Formalism (philosophy of mathematics)In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. 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.
Propositional functionIn propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined. The sentence may contain several such variables (e.g. n variables, in which case the function takes n arguments). As a mathematical function, A(x) or A(x_1, x_2, ...
MetamathMetamath is a formal language and an associated computer program (a proof checker) for archiving, verifying, and studying mathematical proofs. Several databases of proved theorems have been developed using Metamath covering standard results in logic, set theory, number theory, algebra, topology and analysis, among others. the set of proved theorems using Metamath is one of the largest bodies of formalized mathematics, containing in particular proofs of 74 of the 100 theorems of the "Formalizing 100 Theorems" challenge, making it fourth after HOL Light, Isabelle, and Coq, but before Mizar, ProofPower, Lean, Nqthm, ACL2, and Nuprl.
Axiom of reducibilityThe axiom of reducibility was introduced by Bertrand Russell in the early 20th century as part of his ramified theory of types. Russell devised and introduced the axiom in an attempt to manage the contradictions he had discovered in his analysis of set theory. With Russell's discovery (1901, 1902) of a paradox in Gottlob Frege's 1879 Begriffsschrift and Frege's acknowledgment of the same (1902), Russell tentatively introduced his solution as "Appendix B: Doctrine of Types" in his 1903 The Principles of Mathematics.
