Philosophy of Being and Knowledge

Philosophy of Science

Philosophy of Mathematics

Natural science is the subject of the philosophy of science, while the essence and methodology of the humanities are explored by various branches of philosophy (such as the philosophy of history, the philosophy of economics, and the philosophy of culture). However, mathematics falls outside the scope of the philosophy of science, as the methods employed in natural science are unsuitable for mathematics. Natural science is inconceivable without experience: philosophers who study the nature of natural science debate the role of experience (positivists argue that experience is the source of scientific knowledge, while critical rationalists assert that experience can only falsify scientific theories), but all agree that experience is essential for natural science. In contrast, mathematics does not operate with experience in any way. Mathematical truths are fundamentally different from the theories of natural science, primarily because they are not connected to experience. Mathematical truths remain true even if they are not realized in the material world. For instance, Pythagoras' theorem would still be true even if no triangular objects existed in the world. "2+2=4" would be true even if nothing existed at all. Mathematics deals with numbers and shapes, and in the material world, there are neither numbers nor shapes; things have quantity (number) and form, but they are neither number nor form themselves. This means that applying the methods used in natural science to mathematics is inappropriate, and thus mathematics is not a subject of the philosophy of science.

Another distinctive feature of mathematics, setting it apart from natural science, is the absoluteness and immutability of mathematical truths. Laws of nature operate only when nature exists; laws of society operate only when society exists; laws of economics operate only when economics exists. All these truths are valid only when there is something material. In contrast, mathematical truths remain true even when there is nothing in which they could be realized. These truths are eternal and unchanging.

If mathematics—numbers and shapes—is not connected to anything revealed through experience, this means that the existence of mathematics presents a series of questions for philosophy, which constitutes the subject matter of the philosophy of mathematics. These questions include inquiries into what numbers and shapes are, where they come from, whether they actually exist, and what their relation is to the world. The search for answers to these questions constitutes the philosophy of mathematics. It is important to note that the philosophy of mathematics is an older domain of philosophy than the philosophy of science, as mathematics is older than natural science. While natural science, in its current form, is a product of the modern era, mathematics was studied in antiquity. This is why the founder of positivism, Auguste Comte, in his classification of sciences from the most ancient and general to the most recent and specific, placed mathematics first. Important attempts at philosophical reflection on mathematics also emerged in antiquity. Therefore, both mathematics and the philosophy of mathematics were developed fields of knowledge in antiquity. In the field of the philosophy of mathematics, as in most areas of philosophy, general consensus has not been reached, leading to various theories that address the questions posed to this field.

Early philosophical reflections on the nature of mathematics appeared in ancient civilizations such as India, Sumer, Egypt, and others, but complete philosophical theories of mathematics either did not arise or remain unknown to us. There are sufficient grounds to believe that the ancient Egyptians not only possessed advanced mathematical knowledge but also had a philosophy of mathematics, which they passed down orally from older generations of priests to younger ones, without leaving written records. For this reason, their philosophical conceptions of mathematics are unknown to contemporary science. Thus, the history of the philosophy of mathematics must be traced back to antiquity. The foundations of the philosophy of mathematics that remain relevant today were laid by Pythagoras. Like other philosophers of his time, Pythagoras sought to find the underlying principle of all existence, the basis upon which everything originated and could be explained. Pre-Socratic philosophers offered various answers to these questions. For example, the Milesian school (Thales, Anaximander, Anaximenes) saw the foundation of all being in pure and undifferentiated matter. Heraclitus believed that the underlying principle of all existence was a rational principle—Logos. Pythagoras argued that number was the foundation of all being. All material things are subject to number; everything can be counted with it. There is no physical or chemical property of an object that cannot be measured by number (units of measurement). The doctrine of the universality of mathematics, formulated by Pythagoras and developed by his successors, is known as mathematical Platonism. This theory asserts that mathematics underlies the entire material world; it is the rational principle according to which the world is created and without which the world cannot be understood. Pythagoras believed that mathematics was the rational principle used by God as the main means of creating the world; the world was created through mathematics, and thus mathematics is the language of God. To read the message that God left in the world, one must master the language in which the message is written, namely mathematics. This is why the school founded by Pythagoras resembled a monastery more than an educational institution, with members studying mathematics not for knowledge's sake but as a form of meditation. For them, mathematics was not a dry science but a form of mysticism.

Pythagoras exerted a profound influence on Plato, who was captivated by Pythagoras's mathematical Platonism and further developed it. The impact of Plato on philosophy, mathematics, and science was so significant that it led the eminent mathematician Alfred North Whitehead (1861-1947) to assert: "The history of philosophy is nothing but a series of footnotes to Plato." Plato managed to integrate the finest contributions of his predecessors. From Parmenides and Euclid of Megara, he inherited the belief that true being is beyond sensory knowledge, and can only be encountered through spiritual and intellectual elevation above the material and sensory. On this foundation, Plato developed his theory of Forms. True being consists of Forms, or pure abstractions, intellectual realities, while the material world is merely an imperfect reflection of these Forms. For the Forms to be realized in the world, they require an intermediary; this intermediary is mathematics. For instance, to create a circular object, it is insufficient to merely know it must be round; one must also know the diameter of the circle. Similarly, to produce a red object, one must clearly understand the degree of color saturation. Hence, mathematics serves as the medium through which Forms are expressed in the world. Plato thus incorporated Pythagoras’s mathematical Platonism into his own theory. Throughout his extensive creative career, Plato's fascination with Pythagorean mathematics grew. In his later years, he even identified Forms with numbers and spoke of ideal numbers. These ideas became central to his followers, marking the so-called early and middle Platonism.

During the Renaissance, European intellectuals sought to revive everything associated with antiquity. They were particularly intrigued by Plato's philosophy, including his mathematical Platonism. The degree of this fascination is illustrated by the motto of the contemporary philosopher Cardinal Nicholas of Cusa, who claimed that God wrote the book of nature in the language of mathematics. This means that there is no other way to read this book, that is, to understand nature, except through mathematics. The incorporation of mathematics into science, influenced by Plato's impact on Renaissance philosophy and culture, became one of the cornerstones of the scientific and technological progress of the modern era. Since then, neither natural science nor the many other disciplines (sociology, economics, psychology) can be conceived without mathematics.

The Enlightenment era was a time when mathematics came under scrutiny. During this period, the theory of knowledge developed empiricism—the belief that the source of all knowledge is experience. Both English and French Enlightenment thinkers contributed to its development. Some even advocated for sensualism, an extreme form of empiricism, which posits that knowledge is acquired solely through the senses. Mathematics did not fit into this empirical epistemology, as it does not rely on experiential data. This led empiricists to criticize mathematics and exclude it from the list of sciences. However, this criticism proved to be somewhat beneficial for the philosophy of mathematics: it conclusively demonstrated that mathematics is not an empirical science and should thus be studied as a distinct method of knowledge, separate from natural science and other fields of knowledge. The question of the essence of mathematics became particularly pertinent during the rapid development of mathematics in the 19th and 20th centuries.

The recent advances in logic provided another impetus for the development of mathematics. In the 19th and 20th centuries, logicians observed similarities between logic and mathematics, which inspired them to compare these two disciplines. The results of these new trends included the emergence of mathematical logic and the attempt to explain mathematics as a product of logic (logicism).