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Some philosophers of mathematics view their task as giving an account of mathematics and mathematical practice as it stands, as interpretation rather than criticism. Criticisms can however have important ramifications for mathematical practice and so the philosophy of mathematics can be of very direct interest to working mathematicians, particularly in new fields where the process of peer review of mathematical proofs is not firmly established, raising probability of an undetected error. Such errors can thus only be reduced by knowing where they are likely to arise. This is a prime concern of the philosophy of mathematics.
More recently some practitioners have also attempted to relate mathematics to general concerns of philosophy: epistemology and ethics in particular. Those concerns are dealt with at the end of this article.
The philosophy of mathematics has seen several different schools or strains, which primarily focus on metaphysics questions, ie, "Why does it work?". And, the related but logically separate, "Why does mathematics explain the physical world as we see it so well?"
Three schools, intuitionism, logicism and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that (as it stood) mathematics, and analysis in particular, did not live up to the standards of certainty and rigour with which it was over-credited. Each school addresses the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.
As certainty waned, the original foundations problem in mathematics ("which branch of mathematics is the one from which others are derived?") was restated as an open exploration of foundations of mathematics and shared dependency on certain core concepts like order, and then finally as the subset field metamathematicsMetamathematics is mathematics used to study mathematics. It was originally differentiated from ordinary mathematics in the 19th century to focus on what was then called the foundations problem in mathematics. Important branches include proof theory and m which seems simply to be "mathematics useful in doing open-ended metaphysics about mathematics".
The schools are addressed separately here and their assumptions explained:
Mathematical realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel PlatoFor the computing technology, see PLATO System. Plato ( Greek: Platon (c. 427 BC c. 347 BC) was an immensely influential classical Greek philosopher, student of Socrates, teacher of Aristotle, writer, and founder of the Academy in Athens. Plato, who is be's belief in a "heaven of ideas", an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato's view probably derives from PythagorasPythagoras ( 582 BC 496 BC, Greek: Πυθαγρας) was an Ionian mathematician and philosopher, known best for formulating the Pythagorean theorem. Pythagoras, known as "the father of numbers", made influential cont, and his followers the Pythagoreans, who believed that the world was, quite literally, built up by the numbers. This idea may have even older origins that are unknown to us.
Many working mathematicians are mathematical realists; they see themselves as discoverers. Examples are Paul ErdösPaul Erdos ( March 26, 1913 September 20, 1996) was an immensely prolific and famously eccentric mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, and Kurt GödelKurt Godel [ kurˈt godl ], ( April 28, 1906 January 14, 1978) was a logician, mathematician, and philosopher of mathematics, whose biography lists quite a few nations, although he is usually associated with Austria. He was born in Brunn in Austria-. Psychological reasons have been given for this preference: it appears to be very hard to preoccupy oneself over long periods of time with the investigation of an entity in whose existence one doesn't firmly believe. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (eg, for any two mathematical objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than t, might prove undecidable just on the basis of such principles. Gödel suggested quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.
The major problem of mathematical realism is this: precisely where and how do the mathematical entities exist? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? Gödel's and Plato's answers to each of these questions are much criticised.
An important argument for mathematical realism, formulated by Quine and PutnamHilary Whitehall Putnam (born July 31, 1926) is a key figure in the philosophy of mind during the 20th century. After receiving his BA at the University of Pennsylvania (where he was an undergraduate with Noam Chomsky) and PhD at UCLA (under Hans Reichenb, is the Indispensability Argument: mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of some of its epistemic status.
Most forms of logicism (see below) are forms of mathematical realism. For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Maddy's Realism in Mathematics. Intuitionism is the classic example of an anti-realist philosophy of mathematics.
Putnam strongly rejected the term " Platonist" as implying an overly-specific ontology that was not necessary to mathematical practice in any real sense - he advocated a form of "pure realism" that rejected mystical notions of truth and accepted much quasi-empiricism in mathematics - a term that he was involved in coining (see below). An example of a theory that both embraces realism and rejects Platonism is the embodied mind theory (see below).
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