Finitismi

Finitismo, or finitism, is a position in the philosophy of mathematics that asserts that only finite objects and finitistic methods are legitimate in mathematical practice. There are two closely connected strands: ontological finitism, which holds that only finite objects exist, and methodological finitism or the finitistic view, which permits discussions of infinity only as potential or symbolic tools and confines proofs to finitary procedures.

In its methodological sense, finitism rejects the existence of completed infinities and aims to ground mathematical reasoning in finite constructions, finite sequences, and concrete verifications. Proponents argue that finitary reasoning is verifiable by inspection and less susceptible to paradoxes or circular justifications that can arise with infinite totalities.

Historically, finitism is associated with Hilbert’s program in the early 20th century, which sought to secure the foundations of analysis by appealing to finitistic, constructive, and elementary methods to prove the consistency of broader mathematical theories. The later results of Gödel's incompleteness theorems showed limits to achieving that project in its strongest form, influencing how finitist ideals are viewed within the foundations of mathematics.

Finitism stands in contrast to Platonism, which posits a realm of abstract mathematical objects, and to intuitionism, which emphasizes constructive meaning and mental constructions of infinite processes. In modern discussions, finitist ideas influence constructive mathematics and formal proof systems that prioritize finite representations and explicit constructions, while many mathematicians operate with a broader, non-finitistic toolkit.