Philosophy of mathematics

Karl Gauss, in his time, called mathematics the queen of all sciences, giving her a special place in the field of human knowledge. Indeed, quite unlike other sciences, it rather serves them as a language or method of study. Being, perhaps, the most strict of all sciences, it does not have its own strict and generally accepted definition. Throughout its history, transforming itself, the concept of mathematics was also transformed. Scientists, during the whole development of mathematics, were able to compile rather than definitions of mathematics, but a set of aphorisms characterizing it or ideas about it.

“Mathematics is the language in which the book of nature is written” (G. Galileo)
"Mathematics is the science of the necessary conclusions" (B. Pierce)
“Mathematics is a rigorous language, which serves to move from one experimental judgment to another” (N. Bor)
“Mathematics is a hierarchy of formal structures” (N. Bourbaki)
“Mathematics is the science of quantitative relations and spatial forms of the real world” (A. Kolmogorov)

- This is only a small part of the judgments, showing the heterogeneity of ideas about mathematics. In addition to the question of the definition of mathematics, interesting and debatable questions are about its nature (grounds), its methodology, goals, and connection with the real world. The answers to them are also ambiguous and have changed significantly over time, creating various philosophical trends.

The first stage in the development of mathematics as a separate science was the idea of ​​proof, deductive inference, the founders of which were ancient Greek mathematics. The emergence of mathematics as a systematic science had a strong influence on the philosophical thinking of that time, which was reflected in the mystification of mathematics in the teachings of Pythagoras. Pythagoreanism can be considered the first philosophical movement on the basis of mathematics, fully expressed in the thesis of Pythagoras "everything is a number." The Pythagoreans considered mathematics the beginning of all beginnings, the basis of all things. They considered mathematical truths to be inborn, received by the soul in a more perfect world - the world of ideas.

The first crisis of mathematics (incommensurability of segments) struck the philosophy of Pythagoreanism, destroying the harmony of mathematics. A broad and, in a certain sense, complete criticism of Pythagoreanism was given by Aristotle. Mathematics, according to Aristotle, is not knowledge about ideal entities that exist independently of things, but knowledge abstract from things. However, Pythagoreanism influenced for a long time (in some way, still influences) the philosophical understanding of mathematics. The main contribution of the ancient Greek mathematicians was the introduction of rigor in mathematics, especially expressed in the "Principles" of Euclid.

The next significant era in the development of mathematics was the period of "rebirth". With the new needs of science, first of all mechanics, new ideas have emerged that now belong to differential and integral calculus. Mathematics was considered as knowledge of the secondary, experienced, depending on certain external realities. This epoch was accompanied by the second crisis of mathematics, namely the absence of the "rigor of the ancients" in the substantiation of differential calculus. In practice, it gave results, but the use of actual infinitesimal in evidence was too heuristic. In particular, Leibniz, in order to substantiate differential calculus, introduced the controversial concept of “non-Archimedean magnitude”. In the absence of a rigorous justification, various metaphysical and natural-philosophical explanations of the differential began to form.
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The next stage was induced by non-Euclidean geometries (the third crisis of mathematics). Inconsistent with the real world, they were a blow to the classic empiricism of the past era. Non-Euclidean geometries became a subject of heated discussions and were not accepted by many mathematicians for a long time, however, they served as a bifurcation point in the development of mathematics, creating an absolutely new look at it. Now the most important feature of the mathematical theory is consistency, rather than correlation with experience. Although at first there were attempts at a metaphysical explanation of non-Euclidean geometries, later, in many respects by the forces of Poincare, Dedekind, Cantor, Hubert, the equality of mathematical objects related and not related to experience and intuition was recognized. This vision of mathematics is reflected in all of its subsequent philosophy.

Various philosophical and mathematical trends differ mainly in the methods of substantiating mathematics. One of such trends is logicism, which appeared in the spirit of the development of formal mathematical logic. His main task was an attempt to reduce the basis of mathematics - arithmetic to logical tautologies. Her apologist G. Frege did not doubt that logic provides a sufficient basis for finding out the true meaning of all mathematical concepts. However, it turned out that logical rationales, even if they do not lead to paradoxes, still need to attract additional assumptions that are outside the laws of logic. In the idea of ​​the rationale of mathematics lay, first of all, the ideas about the features of logic (formal logic), its primacy, however, this statement is rather doubtful. Poincaré described logicism as "a hopeless attempt to reduce the infinite to the finite."

Another trend was intuitionism. His main point was the belief that some objects of mathematics are certainly clear, and operating with them cannot lead to a contradiction. Appearing to a large extent as a counterbalance to logicism, it was essentially a modification of empiricism. Rejecting many of the principles obtained earlier, he significantly impoverished mathematics, which was one of the reasons for abandoning it.

On the basis of a critical review of all the mathematics justification programs received by that time, Hilbert proposed his own way, which became known as formalism. The main philosophical premise of this trend was that the substantiation of mathematics is only the justification of its consistency. The substantiation procedure proposed by Hilbert consisted, firstly, in formalizing the theory in the symbolic form of the axiom scheme and inference rules, and secondly, in proving its consistency based only on its formal structure. However, this trend was untenable. Two theorems of the mathematical logic of Kurt Godel made a revolution in the substantiation of mathematics. In particular, the second theorem states that the proof of the consistency of any sufficiently rich formal theory is impossible by the means of this theory itself, which makes it impossible for the procedure of substantiation of Hilbert. Thus, any formal theory can only be substantiated by another theory, which leads to the obligatory existence of an unfounded theory or a closed circle of theories substantiating each other.

So, the problem of the substantiation of mathematics, the search for its nature remains open. In my subjective opinion, the answer may be as follows: a mathematical theory remains correct, as long as it seems consistent for people, correct in accordance with our thinking, our logic (something like the anthropic principle, similar to the basis of Karl Menger's theory of value). Thus, the question remains: what is human thinking and logic, what is their nature. Philosophy has also long sought an answer to this question. There were empirical ideas, according to which our thinking is formed through experience, and praxiological ones close to them, which consider thinking as a neural network that learns from its actions, and, let's call them so, sacramental, for example, representing the world of ideas as a separate space. The main feature of the above approach to the justification of mathematics is that by accepting such a principle, we can abstract ourselves from the question of justification, and solve only the problem of the nature of human thought and logic. (although we may never find the answer, because despite the property of reflection that distinguishes us, it is likely that it is not possible to know ourselves)

Literature:

E.A. Belyaev, V.Ya. Perminov "Philosophical and methodological problems of mathematics"