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.

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

 

Our Friends the Integers

What, after all, is a natural number?  There are Frege’s version, Zermelo’s, and von Neumann’s, and countless further alternatives, all mutually incompatible, and equally correct. … There is no saying absolutely what numbers are;  there is only arithmetic.

W.V.O. Quine, “Ontological Relativity” (Journal of Philosophy, 1968)

Randbemerkung:  The repeated reference to the integers in these notes  might mislead the reader into imagining that I accord them the least importance, mathematically or ontologically (let alone theologically).  Not so.  I only recur to them on the rhetorical grounds that the arch-nominalist Kronecker gave them up for free.  Had he instead said:

            Die Mengen hat der liebe Gott gemacht; alles andere ist Menschenwerk

then we should have instead busied ourselves with the necessary Menschenwerk of building up the integers out of set theory in any of the usual ways, pointing out that this new epigram eventually commits you to the integers in any event.  And had he said (oh would that he had):

            Die offenen Mengen hat der Liebe Gott gemacht…. ,

then the touchstone would have been topology.  Yea, had he instead, like certain of our very ethereal contemporaries, posited rather Category Theory at the base, sets and numbers and all the rest to be developed out of that – well, I might personally have balked, because I don’t understand Category Theory.  Still, a donkey does not understand the Goldbach Conjecture, so intellectually that is no objection. 

            The infinities of the Creation  -- and a fortiori, of the Creator – are --- whatever they might be, we may never know, but in any event, nothing particularly to do with whatever a handful of our own species happens, at any particular instant, to latch onto.  The integers are one toenail in one foreleg (or is it hindleg) of the Infinite Elephant.  It matters not what an infinitessimal portion this may be, of that great beast (to Whom even Babar tips his hat), nor how ineptly we manhandle it:  what matters is that the Elephant is Real.  Praise Him!

            Anyhow, the integers are fine, but nothing special.  Frankly, in fact, the Riemann Hypothesis bores me to tears (mainly because I still cannot truly intuit its significance). The Poincaré Conjecture is much  more inciting – though, having been finally, after a century, proved in its entirety, it serves less well than the R.H. as an image of the blaue Blume.  Compare, indeed, the final chapter of Russell’s Introduction to Mathematical Philosophy, for a nice dissing of the integers.  Likewise Wittgenstein (Zettel 706): “Die Zahlen sind der Mathematik nicht fundamental.”

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.]

E8: a Riposte (continued)

[A continuation of this.]

Synge waits until well into his second volume (J. L. Synge, Relativity:  The General Theory (1960), p.  104) to really let rip against Realism; and since he was himself very much a mathematical physicist, rather than an empirical experimenter, his testimony must be respected as coming from within the tent.  He distinguishes “Natural Observations” (NO) from “Mathematical Observations”, and opines:

Between NO and MO  there is a sharp and decisive break.  Only the simplest MO (counting) can be regarded as being NO also … Generally MO involve infinity (irrational numbers, differential calculus, and so on) and so lie outside physics and outside nature.

This is exactly the position of Kronecker (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”).

He then delivers himself of a curious passage, which Irrationalists would seize on with glee (fortunately none of them are reading this):

The peculiar fascination of theoretical physics  lies in the art of forcing meaningful truth out of the meaningless equation NO = MO, which is a symbolic form of the assertion that natural phenomena obey exact mathematical laws.  The true inequality NO =/= MO should not be spoken above a whisper, because it is extremely dangerous.   If believed, it would sever mathematics from physics, and reduce both to sterility through lack of mutual fecundation.  It is whispered here only as an apology to those readers who expect to see the mathematics of relativity [which he presents in great detail] tied to the physics of relativity  by a strong chain of clear thought.  It cannot be done.

These ring like the Night Thoughts of a relativistic physicist, on the eve of taking his own life.

            Despite his conspiritorial tone in that passage, Synge was by no means alone in his reservations. Here is another anti-Realist view from the world of physics:

A. D’Abro, The Rise of the New Physics (1939), vol. II, p. 728:

A hyperspace is obviously a mathematical fiction; and waves that can be represented only in a fictitious space  must themselves be unreal.

Now here indeed is a statement that has been overtaken by events.  In the view of string theory, this hyperspace, far from a mathematical fiction, is a physical fact, the one we live in;  indeed, we must beware lest those compactified but very real extra dimensions someday unfurl in our faces.  --  The point here being, not to make any point whatsoever about cosmic geometry, let alone to proclaim the truth of string theory:  but simply to counsel against that “obviously”, when dismissing the Realist picture.

[concluded here]

Theologia Mathematica: I

THEOLOGIA MATHEMATICA

 [Synopsis:

            Beginning with a parsimonious outset of only two Postulates,

            (1) Die ganzen Zahlen hat der liebe Gott gemacht … [Kronecker]

            (2) …visibilium omnium et invisibilium [the Credo]

we conclude to the Realist position in mathematics, associated with Cantor and Gödel. We note the nice fit with theism.]

I

The great Laplace, having presented his Mécanique Céleste, and having been asked by the Emperor, "Mais où est Dieu dans tout cela ?", notoriously replied:

"Sire, je n'ai pas eu besoin de cette hypothèse."

The cheese-eating atheist, feeling pleased with himself

(I am reminded of the austere style of his countryman Lagrange, who boasted that his Mécanique analytique contained no pictures to help make things plain; and later their compatriot Dieudonné, who in the preface to his celebrated Foundations of Modern Analysis, warns his audience that the tome will contain no such sweetmeats as pictures or diagrams, as that would only encourage the reader.)

            Perhaps people read too much into that oft-quoted remark (or in a sense, too little).  It is often taken, I suspect, as a dismissive, not to say smart-alecky reply:  a snook cocked at theists.  But really the context is both richer and more narrow.  The famous remark is more austerely analytical, I believe, and comparable to Newton’s celebrated “hypotheses non fingo”.

            Newton discovered the basic clockwork of the planetary scheme, but it was not at the time apparent, whether that system were ultimately stable under the various perturbations that mass is heir to.   And if it turned out in fact not to be so, then how to explain its evident stability over all geological time (itself of a vastness only recently appreciated – long enough to let *us* evolve, for instance)?  Before inertia was discovered (or, again rather, in a way more like posited, but still: based upon a more systematic survey of the phenomena), a  traditional perspective had angels impelling the planets in their paths by constantly puffing on them from behind (a pleasant thought); now it seemed as though these same angels might have to be called out of retirement, not to impel the planets, but to herd them from time to time, lest they wander off like lost sheep.  It was Laplace’s great achievement to prove stability by sheer mathematical means – as noble a use of the imaginative faculty  as ever graced the Sistine Chapel, and by no means a snub to the Watchmaker.  Laplace, in demonstrating that planets (‘wanderers’ in Greek) were (like the a-toms) misnamed,  proved we may dispense with the hypothesis of shepherding angels –  for this.

            But there is more beneath the firmament than the placid planets.  Beyond the deductive system of classical mechanics, there is the inductive panoply of actual life.  This too Laplace addressed, in his Essai philosophique sur les probabilités.  And again, he made no use of the supernatural.  But perhaps he took too much comfort from the pleasing and positive example of the stability of the solar system.  There was an overconfidence, a sunny bumptiousness, taken to task at length by Keynes in his Treatise on Probability (a work too little known, and which I commend to your attention).  Keynes descries a witless wizard behind the mathematical manipulations that pretend to deduce so much, and who ultimately throws us back -- surreptitiously -- on human intuition to make sense of events.  (Pay no attention to the man behind the screen.)

            Since Keynes’ time, the problems of induction and prediction  have only grown worse – or rather, they remain the same as ever, but our awareness of their depth and paradox has grown.  We have now become familiar with examples of dynamical chaos, even within the heart of the classical theory; and learned that such systems are, theoretically, rather the rule than the exception.  Even that simplest paradigm of all, the fabled billiard table, affords examples.  There are tables so shaped that, given a desired degree of knowledge of the trajectory by some future time, one may attain it by sufficiently precise initial conditions, whose precision is in some sense reasonably related to that of the required precision of prediction;  and there are tables so shaped that one may not.  (That is, the relation of the intransigent epsilon to the hapless delta, is in one case that of a banker, with perhaps rather stiff interest rates, and in the other, that of a highway robber.) This directly refutes the Laplacian determinist vision, which once extended to the universe as a whole, and which now fails right in the pool hall.

[A note, since this essay is going out to a diverse audience:  The time is past, when an author need pull his punch, for fear some member of his audience may not have met this or that notion.  You are online, or you wouldn’t be reading this.  So if “dynamical chaos” is an unfamiliar term, simply Google it up in Wikipedia.  Wikipedia knows all that can be known to Man.  – Though incidentally this does *not* mean that we should actually *pray* to Wikipedia.  Just so you know.]

            So we are back to a paradox.  System after system is either itself chaotic, or is in contact with, and thus influenced by, chaotic systems:  The parson is a placid man, but he lives within the weather.  Why then does the center apparently hold?  Why does it not all eventually – spin out, or bubble down to a sort of porridge?  Does it, like the imagined planets, need a nudge from time to time, to get it back on track?  -- Well, whether it needs them or not, it gets them, but: from us.  For the nonce, let us leave God and the seraphim aside.  [Note: In what follows, we assume the three-level system of natural description which C.S. Lewis expounded in Miracles. Roughly: subnatural = quantum; natural = classical (including relativistic etc.);  supranatural = involving free will.]  For supra-natural intervention, we need look no farther than our own fingers – righting the wineglass that had started to topple, or pulling the baby back from the edge of the stage. (The paradigm case of supra-natural intervention is supernatural intervention, by ghosts or by God; but as a logical problem, this differs little from the intervention of human will, and thus may be dispensed with where parsimony suggests.)

            Most of what goes on the the universe is (let us call it) naturalistic – whether ruled by the (classical, relativistic) laws of the natural, or the (quantum, aleatory) laws of the subnatural, or some complexus of both. But most of what *we* experience, day to day – that is, experience in consciousness, as opposed to this or that enzyme oozing about – is generously admixed with the supra-natural.  I mean this in its familiar, almost ho-hum sense (except that, by dint of our ongoing ho-humming, we have become dulled to the fact that it is, strictly, miraculous.  Our simplest Saturday afternoon crackles with miracles like a fourth-of-July sky.)

            Now, this in itself need not point to God (let alone prove Him), any more than this or that billiard-ball impact proves classical mechanics, or some passing photon yields Maxwell’s equations.  Indeed, in so far merely as itself, it does not so much as indicate that this supra-natural capacity in ourselves is even rational or good:  picture (though only for a dreadful moment) a universe peopled exclusively by madmen and sociopaths, the free-willed equivalents of scorpions.  That is to say, an outside observer of our universe might detect its departures from plain (non-quantum, non-noetic) determinism, whether from sub-natural or supra-natural inputs; but lacking internal access to the lived experience, he could not tell whether these departures made sense.  Indeed, our hunch now is that the subnatural inputs do not “make sense” – that is, no moral sense, no sense beyond themselves.  It will all (within its own world) dutifully trot along in the path laid out by the Schroedinger equation, while its inputs to our world look to us like clowns piling out of an infinite Volkswagon: but it will not, pace philosophers from Protagoras to Penrose, supply or even heighten our humanity, our morality, our free will.  Indeed, from our present vantage, the subnatural is somehow even more alien to the noösphere, than is the shadow play of Newton, or the passion play of Darwin.  He who would seek the key to our humanity there, seeks the stars in a mudpuddle.

            Whereas we, in our priviliged observatory of our own shared experience (though of nothing else), can report:  Yep, it makes sense.  It’s often in practice too f***ed-up for words, but we’re not just ensouled scorpions, we’re … possibly fallen, anyhow substantially tattered angels.

Addendum:

 A distant kin to the imagery of the ushering angels, impelling the planets and keeping them on course, cropped up again in 1925, with de Broglie’s idea of “pilot waves”, guiding the electrons in their rounds while orbiting the proton; revived again in another context by David Bohm, in his hidden-variable theory of quantum mechanics.

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