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.]

Truth Decay

The empirical character of a very successful theory  always grows stale, after a time.  We may then feel (as Poincaré did with respect to Newton’s theory) that the theory is nothing but a set of implicit definitions or conventions …

-- Karl Popper, “Truth, Rationality, and the Growth of Knowledge”, repr. in Conjectures and Refutations: the Growth of Scientific Knowledge (1962; page refs. to the Harper paperback reprint), p. 240

I’m currently reading The Shape of Inner Space, by Shing-Tung Yau -- the Yau of Calabi-Yau, which lies at the heart of string theory.    Unlike Smolin’s The Trouble with Physics, let alone Woit's Not Even Wrong, the author is not out to debunk the theory in any way, especially as he was one of its mathematical progenitors:  nor to puff it, like Brian Greene, since  unlike those contentious authors, Yau is not a physicist by trade, but a pure geometer.  But towards the end of the book, he is led to exclaim: 

Given that much of string theory now hinges on compactifications of Calabi-Yau manifolds, which have these moduli with their associated massless scalar fields  and particles that don’t appear to exist, is string theory itself doomed?

And he quotes physicist Burton Richter against those quasi-nihilistic latitudinarians who have (as a recourse of despair) embraced “the landscape” (short version:  Anything Goes):

To them the reductionist voyage that has taken physics so far has come to an end.  Since that is what they believe, I can’t understand why they don’t take up something else -- macramé, for example.

There have, throughout history, been repeated instances of premature prophecying of “the End of Physics”:  but  there   the idea was that we were close to having solved everything;  never, that physics might one day become permanently stuck.

*

It was with these passages ringing in my mind, that I picked up Jonah Lehrer’s essay in the current issue of The New Yorker:  “The Truth Wears Off”.   I won’t summarize it -- it is brilliantly written, go read it -- but merely comment that the widespread scientific predicament there depicted -- in physics and biology and psychology and beyond -- seems even worse that the slightly softened interpretation that the author tries to present as a possible explanation or compromise.   Neither “happenstance” nor unconscious bias  can explain the range of what is supposedly going on.  E.g. the researcher who repeatedly failed to replicate his own results, not for want of trying, should  if anyone  have been prey to such bias.

[Cultural comment, to be developed if anyone’s interested:  the scientific standing of Rhine’s experiments in ESP.
-- Indeed, this just in:
http://www.nytimes.com/2011/01/06/science/06esp.html?ref=global-home&pagewanted=print  ]

*

As Pauli wrote to Kronig in 1925:
"At the moment, physics is again terribly confused.  In any case, it is difficult for me, and I wish I had been a movie comedian or something of the sort, and had never heard of physics."
Or cosmologist Edward Harrison in 1975, saying that endless expansion "would make the whole universe meaningless.  If that were true, I would quit, and spend my life raising roses." 

*

There is a thread, a thought, that links these observations;  but  once again -- Wovon man nicht sprechen kann, darüber muss man schweigen.

*

The Shape of Inner Space is something of an intellectual autobiography, in addition to an overview of Calabi-Yau.  And quite an engagingly modest one, as these things go.   By the end of it, you have a comfortable familiarity with the author, and are full of friendly feelings.

Immediately after finishing it, I happened to re-skim George Szpiro’s equally well-written volume, Poincaré’s Prize , and was startled to see that the sinister puppet-master depicted in the chapter “The Gang of Four, Plus Two”, is none other than Shing-Tung Yau.  When I originally read the book, the name meant nothing to me; but now… it was like rounding the corner, and encountering your own cousin brandishing a knife.

Now… in the grander scheme of things, so what;  we all have our faults.  But this account of academic rivalries  gibed with my surprise at searching Shape’s index for the name of Woit or Smolin, and not finding them.  Whatever their merits or demerits, these are popular recent book-length treatments of string-theory, and one might expect that Yau would reply to them, if only to rake them over the coals.  But only towards the end of his book  does he mention theirs, in briefest passing:  and then, without uttering their names (as one might be forced to allude to Mein Kampf, but draw the line at penning the name of its author).

Yau does, however, offer a kind of reply, as he demonstrates, pretty convincingly, that String Theory, the beneficiary of much math, has in turn created, and inspired, solid math in its own right, which will survive independently of whether the particular universe of our own sojourn  happens to embody the physics.   Indeed, the fact that Ed Witten received a Fields Medal, rather than a Nobel Prize for Physics, pretty much demonstrates that.

*

Addendum:

Pauli was not alone in his lament.   Schrödinger to Bohr, 1926:

If we are going to stick to this damn quantum-jumping, then I regret that I ever had anything to do with quantum theory.

And Einstein in 1924, re the idea of renouncing strict causality:

I would rather be a cobbler, or even an employee in a gambling house, than a physicist.

This is rather a rarified kind of job-dissatisfaction.  It is as though a carpenter, towards the end of a long and successful career, were to cry out, “Had I known that wood is mere cellulose, rather than a habitation for dryads,  I would have preferred to have been anything -- a plumber or a physicist -- rather than this!”

But the phenomenon does exist.  The author in which I found those two quotes -- J. C. Polkinghorne, The Quantum World (1984), p. 53 -- after a distinguished career as a particle physicist, resigned his chair, and became a vicar in a country parish in Kent.  It was from this Father-Brown-like living that he published his popular little volume with Princeton University Press.

Similar instances could be cited, notably Alexander Grothendieck.  (Motif: “Goodbye to All That.”)

 

[Update Feb 2014]  A statistical reason for “truth decay”, with examples:

Most scientists would look at a P value of 0.01 and “say that there was just a 1% chance” of the result being a false alarm. “But they would be wrong.”

Rather than being convenient shorthand for significance, the P value is a specific measure developed to test whether results touted as evidence for an effect are likely to be observed if the effect is not real. It says nothing about the likelihood of the effect in the first place.

http://www.nature.com/news/scientific-method-statistical-errors-1.14700