Philosophy of Math

The philosophy of math is a branch of philosophy about mathematical knowledge, raising ontological and epistemological questions. The philosophy of math is concerned with two main issues: the meaning of mathematical sentences, and whether abstract objects exist. Hence, developing a semantic theory for the language of mathematics is the main ambition in mathematical philosophy. During the nineteenth century, the philosophy of math was traditionally separated into four schools, a time when the general philosophical perspective leaned toward the empirical, rather than the platonistic aspects of mathematical theories. The first school is logicism, which attempts to reduce math to logic, an idea originally formulated by Leibniz and later detailed by Dedekind, Peano, and Frege on the principles of mathematical theories and logic. However, Frege’s derivation of his logical principle, Basic Law V, was flawed. This led to Russell pointing out a contradiction, which was known as Russell’s paradox. The second school is intuitionism, which originated from the work of mathematician Brouwer, who was inspired by Kant’s view of what objects are. According to intuitionism, mathematics is a sort of construction, where numbers, proofs, theorems, mathematical meanings, etc., are all mental constructions created by the “ideal” mathematician. Nevertheless, the “ideal” mathematician still remains as a finite being, which means that intuitionism rejects certain abstract concepts such as the existence of the actual (completed) infinite, and only the potential infinite exists. The third school is formalism, which was first thought of by Hilbert, who agreed with intuitionists that natural numbers are basic objects in math. However, he did not consider natural numbers to be mental constructs, but rather, as symbols that embodied concrete, yet abstract, objects - referred to as “quasi-concrete” objects. Last but not least, the fourth school is predicativism, which originated from the works of Russell and his paradox. In an attempt to reassert logicism, he defined what is called the “vicious circle principle”, in which an entity that cannot be defined without referring to a class to which it belongs to cannot be accepted. Definitions that violate the vicious circle principle are called “impredicative” and correct definitions, which is that a class only refers to entities that exist independently from the defined class, are called predicative. During the 20th century, there was a renewed interest in the platonistic perspective in the philosophy of math. Gödel, a platonist, viewed that there is a strong parallelism between theories of mathematical objects and concepts and of physical objects, known as Gödel’s platonism. Our mathematical intuition with mathematical objects and concepts are analogous to our perceptual relation to physical objects, and hence provides intrinsic evidence for abstract, mathematical principles. With such a wide variety of theories, the Philosophy of Math continues to be greatly researched and examined.