History of Mathematics met two great revolutions. Both of them completely changed its exterior aspect as wall as its social function. Initial momentum to these revolutions was provided by the impossibility to keep living according to the old canon. That is, to be unable to express and to face in exact form the fundamental problems of the natural sciences using the existing mathematical language. The first is linked with the name of Descartes and the second to those of Newton and Leibniz. Yet both of them are much more than such great names. According to Gibbs, Mathematics is a language, and the essence of these two revolutionary breakthroughs was the creation of new far-reaching mathematical languages.
Indeed Greek mathematics spoke in the language of Aristotelian logic which formalized with due rigor the everyday language. The result of the first revolution was that Mathematics started to speak in the language of Commutative Algebra, while the second taught to speak in the language of Differential Calculus.
Newton’s classical treatise “Principia Matematica Naturalis Philosophiae” is in essence the first textbook about that part of the grammar of the language of the nature of “Differential Calculus” which he was able to decipher. Obviously this first book allowed to understand only the most simple phrases of the Nature. Subsequent generations of mathematicians and physicists, improving their ability to speak these languages, managed to understand even more complicated sentences, and in parallel new more elaborated and completed versions of the Newtonian language were written.
From its very birth, Differential Calculus is the mother language of Classical Physics. Thanks to Differential Calculus, Maxwell was able to discover by his mere pencil the foundations of modern civilization: electromagnetic waves. And Einstein was able to describe the geometrical shape of our Universe. On the other hand, a century-long of experience has made up our minds that it is not possible to describe with this language the phenomena of Quantum Physics. Past century’s physicists looked for tools to describe the quantistic world within mathematics, and in some cases they were forced to invent their own mathematics. Remarkable examples of such inventions are the Dirac’s “delta function” and the Feynmann’s “path integral”. From all that was found by physicists within contemporary mathematics or made up by their own, a strange slang aroused, in which elements of Differential Calculus are mingled with Hilbert spaces, measure theory, and operator-valued generalizations of functions, and so on. This is nothing but a slang, not a consistent natural language in which it would be possible to formulate adequately the essence of quantum phenomena. Inability of modern mathematics to face this challenge is brightly shown by the fact that the existing theory of quantum fields and its generalizations that claim to describe the most fundamental principles of Unvierse, are mathematically based on the “theory of perturbations”. A basis far from solid.
Therefore this manifest mathematical inconsistency of present quantum theories unequivocally points out that the mathematical principles of the quantistic component of the natural philosophy are yet to be settled. Thus the quest for the natural mathematical language for quantum theories is one of the deepest problems of modern mathematics. Such a challenge is absolutely independent of the current problems of theoretical physics and continuous seasonal changes of leading trends. Yet such an exposition of the problem would be just a pointless philosophical exercise if it was not reformulated in a concrete way and turned into a well-posed scientific problem. The state of mathematics 30-40 years ago would not allow that. But the recent advances in understanding the nature of non linear PDEs open new possibilities to us.