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These schools maintain that only mathematical entities which can be explicitly constructed have a claim to existence and should be admitted in mathematical discourse.
A typical quote comes from Leopold Kronecker: "The natural numbers come from God, everything else is men's work." A major force behind Intuitionism was L.E.J. Brouwer, who postulated a new logic different from the classical Aristotelian logic; this intuitionistic logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected. Important work was later done by Errett Bishop, who managed to prove versions of the most important theorems in real analysis within this framework.
In Intuitionism, the term "explicit construction" is not cleanly defined, and that has lead to criticisms. Attempts have been made to use the concepts of Turing machine or recursive function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing.
See also: Mathematical constructivism, Mathematical intuitionism
These theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of number springs from the experience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.
The physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation. However, the human mind has no special claim on "reality" or approaches to it built out of math; If such constructs as Euler's Identity are "true" then they are true as a map of the human mind and cognition, not as a map of anything it "sees".
The effectiveness of mathematics is thus easily explained: mathematics was constructed by the brain in order to be effective in this universe.
The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. Núñez. (Since this book was first published in the year 2000, it may still be one of the only treatments of this perspective.) For more on the science that inspired this perspective, see cognitive science of mathematics.
This theory sees mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly compared to reality and may be discarded if they don't agree with observation or prove pointless. The direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it.
To some mathematicians, this theory seems intuitively wrong given their seeming permanence of mathematics. But this permanence is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous finished mathematics is cast into doubt, and is re-examined and corrected only to the degree it is required or desired by the needs of current applications and groups. Errors occur and persist, sometimes for generations, and notational bias is common. Finished mathematics is often accorded too much status, and folk mathematics not enough, due to an over-belief in axiomatic proof and peer review as practices.
Mathematics also has subcultures. Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Each speciality forms its own epistemic community and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics.
If the social process of 'doing mathematics' is seen as actually creating the meaning, the term social constructivism is more appropriate. If deficiency of human capacity to abstract, human cognitive bias, or lack of sufficient collective intelligence is seen as preventing the comprehension of a 'real' universe of 'mathematical objects', the term social realism is more appropriate.
Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless. Some social scientists also argue that mathematics is not real or objective at all, but is affected by racism and ethnocentrism. Some of these ideas are close to postmodernism.
Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko . Some consider the work of Paul Erdös as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g. via the Erdos number. This strongly influenced work on measuring reputation but has had little impact on mathematics as such.
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