Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy where infinite mathematical objects (e.g., infinite sets) are accepted as legitimate within the Platonic universe of mathematics.
Contents

Main idea 1

History 2

Classical finitism vs. strict finitism 3

Views regarding infinite mathematical objects 4

Other related philosophies of mathematics 5

See also 6

References 7

Further reading 8

External links 9
Main idea
The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets. While all natural numbers are accepted as existing, the set of all natural numbers is not considered to exist as a mathematical object. Therefore quantification over infinite domains is not considered meaningful. The mathematical theory often associated with finitism is Thoralf Skolem's primitive recursive arithmetic.
History
The introduction of infinite mathematical objects was a development in mathematics that occurred a few centuries ago. The use of infinite objects was a controversial topic among mathematicians. The issue entered a new phase when naive set theory and used it as a base for his work on transfinite numbers. When paradoxes such as Russell's paradox, Berry's paradox and the BuraliForti paradox were discovered in Cantor's naive set theory, the issue became a heated topic among mathematicians.
There were various positions taken by mathematicians. All agreed about finite mathematical objects such as natural numbers. However there were disagreements regarding infinite mathematical objects. One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.
Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects, and accepting ideal mathematical objects does not cause a problem regarding finite mathematical objects. More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them. Therefore allowing infinite mathematical objects would not cause a problem regarding finite objects. This led to Hilbert's program of proving consistency of set theory using finitistic means as this would imply that adding ideal mathematical objects is conservative over the finitistic part. Hilbert's views are also associated with formalist philosophy of mathematics. Hilbert's goal of proving the consistency of set theory or even arithmetic through finitistic means turned out to be an impossible task due to Kurt Gödel's incompleteness theorems. However, by Harvey Friedman's grand conjecture most mathematical results should be provable using finitistic means.
Hilbert did not give a rigorous explanation of what he considered finitistic and refer to as elementary. However, based on his work with Paul Bernays some experts such as William Tait have argued that the primitive recursive arithmetic can be considered an upper bound on what Hilbert considered finitistic mathematics.
In the years following Gödel's theorems, as it became clear that there is no hope of proving consistency of mathematics, and with development of axiomatic set theories such as Zermelo–Fraenkel set theory and the lack of any evidence against its consistency, most mathematicians lost interest in the topic. Today most classical mathematicians are considered Platonist and believe in the existence of infinite mathematical objects and a settheoretical universe.
Classical finitism vs. strict finitism
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 discovered the hierarchy of transfinite cardinals in the end of the 19th century.
Views regarding infinite mathematical objects
Leopold Kronecker remained a strident opponent to Cantor's set theory:^{[1]}
God created the natural numbers, all else is the work of man.^{[2]}
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 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 neverending 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
Other related philosophies of mathematics
Ultrafinitism (also known as ultraintuitionism) has an even more conservative attitude towards mathematical objects than finitism, and has objections to the existence of finite mathematical objects when they are too large.
See also
References

^ Eriksson, K., Estep D., and Johnson C. Applied Mathematics: Body and Soul. Volume 1. Springer, 2004, p. 230232.

^ From an 1886 lecture at the 'Berliner NaturforscherVersammlung', according to H. M. Weber's memorial article, , in Jahresbericht der Deutschen MathematikerVereinigung, Vol. 2 189192Leopold Kronecker
Further reading

Feng Ye (2011). Strict Finitism and the Logic of Mathematical Applications. Springer Science & Business Media.
External links
This article was sourced from Creative Commons AttributionShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, EGovernment Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a nonprofit organization.