Pucker Up

The world took notice when some old mathematical conjectures  were finally solved in our own lifetimes:  Fermat’s Last “Theorem”, and the Poincaré Conjecture.   To somewhat less fanfare, a conjecture concerning sphere packing, dating back to Johannes Kepler in the seventeenth century (yes, that Kepler -- the planet guy) was finally solved in the twenty-first.   A breakthrough came in 1998, but a formal proof was not recognized until this very year.

George Szpiro wrote an unusually readable account in his book Kepler’s Conjecture (2003).   It is an example of a sphere-packing problem,  which is a global problem, with applications stretching from the fruiterer’s tray to error-correcting codes. Among the related matters is the kissing number problem, which is a local problem:  how many spheres (or, in two dimensions, discs) can you pack around a given central sphere.  The answer is the Newton number or kissing number for that dimension.   

In two dimensions, the answer is six, as you can verify by ringing a penny with six of its fellows;  that no further coin could butt in, can be seen “by inspection”.  (That, incidentally, is the only known use for pennies, in our own day.)   For three dimensions (the everday space we live in), the answer turns out to be 12.  That is more difficult to see, though you can accomplish it using a baker’s dozen of beachballs and a dozen undergraduates.

Somewhat surprisingly, for dimensions four, five, six and seven, the answer is not known exactly, only upper limits.  Very surprisingly, we then suddenly get a break :

In eight dimensions, all of a sudden the Newton number is known exactly:  240 white balls can kiss the black ball in the center.  Why is that number known precisely?  In this case the upper bound came out to be 240;  but a certain well-known lattice arrangement, called E₈, also allows 240 balls to touch the central sphere.   Since the actual kissing number of E₈ coincides with the theoretical upper bound, 240 must be the highest kissing number. -- From dimensions nine to twenty-three, again, only bounds are known.   (p. 96)

What is so gratifying about this is that what is in essence a simply, concretely conceptualizable problem  should be solved by the testimony of so fiercely abstract an algebraic object as E_8.  (Szpiro calls it “well-known”;  depends what circles you move in.  I just asked the fellow next to me in the pub, and the poor fool confused it with the Leech lattice.)  I first learned about this enormous structure from a 2007 article in the New York Times.   Or rather, learned virtually nothing about it at all, beyond its mere name, since the journalist, and all hands asked, concurred that the thing is indescribable.  A sketch of that encounter, back from when we were shilling for Platonism, can be read here.

Language note:  This metaphor of ‘kissing’ (for tangency) is used elsewhere in mathematics: osculating plane, osculating circle.

Beauty is Truth, Truth Beauty—NOT

In science and mathematics, theism is a prophylactic (against narrow empiricism), but not a guide (so far as I know).   It is thus to be contrasted with Beauty (the capitalization is here sarcastic), which some scientists, and all popular science writers, have contended  may legitimately guide physical research.  But it is a slippery cicerone of a guide.   Few more beautiful notions have been put forth  than Kepler’s structuring of the solar system upon the quintet of Platonic solids.  For that matter, circular orbits may seem prettier than bulgy ones;  and the vision of the planets being chivvied along on these  by attendant angels, like the blowing winds at the four corners of medieval maps, is charming.  The successors to these pleasing pictures  have their own, more austere, beauty:  but only in hindsight, or to their inventors.  At any given stage, our aesthetics are too underdeveloped to be trusted to guide us aright.

Scientists have indeed sometimes had Beauty in mind, at some point, during their pursuit of an intriguing hypothesis.  The cases in which the hypothesis proved correct, are the ones we hear about.  No-one pens a memoir boasting how the pursuit of pure Beauty led him into a scientific dead end.

But the reason for the prevalence of paeans to Beauty in popular science writing  probably owes less to the former class of experience, than to mere expediency.  The man on the omnibus  figures he knows a bit of Beauty when he spots some, just don’t bother him with a lot of messy maths.   Kekulé  dreams of the Ouroboros, and next thing you know  has figured out the carbon ring.  A cinch!  So the author can flatter such readers’ fancy, while sparing them brain-crunching labor, doling out little toy versions of Black Holes, String Theory, or what have you.

*

It must be allowed that physicists do sometimes speak in such terms, even when talking quietly among themselves, rather than attempting to stun the public into goggle-eyed Gawsh-Paw stupefaction.   Here is one instance, deliberately cited from a textbook aimed at physics majors, rather than from public lectures or popularizations:

R. Adler, M. Bazin & M. Schiffer, Introduction to General Relativity (1965), p. vii, 1:

Since there are numerous works available which deal with the general theory of relativity, some of them masterful and even classical, it seems necessary to explain the specific intention of the present book.  … Our principal aim has been to show the close interaction of mathematical and physical ideas, and to give the reader a feeling for the necessity and beauty of the laws of general relativity. … General relativity … represents a fusion of mechanics and the theory of gravitation, on the one hand, and of geometry, on the other.  The combination … will result in great formal beauty and mathematical elegance.

(“Elegance” in particular  is a favorite mathematician’s word, less gushy than “beauty”.)

Steven Weinberg, Dreams of a Final Theory (1992), p. 6:

When it turns out that mathematically beautiful ideas are actually relevant to the real world, we get the feeling that there is something behind the blackboard, some deeper truth foreshadowing a final theory that makes our ideas turn out so well.

The Platonic thought is congenial, but let’s look closer at this epithet “beauty”. 

For the actual discoverer, such a frisson is no doubt felt. Platonic Forms get good reviews, from those who have been privileged to glimpse them.  And these essays have tended to a Realist view of these Forms (in tune with a background assumption of theism).  But that much leaves open, where and whether and to what extent these Forms may manifest themselves in the actual rough and tumble of this world.   Our life here below is littered with broken symmetries and broken hearts.

Weinberg is a particularly stellar theoretical physicist, and thus equipped to say such things if anyone is.   But note that scientists who talk like that  tend to be mathematicians or physicists, not chemists or stock-breeders.  (Or syntacticians, for that matter.  Despite the increasingly abstract and structured nature of one well-known line of inquiry, its proponents have never been guilty of marketing it for its “beauty”.)  And the pulchritude alluded to tends to be the clarity of the blackboard, not the messiness of the lab.

In the face of testimony such as that quoted, beware too the selection effect:  What makes it onto the printed page are Winner’s Narratives.  Basking in his Nobel, the lionized scientist allows as how “I gazed on Beauty bare, and she did spread for me  the doors of Truth”, much as the (perhaps accidentally) successful investor will wink and share his Winning Formula ("Always go with your gut" or whatever).  Meanwhile there have been a great many first-rank scientists who thought they had a truly beautiful idea, but you don’t hear about it, because it didn’t pan out (Lord Kelvin with his vortices in the ether, Karl Pearson with his ether squirts).

Aestheticism in general is not notably congenial to the scientific enterprise.  That same Keats who perpetrated the (too-)oft-quoted jingle “Beauty is Truth, Truth Beauty” (adding, in a slogan worthy of Big Brother, “That’s all you know, and all you need to know”) once proposed a toast “to the confusion of Newton” for having explained the rainbow (all this, one imagines, shortly before his cortex melted into a syphilitic soup, the effect of having embraced one beauty too many in Drury Lane).

           

There is some potentially valid content to this “beauty” motif:  basically, the focus on structure rather than number, clean architecture rather than the kludge.  (Though if Nature herself prove a kludge, we’re out of luck.)  Whether the deeper understanding we may thus arrive at is best described as “beautiful” rather than “sexy” rather than “chilling” rather than “scrumptious” rather than “word, dude!”, is unimportant.  The practical reason for all this “beauty” talk  is that it sells books.  And the reason for that is a matter of bad faith:  John Q. Public (commendably) approaches the altar of science, presided over by a handful of high priests; but then his strength fails him; will this be hard, like calculus? and at once comes the coo of reassurance:  “Nooooooo,  wee  ah-rin thee land of byooooooty, all yooo have too doo is feeeeel”…  Yeh, right.  That is the attitude which plastered Weinberg’s perfectly clear and level-headed book with the gooey title, “Dreams….”  (presumably that was whelped by some gnome in marketing, not by the author himself).

If string theory should eventually turn out to have been a dead end for physics, then the mathematical Beauty that  for so many years  mesmerized its devotees  may be said to be that of the Sirens.

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[Further notes]

Cf.  Daniel Silver, “Knot Theory’s Odd Origins”, in American Scientist, March 2006, ironically imagining the mental state of  Peter Tait and Lord Kelvin as they put forth their theory that “chemical elements were knotted tubes of ether”:

No cumbersome hypotheses would be needed to explain chemical properties;  they were a result of topology.  It was simple and beautiful -- it had to be true.

A healthier attitude, cited by George Szpiro Poincaré’s Prize(2007), re the reception of G. Perelman’s proposed proof of the conjecture:

They both found the papers  beguiling in their beauty.  But somebody needed to check the nuts and bolts.

 

Arthur Koestler, The Act of Creation (1964), p. 213:

All through his life  Kepler hoped to proved that the motion of the planets round the sun obeyed certain musical laws, the harmonies of the spheres. … Kepler never discovered that he was the victim of a delusion.

Arthur Koestler, The Act of Creation (1964), p.330:

False inspirations and freak theories  are as abundant in the history of science  as bad works of art.

~

Hadamard is an adherent of the beauty criterion, at least for math, and at least for the practice of math (rather than as a criterion of truth for the results):

These examples are a sufficient answer to Wallas’s doubt on the value of the sense of beauty as a “drive” for discovery.  On the contrary, in our mathematical field, it seems to be almost the only useful one.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 130

~

Just the existence of such a mathematically elegant unifying picture  appears to be telling us something deep about the mathematical underpinnings of our physical universe.

-- Roger Penrose,  The Road to Reality (2004), p. 471

But a few pages later, in a more cautious mood:

It is important not to allow ourselves to be carried away by the beauty and seeming finality of such apparently tightly-knit mathematical schemes.  Nature has had a habit, in the past, of tempting us to a euphoric complacency  by the power and elegance of the mathematical structures that she appears to force us to accept as guiding her world, but then jolting us, from time to time, out of our conceptual torpor by showing us that our picture could not have been correct, after all!  Yet the shift has always been a subtle one  which leaves the previous edifice still standing proud, despite the fact that the foundations on which it had stood  have now been completely replaced.

-- Roger Penrose,  The Road to Reality (2004), p. 485

The reference is to e.g. Newtonian physics surviving as the low-energy limit, or macroscopic arena, of phenomena covering in more particularity (and in a mathematically radically different way) by relativity and by quantum mechanics.  That is to say, the earlier account is not refuted or abolished, but sublated -- aufgehoben.

[Update]  For a thoughtful essay by a first-rank physicist, try “Beauty and the Quest for Beauty in Science”, a 1979 lecture reprinted in S. Chandrasekhar,  Truth and Beauty (1987), p. 59f. ]