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In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. Another form of finitism was pursued by Hilbert and Bernays.
In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who do not allow countably infinite objects as strict finitists. Historically, the written history of mathematics was thus classically finitist until Cantor invented the hierarchy of transfinite cardinals in the end of the 19th century. Leopold Kronecker remained a strident opponent to Cantor's set theory:
God created the natural numbers, all else is the work of man.
In 1923, Thoralf Skolem published a paper in which he presented a semi-formal system, which is now known as primitive recursive arithmetic, that is widely taken to be a suitable background for finitist mathematics. This was adopted by David Hilbert and Paul Bernays as the "contentual" finitist system for metamathematics, in which a proof of the consistency of other mathematical systems (e.g. full Peano Arithmetic) was to be given. (See Hilbert's program.)
Reuben Goodstein is another proponent of finitism. Some of his work involved building up to analysis from finitist foundations. Although he denied it, much of Ludwig Wittgenstein's writing on mathematics has a strong affinity with finitism. If finitists are contrasted with transfinitists (proponents of e.g. Georg Cantor's hierarchy of infinities), then also Aristotle may be characterized as a strict finitist. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity. (Note that Aristotle's actual infinity means simply an actualization of something never-ending in nature, when in contrast the Cantorist actual infinity means the transfinite cardinal and ordinal numbers, that have nothing to do with the things in nature):
But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.—Aristotle, Physics, Book 3, Chapter 6
See also 
- Eriksson, K., Estep D., and Johnson C. Applied Mathematics: Body and Soul. Volume 1. Springer, 2004, p. 230-232.
- From an 1886 lecture at the 'Berliner Naturforscher-Versammlung', according to H. M. Weber's memorial article, Leopold Kronecker, in Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 2 1891-92