Some French mathematicians used to manage to write in a more fluent style without stating too many definite theorems. This was more agreeable than the present style of the research papers or books which have so much symbolism and formulae on every page. I am turned off when I see only formulas and symbols, and little text. It is too laborious for me to look at such pages not knowing what to concentrate on. I wonder how many other mathematicians really read them in detail and enjoy them.
There do exist, though, important, laborious and inelegant theorems. For example some of the work connected with partial differential equations tends to be less "beautiful" in form and style, but it may have "depth," and may be pregnant with consequences for interpretations in physics.
How does one arrive at a value judgment nowadays?
Mathematicians, whose job in a sense is to analyze the motivation and origin of their work, fool themselves and may be remiss when they think their main business is to prove theorems without at least indicating why they may be important. If left entirely to aesthetic criteria, doesn't it compound the mystery?
I believe that in the decades to come there will be more understanding, even on a formal level, of the degree of beauty, though by that time the criteria may have shifted and there will be a super beauty in unanalyzable higher levels. So far when anyone has tried to analyze the aesthetic criteria of mathematics too precisely, whatever was proposed has seemed too narrow. It has to appeal to connections with other theories of the external world or to the history of the development of the human brain, or else it is purely aesthetic and very subjective in the sense that music is. I believe that even the quality of music will be analyzable — to an extent only, of course — at least by formal criteria, by mathematizing the idea of analogy.
Some of the old problems, unsolved for many years, are being settled. Some are solved with a bang, and others with a whimper, so to speak. This applies to problems seemingly equally important and a priori interesting, but some, even famous classical ones, are solved in such a specific way that there is nothing more to be asked or said. Some others, less famous, immediately upon solution become sources of curiosity and activity. They seem to open new vistas.
As for publications, mathematicians nowadays are almost forced to conceal the way they obtain their results. Evariste Galois, the young French genius who died at the age of twenty-one, in his last letter written before his fatal duel, stressed how the real process of discovery is different from what finally appears in print as the process of proof. It is important to repeat this again and again.
On the whole and in the large lines there does seem to exist a consensus among working mathematicians about the value of individual achievements and the value of new theories. There must therefore be something objective if not yet defined about the feeling of beauty which mathematics offers, dependent sometimes also on how useful it turns out to be in other branches of itself or of other sciences. Why mathematics is really so useful in the description of the physical world, for me at least remains philosophically a mystery. Eugene Wigner once wrote a fascinating article on this "implausible" usefulness of mathematics and titled it "The Unreasonable Effectiveness of Mathematics."
It is, of course, one very concise way of formalizing all rational thought.
It also has manifestly, in elementary, secondary and advanced schools the value of training our brain, since practice, just as in any other game, sharpens the organ. I cannot say whether a mathematician's brain is today sharper than it was in the time of the Greeks; nevertheless on the longer scale of evolution it must be so. I do believe mathematics may have a great genetic role, it may be one of the few means of perfecting the human brain. If true, nothing could then be more important for humanity, whether to arrive at some new destiny as a group or for individuals. Mathematics may be a way of developing physically, that is anatomically, new connections in the brain. It has a sharpening value even though the enormous proliferation of material shows a tendency to beat things to death.
Yet every formalism, every algorithm, has a certain magic in it. The Jewish Talmud, or even the Kabbalah, contains material which does not appear particularly enlightening intellectually, being a vast collection of grammatical or culinary recipes, some perhaps poetic, others mystical, all rather arbitrary. Over centuries thousands of minds have pored over, memorized, dissected, and classified these works. In so doing people may have sharpened their memories and deductive practice. As one sharpens a knife on a whetstone, the brain can be sharpened on dull objects of thought. Every form of assiduous thinking has its value.
There exist in mathematics propositions, such as the one called "Fermat's Great Theorem," which, standing by themselves, seem special and unrelated to the main body of number theory. They are very simple to state but have defied all the efforts of the greatest minds to prove them. Such statements have stimulated young minds (my own included) to more general wonder and curiosity. In the case of Fermat's problem, special or irrelevant as it is by itself, it has stimulated through the last three centuries of mathematics, the creation of new living objects of mathematical thought, in particular the so-called theory of ideals in algebraic structures. The history of mathematics knows a number of such creations.
The invention of imaginary and complex numbers (which are pairs of real numbers with a special rule for their addition and multiplication) beyond the immediate purpose and the use to which they were put, opened new possibilities and led to the discovery of miraculous properties of the complex variables. These analytic functions (the examples of which are, to mention the simplest, z = √w, z = ew, z = log w), possess unexpected, simple and a priori unforeseen properties deriving from the few general rules which govern them. They have convenient algorithms and rather deep connections with the properties of geometrical objects and also with the mysteries concerning the seemingly so familiar natural numbers, the ordinary integers. It is as if some invisible different universe governing our thought became dimly perceptible through it, a universe with some laws, and yes, facts, of which we become only vaguely aware.
The fact that some seemingly very special functions, like the Riemann Zeta function, have such deep connections with the behavior of integers, of prime numbers, is hard to explain a priori and in depth. This is really not well understood to this day. Somehow these entities, these special analytic functions defined by infinite series, have been generalized more recently to spaces other than the plane of all complex numbers, such as to algebraic surfaces. These entities show connections between seemingly diverse notions. They also seem to show the existence (to make a metaphor stimulated by the subject itself) of another surface of reality, another Riemann surface of thought (and connections of thought) of which we are not consciously aware.
Some of the properties of the analytic functions of the complex numbers turn out to be not merely convenient, but very fundamentally tied to physical properties of matter, in the theory of hydrodynamics, in the description of the motions of incompressible fluids such as water, in electrodynamics, and in the foundations of quantum theory itself.
The creation of a general idea of a space, abstracted to be sure, but not really completely dictated or uniquely indicated by the physical space of our senses, the generalization to the n-dimensional space where n is greater than three and even to infinitely many dimensions, and so useful at least as a language for the foundations of physics itself, are these marvels of the power of the human brain? Or is it the nature of the physical reality which reveals it to us? The very invention, or is it "discovery," that there are different degrees or different kinds of infinity has had not only a philosophical, but beyond that, a striking psychological influence on receptive minds.