Condescension is the mother of mediocrity. A fresh product of a mediocrity is called a banality. Time makes banal the most splendid achievements, seminal theories, and challenging problems. However, everyone must try and abstain from producing brand-new banalities.
What is primary in science: abstract theories or particular problems? The style of this question reminds the disputes about priority between the egg and the chicken. Free choice is the main cornerstone of science. Therefore, it seems reasonable to understand what problems and theories we should choose to avoid banality.
D. Mumford, one of the most beautiful mathematical minds of today, remarked once that he honestly carried out some ghastly but wholly straightforward calculations while checking something. “It took me several hours to do every bit and as I was no wiser at the end... I shall omit details here.” Narrating this episode, another outstanding mathematician, Yu. I. Manin, concluded: “The moral: a good proof is one which makes us wiser.”
The following slight abstraction of this thesis transpires: IN SCIENCE WE APPRAISE AND APPRECIATE THAT WHICH MAKES US WISER. The notions of a good theory open up new possibilities of solving particular problems. Rewarding is the problem whose solution paves way to new fruitful concepts and methods.
Indispensability is the most important quality of a good problem or theory. The poetic image of “the intractable problems that boggle our mind” by S. Nadson, a famous Russian poet, brings about the flavor of the essence and traits of the meaning of the indispensable.
The greatest minds create indispensable scientific concepts and ponder them over. They pose indispensable scientific problems and contemplate over their solutions. The indispensable theories and problems propel science. The best scientists propounded not only indispensable theories and addressed not only indispensable problems. But only indispensable theorems and problems make these scientists great.
A good theory enables us to settle some indispensable problems. We know many classical examples of fruitful and powerful theories. Euclidean geometry and differential calculus were gigantic breakthroughs in the understanding and mastering the reality. Centuries witness the strength and power of these theories yielding everyday's solutions of uncountably many practical problems.
Solution of an indispensable problem is a grind stone for a good theory. The example is magnificent of D. Knuth, the author of the famous three volumes of The Art of Computer Programming, who demonstrated the vitality of his theoretical views with the stunning programs of TeX and Metafont. Good problems help us to develop good theories. As a rule, solution of an indispensable problem requires a new conceptual technique and revision of the available theoretical gadgets. Squaring the circle, the variational principles of mechanics, and the majority of the Hilbert problems provide examples of the questions that brought about sweeping changes in the theoretical outlooks of science.
We must not narrow and simplify the concept of a problem. Science endeavors to make the complex the simple. Therefore, always actual are the reconsideration and inventory of the available theories as well as their simplification, generalization, and unification. The history of science knows many examples of the perfection, beauty, and practical power of the theories that arose by way of abstraction and codification of the preceding views. The success of a new theory proves that this theory was indispensable. For instance, the last, twenty-third, “nonconcrete” problem in the Hilbert list proves to be an indispensable forerunner of the renewal of variational calculus which is implemented in the modern theories of optimization, control, programming, and operations research.
Freedom in science is the consciousness and appreciation of the indispensable, a vaccine against banality.
October 25, 2005