Monday, June 18, 2007

Erudition, Eloquence, and Elegance in Mathematics

Frank Luger headshot by Frank Luger

In any field of human intellectual endeavor, the 'sine qua non' of 'eternal' excellence are the three classical hallmarks known as erudition, eloquence, and elegance. All these three, in turn, entail various qualities, as it will be mentioned below. They are the indispensable legs of the tripod on which true quality of the timeless transcendental kind rests, which may be best expressed by the single word: excellence.

Mathematics is a rather unique intellectual endeavor. Unquestionably one of the greatest triumphs of the human intellect -- and by the same token, a truly great tribute to it -- mathematics enables one to go 'where no one has gone before' or to borrow a literary phrase, 'where even angels fear to tread'. Now, in this essay, I don't wish to get drawn into such disputes as Platonic vs. Aristotelian mathematics, or whether mathematics is discovered or created or both; and I have no intention either to discourse on merits and shortcomings, or to engage in any kind of sermon for or against mathematics. Quite simply, the purpose of this paper is to draw attention to just what constitutes excellence in mathematics, regardless of the idiosyncrasies of any particular mathematician, whether still living or already standing in the Pantheon, frozen in lofty marble among eternal geniuses. In other words, don't expect here a cookbook recipe for winning the Fields Medal1, quite regardless of how smart and (mathematically) knowledgeable you might be. You need much originality in this field, despite a huge amount of indispensable basic knowledge and rapid developments in every branch; and it is fair to say that no matter how impersonal mathematics appears, your own cognitive style and pattern-recognition gifts leave plenty of room for individuality in this special, infinite playground of the intellect.

However, none of the above justify the first sentence of the previous paragraph. Mathematics is unique because the man-made and the nature-made get intimately intertwined in it; and both components are present in every branch of mathematics, no matter where you look. But the proportions may be very different. In probability and statistics, number theory, and the like, the man-made aspect predominates so much that the nature made component can only be discerned with specific effort; whereas in most areas of mathematical physics, the nature-made aspect is not only conspicuous, but must even be given priority2, at least according to the vast majority of (theoretical) physicists.

Also, there have been shifting emphases on these components throughout history; but it is only in relatively recent times, that mathematics has gradually become independent of natural philosophy at first and from physical significance at last. This emancipation has taken about a century, roughly between the non-Euclidean geometries of Gauss, Bolyai, and Lobachevsky in the early XIXth century until the Quantum Mechanics of Planck, Bohr, Schrödinger, Heisenberg, Born, Dirac, et al. around the 1930s. Dirac, in particular, in a famous quote has gone as far as asserting that if there is a discrepancy between experiment and mathematics, one ought to jettison the experiment and retain the mathematics. Today, maybe 10% or so of advanced mathematics has physical meaningfulness; and mathematical research merrily proliferates following its own recipes, in disdainful disregard of physics or even philosophy except perhaps for the part of philosophy which belongs to logic in general and mathematical logic in particular. In other words, the man-made component has become overwhelmingly, maybe to 90%, predominant over the nature-made component; and this is precisely why mathematics is so unique, considering that no matter where we look, it works.

A word of qualification is in order. It works, and ubiquitously at that; but this is still in our world of human sense-perception. How it may or may not work independently of the 'bubble' of virtual reality within which its human creators perforce live, remains an open question. If there is such thing as ultimate reality, it may or may not be adequately handled by our mathematical sophistication. One might conjecture, that human limitations can but result in projections of those limitations into other worlds, assuming that such worlds exist, quite independently of us. Also, with regards to extraterrestrial life, no matter how probable it is that such life is intelligent, there's no reason whatsoever, to suppose that those life-forms evolved intelligence along human lines. To be sure, there are certain mathematical things that we have reason to believe, are universal; for example the prime numbers, as it was eloquently emphasized in the late Cornell astrophysicist Carl Sagan's masterpiece: "Contact". But unless and until some way is found for interstellar communication and travel, we have no means to confirm or disconfirm such conjectures. Perhaps some future discovery will rob the 'Queen of Sciences' of her crown, but until then, mathematics reigns supreme, and we ought to bow to her.

Erudition is the first and perhaps most obviously important requirement for mathematical excellence. Little comment is needed. One must be intimately familiar with advanced mathematics in order to attain mastery. In fact, no attainment of mastery is possible before every detail has become so intuitively evident, that one can run circles around it and devise alternatives. It's just that the rapid growth of every branch of mathematics makes it increasingly difficult to master but ever narrower segments of specialities. A curious situation arises whereby the more one knows about specifics, the less one is able to keep sight of the whole, all the way until the absurd predicament of knowing everything about almost nothing. Already a hundred years ago this was well stated in one of Poincaré's theorems, according to which the more one approximates a mathematical truth, the more elusive it becomes. This recent proliferative trend has produced more and more 'specialist barbarians', which, of course, is at the expense of erudition, because erudition requires much general knowledge in addition to specialized competence in whatever narrow aspect of mathematics. This applies to both pure and applied fields of mathematics, although perhaps attains greater importance in the pure fields, because it is here that the specialization tendencies are the most pronounced. It is fair to say that today's mathematicians are less erudite in a general sense than ever, no matter how competent they may be in some highly specialized area. The only solution to this predicament is synthesis, whereby many seemingly disparate aspects are brought to common denominators; and the resultant simplification gives rise to generalization, which then makes room for new growth cycles. Generalization is one of the most important aspects of the growth of mathematics, being the key to usefulness. Interestingly, the greater the generality, the greater the simplicity. This is one of the main reasons why advanced mathematics is easier than the less advanced parts. Simplicity also greatly facilitates the other two requirements of mathematical excellence: eloquence and elegance, by getting rid of unwanted or unnecessary information and drawing attention to the important facts. Simplification by generalization was most eloquently illustrated by David Hilbert, the second 'Prince of Mathematicians' (the first was Gauss) in 1890, when he proved Gordan's 1868 theorem by throwing away 90% of Gordan's premisses and putting the rest into the form which is known as Hilbert's Finite Basis Theorem. This far-reaching and profound theorem shows eloquently and elegantly that greater generality and greater simplicity are practically inseparable.

Eloquence, as the second requirement for mathematical excellence, might strike a strange note. That's because eloquence is traditionally associated with rhetorics and the fluent, polished, and effective use of language, especially in public speaking. Yet the same argument or proof may be presented clumsily or eloquently. An eloquent proof immediately appears as smooth, almost natural flow of ideas, without even a trace of unnecessary or cluttering information. Obviously, a prerequisite of eloquence is thorough mastery of the field in general and the problem in question in particular. Yet, technical mastery is not enough. One needs a certain creativity and imaginative playfulness, as well as originality and style. This brings us to the third leg of the tripod: elegance.

Elegance, as the third requirement for mathematical excellence, is the truly artistic aspect. Its hallmarks are grace and refinement, ingenuity and simplicity, extraordinary effectiveness and efficiency. To explore, to discover patterns, to explain the significance of each pattern, to invent new patterns similar to those already known, are among the normal activities of what mathematicians do. How they do what they do depends on the quality of the mathematician. An excellent mathematician has an almost inimitable style, but despite much idiosyncrasy, the style will invariably be elegant.

The history of mathematics is marked by alternating contractions and expansions, analyses and syntheses, unifications and generalizations. If all of mathematical knowledge could be expressed in two principles, the excellent mathematician would not rest until s/he could demonstrate that the two are rooted in a single one. But that would give rise to new problems, and new cycles of expansions-contractions-expansions. Such pulsation has been characteristic of the growth of mathematics throughout history in erudite, eloquent, and elegant ways. As mathematics is both science and art3, it may perhaps be fairly said, that erudition stands for science, elegance for art, and eloquence bridges the two of them.

1 Equivalent of the Nobel Prize in mathematics, except that the Fields Medal is conferred upon its recipient only once every four years, in contrast to the Nobel laurels which are and have been awarded every year.

2 There have been many disputes around this point. Some famous people, such as Bohr, Dirac, etc. argued in favor of throwing out those parts of microphysics which 'deviated' significantly from mathematics; whereas Einstein et al. insisted on the priority of physics and the experimental validation of mathematical theories. Pure mathematicians in the vein of Gauss, Hilbert, Poincaré, Hardy, and many others, simply could not care less either way; for them the intrinsic esthetics and consistency of pure mathematics was far more important and normative than whether physicists happened to find any pragmatic use for beautiful mathematical theorems.

3 cf. Luger, F. Necessitas Mathematicae, in Commensal, No. 100, March 2000, pp. 20-24; also in Telicom, Vol. XV, No.1, Oct./Nov. 2000, pp. 66-71; Gift of Fire, Issue 122, Jan. /Feb. 2001, pp. 36-41; PhiSIGma, No. 23, Sept./Oct. 2001, pp. 20-25.

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