Philosophy of Mathematics

imageKarl Gauss, at one time, called mathematics the queen of all sciences, giving her a special place in the field of human knowledge. Indeed, completely unlike other sciences, it most likely serves them as a language or a method of study. Being perhaps the most stringent of all sciences, it does not have its own strict and generally accepted definition. Throughout its history, transforming itself, the concept of mathematics was transformed. Scientists, throughout the development of mathematics, were able to make up 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. Galilei)
“Mathematics is the science of the necessary conclusions” (B. Pierce)
“Mathematics is a strict language that serves to move from one experienced 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 controversial are the questions about its nature (foundations), its methodology, goals and connection with the real world. The answers to them are also ambiguous and significantly changed over time, creating various philosophical trends.

The first stage in the development of mathematics as a separate science was the idea of ​​proof, a deductive inference, the founders of which were ancient Greek mathematicians. The emergence of mathematics as a systematic science strongly influenced 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 course on the foundation of mathematics, which is fully expressed in the thesis of Pythagoras "everything is number." The Pythagoreans considered mathematics to be the beginning of all principles, the foundation of all things. They considered mathematical truths innate, received by the soul in a more perfect world - the world of ideas.

The first crisis of mathematics (the incommensurability of segments) dealt a blow to the philosophy of Pythagoreanism, destroying the harmony of mathematics. Broad and, in a sense, a 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 abstracted from things. However, Pythagoreanism for a long time influenced (in some way, still affects) the philosophical understanding of mathematics. The main contribution of ancient Greek mathematicians was the introduction of rigor in mathematics, especially expressed in the "Principles" of Euclid.

The next significant epoch in the development of mathematics was the period of “revival”. With the new needs of science, primarily mechanics, new ideas have appeared that now relate to differential and integral calculus. Mathematics began to be regarded as secondary, experimental knowledge, depending on some external realities. This era was accompanied by a second crisis of mathematics, namely the absence of the “strictness of the ancients” in the justification of differential calculus. In practice, it yielded results, but the use of relevant infinitesimals in evidence was too heuristic. In particular, Leibniz, to justify differential calculus, introduced the contradictory concept of "non-Archimedean value." In the absence of a rigorous justification, various metaphysical and natural philosophical explanations of the differential began to form.

The next stage was induced by non-Euclidean geometry (the third crisis of mathematics). Incomparable with the real world, they became a blow to the classical empiricism of the past era. Non-Euclidean geometries became the subject of heated debate and were not accepted by many mathematicians for a long time, however, they served as the bifurcation point in the development of mathematics, creating a completely new look at it. Now, the most important feature of mathematical theory has become consistency, not 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, Hilbert, the equality of mathematical objects related and not related to experience and intuition was recognized. Such a vision of mathematics was reflected in all of its subsequent philosophy.

Various philosophical and mathematical trends differ mainly in the methods of substantiating mathematics. One of these currents is logicism, which appeared in the spirit of the development of formal mathematical logic. His main task was to try to reduce the basis of mathematics - arithmetic to logical tautologies. Her apologist G. Frege did not doubt that logic provides a sufficient basis for elucidating 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 involve additional assumptions that are outside the laws of logic. The idea of ​​the logical foundation of mathematics lay, first of all, the idea of ​​the features of logic (formal logic), its primacy, however, this statement is rather doubtful.

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. Having appeared to a large extent as a counterbalance to logicism, it was essentially just a modification of empiricism. Refusing many of the principles obtained earlier, he significantly impoverished mathematics, which served as one of the reasons for rejecting it.

Based on a critical review of all mathematics justification programs that had been received by then, Hilbert proposed his own path, which became known as formalism. The main philosophical premise of this trend was that the justification of mathematics is only the justification of its consistency. The substantiation procedure proposed by Hilbert consisted, firstly, of formalizing the theory in a symbolic form of the scheme of axioms and inference rules, and secondly, of proving its consistency based only on its formal structure. However, this trend turned out to be untenable. Two theorems of mathematical logic by Kurt Gödel revolutionized the foundation 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 to justify Hilbert. Thus, any formal theory can be justified only by another theory, which leads to the obligatory existence of an unjustified theory or a vicious circle of theories substantiating each other.

So, the problem of substantiating mathematics, searching for its nature remains open. In my subjective opinion, the answer may be this: a mathematical theory remains true as long as it seems consistent for people, true in accordance with our thinking, our logic (something like an anthropic principle, similar to the basis of Karl Menger's value theory). Thus, the question remains what is human thinking and logic, what is their nature. Philosophy has also been looking for an answer to this question for a long time. There were empirical ideas, according to which our thinking is formed through experience, and praxiological ones that are close to them, considering thinking as a certain neural network, learning from their 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 substantiation of mathematics is that by adopting such a principle, we can abstract from the issue of justification, and solve only the problem of the nature of human thought and logic. (although we may probably never find the answer, because despite the distinguishing property of reflection, it is likely that it is not possible to know ourselves)

References:

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

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