finitists

Finitists are mathematicians and philosophers who adhere to finitism, a philosophical position concerning the nature of mathematical objects and reasoning. At its core, finitism asserts that mathematical objects exist only if they can be constructed in a finite number of steps. This implies that certain mathematical concepts, particularly those involving infinite sets or non-constructive existence proofs, are viewed with skepticism or outright rejected.

The primary concern for finitists is the avoidance of impredicative definitions and proofs that rely on the principle of excluded middle for infinite collections. An impredicative definition is one that defines an object in terms of a totality to which the object itself belongs. For example, defining "the smallest number greater than all numbers in set X" would be impredicative if X contains that number. Finitists generally favor constructive methods of proof, where the existence of a mathematical object must be demonstrated by providing an explicit construction.

This stance leads finitists to reject or reinterpret much of classical mathematics. For instance, the axiom of choice, which allows for the selection of elements from an infinite number of sets without specifying a rule, is often problematic for finitists. Similarly, proofs that rely on the law of excluded middle (a statement is either true or false) applied to infinite sets are not always accepted. Instead, finitists often work within formal systems like intuitionistic logic or constructive set theory.

Notable figures associated with finitist ideas include Leopold Kronecker and, more prominently, Haskell Curry and certain schools of intuitionism influenced by L. E. J. Brouwer. While finitism represents a minority viewpoint within mainstream mathematics, it has had a significant impact on the philosophy of mathematics and the development of constructive mathematical approaches.