Editing Philosophy of mathematics (section)

===Social constructivism<!--'Postmodern mathematics' redirects here-->===
{{Main|Social constructivism}}
[[Social constructivism]] sees mathematics primarily as a [[Social constructionism|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 evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that 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. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints—the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated—that work to conserve the historically defined discipline.

This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as [[mathematical practice]] evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and [[mathematical folklore|folk mathematics]] not enough, due to an overemphasis on axiomatic proof and peer review as practices.

The social nature of mathematics is highlighted in its [[subculture]]s. 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. Social constructivists argue each speciality forms its own [[epistemic community]] and often has great difficulty communicating, or motivating the investigation of [[unifying conjecture]]s that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's [[cognitive bias]], or of mathematicians' [[collective intelligence]] as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless.

Contributions to this school have been made by [[Imre Lakatos]] and [[Thomas Tymoczko]], although it is not clear that either would endorse the title.{{Clarify|date=August 2010}} More recently [[Paul Ernest]] has explicitly formulated a social constructivist philosophy of mathematics.<ref>{{cite web |first=Paul |last=Ernest |url=http://www.people.ex.ac.uk/PErnest/pome12/article2.htm |title=Is Mathematics Discovered or Invented? |publisher=University of Exeter |access-date=2008-12-26 |archive-date=2008-04-05 |archive-url=https://web.archive.org/web/20080405035604/http://www.people.ex.ac.uk/PErnest/pome12/article2.htm |url-status=live }}</ref> Some consider the work of [[Paul Erdős]] 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 [[Erdős number]]. [[Reuben Hersh]] has also promoted the social view of mathematics, calling it a "humanistic" approach,<ref>{{cite interview |first=Reuben |last=Hersh |interviewer=John Brockman |url=http://edge.org/documents/archive/edge5.html |title=What Kind of a Thing is a Number? |publisher=Edge Foundation |date=February 10, 1997 |access-date=2008-12-26 |archive-url=https://web.archive.org/web/20080516103111/http://edge.org/documents/archive/edge5.html |archive-date=May 16, 2008 |url-status=dead }}</ref> similar to but not quite the same as that associated with Alvin White;<ref>{{cite web |title=Humanism and Mathematics Education |work=Math Forum |url=http://mathforum.org/mathed/humanistic.math.html |publisher=Humanistic Mathematics Network Journal |access-date=2008-12-26 |archive-date=2008-07-24 |archive-url=https://web.archive.org/web/20080724071728/http://mathforum.org/mathed/humanistic.math.html |url-status=live }}</ref> one of Hersh's co-authors, [[Philip J. Davis]], has expressed sympathy for the social view as well.