50

The Science of Significant Form

lan Stewart

What is the relation between mathematics and its applica- tions? What should it be? The Intelligencer has given us two views, those of Kac [12] and Hermann [10]. Both seem to agree that some kind of gulf exists and should be bridged, though they disagree on the precise treatment. I 'd like to put a third viewpoint, disagreeing with both of them. Kac does ask for "a debate or even a confronta- tion". I 'd prefer it to be a debate.

The topic is of course vast, and each mathematician's view of it is constrained by his own background and interests. It is important to understand right from the beginning that the topic is so vast that no one person has any real chance o f encompassing the whole thing. Mathe- matics alone is too big; but we are here discussing its rela- tionship to the whole o f science. Any personal view (in- cluding this one) is bound to be partial and imperfect.

Core Mathematics

Hermann asserts that applications of mathematics do not happen by "pure chance", and says:

"Mathematicians seem to be most comfortable with the belief that the applications happen randomly, as a Pois- son process. I don ' t think so."

The short answer to this is that he has so oversimplified the position that he wishes to attack that it is hardly worth the effort of an assault. But before arriving at what I hope will be a more reasonable position, let me review three case-histories. It would be easy to find dozens more.

Number theory is generally considered to be among the "purest" areas o f pure mathematics (whatever that is). Probably wrongly so: see Mackey [14] for a historical dis-

About the author

Ian Stewart has been on the faculty at the University of Warwick since completing his Ph.D. there in 1969. His thesis was on Sub- ideals of Lie Algebras, supervised by Brian Hartley. Since then, his mathematical int- erests have spread from Lie algebras to singularity theory and subversive activities such as catastrophe theory. His book with Tim Poston, "Catastrophe Theory and its Applications" was listed among Choice mag- azine's selection of Outstanding Academic Books, 1978. His most admired (dead) mathe- matician is Poincar~, which is odd because he has a prejudice against Frenchmen. His favorite theorem is the Hairy Ball Theorem; his favorite example: Q(e2~i723) does not have class number i. The open problem he mentions might also be considered subversive by some: he hopes that nonstandard analysis can be used for something sufficiently spec- tacular to force its import into the main- stream. Ian Stewart's nonmathematical act- ivities include playing cricket and the guitar, making wine and drinking wine, pain- ting scenery, and carpentry. He also writes Science Fiction, and stories have appeared in Analog and OMNI. His wife, who is a nurse by training, currently runs the village playgroup. They have two small sons.

51

cussion of the influence of number theory upon physics, largely unperceived by either number-theorists or physicists since it happened through the intermediary of harmonic analysis and a continuous/discrete analogy. But let that pass. Consider the Fermat Theorem - not the "Last" one, but the "lit t le" one:

a p -1 ~ 1 (rood p).

If ten years ago anyone had predicted a military applica- tion of that, I doubt he would have been taken seriously. However, it forms the basis of one of the first " trapdoor codes". A trapdoor code has a simple encoding algorithm whose inverse decoding algorithm is inordinately complex (in the absence of a small amount of 'key ' information); it has the startling feature that the encoding algorithm can be made public. The application had enough military significance for the Pentagon to try to slap a security blanket on publication. (Incidentally, this is strong evi- dence that the military did not see this application, or in- deed the application of any of the other branches of com- binatorial mathematics involved, coming in advance.)

Next: General Relativity. It is well-documented that Einstein first decided, on mostly aesthetic and philosophi- cal grounds (but of course based on a few crucial experi- ments), what sort of mathematics he needed to know to set it up. He then went to consult the differential geometers to learn it - and he didn't find it easy. But it did already exist. The theory of differential invariants arose in part as a natural extension of that of algebraic invariants by Cayley, Sylvester, Gordan, and the rest: these had been developed as an interesting gadget with no obvious practical uses. Connections between differential geometry and gravitation were not widely expected - though there are a few remark- able anticipations, such as Clifford [5].

Third: matrices. When Cayley developed their proper- ties, he wrote (somewhere: I haven't found the reference) to the effect that "here at least is something that will not find any practical applications". And indeed linear algebra was not taught outside mathematics departments until well into this century.

These are three unexpected applications of mathematics that had been developed for a quite different purpose: in at least two cases, apparently for the sheer joy of the thing.

We have here the seeds of the idea that applications are "random". But a bald statement like that mis-states the historical message. The evidence does not suggest that any particular piece of mathematics has much the same chance of becoming applicable as any other if you wait long enough. Almost certainly, a large proportion of the mathe- matics being studied today will never acquire useful appli- cations. The snag is that it does not appear possible to decide in advance which bits will: in particular applicabil- ity when produced is an unreliable guide.

The physicist Freeman Dyson [6] relates a conversation between Jeans and Veblen, in the early 1900's, about

group theory (including finite and Lie groups). "We may as well cut out group theory," said Jeans. "That is a sub- ject which will never be of any use in physics." I t 's a good job that science did not follow Jeans's advice: most of particle physics would not exist today if it had. But the- point is not that Jeans was a stupid, unimaginative, pre-

judiced fool. His view was the orthodox, considered view of the physicists of the period. Dyson [6] says: "Very few men at that time had the slightest inkling of the fruitful- ness that would result from the marriage of physics and group theory."

Nor, in fact, should it be necessary to at tempt to decide in advance which parts of mathematics are likely to prove applicable in the long run, even i f that is your criterion for them to be worth supporting or pursuing (which, to save an entirely separate argument, I am happy to assume here). And the reason is something that neither Hermann nor Kac take sufficiently into account. It is the unity of mathematics.

There is some sort of spectrum of types of mathematics. At one end are the bits that have immediate and obvious applications: say, numerical weather-prediction. At the other are items whose potential for application is dubious and in no way visible: dehydrated elephants, in Kac's colour- ful terminology. (But see Poston [ 16] if you ' re an elephant fan.) In between are things like p-adic algebraic groups (cited by Hermann with what I detect to be skepticism) or eigenvalues of the Laplacian (with important theoretical uses, hence more "practical"), or the curvature of diffeo- morphism groups.

But - and this is something that perhaps the "pure" mathematician perceives more strongly than the traditional "applied" one - the whole spectrum is a single, united struc- ture. I don' t want to overstate this: first, at any given time, the structure is unfinished, and not all of the unity is apparent; second, there's a good bit of junk around which is manifestly trivial or insignificant and cannot possibly make a contribution. But the underlying structure of the significant parts, the mainstream, what Steen [20] calls the "core", is that of a single, organic whole. And hence any application of part of that structure tends to justify the entire thing.

The crucial factor, of course, is significance. It is almost as hard to define that as it is to decide in advance whether a given idea will one day prove applicable to science; but any mathematician worth his salt ought to recognise that it exists (though his actual judgement whether it is present will be questionable outside his own field of competence). A mathematical idea is significant if it lends extra power to the mathematical elbow, if it makes, or looks as if it may in future make, a substantial contribution to the mainstream.

Fermat 's theorem, Riemannian manifolds, linear algebra: all are significant pieces of mathematics, and were taught as such by mathematicians, sometimes for centuries, before applications were found.

52 The unity of mathematics makes it imperative that

judgements of significance are not made on the basis of 'What can it do for me, now?". Hermann says:

"I couldn't care less about counting how many trees there are on a graph or a new algorithm for a numerical analysis in n instead of 2n multiplications until I see the relevance to the applications in which I am interested."

I think he is perfectly entitled to make his own judgement of what he wants to do on such a basis; but I don't recom- mend it as a way to decide what anyone else should do. (However, I hold no personal brief for tree-counting or pedantic bound-improving, which must stand or fall not on whether they are of any use to Robert Hermann, but on whether they have any real significance for mathe- matics.)

Good, competent, useful mathematicians mostly con- tribute significant things to the core. Various of-them- selves-insignificant things get produced along the way: large numbers of Ph.D. theses contain only results that any expert in the field would conjecture and prove within a week - but we have to train research mathematicians somehow. It is a terrible pity that a lot of this stuff gets published - but that has been forced on us by university administrators who insist on counting publications to decide promotions or job-applications; it's not a structural fault within mathematics.

Poor, incompetent, or unimaginative mathematicians do not make significant contributions. We could save trees by preventing them publishing; but it is a fallacy to assume that we can somehow redeploy them into more useful areas; because they will largely remain poor, incompetent, and unimaginative in their new field - and uncomfortable too, because they didn't choose to work in that area anyway.

Between these two extremes are a large number of sound, competent, but uninspired mathematicians: they make some kind of contribution, seldom large, seldom crucial, but occasionally accumulating to give something worth having. You won't change them much by redeployment either: you won't create brilliant researchers out of nothing. And they do perform one very important task: they help to keep mathematics alive. If every student had to learn from a textbook alone, the subject would slowly die out altogether - except for the odd genius. And in our present society, we can't afford to let this happen.

Technician versus Creator

Now let me play Devil's advocate for a paragraph. In the example of trapdoor codes, the cryptographers would no doubt have discovered the Fermat theorem when the need became apparent, if it had not akeady existed. (Though it is difficult to envisage a civilization that could develop electronic computers without having first noticed the

Fermat theorem.) And they found the application because they were cryptographers.

Fine. Whichever breed of scientist f ound the applica- tion makes little difference: it was found nonetheless. I don't think it is seriously suggested that it is the mathe- maticians who will find the unexpected applications: their job is to create the mathematics. And the fact remains that the Fermat theorem, and the various combinatorial gim- micks that have been ranged alongside it since, had all been discovered before the cryptographers started on the problem; and they created the right kind of mental background for them to think against.

Oh, one minor point: actually they weren't crypto- graphers at all. One was an electrical engineer and one worked in artificial intelligence. They got interested in cryptology, certainly: but what made the breakthrough possible was that their particular talents, developed for entkely different purposes, came together at the right moment. Kac tells us that contributions made to biology by physicists happened because they became biologists; but I think it is far more important to realise that, had they been biologists all along, they would never have made those contributions - because they did not know the necessary physics!

Obviously, in order to develop an application to sub- ject X, it is necessary to learn something about subject X; but this tautology tells us nothing at all about the only real mystery: why is it that time and time again the big breakthrough in subject X came from outside the field entirely? For example, recent developments in the theory of turbulence have depended strongly on two things from outside fluid dynamics, designed without any thought of application to turbulence: strange attractors in topological dynamics (e.g. Ruelle and Takens [19]) and the laser as an experimental tool (Swinney, Fenstermacher, and Gollub [21 ]). The laser was often described as "a solution looking for a problem" in its early days, by well-meaning scientists who felt it was a waste of time: in the event, their opinion has proved singularly unimaginative and unhelpful. Applied mathematicians got a head start on the pure fraternity, as regards strange attractors and chaotic dynamics, in the pioneering paper of Lorenz [ 13] which developed out of numerical weather-prediction: they appear not to have taken the slightest interest in it until Smate's school (in initial ignorance of Lorenz's work) realised the enormous significance of the idea. (Of course you can track it back to Poincar6; but he never made any use of it; he just ex- pressed his horror that it could occur.)

More striking still is the breakthrough that does not, in any effective sense, solve a well-known problem. General Relativity I is an obvious case: Dyson [6] says:

1 It is questionable whether astronomers were really worried about the precession of the perihelion of Mercury: it was a nice piece of evidence later on, but not a major problem in the field.

"General Relativity is the prime example of a physical theory built on a mathematical "leap in the dark". It might have remained undiscovered for a century if a man with Einstein's peculiar imagination had not lived."

The difficulty that Einstein had in learning differential geometry strongly suggests that, had the mathematics not already been available, he would not have been able to develop it himself (though with Einstein, who can tell?). It is of course perfectly fair to argue that the reason that differential geometry existed at all can be traced back to physics: the physical intuitions of Riemann and the geode- tic work of Gauss. But the vehicle for Riemann's work on manifolds was his Habilitationsvortrag: On the hypotheses which lie at the foundations o f geometry. The basic idea is a purely mathematical reflex: generalize Gauss from 2 to n dimensions. Riemann's feeling for mathematical form guided his thoughts - though he had some physical intui- tions too. A solid application had to wait for Einstein.

Geometry, of course, is based on spatial intuition, hence is of physical origin. But to trace the "applied" aspect back this far is surely to beg the question: almost all of mathe- matics is based on geometry or number, so is "rooted in reality" in some sense.

Einstein is a good counterexample to Kac's view (which I find sterile and depressing), of the role of mathematics in scientific discovery. It appears to be "the mathematician as technician", or may be

"Ours not to reason why, Ours but to do or die."

Specifically:

"It seems self-evident that mathematics is not likely to be of much help in discovering laws of nature."

That was Kac [12]. Here's Dyson [6] again:

"One factor that has remained constant throughout all the twists and turns of the history of physical science is the decisive importance of mathematical imagination... in every century in which major advances were achieved the growth in physical understanding was guided by a combination of empirical observation with purely ma- thematical intuition. For a physicist mathematics is not just a tool by means of which phenomena can be cal- culated; it is the main source of concepts and principles by means of which new theories can be created."

Their positions aren't quite as far apart as they may seem, because Kac has misstated his case anyway. He says that mathematics will not be of much use in discovering new physical laws; but his argument is that mathematicians will not be much use. Part of the argument can be ignored: the Catch-22 proof-by-definition that, should they succeed, they will have become physicists and not mathematicians anyway. That line is neither helpful nor informative. But

53 what I think Kac means is that mathematicians will not be much use in the actual process of discovering new theories of nature. (I shrink from the word "laws".) That is, not the mathematical run-up to the discovery, but the actual discovery itself.

That is: Einstein discovered General Relativity because he was a physicist.

It's probably true. But is it fair to say that Riemann's contribution, or that of the Italian geometers, was "not much help" in the discovery? Mathematicians contribute to the discovery o f new theories o f nature by providing the mathematical concepts necessary to the formulation o f those theories. Where would Kepler have been if the Greeks had not studied ellipses? How could particle physi- cists have discovered SU(3) symmetry if nobody had ever looked at a group? (The applied mathematicians had used "symmetry" arguments for centuries, never seeing them as more than a useful trick: it was the systematic study of the concept for its own sake, because it looked significant, that provided the muscle. Applied mathematicians still fail to extract a proper amount of juice from symmetry, because most of them have never learned anything about group representations. The physicists have, and they make good use of them.) How far would plane fluid dynamics have got without complex analysis? Or electrical engineer- ing without complex numbers? How can physicists recog- nise the essentially topological nature of solitons, in the absence of topology? How can anyone hope to come to grips with turbulence, or nonlinear phenomena in general, without topological dynamics?

There is also the other side to the coin. Newton created his mathematics in order to do physics (and was much abused for so doing: see Hall [8].) Complex analysis was as much aided by fluid dynamics as vice versa. The trade goes both ways: what I find objectionable is the belief that, at the level of concepts applicable to nature, it goes only one way, from physics into mathematics.

Dyson also gives a warning:

"In the process of theory-building, mathematical intui- tion is indispensable because the 'evasion of unnecessary thought' gives freedom to the imagination; mathematical intuition is dangerous, because many sciences demand for their understanding not the evasion of thought, but thought."

To some extent Kac and Dyson agree here: to develop a theory into a successful technique requires a detailed know- ledge of the subject. It would be arrogant to assert the con- trary. But there is a difference between Kac and Dyson. Kac seems only to be thinking about the development of something already reasonably established, whereas Dyson is interested in the infancy of a new idea as well. Kac wants mathematicians to stay away from the frontiers of natural science because their imagination is dangerous; Dyson wants them to be free to exercise their imagination, pro-

54 vided everybody else recognises the potential dangers if they get it wrong. Kac wants to play safe, stick to the tra- ditional framework, and develop; Dyson wants to take intel-

lectual risks and create. Of course, science needs both; and it probably spends

99% of the time developing. We need mathematicians who will follow Kac's advice.

But please, dear Lord, not all of them. Another point that Kac has missed is that the initial

mathematical idea that sparks off a new development is not necessarily helped by extensive knowledge of the sub- ject it is applied to. For example, a really powerful new result on solutions to nonlinear wave equations is bound to have applications to fluid mechanics and engineering; and you don't need to know anything much about those subjects to realise it.

An interesting case is when the mathematician spots a possible connection, but lacks the expertise to follow it up; and the workers in the field concerned are too unfami- liar with the mathematics to see the connection themselves. What should he do? If I interpret what Kac says in the right spirit, he shouldn't ever be in this position, but if he is, he is obliged to "play by the rules"; that is, adopt the stan- dard paradigm within the applied field. If his suggestion is genuinely new, that means he has to shut up. I don't call that a recipe for progress.

A topical case in point is the application of singularity theory to bifurcation theory. It has taken a decade of effort to convince a largely (but with honourable excep- tions; see, for example, [4]) antipathetic group of bifurca- tion-theorists that powerful methods from differential topology will prove useful in their subject (see Marsden [ 15 ]). I 'm not blaming the bifurcation-theorists: there was nothing in their training that equipped them to evaluate singularity theory in terms of its potential - only its results.

In the end the mathematicians had to learn enough bifurca- tion theory to get results, that is, they had to do the bifur- cation-theorists' job for them. Fortunately they stuck to their guns, despite resistance from bifurcation theory and outright hostility elsewhere.

Since not everyone will agree that the above is an example, however, let me take two more: to be absolutely safe, I've taken them from Kac's and Hermann's articles.

Hermann says that he has been looking at "linear sys- tems" and can see some areas for future work:

"A systematic mathematics does not exist, but I believe it could be created - based on algebraic geometry, which is usually thought of as the most abstract and "useless" part of mathematics."

If so, he'd better get on with the job: there are precious few engineers versed in algebraic geometry. (Incidentally, how does he reconcile this example with his assertion that significant applications of mathematics arise only from mathematics that is developed with applications in mind?

I don't recall Grothendieck writing much on electronic engineering.)

Kac says:

"Within the last year or s o . . . we have witnessed a new promising and exciting confluence of mathematical and physical ideas. I speak, of course, of modern differen- tial geometry on the one hand and the theory of Gauge Fields on the other. Miraculous as it may seem, fibre bundles, homotopy, and Chern classes are becoming as much parts of physical terminology as instantons, gauge fields, and Lagrangians are becoming part of the mathe- matical one."

This "miraculous" confluence did not come about by the mathematicians sitting around waiting for physicists to ask them about Chern classes. The mathematical tools were ready to hand, not because of expected applications to Gauge Fields, but because they had seemed significant objects to mathematicians.

Nor did it come about by physicists sitting and waiting to be told about a new mathematical method for studying instantons. It required both parties to make contact with the other, recognise the possibility of fruitful interaction, and work very hard to develop a common language.

None of this would have occurred unless both had already developed the basis of the common language in their own fields. It takes two to tango.

One of the uses physicists hope all this work will achieve is a reformulation of quantum mechanics a curious aim if "mathematics is not likely to be of much help in discovering laws of nature".

What Makes Mathematics Useful?

I mentioned the unity of mathematics: Steen [20] makes a similar point:

"Core mathematics is the science of significant form. It is nourished both by internal e n e r g y . . , and by new fuel supplied by the outer layers that are in closer con- tact with human problems. Layers near the core employ sophisticated techniques in the service of external objec- tives. Theories in those layers are directed more towards solving problems than discovering basic form. Layers remote from the core employ mathematics more as metaphor than as theory: applications blend with tech- niques so thoroughly that a totally different discipline emerges. Theory and problems diffuse through the ill- defined boundaries between these layers, each enriching the other and nourishing both mathematics and science."

One of the reasons that the core is powerful is that it is a machine for seizing upon significant facts and generating significant implications. It differs from the more practi- cally oriented disciplines primarily in its conceptual depth

55

and breadth, and - I think a very important feature - its ability to generalize from a single 'typical' example. If an applied mathematician, an engineer say, comes across a

matrix ( a 0 bc) and l~176 at entry a' what d~ he see?

I'd guess he probably sees an eigenvalue. I'd be surprised if he saw a character of a Borel subgroup. But for many problems it is a grasp of the structure at the sort of depth inhabited by characters of Borel subgroups that opens up the really fruitful lines of attack - e.g. the whole machinery of representation theory becomes available.

Quite a nice example - if you have the imagination to appreciate it - is given by Arnol'd [ 1 ], page 341. It con- cerns an estimate of the period of time over which a long- term dynamical prediction of the weather is possible. Re- call that numerical weather prediction is well to the "prac- tical" end of our spectrum of mathematics.

By reformulating the problem, Amol 'd turns it into one about the curvature of the (infinite-dimensional Lie) group of volume-preserving diffeomorphisms of the sphere. There is an analogy with finite dimensions: here negative curvature causes exponential instability of geodesics. The same should hold good in infinite dimensions; and geodesics on the diffeomorphism group provide one reformulation of the flow of an ideal fluid.

It would be hard to find anyone working with numerical weather prediction who had ever heard of S O Diff S 2 , let alone with any interest at all in its curvature. If you ex- plained to them how it reformulated fluid flow, they would most likely place this firmly in the "man-made" category of reformulations (in Kac's classification): sterile and use- less.

However, Arnol'd uses it to show that to predict weather two months in advance, one needs data accurate to five more figures than the desired accuracy of prediction. This is a practical impossibility. (He does have to replace the spherical Earth S 2 by a toms; this is because he knows how to computeS o Diff T z, and not because he is pig- ignorant about the shape of the Earth, so the exact figure may be wrong. But - and I say this with tongue only partly in cheek - how many numerical-weather-predictors realise that the future of their entire subject depends on the cur- vature orS o DiffS2?)

Another example, of very recent vintage, is the work of Golubitsky and Langford [7] on degenerate Hopf bifurca- tion. The original Hopf theorem provides conditions under which a dynamical system will bifurcate from a sink to a limit cycle. The essentially "dynamic" nature of this bifur- cation appears to be quite different from the "static" bifurcations of elementary catastrophe theory or singular- ity theory. Thus, to quote Smale [18]:

"Catastrophe theorists often speak as if C T . . . was the first important or sys temat ic . . , study of discontinuous phenomena via calculus mathematics. My view is quite

the contrary and in fact I feel the Hopf bifurcation (1942) for example lies deeper than CT."

Golubitsky and Langford show that not only can the Hopf theorem be reduced to singularity theory; and proved that way: it can be generalized to give a theory of unfoldings of degenerate Hopf bifurcations. The technique uses little more than the Lyapunov-Schmidt reduction process (the applied mathematicians' name for the Implicit Function Theorem) and the Z 2-equivariant version of elementary catastrophe theory. That is, the Hopf theorem is certainly no deeper than, and arguably less deep, than catastrophe theory; and it is a straightforward consequence of it.

The moral: even experts can be wrong; and a mathe- matical idea can be applied in more than one way.

It is helpful to reformulate a problem in several ways, because each reformulation makes it accessible to different parts of the mathematical machine. Certainly this industry has at various times been over-extended. Carath6odory's reformulation of thermodynamics is, as Kac says, largely useless, but not because of any kind of "god-given" or "man-made" distinction: it is useless because it does not open up a channel that permits any powerful machinery to be brought to bear. I agree that this kind of thing is pointless, but it is a matter of what new approaches a reformulation opens up, not how "natural" it is. (Thermo- dynamics cries out to be formulated sensibly and coherent- ly, at least to the extent that the literature stops arguing about difficulties due entirely to poor formulation. Differ- ential forms would be a great help: Jauch [ t 1 ] began such a reformulation. The language of differential forms is one of mathematical structures, many of which have no sensible physical interpretation.)

In an ideal world, and if everyone took their adjectives seriously, the job of applied mathematicians ought to be to act as a link between pure mathematicians (tool-build- ers) and physicists (tool-users). That is, they should find out from the physicists what the problems are, and tell the (pure) mathematicians, so that they can think about solv- ing them; or take a method the (pure) mathematicians have already developed, maybe brush it up a little, tailor it to the problem at hand, and pass it on to the physicists. I don't mean this in a menial way, but surely the job of the applied mathematician should be to apply mathematics? And the job of the pure mathematician should be to create the mathematics that is, may be, or will be required, taking proper account o f the in ternal needs o f the tool-building task as well as the problems that the tools are intended to solve.

I don't think it always works out that way, partly because the applied mathematicians are not communicat- ing properly. Hermann reports that an increasing number of engineers and scientists are bypassing mathematics alto- gether, though in a rather short-term pragmatic way. And there is a definite trend now whereby physicists and pure

56 mathematicians talk directly to each other, bypassing the "applied" fraternity. They have to: the "applied" mathe- maticians have not equipped themselves either with the mathematical techniques and concepts required, or the physical knowledge. Some of the developments in Gauge Field Theory go this way. Few of the more traditional applied mathematicians understand either Gauge Field Theory, or topology: their horizons, both mathematical and physical, are firmly bounded by the nineteenth cen- tury.

The next item is of course easier to discuss with hind- sight; but I don't think it is at all "miraculous" that fibre bundles or Chern classes are showing up in physics. I 'm astonished at the precise place they have shown up; but topology accounts for something like one third of all mathematical research, and the reason for this effort is not just academic gameplaying, but a very clear percep- tion of the importance of topology for the mainstream. Mathematicians study topology today because they can- not do their subject without it. In addition, topology had its origins in physical problems, and in some sense the sub- ject has never forgotten this, even though some workers within it may have. It has taken topology something like fifty years of hard struggle to solve its own internal devel- opmental problems to the point at which it becomes useable. It is hardly surprising if, now that this position is reached, applications begin to appear.

Hermann, I 'm sure, would argue that the occurrence of fibre bundles in physical theories is evidence of the con- tinuing genius of Elie Cartan. I 'm sure it is. But Cartan was thinking about relativity: what has that got to do with quantum field theory? It is Cartan's mathematical genius that we have to thank - and that of those who took his rather intuitive and ill-defined ideas and formulated con- cepts such as fibre bundles cleanly, to the point where it became possible to be sure that proofs involving them really were proofs. And I am very suspicious of the argu- ment: "if it was designed to do one application it's no surprise if it applies elsewhere". Which of the following is the more mystical?

(a) An idea inspired by General Relativity turns out to be useful in Quantum Field Theory, although there is no physical connection between the two fields;

(b) A powerful mathematical technique, developed to handle problems inspired by General Relativity, turns out to be useful elsewhere.

My feeling: (a) is inexplicable, (b) makes good sense. But once (b) makes good sense, what's so surprising about:

(c) A powerful mathematical technique, developed to handle important internal problems of mathematics,

turns out to be useful elsewhere?

Similarly, Gauss's ideas about surfaces inspired Rie- mann's theory of manifolds. While I can see no sensible

connection between geodesy and gravitational theory - which makes the development inexplicable on the level of "what is it good for to me, now?" - there is an immedi- ate structural and conceptual link between a 2-manifold and an n-manifold, and the fact that Gauss found uses for the former suggests that the latter, if significant, will also prove useful. And eventually it did. (And its significance was appreciated by mathematicians long before.)

I don't believe in any kind of "conservation of appli- cability". If it is unsurprising that a gadget invented to do relativity turns up in quantum field theory, it is no more surprising if one invented to do homotopy theory does.

Or even p-adic groups.

Mathematics and Science

The unity of mathematics has implications for its relation to the sciences, which I think is considerably more subtle than either Kac or Hermann would lead us to believe. Her- mann's prescription is that each individual mathematician should ask himself "What practical use is the work that I am now doing to have?", and if he can't give a specific answer, he should stop doing it and find something more immediately useful instead. Kac, on the other hand, doesn't to quite so far: he has no objection to mathema- ticians doing what they want to, as long as they don't try to import their results into areas that they have not studied in depth.

I recently spent a few days at Bristol University, talking to some physicists. Among the things I brought back were two offprints [3, 9]. One applied the Weierstrass continuous- but-nowhere-differentiable function to quantum mechanics; the other used Jacobi symbols and Gauss sums to study certain properties of diffraction gratings. Fortunately, neither Weierstrass, Jacobi, nor Gauss had followed Her- mann's advice.

The trade goes the other way too. The computation of Wigner functions in quantum optics led Berry and Balazs [2] to study the iteration of a curve under a generic area- preserving map on the disc. This is a fascinating problem mathematically: while topological dynamicists have studied iteration of points extensively, they seem to have missed this generalization; and it looks very interesting. It has potential applications elsewhere: a trivial example is the patterns you get on top of a cup of tea when you stir it with a spoon.

The point is that while a lot of this trade in ideas is fairly predictable and routine stuff, every so often it is not. Nobody is surprised if a new theorem about the wave equation has applications to waves, or if a new experimental observation about waves has implications for the wave equation. But I, for one, am surprised to find Gauss sums and Jacobi symbols telling me something about diffraction gratings; or an attempt by a physicist to calculate some-

57 thing that I have never heard of, suggesting deep and beau- tiful problems in topological dynamics.

I don ' t think that Kac's or Hermann's prescriptions are especially dangerous unless either o f them develops the political clout to try to force, e.g., the NSF to adopt their criteria when assessing a proposed research project. They are not a danger precisely because plenty of work gets done that is not restricted to narrow or even clearly defined objectives. But - and this is important - if in the past the organization of science had been different from what it actually was, and if it had been along the lines they are sug- gesting, science would be a very poor shadow indeed of its present self. And, what is in some ways worse, we would probably be congratulating ourselves on what a marvellous and satisfying structure we had evolved - because we would be quite unaware of what we were missing.

I hope this doesn't sound complacement, because it is not meant to be. It would be easy to use what I am saying as an excuse to promote all kinds of futile nonsense. But this can only happen if science, and mathematics, loses its collective sense of good judgement. The opinions of an individual, or even a pressure-group, count for little in science in the long run: the success of science as an activ- ity lies in the extent to which it has developedprotective mechanisms to circumvent some of the worse tendencies of the human psyche: emotional sloganeering, empire- building, wishful thinking, lack o f imagination, over-ima- gination, prejudice, conservatism, innovation for its own s a k e . . . At any given time, in parts of the structure, all of these are present to some extent; but they have not yet succeeded in subverting the entire enterprise.

I don' t care a fig whether a particular mathematician tries to relate his work to practical ends, or the natural world, or not. I do consider it essential that mathematics as a whole, the collective mainstream, maintains a proper two-way trade with the sciences, because the historical evidence is that this trade is responsible for the health of both.

My impression is that nowadays "pure" mathematicians are, in fact, increasingly looking towards applications. One reason for this is less than praiseworthy: it is that they thereby protect their positions at a time when society is increasingly demanding immediate practical payoff. But the other is that, over the last century, "pure" mathe- matics has evolved some very powerful techniques, with great potential for applications - and these have not achieved application because workers in applied subjects are unaware of their existence. There is a communication barrier here. I won' t call it a breakdown because that tends to suggest it is somebody's fault , and that's not the point. The point is that, say, the theoretical engineer studying the yon Kfirmfin buckling plate equation at a double eigenvalue is asking a mathematical question whose answer [17], at the present state of the art, depends on deep results about the action o f the ring of invariant germs

(for Z 2 x Z 2 acting on R 2) on the module of equivariant germs. These are not concepts encountered in the average engineering course. But they are also not pointless abstrac- tions imported into engineering by topologists seeking to avoid unemployment: they are a natural way that the mathematical problem "wants" to go, in the context of our present knowledge. It is vital that this type of communi- cation gap be bridged: to do it involves going out and talk- ing to the people on the other side, even i f they don' t want to listen. It does o f course add a certain strain if something like Newsweek tries to get in on the act and blows every- thing up out of all proportion; but that is hardly an adequate reason to disown the whole process. Or to bite the heads off those who are trying to communicate, for ever daring to set foot on somebody else's territory with- out first joining the club and agreeing to abide by all the old club rules.

Thus, while I think that individual mathematicians should be encouraged to bear connections with science in mind (and the courses we teach our students ought to make more attempt to do this) I don' t see any great need to insist on this, or to contemplate massive changes in the way that mathematics gets done. All scientific research involves a lot of groping around at the frontiers of what the human mind can understand: a lot of that work, at the end, may seem 'wasted', but it is part of the process as a whole and it is optimistic in the extreme to imagine that narrowing the criteria for pursuing a line of research will increase the overall efficiency of the process.

Nor is it always sensible to be seduced by "direct" approaches to obviously practical problems. I have no idea how much money and effort has been spent on the direct

numerical integration of PDEs for the world's weather, from observational data, in the hope of making long-term

predictions; but it looks as if the only possible benefits from such a scheme are a matter of spin-off, because the inherent instability of the phenomena concerned will prob- ably prevent any such attempt from succeeding. Indirect methods are perhaps another matter.

To sum up: the relation between science and mathe- matics functions on a collective level, not an individual one. It is often indirect and unpredictable, rendering the more obvious significance criteria (what is it good for now?) naive and simplistic. And the breadth o f modern mathe- matics is a strength, not a weakness. We should resist all attempts, however well-intentioned, to narrow it - and hence weaken it.

References

1. V. I. Arnol'd: Mathematical methods o f classical mechanics, Springer, New York, 1978

2. M. V. Berry, N. L. Balazs: Evolution of semiclassical quantum states in phase space, J. Phys. A: Math. Gen. 12 (1979) 625 -642

58 3. M. V. Berry, Z. V. Lewis: On the Weierstrass-Mandelbrot frac-

tal function,Proe. R. Soc. Lond. A 370 (1980) 459 484 4. S.-N. Chow, J. K. Hale, J. Mallet-Paret: Applications of

generic bifurcation I, l l ,Arch. Rat. Mech. Anat. 59 (1975) 159-188; 62 (1976) 209-236

5. W.K. Clifford: On the space theory of matter, lecture to the Cambridge Philosophical Society 1876, reprinted in The Worm o f Mathematics (ed. J. R. Newman), Vol. 1, Simon and Schu- ster, New York, 1956, pp. 546-547

6. F . J . Dyson: Mathematics in the physical sciences, in The Mathematical Sciences, M[T Press, Cambridge, MA, 1969, pp. 97-115

7. M. Golubitsky, W. F. Langford: Classification and unfoldings of degenerate Hopf bifurcations, preprint, Univ. of Warwick, 1980

8. A. R. Hall: Philosophers at War: the quarrel between Newton and Leibniz, Cambridge Univ. Press, 1980

9_ J. H. Hannay, M. V. Berry: Quantization of linear maps on a torus - Fresnel diffraction by a periodic grating, preprint, H. H. Wills Physics Laboratory, Univ. of Bristol, 1980

10. R. Hermann: A view of applied mathematics, Math. lntelli- gencer 1 (1978) 145 147

11. J.-M. Jauch: Foundations o f modern quantum mechanics, Addison Wesley, Reading MA, 1968

12. M. Kac: Math. InteUigencer 1 (1978) 97-98 13. E.N. Lorenz: Deterministic nonperiodic flow, J. A toms. Sci.

20 (1963) 130 141

14. G.W. Mackey: Harmonic analysis as exploitation of symmetry - A historical survey, BulL Amer. Math. Soc. 3 (1980) 543-698

15. J. E. Marsden: Review of Treatise on Analysis by Jean Dieu- donn6, Bull. Amer. Math. Soc. 3 (1980) 719-724

16. T. Poston: Math. lntelHgencer 1 (1978) 249 17. D. Schaeffer, M. Golubitsky: Boundary conditions and mode-

jumping in the buckling of a rectangular plate, Commun. Math. Phys. 69 (1979) 209 236

18. S. Sma!e: Review of Catastrophe theory: selected papers 1972-1977, by E. C. Zeeman, Bull. Amer. Math. Soc. 84 (1978) 1360 1368

19. D. Ruelle, F. Takens: On the nature of turbulence, Commun. Math. Phys. 20 (1971) 167-192

20. L.A. Steen: Mathematics Today, inMathematics Today - 12 informal essays, Springer, New York, 1978, pp. 1 -12

21. H. L Swinney, P. R. Fenstermacher, J. P. Gollub: Transition to turbulence in a fluid flow, in Synergetics (ed. H. Haken) Springer, Berlin Heidelberg New York, 1977, pp. 60-68

lan Stewart Mathematics Institute University o f Warwick Coventry, CV4 7AL England

Mathematical Logic

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