The philosophy of mathematics is the research field of philosophy, in which the foundations of mathematical knowledge, the place of mathematics in the knowledge system, the ontological status of mathematical objects, methods of mathematics are revealed. This philosophy of mathematics is an essential part of almost all philosophical systems. Almost every philosopher tried to express his attitude to mathematics and determine the place of this area of knowledge. When solving actual mathematical problems, the philosophical justification of mathematical knowledge is necessary for clarifying the conditions for the reliability of the latter; this served as an occasion to appeal to the philosophy of many leading mathematicians. Also, the philosophy of mathematics involves the use of mathematics to clarify or justify a philosophical position. Many philosophical studies contain mathematical examples, serving either as an illustration of philosophical reasoning, or even as a methodological basis for philosophical research. The most vivid example of the use of mathematics in philosophy is given by Nikolai Kuzansky, for whom geometric images are perhaps an adequate means of conducting philosophical reasoning.
Plato first substantiated the importance of mathematics for philosophy. He considered numbers and geometric figures as Eidos and paradigms, i.e., principles and the beginnings of things, through which the latter acquire semantic certainty and become involved in being. Mathematics studying eidos is important for Plato primarily because it reorients the mind from consideration of the transient and becoming being to a truly existent, stable and self-determined. It, therefore, turns out to be the preparatory stage for dialectics; direct knowledge of the idea of the Good – the higher reality, the participation of which gives the existence of mathematical objects.
Aristotle occupies a different position concerning mathematics, according to which numbers and geometric figures (more precisely: lines, surfaces and bodies) are only the results of abstraction from sensible things, their specific properties. Mathematics, like any other science, studies the essence (see Essence and phenomenon), but not comprehensively, but only highlighting the quantitative aspect of interest to it. Numbers and quantities are that aspects of the existence of a thing that mathematics draws attention to.
In the philosophy of modern times, we can distinguish two – in many ways contrary – approach to mathematics, which developed within the framework of rationalism and empiricism. In rationalism, mathematics was regarded as the most reliable foundation of all knowledge, whereas empiricism tried to derive it from experience. Characteristic in this respect are, for example, the views of Descartes, on the one hand, and Berkeley on the other. Descartes proceeds from the premise that all knowledge should be based on the foundation of a clear and immediate intellectual contemplation, intuition, which gives the possibility of direct discretion of the truth. Such direct discretion is possible, however, only when we are dealing with the most simple and at the same time fundamental concepts – those that are inaccessible to any analysis and representation through the other. As such a fundamental and directly clear concept Descartes indicates the extent. This immediately makes science, which studies extended geometries, the basis for all other sciences. It is the reduction to the extent that the truth of all scientific concepts must be justified. Geometric intuition (contemplation of extended values) serves as the basis for mathematics itself. With the help of the ratio of quantities, Descartes introduces numbers and numerical relations, and algebraic equations acquire meaning because they are regarded as line equations. Possibilities of geometrization in cognition of nature Descartes considered almost limitless. He not only tried to build on this basis almost all natural science disciplines (including, for example, physiology) but also did not exclude the possibility of using his universal method and to explain human behavior.
The evaluation of mathematics in Berkeley’s philosophy is contrary to Descartes’s position in the sense that he not only does not consider mathematical concepts as the basis for knowledge but, on the contrary, tries to show that mathematics, like no other science, is prone to errors and contradictions. Berkeley largely anticipated discussions about the foundations of mathematics at the beginning of the 20th century, pointing out that it is necessary to approach with special care the procedure for the formation of mathematical concepts to avoid mistakes and paradoxes in this science. Correctly educated Berkeley believed that the concept, which directly expresses the data of the senses. There is only that which is perceived, and everything else is a way of representing the perceived. The number and geometric figure are just such representants. However, combining different representations, a mathematician can go very far from their base and construct such abstract constructions that no sense corresponds to. Berkeley proposed to clear mathematics from baseless abstractions, criticizing primarily the calculus of infinitesimal, which he found contradictory and also completely useless.
The position of those who, like Descartes, considered mathematics the basis of all scientific knowledge, is more advantageous from the development of mathematical natural science, since it explains the extraordinary effectiveness of mathematics in the study of nature. However, the criticism of mathematical knowledge from the standpoint of empiricism (in which Berkeley seems to have succeeded more than others) offered a more sober attitude to mathematics, contrasting rationalist enthusiasm with the intention to establish the limits of its applicability. Both of these intentions were realized by Kant, who, on the one hand, set the task of substantiating the use of mathematics in natural science, and on the other – clearly define the boundaries of both mathematics and the whole of natural science in general. Kant defined number and magnitude as a priori forms of knowledge, in addition to which reason can not think of any phenomenon at all. Knowledge of nature consists in the construction of natural objects by the rules of reason, and since number and magnitude specify such rules, any object is primarily mathematical. Everything in nature is measurable and quantifiable – in another way, we simply can not think it. At the same time, mathematics always remains in the sphere of sensuality. Its concepts are applicable only to what is available to immediate contemplation, which can only be sensual (and not intellectual, as Descartes believed). This approach to mathematics almost does not cause difficulties, if we are talking about Euclidean geometry, algebra, and arithmetic. However, the problems of the calculus of infinitely small Kant, unlike Berkeley, almost did not a concern.
The philosophical substantiation of mathematical knowledge was constantly discussed not only by philosophers but also by mathematicians. However, the peak of the preoccupation of leading mathematicians with philosophical problems occurred at the beginning of the 20th century and was associated with a crisis that had broken out at the time. The directions that appeared at that time in mathematics (they are usually distinguished by four: logicism, intuitionism, formalism and set-theoretic direction) differ primarily by philosophical attitudes, which in turn influenced the structure of the mathematical discourse developed by them. However, the position of each direction was closely linked with the philosophical classics.
Russell, who formulated the philosophical basis of logic, in many respects, solidarized with English empiricism. He proceeded from the fact that the foundation of mathematics lies outside of it and all mathematical knowledge must be grounded in non-mathematical premises. Their reduction reveals the truth of mathematical judgments to the simplest and directly established judgments about reality, i.e., empirical facts. Russell was convinced that mathematics would make sense (and get rid of contradictions) when it will be shown that it reflects some real state of affairs. The greatest difficulty in his conception was the explanation of what exactly this real state of affairs means, that is, what should be called facts and how to establish them.
The opposite position was taken by the founder of the intuitionistic school, Brouwer. He considered mathematics to be a completely self-sufficient discipline, the foundations of which lie within herself. Moreover, according to Brouwer, mathematics is the purest expression of the fundamental intuitions underlying any cognitive activity. Speaking of intuition, he, first of all, had in mind the intuition of a numerical series, which, being directly clear, sets the a priori principle of any mathematical (and not only mathematical) reasoning. The latter he represented as a sequence of constructive actions, carried out one after another according to a certain law. The validity of mathematical concepts was therefore identical with their constructiveness. According to Brauer, all non-constructive abstractions (above all abstraction of actual infinity) must be eliminated from mathematics. The idea of constructiveness was also used by Hilbert, who proposed a formal program of substantiating mathematics. His project included two main points:
- the axiomatization of basic mathematical disciplines; and
- the proof of the consistency of axiomatically given theories within metamathematics. The first point meant a special interpretation of the ontological status of mathematical objects. They were considered only as symbols or their combinations, which have no essence and definition.
Their certainty arises only from the place in the formulas of the theory, i.e. thanks to the complete set of relationships in which they participate. The second point of the Hilbert program proposed to treat mathematical reasoning in the same way as the object of the theory. Proof of a mathematical theorem, just like mathematical objects, is a certain combination of symbols, i.e. an object constructed according to specified rules. The completeness and regularity of such objects should be a guarantee of their consistency. Gilbert considered it especially important that any mathematical reasoning is finite and accessible to direct sensual contemplation. Here Gilbert directly associates with Kant. Moreover, the Hilbert program can be viewed as a kind of apology for Kantianism precisely where the positions of the latter are most vulnerable – in those areas that do not deal with contemplated objects. The fact is that in reasoning (i.e. axiomatic theory) any infinite object is still only a directly contemplated symbolic construction.
In general, there are several main problems, on which the philosophy of mathematics is concentrated continuously. First, it is a problem of intuition or immediate sensual or intellectual contemplation. It is the clarity and simplicity of contemplation that proves the validity of mathematical knowledge. The second problem is where to look for the possibility of such contemplation: does it come from mathematics itself, or does it lie in other areas from which mathematics should be derived. Both problems remain the focus of the philosophy of mathematics and continue to determine the content of contemporary discussions largely.