Ultrafinitism

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In the philosophy of mathematics, ultrafinitism, or ultraintuitionism, is a form of finitism.

Ultrafinitists deny the existence of the infinite set N of natural numbers, on the grounds that it can never be completed. In addition, ultrafinitists are concerned with our own physical restrictions in constructing mathematical objects. Thus some ultrafinitists will deny the existence of, for example, the floor of the first Skewes' number, which is a huge number defined using the exponential function as exp(exp(exp(79))), or

$e^{e^{e^{79}}}\mbox{.} \!$

The reason is that nobody has yet calculated what natural number is the floor of this real number, and it may not even be physically possible to do so.

Ultrafinitism is a form of constructivism, but even constructivists generally view the philosophy as unworkably extreme. The logical foundation of ultrafinitism is unclear; in his comprehensive survey Constructivism in Mathematics (1988), the constructive logician A. S. Troelstra dismissed it as "no satisfactory development exists at present". This was not so much a philosophical objection as it was an admission that, in a rigorous work of mathematical logic, there was simply nothing precise enough to include.

Serious work on ultrafinitism has been led, since 1959, by Alexander Esenin-Volpin.

Other considerations of the possibility of avoiding unwieldily large numbers can be based on complexity theory, including the notion of feasible number.

An objection has been raised that ultrafinitism is a nonsensical stance, being described by the absolute (without reference to a place) non-existence of a certain entity. According to this view it is as nonsensical to talk of existence without a place as it is to talk of a proof without assumptions. Actually, this objection, which originated with Carnap, could be applied to Ontology on the whole (and hence to Platonism, intuitionism, and finitism).

Ultrafinitism can be seen as an alternative term to strict finitism.