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? Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ... Recursion theory, or computability theory, is a branch of mathematical logic dealing with generalizations of the notion of computable function, and with related notions such as Turing degrees and effective descriptive set theory. ... Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will... Proposition is a term used in logic to describe the content of assertions. ... When someone sincerely agrees with an assertion, they might claim that it is the truth. ...

One common mathematical paradigm 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. Since the late 1960s, the word paradigm (IPA: ) has referred to a thought pattern in any scientific discipline or other epistemological context. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ...

This formalistic approach 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. In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. ...

The above-mentioned notion of formalistic truth could also turn out to be rather pointless: it is certainly possible that all statements can 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. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...

In mathematical realism, sometimes called Platonism, the existence of a world of mathematical objects independent of humans is postulated; 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? Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will... Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

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. Theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. ... In the philosophy of mathematics, mathematical practice is used to distinguish the working practices of professional mathematicians (eg. ... In sociology, a group is usually defined as a collection consisting of a number of people who share certain aspects, interact with one another, accept rights and obligations as members of the group and share a common identity. ... The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science. ...

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. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999 in the...

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. Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will... Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ...

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. Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist (according to our present understanding of physics). ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Frege is contradictory. ... In mathematical logic, a formal system is said to be consistent if it doesnt contain a contradiction, or, more precisely, for no proposition are both and provable. ...

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, who he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time. The word formalism has several meanings: A certain school in the philosophy of mathematics, stressing axiomatic proofs through theorems specifically associated with David Hilbert. ... David Hilbert David Hilbert (January 23, 1862, Wehlau, East Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Hilberts Program was to formalize all existing theories to finite real complete set of axioms, and provide a proof that these axioms were consistent. ... In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ... In the philosophy of mathematics, 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. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ... The Mathematische Annalen is a German mathematical research journal published by Springer-Verlag. ...

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 mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... Kurt Gödel (IPA: ) (April 28, 1906 Brno, then Austria-Hungary, now Czech Republic – January 14, 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher of mathematics. ... Arithmetic is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...

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), one knows how to avoid running into them. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...

References

• 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, S.C. (1971), Introduction to Metamathematics, North–Holland Publishing Company, Amsterdam, Netherlands.

• 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).

• 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.

Reuben Hersh (December 9, 1927 - ) is an American mathematician, now an emeritus professor of the University of New Mexico. ... David Hilbert David Hilbert (January 23, 1862, Wehlau, East Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Stephen Cole Kleene (January 5, 1909 – January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... Hilary Whitehall Putnam (born July 31, 1926) is a key figure in the philosophy of mind during the 20th century. ... A. Thomas Tymoczko (1943-1996) was a philosopher specializing in logic and the philosophy of mathematics. ... A. Thomas Tymoczko (1943-1996) was a philosopher specializing in logic and the philosophy of mathematics. ... Hermann Weyl Hermann Weyl (November 9, 1885 - December 8, 1955) was a German mathematician. ... Raymond Louis Wilder (3 Nov 1896, Palmer, Massachusetts - 7 July 1982, Santa Barbara, California) was an American mathematician, who specialized in topology and gradually acquired philosophical and anthropological interests. ...

See also

• Euclid's Elements
• Kaina Stoicheia
• Liar paradox
• New Foundations
• Philosophy of mathematics
• Principia Mathematica
• Quasi-empiricism in mathematics
• The Simplest Mathematics

The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates... Kaina Stoicheia (Καινα στοιχεια) or New Elements is the title of several manuscript drafts of a document that Charles Sanders Peirce wrote circa 1904, intended as a preface to a book on the foundations of mathematics. ... In philosophy and logic, the liar paradox encompasses paradoxical statements such as: Analyzing the statement I am lying now, if what the speaker says is true, then the statement I am lying now is false, that means when the statement was said, the speaker was actually lying. ... In mathematical logic, New Foundations (NF) is a candidate set theory proposed by Willard van Orman Quine, obtained from a streamlined version of the theory of types of Bertrand Russell. ... Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will... The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ... Quasi-empiricism in mathematics is the movement in the philosophy of mathematics to direct philosophers attention to mathematical practice, in particular, relations with physics and social sciences, rather than the foundations problem in mathematics. ... The Simplest Mathematics is the title of a paper by Charles Sanders Peirce, intended as Chapter 3 of his unfinished magnum opus, the Minute Logic. The paper is dated January–February 1902 but was not published until the appearance of his Collected Papers, Volume 4 in 1933. ...

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