Classical mathematics

In a broad sense, the term classical mathematics designates the settled part of mathematics, material taught at almost all school and college classrooms, as opposed to the "modern" topics the professional mathematicians are working on.^[1] The term has been used in this sense since at least the 19th century^[2] and continues to be used in the 21st century.^[3][4]

After the foundational crisis of mathematics in the early 20th century, the term "classical mathematics" was appropriated by the adepts of constructive mathematics to designate its traditional (and still very widely supported^[1]) counterpart of the foundations of mathematics, ostensibly to avoid using the "nonconstructive mathematics" label with its negative connotations (the constructive math does not deny the correctness of the theorems with proofs based on "infinite search").^[5] Classical mathematics in this sense is being used by almost all practicing mathematicians, as it offers an easier path to results, and, consequently, yields more discoveries.^[6]

The classical foundations of mathematics is the mainstream approach, pretty much unchanged since the 1920s, and includes the mathematical logic,^[7] with a set theory based on "Zermelo–Fraenkel with choice" axiom system (ZFC set theory), or using a very similar Von Neumann–Bernays–Gödel set theory as an option.^[8][9] It stands in contrast to other types of mathematics such as constructive mathematics or predicative mathematics. In practice, the most common non-classical systems are used in constructive mathematics.^[10]

Classical mathematics is sometimes attacked on philosophical grounds, due to constructivist and other objections to the logic, set theory, etc., chosen as its foundations, such as have been expressed by L. E. J. Brouwer. Almost all mathematics, however, is done in the classical tradition, or in ways compatible with it.

Defenders of classical mathematics, such as David Hilbert, have argued that it is easier to work in, and is most fruitful; although they acknowledge non-classical mathematics has at times led to fruitful results that classical mathematics could not (or could not so easily) attain, they argue that on the whole, it is the other way round.^{[citation needed]}

• Constructivism (mathematics)
• Finitism
• Intuitionism
• Non-classical analysis
• Traditional mathematics
• Ultrafinitism
• Philosophy of Mathematics

• Mandelkern, Mark (1989-02-01). "Brouwerian Counterexamples" (PDF). Mathematics Magazine. 62 (1): 3−27. doi:10.2307/2689939. JSTOR 2689939.
• Ball, W.W.R. (1889). A History of the Study of Mathematics at Cambridge. Landmarks of science. University Press. Retrieved 2024-07-17.
• Griffiths, H.B.; Hilton, P.J. (2013). A Comprehensive Textbook of Classical Mathematics: A Contemporary Interpretation. Springer New York. ISBN 978-1-4612-6322-7. Retrieved 2024-07-17.
• Hofmann, J.E. (2003). Classical Mathematics. Barnes & Noble Books. ISBN 978-0-7607-4746-9. Retrieved 2024-07-17.
• Chan, Yuen-Kwok (2023-04-30). "A Leisurely Random Walk Down the Lane of a Constructive Theory of Stochastic Processes". Handbook of Constructive Mathematics. Cambridge University Press. pp. 333–356. doi:10.1017/9781009039888.014. ISBN 978-1-009-03988-8.
• Bridges, D.; Richman, F. (1987). Varieties of Constructive Mathematics. Lecture note series. Cambridge University Press. ISBN 978-0-521-31802-0. Retrieved 2024-07-17.
• Ferreirós, José (2010-12-31). "The Crisis in the Foundations of Mathematics". The Princeton Companion to Mathematics. Princeton University Press. doi:10.1515/9781400830398.142. ISBN 978-1-4008-3039-8.
• Blackburn, Simon (2016). A Dictionary of Philosophy (3rd ed.). Oxford University Press. ISBN 9780198735304.

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