si nemo a me quaerat, scio …”

~

Even in a universe which (like Wolfram’s) abjures the continuum, the continuum might turn out to be mathematically indispensible for its treatment.   Cf. the indispensible role of “imaginary” numbers in electromagneticsm or quantum mechanics, even though all observables must be real-valued.