Philosophy of Being and Knowledge

Philosophy of Science

Mathematical Theories

The central question that the philosophy of mathematics grapples with is the nature of mathematics itself: whether numbers and forms exist independently of the human mind or are merely constructs of it.

• The earliest response to this question is known as mathematical realism, which holds that mathematics exists as an independent domain of reality, unaffected by human cognition. According to realists, mathematics exists in the same way that the world, natural law, and morality do. Plato provided a classic exposition of mathematical realism, asserting that mathematics represents a separate level of existence that neither derives from the material world nor from the human mind but rather shapes both. Mathematical realism remained foundational in mathematical theory until today. Critics of mathematical realism have argued that if mathematics is an independent reality, then where do humans acquire mathematical knowledge? One answer proposed by Austrian Kurt Friedrich Gödel (1906-1978) is that humans are born with an innate mathematical sense, akin to an aesthetic sense, with mathematical knowledge being a priori and intuitive.

For millennia, mathematical realism was an unchallenged axiom. It was only in the 20th century, with the rise of postmodern tendencies rejecting absolutes and unchanging truths, that attempts were made to reject realism in mathematics. Realism, once an axiom, became just one of many theories. Those who rejected realism were faced with the task of explaining the nature of mathematical judgments. Without its uniqueness, mathematics had to be reduced to something else, leading to numerous relativistic theories of mathematics.

• At the beginning of the 20th century, the rapid development of logic, philosophy of language, and philosophy of science influenced the theory of mathematics, leading to attempts to refute mathematical realism. Some logicians and mathematicians proposed that mathematics is a product of logic, a stance known as logicism. One of the most influential proponents of logical positivism and a member of the Vienna Circle, Rudolf Carnap (1891-1970), who sought to combine logicism and positivism, argued that all judgments can be divided into two groups: those with meaningful truth-value and those without. Neopositivists assert that truth-value is only present when it is evident or empirically verifiable. If a judgment’s truth-value cannot be established, it is deemed meaningless. Judgments with truth-value derive from states of affairs (e.g., “The sun shines,” “The moon is larger than the sun”) or from their linguistic-logical structure (e.g., “A square has four sides,” “A square is round”). These judgments can be either true or false, but crucially, their truth-value is verifiable. Since mathematical judgments evidently lack empirical connection, neopositivists associate them with logic. The founder of logicism was German logician and mathematician Gottlob Frege (1848-1925), who sought to reduce all mathematical judgments to logical principles, a view later developed by Bertrand Russell (1872-1970).
• Another theory explaining the origin of mathematical judgments is formalism. Adherents to this theory liken mathematics to a game, with judgments analogous to bets in the game. A successful bet means winning, while an unsuccessful one means losing. The rules of this mathematical game have been set by Euclidean geometry, and mathematicians (at least before the advent of non-Euclidean geometries) are bound by these rules. Pythagoras, by stating his theorem, made a successful bet within the framework of Euclidean geometry and won the game. Therefore, the foundation of mathematics lies in axioms, which are universally accepted truths that no one disputes or proves, much like the rules of a game. One of the most prominent advocates of formalism was David Hilbert (1862-1943). For formalists, mathematical judgments are not absolutely true.
• Deductivism is a mathematical theory similar to formalism, but without invoking the concept of a game. Deductivists view mathematics as a product of human culture, with certain mathematical axioms being formed in the past but not as absolute truths. These axioms are universally agreed upon, and their truth derives from this consensus, not the other way around. All mathematical judgments derive from and are justified by these axioms. Mathematical axioms are quite stable, like many other universally accepted ideas, but they are not absolute.
• French mathematician Jules Henri Poincaré (1854-1912) proposed a theory known as conventionalism. According to this theory, mathematical judgments are not absolute; they are true only because people agree to consider them true. Euclidean geometry is true only within a certain agreement (convention).
• The theory of intuitionism, developed by Dutch mathematician L.E.J. Brouwer (1881-1966), was grounded in Immanuel Kant's philosophy, which occupied a middle position between rationalism and empiricism, asserting that both experience and mental activity are necessary for acquiring knowledge. The mind transforms experiential data into knowledge: without experience, the mind has nothing to process, and without the mind, experience remains merely an accumulation of information about individual facts. Kant believed that the mind contains mechanisms for processing data and turning individual facts into general theories. Brouwer equated these a priori principles of the mind with mathematics. For a scientific theory to emerge, empirical data are necessary; otherwise, science would have nothing to process. Mathematics serves as the algorithm for processing experiential data, organizing them into universal theories. Thus, in intuitionism, mathematics has its roots in the human mind.
• American mathematician Stuart Shapiro (b. 1951) founded mathematical structuralism, applying Ferdinand de Saussure's ideas to mathematics. Structuralists view mathematics as a coherent structure, with mathematical judgments gaining their truth-value only within this structure. Outside the structure, the truth of mathematical judgments cannot be established.
• Psychologism, an exaggerated belief that psychology can explain everything, also infiltrated mathematical theory. Some mathematicians, deeply fascinated by psychology, treated mathematics as a product of psychological processes, viewing mathematical laws as expressions of the psyche. Psychologism advocates suggested explaining mathematical regularities through psychology.

All the aforementioned positions, except mathematical realism, are consequences of rejecting realism. Realists believe that mathematical truths are absolute, eternal, and unchanging, and that mathematics is an independent reality. Rejecting realism entails denying mathematics as an autonomous entity, necessitating an explanation of mathematics based on some other foundation. This led to attempts to explain mathematics as derivative from logic, psychology, games, intuition, etc. These concepts challenged the absoluteness of mathematics. The rejection of mathematical realism had certain benefits: it dismantled the inviolable bastion of traditional mathematics, revealing new possibilities such as non-Euclidean geometry. On the other hand, rejecting realism generally undermines the value of mathematics: if mathematical judgments are relative and do not reveal truth, their value is diminished. Besides mathematical realism and relativism (all theories denying the absolute truth of mathematical judgments), intermediate positions also emerged.

• The position that merges mathematical realism with relativism is empiricism, as developed by Willard Van Orman Quine (1908-2000) and Hilary Putnam (1926-2016). These scholars believe that mathematics is an independent and autonomous reality, aligning with the realist position. However, empiricists focus on resolving the main critique of realism: if mathematics is an independent reality, how do people acquire mathematical knowledge? Empiricists argue that, despite mathematics being an independent reality, knowledge of mathematical truths is only possible through experience. Mathematical truths exist independently of anything, but without experience, they would be unknown to people. For example, "2+2=4" is true regardless of whether people or objects to count exist. However, without objects to count, or applied mathematical experience, knowledge of mathematical truths would never be acquired. This distinction differentiates empiricism from realism. Plato believed that mathematical judgments are not only independent and absolute but also innate knowledge. Rationalists, including Plato, hold that people are born with knowledge that is organized through learning. Empiricists contend that mathematical truths are known through experience.