Finitism

Finitism is the position in philosophy of mathematics that only finite mathematical objects and procedures are legitimate — that mathematics should not posit or reason about actually infinite collections, quantities, or processes. The finitist holds that mathematical existence is constructive and bounded: a number, set, or structure exists only if it can be built up in a finite number of steps from acknowledged starting points.

Finitism was the methodological foundation David Hilbert demanded for his consistency proofs: the Hilbert Program required that mathematics prove its own consistency using only finitistic reasoning — reasoning about concrete, surveyable, finite objects — to avoid circularity. If consistency could be established finitistically, it would rest on a foundation even the most skeptical critic must accept.

Gödel's second incompleteness theorem terminated this program: no consistent finitistic system sufficient to express basic arithmetic can prove its own consistency. The consistency proof for any system of a given strength requires a stronger system — and that stronger system requires a yet stronger one, in a sequence that terminates only at the transfinite.

There is a strict and a liberal variant of finitism. Strict finitism (advocated by Alexander Esenin-Volpin) denies not only actual infinity but also arbitrarily large finite numbers: there is some largest surveyable number, and mathematics beyond it is suspect. Constructive mathematics is a more liberal cousin, accepting potential infinity (processes that can always be extended) but rejecting actual infinity (completed infinite totalities).