Formalism (philosophy of mathematics)

A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess.

"[1] According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they are not "about" anything at all.

In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.

Along with realism and intuitionism, formalism is one of the main theories in the philosophy of mathematics that developed in the late nineteenth and early twentieth century.

[2] The early mathematical formalists attempted "to block, avoid, or sidestep (in some way) any ontological commitment to a problematic realm of abstract objects.

"[1] German mathematicians Eduard Heine and Carl Johannes Thomae are considered early advocates of mathematical formalism.

[10] Hilbert was initially a deductivist,[citation needed] but he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic.

Bertrand Russell has argued that formalism fails to explain what is meant by the linguistic application of numbers in statements such as "there are three men in the room".