Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 noun mathematics An extreme form of
constructivism , according to which a mathematical object does not exist unless it can be constructed fromnatural numbers in afinite number of steps.
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
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Examples

The first, and perhaps most definitive, indication that the later Wittgenstein maintains his finitism is his continued and consistent insistence that irrational numbers are rules for constructing finite expansions, not infinite mathematical extensions.
Wittgenstein's Philosophy of Mathematics Rodych, Victor 2007

A second strong indication that the later Wittgenstein maintains his finitism is his continued and consistent treatment of
Wittgenstein's Philosophy of Mathematics Rodych, Victor 2007

But Lavine has developed a sophisticated form of settheoretical ultrafinitism which is mathematically nonrevisionist (Lavine 1994).
Philosophy of Mathematics Horsten, Leon 2007

For example, a prooftheoretic analysis may contribute to establish if a certain theory complies with a given mathematical framework (e.g., predicativity, finitism, etc.).
Set Theory: Constructive and Intuitionistic ZF Crosilla, Laura 2009

Nor is the “finitism” characteristic of Hilbert and Bernays 'later work present in Dedekind (an aspect developed in response both to the settheoretic antinomies and to intuitionist challenges), especially if it is understood in a metaphysical sense.
Dedekind's Contributions to the Foundations of Mathematics Reck, Erich 2008

Such finitism might have been acceptable to Dedekind as a methodological stance; but in other respects his position is strongly infinitary.
Dedekind's Contributions to the Foundations of Mathematics Reck, Erich 2008

Although this idea was later adopted by the other structuralistic programs, it plays a unique rÃ´le within Ludwig's metatheory in connection with his finitism.
Structuralism in Physics Schmidt, HeinzJuergen 2008

This leads to a position that has been called ultrafinitism.
Philosophy of Mathematics Horsten, Leon 2007

Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986, 300301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite
Wittgenstein's Philosophy of Mathematics Rodych, Victor 2007

On Wittgenstein's intermediate finitism, an expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number n has a particular property.
Wittgenstein's Philosophy of Mathematics Rodych, Victor 2007
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