Foundations of mathematics

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Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. The search for foundations of mathematics is however also a central question of the philosophy of mathematics: On what ultimate basis can mathematical statements be called true?

The foundational philosophy of Platonist mathematical realism, as exemplified by mathematician Kurt Gödel, proposes the existence of a world of mathematical objects independent of humans; the truths about these objects are discovered by humans. In this view, the laws of nature and the laws of mathematics have a similar status, and the effectiveness ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. The obvious question, then, is: how do we access this world? (cf Anglin 1991 p. 218)

The foundational philosophy of formalism, as exemplified by David Hilbert, is based on axiomatic set theory and formal logic. Virtually all mathematical theorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is then nothing but the claim that the statement can be derived from the axioms of set theory using the rules of formal logic (cf Anglin 1991 p. 218).

Formalism does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why "true" mathematical statements (e.g., the laws of arithmetic) appear to be true in the physical world, and so on. Formalistic truth could also turn out to be rather pointless: it is possible that all statements could be derived from the axioms of set theory. Moreover, as a consequence of Gödel's second incompleteness theorem, we can never be sure that this is not the case.

The foundational philosophy of intuitionism or constructivism, as exemplified in the extreme by Brouwer and more coherently by Stephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of non-existence. For example, as a consequence of this the form of proof known as reductio ad absurdum is suspect (cf Anglin 1991 p. 218).

Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a social group. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.

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[edit] Foundational crisis

The foundational crisis of mathematics (in German: Grundlagenkrise der Mathematik) was early 20th century's term for the search for proper foundations of mathematics.

After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.

One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell's paradox) and to be inconsistent: an undesirable situation in which every mathematical statement that can be formulated in a proposed system (such as 2 + 2 = 5) can also be proved in the system.

Various schools of thought on the right approach to the foundations of mathematics were fiercely opposing each other. The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a formal system proved sound by metamathematical finitistic means. The main opponent was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols [citation needed]. The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time.

Gödel's incompleteness theorems, proved in 1931, showed that essential aspects of Hilbert's program could not be attained. In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system – such as necessary to axiomatize the elementary theory of arithmetic – a statement that can be shown to be true, but that does not follow from the rules of the system. It thus became clear that the notion of mathematical truth can not be reduced to a purely formal system as envisaged in Hilbert's program. In a next result Gödel showed that such a system was not powerful enough for proving its own consistency, let alone that a simpler system could do the job. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means. Meanwhile, the intuitionistic school had failed to attract adherents among working mathematicians, and foundered due to the difficulties of doing mathematics under the constraint of constructivism.

In a sense, the crisis has not been resolved, but faded away: most mathematicians do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided.

Platonism
”Platonists, such as Kurt Gödel, hold that numbers are abstract, necessarily existing objects, independent of the human mind” (Anglin (1994) p. 218)
Formalism
“Formalists, such as David Hilbert (1862-1943), hold that mathematics is no more or less than mathematical language. It is simply a series of games...” (Anglin (1994) p. 218)
Intuitionism
”Intuitionists, such as L. E. J. Brouwer (1882-1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them." (Anglin (1994) p. 218)

[edit] References

  • W. S. Anglin, Mathematics: A Concise history and Philosophy, Springer-Verlag, New York, 1994. Chapter 39 Foundations contains concise descriptions, for the 20th century, of Platonism (with respect to Gödel), Formalism (with respect to Hilbert), and Intuitionism (with respect to Brouwer).
  • Goodman, N.D. (1979), "Mathematics as an Objective Science", in Tymoczko (ed., 1986).
  • Hart, W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
  • Hersh, R. (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986).
  • Hilbert, D. (1922), "Neubegründung der Mathematik. Erste Mitteilung", Hamburger Mathematische Seminarabhandlungen 1, 157–177. Translated, "The New Grounding of Mathematics. First Report", in (Mancosu 1998).
  • Kleene, Stephen C. [1952] (1991). Introduction to Meta-Mathematics, Tenth impression 1991, Amsterdam NY: North-Holland Pub. Co. ISBN 0-7204-2103-9.
In Chapter III A Critique of Mathematic Reasoning, §11. The paradoxes, Kleene discusses Intuitionism and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician.
  • Mancosu, P. (ed., 1998), From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, Oxford, UK.
  • Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5–22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).
  • Putnam, Hilary (1975), "What is Mathematical Truth?", in Tymoczko (ed., 1986).
  • A. S. Troelstra (no date but later than 1987), "A History of Constructivism in the 20th Century", http://staff.science.uva.nl/~anne/hhhist.pdf, A detailed survey for specialists: §1 Introduction, §2 Finitism & §2.2 Actualism, §3 Predicativism and Semi-Intuitionism, §4 Brouwerian Intuitionism, §5 Intuitionistic Logic and Arithmetic, §6 Intuitionistic Analysis and Stronger Theories, §7 Constructive Recursive Mathematics, §8 Bishop's Constructivism, §9 Concluding Remarks. Approximately 80 references.
  • Tymoczko, T. (1986), "Challenging Foundations", in Tymoczko (ed., 1986).
  • Tymoczko, T. (ed., 1986), New Directions in the Philosophy of Mathematics, 1986. Revised edition, 1998.
  • Weyl, H. (1921), "Über die neue Grundlagenkrise der Mathematik", Mathematische Zeitschrift 10, 39–79. Translated, "On the New Foundational Crisis of Mathematics", in (Mancosu 1998).
  • Wilder, Raymond L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.

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