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. ]
A good example might be the sometimes condescending use of the musical analogies in discussing Riemann's Hypothesis, whether it refers to a Fourier-like process to build up the "signal," a similar process with respect to the (imaginary part of the) zeros of the Zeta function (whose real part is [?] 1/2), or attempts at prime-based music. See comment, Wolfram's MathWorld, Interprime; see also search results, Google, "prime music." Often, if not typically, the discovery of "beauty" coincides with an inabilty to wring any more math out of a situation.

ReplyDeleteThe avoidance of narrow empiricism in math can also be a dangerous habit. Erdos' Big Fascist often rewards relentless calculators and shrewd hoarders of numerical detail with insights that elude those who strive for big results. Not only is there no royal road to mathematics, the shortest way is almost always the long way, which arguably begins: 1,2,3,...

ES-VM's famous line about Euclid is I think simply in error. She never saw that photo of Dree Hemingway.

What about beauty being a side product of something else. For example, parsimony tends to result in usually well-structured, but always "slim" models. These models also, most commonly,y come with an aesthetic appeal, but this aesthetic appeal is actually not the reason for why parsimony is used. Rather, it generates hypothesis that can readily be falsified (less assumptions -> higher efficacy of testing), so the aesthetic shivers we feel when applying this method are not the reason for why it still is used.

ReplyDeleteMy 2 Euro cents.

Euro pence cheerfully accepted!

ReplyDeleteWe may be next with our hat out, behind Greece..

> aesthetic appeal is actually not the reason for why parsimony is used. Rather, it generates hypothesis that can readily be falsified (less assumptions -> higher efficacy of testing),

ReplyDeleteWaitaminit waitaminit...is that *true* ???

OK it's late, not thinking straight, but:

(Case A) A hundred ad-hoc hypotheses. Easy to test, and to pick one off.

(Case B) "The Universe is ruled by the principle: symmetry laws = group-theoretic transformations". Elegant but... elusive.

(Case C) "All is One". As simple as possible -- and unrefutable, and unproveable.

No, I think the aesthetic element is really fundamental.

In fact, I think it is more fundamental than *either you or I realize*.