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.

Thoughts 'n' Things

  Rudy Rucker, Infinity and the Mind, p. 38:

(**) Just as a rock is already in the Universe, whether or not someone is handling it,  an idea is already in the Mindscape, whether or not someone is thinking it.

This is itself a pleasant thought, recalling the ditty about God-in-the-quad; but in actual fact – I don’t think so.

(So you see—I am not an uncritical Platonist.  Platonic heaven must be so gerrymandered, as to exclude such things as cheese doodles and Sponge Bob Squarepants.)

The actual universe has (for example) -- whatever geometry it has:  regardless of whether there are rational creatures capable of understanding it, let alone deriving it.  Likewise the landscape of math.  But particular formulations of physics, and perhaps even of math – matrix mechanics v. wave mechanics, Cauchy analysis vs. non-standard analysis – do not exist in complete independence from their proponents.  They are, one might say, propositions, not objects.  The objects (or patterns, or whatever they are)  exist  even in the absence of  a person to spout propositions about them; but the propositions require a proposer.  – Nothing specially abstract here; the same thing is true of rocks.  This rock exists independently of any finite mind, but: “There lies a rock” and “Behold that rock!” and “What a rock that is!” must come out of some actual someone’s mind or mouth.

            The unbridledly idealistic view in (**) conjures up a skyscape of untethered thought-balloons.  It is pleasant to contemplate, in a comic-strip sort of way, but not to be taken too seriously.  For one thing, unlike the situation with mathematical truths, where anyone at any place or time might discover them, there is no way for a rational creature in another galaxy or dimension to reach out and grab one of those thought-balloons by the tail;  he is required to blow his own bubbles.  Whereas the structures of mathematics are like fixed landmarks, which one encounters again and again, from different approaches.  For instance:  Yang-Mills gauge theories, discovered by the physics expedition; and connections on fibre-bundles, discovered by the math team; and lo, they meet in the middle.  Likewise group-theory.  Different body-parts of this have been grabbed onto by matrix theory, algebra (symmetries of solutions to equations), geometry (the Erlangen program), particle physics (glad you could get here; meet Sophus Lie), and in time it becomes clear that it’s all part of the same elephant.  Whether they come from physics, or mathematics, or computer science, two such explorers may not realise that they have come upon the same mountain, till they have circled around it a bit and compared notes.  And this happens repeatedly.  We may summarize in an epigram:  The mindscape of mathematics is a multidimensional torus:  whatever direction you set off in, you eventually wind up back at Hilbert’s Hotel.

It turns out that Shing-Tung Yau likes this montane metaphor as well.  Cf. The Shape of Inner Space (2010), p. 103:

A mathematical proof is a bit like climbing a mountain.

And he nicely outlines the Yang-Mills case (p. 290):

The physicist Chen Ning Yang was similarly astonished to find that the Yang-Mills equations, which describe the forces between particles, are rooted in gauge theories in physics  that bear striking resemblances to ideas in bundle theory, which mathematicians began developing three decades earlier, as Yang put it, “without reference to the physical world”.  When he asked the geometer S. S. Chern how it was possible that “mathematicians dream up these concepts out of nowhere,” Chern protested, “No, no.  These concepts were not dreamed up.  They were natural and real.”

            Contrast the case with “thoughts”.  Supposititious entities of the mindscape, even some popular thought-balloon, tethered to a billion different heads, need never be rediscoverable by another explorer, nor acknowledged as real should he simply be grabbed by the lapel by one of the thinkers, and treated to an exposition of same.  For example, the notion held dear by countless generations of schoolboys around the globe, of the uniquely funny nature of flatulence, will never appear among the gravely ellipsoidal thought-balloons of the solons of Fdrmrphlandia; even “funny”, for them, is not well-defined, and not particularly worth defining.

Now, probably Rucker meant to restrict the realm of “ideas” to just some of them.  Not, “Wouldn’t it be fun to dip Suzy’s pigtail into the inkwell!”, but things like “The square of the hypotenuse is equal to the sum of the squares on the other two sides.”  Fine; but careful, here.  The Pythagorean theorem has  as its basis  a fact about Euclidean geometry, in every possible world; just as Fermat’s Last Theorem expresses (in a possibly somewhat contingent and imperfect way) a fact about the natural numbers.   But a fact is not the same thing as an idea.  As a matter of fact, there is a coffee stain on this shirt; but “the idea of this coffee-stained shirt” is no strut or girder of God’s architectonics.  An idea concerning a fact of mathematics, in a finite mind,  may bear – must bear -- but an imperfect relation to the fact itself (‘fact’ here used broadly: it may refer to a wildly transfinite complexus of relations, some of them perhaps perceptible only to angels).   Most people’s ideas of mathematical truths bear as much relation to the truths themselves  as does a crayon scribble to the Sistine Chapel  which it might (based merely upon memory of a fleeting ill-lit glimpse) attempt to depict.  To posit that all truths of mathematics exist as Ideas in God’s mind, is logically allowable, but really adds nothing, and is in any case unknowable. To identify these truths with the neuronal states of the pitiful meat-wads sloshing around in our half-cracked crania, is to add nothing at all, but is rather to detract.

[Appendix]  Karl Kraus apparently entertained a notion of independent or pre-existent thoughts.  He speaks of someone being

von der Präformiertheit der Gedanken  überzeugt, und davon daß der schöpferische Mensch  nur ein erwähltes Gefäß ist; und davon, daß die Gedanken und die Gedichte da waren  vor den Dichtern und Denkern.

-- “Heine und die Folgen”, reprinted in J. Franzen, The Kraus Project, p. 88

The whole ‘meme’ idea (itself a meme) is similar -- not that the various Chiclet-thoughtlets were truly Platonically pre-existing, but that, once hatched, they lead a promiscuous existence, wandering into people’s minds  like pollen into our air-passages.

~

Footnotes from the 19^th century:

Dedekind … allowed his philosophy of mind  much reign, with a ‘proof’ that “there are infinite systems”;  for he gave  as evidence “the totality S of all things, which may be objects of my thought”, since  as well as any of its elements s,  it contained also “the thought s’ that can be the object of my thought …This ‘proof’ did not gain a good reception.”

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 105

For Frege,

In contrast to subjective ‘ideas’ (Vorstellungen), ‘thought’ was intended in an objective sense, rather like state of affairs, sharable among thinkers  and indeed independent of anyone thinking then.

-- I. Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 190

Welcome to Hilbert’s Hotel

[This is a continuation of a thread begun here.]
   

        To make any sense of the structure of the natural numbers and all they entail, we have argued (so far without proof), requires putting to work the whole apparatus of mathematics. Now, possibly anticipating such a move on our part, you (Ned the Nominalist), grudgingly grant Kronecker the strict construction of his text, and no more:  you draw the line at the positive integers as simply given, frozen, inert.  God, you concede, made the natural numbers (in a fit of prodigality which He no doubt later came to regret), “but He did not say you might actually add them, thus allowing them to become bigger than their britches, and to stand on one another’s heads; let alone to divide them (thus yielding the by-blow of the fractions), let alone to subtract them (thus yielding the aptly-named “negative” integers; as in, “Negative on that, Mac.”), let alone take their square roots (thereby whelping the well-dubbed “irrationals” – “wackos” would be more like it); let alone – O, let remotely alone – the square root of a negative integer, thus coughing forth the demon spawn that has no name in polite company.  You start messing with any of that, Mac, you’re ontologically on your own.”

            Fair enough.  Adding two to two is indeed to play with fire (the divine, the Heraclitean fire), and not much less so than calculating co-products on fibre bundles.  So let us indulge in no calculations with numbers.  Each is inviolable, isolated, vestal.  Let us simply put them into pots and take them out again, like Eeyore with his balloon.  How many go into a six-pack?  Six!  How many eggs in a dozen?  Twelve!   And how many integral guests in Hilbert’s Hotel, which was built to house all of them?  Why, all of them!

            And so we come (weary travelers) at last to Hilbert’s Hotel.  In austerity it somewhat resembles the castle of Kafka’s parable; but it is much more inviting than that somewhat dour institution, welcoming any new arrival if he can possibly be accommodated.  And on this dark evening, indeed, the hotel is is full, every one of its innumerable rooms filled to capacity, since each one can accommodate but a single integer, and innumerable integers have already taken up lodging for the night. And yet, lo, one more integer shows up, bindle over shoulder.  Whatever shall we do?

            The tale has been well told by Rudy Rucker in Infinity and the Mind, so I’ll do no more than to abbreviate the beginnings of that Thousand and One Nights account of all the goings-on at that inn, with more running about in the corridors  than in a farce by Feydeau, and refer the reader to that estimable work. [If you haven’t read it, do.  A fun read, with no pre-requisites, and it itself is a sort of pre-requisite to later parts of this ongoing essay.]  Put briefly: the concierge simply shifted the guest in room 1, to room 2; the guest in room 2, to room 3; and so on; so that the newcomer moved into room one and had a restful sleep. More awkward was the next night, when the still-full hotel saw the arrival this time of infinitely many guests.  But with a bit of baksheesh, the concierge found a way: guest 1 moves into room 2, guest 2 into 4, guest 3 into 6, and so on; while the grateful newcomers troop into the thus-vacated rooms 1, 3, 5, 7, 9, ….

            It gets hairier than that:  I refer you to Rucker, who doth a tale unfold, whose lightest word will make thy knotted and combinèd locks to part, and thine each particular hair to stand on end… culminating in the hilarious contretemps, when Groucho, arriving too late, discovers his stateroom to be already occupied by a decidedly fretful porpentine….

            So, welcome to the Hotel!  Enjoy your visit!

Credo

[Theologia Mathematica, ch. 2.   Continues this.]

        

There is a phrase within a clause within a prayer, of which I am particularly fond, and upon which I ponder ceaselessly:  “…the Father almighty, Maker of Heaven and Earth,

and of all things visible and invisible…”

… visibilium omnium et invisibilium.  Doubtless each line of the credo might profitably exfoliate into a tome,

http://en.wikipedia.org/wiki/Mass_%28music%29#III._Credo

http://www.youtube.com/watch?v=fjtxKpHSXzg&feature=related

but let us now, here, pause for a moment at this one.

           

            The phrase is no mere afterthought.  There are in fact a whale of a lot of invisibles out there, and it’s not just ghosts, or disembodied ectoplasm, or ethereal unstructured mush.  Nor do I mean “dark matter” or “dark energy”, though apparently there are gobs & gobs of that as well: more than of ordinary matter, which now rattles about in the cosmos like spare change.  (Or, to coin a phrase, like angry candy.) No, dark-whatever, doubtless all jolly stuff, and quaint in its way, but no more inherently fascinating than, say, rabbits.  It’s merely the library-paste and plasticine that happened to be lying about when God got around to making this particular universe on some particular day, possibly with leftovers from some earlier practice project; and now it hangs about, drifting moodily hither and yon, like so much unemployed blancmange.  Being invisible doesn’t make it significant or interesting.  Let us have no fetish about the invisible.  If for some reason the credo had said, “… and of all things probable and improbable,” and if the improbable had somehow been mostly ignored, yet contained most of what was of interest in the universe, then we’d be talking about the improbable; or the fantastical; or the ironical.  The entities  I shall be getting at here  are not significant because they’re invisible; I’d be even willing to concede that they’re significant despite being invisible, that visibility would be one further and delightful perfection, one which we may someday hope to glimpse.  In any event, what is meant here is the mathematical scaffolding, on which the sun and the moon and the quarks hang  like so much laundry.  That is, the plan of the thing, so much more permanent and pervasive than the things themselves.  I mean the symphonic score  from which our ephemeral melodies derive.

            Properly apprehended, it is a structure of – crystalline palaces, transparent and thus largely invisible to the untutored gaze, save as the light of insight  glances off them at an angle, and so catches the inner eye.  These ideal edifices are as hard and as chiseled and as real, as our own makeshift dungeons of stone: nay, more real, for these intricate perfections are the prototypes, whereof our own poor earthbound shantytown is but the fallen, partial, semi-crumbled, quasi-scrambled, half-forgotten misremembered afterimage.  They are, it is true, invisible: but in part (in increasing part) -- not unimaginable.  Through intense and lifelong study, we may – by luck, or grace, or mental sweat – eventually acquire a glimpse of their upper ramparts, from which turrets rise, from whence pennants flutter – flutter in a plenitude, an infinitude of dimensions, one upon the other like palace halls; so that our own most swirling ballet or crashing waves  are but as the slogging of an ant  trapped between the narrow glass walls of the ant-farm.

            These diaphanous entities, being (as we shall argue) a part of the Creation, display a different side of God, from what we customarily encounter.  Or rather, as it may be, many different sides: the mystery of the Trinity becomes the mystery of the Infinity.  For we must not think of “Math” as just some subject in school, or as a section at the bookstore, beyond “Gardening” and next to “Pets”.  For one thing, there is just so much of it,  acres of math like fields of wheat, with more unfolding with each passing day, and much which, when first met with, seems qualitatively, drastically diverse:  not like different species, say a wolf and a fox, but like different phyla – a microbe and a mastodon.  And even as science has discovered some of the commonalities between mastodons and microbes, in the process deepening our appreciation of each, so too does the steady, then accelerated, and finally springing advances of our collective understanding – as it might be, the Mathematical Overmind – deepen and widen and heighten and… beyonden  our sense of the unitary structure of All There Is.  The whole enterprise is so fantastic, with such unity-in-diversity (again, compare the Trinity) that whole new fields have evolved at a metalevel, just to keep tabs on it all:  Set Theory and Proof Theory, to police our reasonings, and Category Theory, to provide display cases for all the genera of the menagerie, in the museum of the mind. 

            At the bookstore-cum-giftshop in Hilbert’s celebrated Hotel, you will find aisles for:  History; Fiction (including Astrology and Economics); Physics ‘n’ Chemistry; Biology; Number Theory; Point-set Topology; Algebraic Topology; Algebraic K-Theory; Topological K-Theory; Real Analysis; Complex Analysis; The Riemann Hypothesis; Sheaf Theory; Topos Theory; The Poincaré Conjecture.;  and Miscellaneous (i.e., gardening, computers, self-help, sports, celebrities, stamp-collecting, and all the rest).  There is no separate section for Theology, since that overlaps all of them.

            Now, none of this is exactly new: it is paleo/retro-NeoPlatonism. A retread, you may say, and twice-refried.  Yet there is now much more concrete substance to the view, than was available to Plato or Plotinus. They might imagine the cube and the icosahedron, and Kepler might attempt to stuff the planetary orbits into such homely and visible (risible) boxes, but they never encountered a Riemann manifold – or rather, they did, because we live in one, but they couldn’t see it  so they couldn’t imagine it, any more than they imagined fibre bundles or E8. 

            The Atheist, viewing a world charged with the grandeur of God -- which is difficult to ignore, since it will flame out, like shining from (to coin a figure) shook foil -- sniffs and dismisses it as tinsel: seeing the shook foil but not the Shaker.  So too the Nominalist, beholding or rather failing to behold  the serried ranks of theorems rising like seraphim beyond sight, regards these as a mere medley of contingent things, simply frothed out of someone’s brain, and which might, like a limerick or a pop-tune, have frothed out into something quite different.  (This is if anything the more charitable of contemporary dismissals, vice the dismissal of math and science as being merely the Eurocentric patriarchal dogma of the racist sexist agist ablist ruling class…)

*

            The nominalist viewpoint was given epigrammatic utterance (1886) by Kronecker, thus:

“Die ganzen Zahlen hat der liebe Gott gemacht; alles andere ist Menschenwerk.”

(“Ganzen Zahlen” means integers, but he may have meant only the non-negative integers, or “natural numbers”.)

Kronecker, rueing the day that ever he denied the transcendence of higher mathematics

(Randbemerkung:  A delightful swipe at Kronecker, well below the belt, and delivered with punch by a pugnacious Platonist, can be savored here:  The Continuum. )

Stephen Kleene translates, “God made the integers, all the rest is the work of man”, and glosses:

We cannot expect that the cognizance of the natural number sequence  can be reduced to that of anything essentially more primitive than itself.

(Introduction to Metamathematics, p. 19.)

The natural numbers – non-negative, non-zero whole numbers -- were, indeed, the only numbers recognized by the Pythagoreans.  One reads somewhere of a Pythagorean  casting himself into the sea in despair, upon encountering the proof of the non-rationality of the square root of 2; in fact, such self-drowning might just as well have been prompted by the sight of half an apple, since once you accept fractions, you are heading straight for E8.

            For in admitting the integers as being in no wise contingent – the work, indeed, of the Necessary Being – while desiring to dismiss the rest (“Menschenwerk” sounds even more like a kindergartner’s art project than the more Biblical “the work of man”), Kronecker has left the castle of mathematical agnosticism unguarded, by leaving open its postern gate.  Suppose we were – setting aside centuries of other riches – to begin by restricting ourselves to the laws of the natural numbers.  We would notice (as Euclid noticed) the primes, and require, for their adequate handling, great heaps of Number Theory.  Now, this already is no small thing.  You could fill a succession of lifetimes with nothing but Number Theory.  New discoveries emerge daily – some of them with such grave implications that they are actually classified, and at compartmented levels well beyond Top Secret, which does well enough for the design of an airplane or the movement of troops.  But the wealth is not just quantitative.  For to adequately handle nothing more than the primes, you need all of Number Theory, including even Analytic Number Theory, which brings in the continuum and the charmingly designated “imaginary” numbers (now much easier to imagine, though they remain as invisible as the number “23”), including indeed the Riemann Hypothesis, which already brings us to the frontier of knowledge with its outstanding unsolved problems.  Soon the whole of mathematics would come tumbling in through the unguarded door. 

[Note:  It’s never that simple.  I am aware that Kronecker himself was even more nominalistic that the famous quotation might suggest, as he did not accept the integers as a finished totality.  He went to some effort to derive results in a way that makes no use of such a totality.  Sort of neat if you can pull it off, like building a castle entirely out of toothpicks.  But if the idea of an actual infinity is problematic, that of the integers somehow running out of breath is even moreso.  I shall accept the natural numbers as given, and shall, for polemical purposes,  portray, as our foil, a sort of idealized Kronecker (who may even now be repenting of his nominalism, in some warm place) – the Kronecker of the quote – as accepting them as well.]

[As to the actual forked-radish of that name,  Joseph Dauben remarks (Georg Cantor, p. 66):

No-one could have been more opposed to Cantor’s ideas, nor have done more to damage his early career, than Leopold Kronecker.

Really, were it not for his key concession that the natural numbers are God-given, Kronecker might well be the villain of the piece.  As George Szpiro wrote of him (in Poincare’s Prize):  “Kronecker would not accept anything if it had not been invented by himself.”   Actually that can’t be quite accurate, since on that account he would not accept the integers… But anyhow, a perfect summary of the solipsist/nominalist epistemology.]




[The essay continues here.]