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, if any, do mathematical entities such as numbers exist?" and "why and how are mathematical statements true?". Some philosophers of mathematics view their task as being to give an account of mathematics as it stands, as interpretation rather than criticism. However, others' conclusions can have important ramifications for mathematical practice and so the philosophy of mathematics can be of very direct interest to working mathematicians.
The philosophy of mathematics has seen several different schools which will be presented in this article. Three of these, 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 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.

Mathematical Realism, or Platonism
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 Plato's belief in a "heaven of ideas", an unchanging ultimate reality that the everday world can only imperfectly approximate. Plato's view probably derives from Pythagoras, 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 Erdös and Kurt Gödel. 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 2 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 hypothesis, might prove not be decidable just on the basis of such principles. Gödel suggested quasiempirical 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 Putnam, is the Indispensability Argument. It either offers convincing answers to such questions or allows us to dispense with them entirely, but does so by stripping mathematics of some of its epistemic status.
The Indispensability Argument is as follows: 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. Unlike more traditional versions of realism it does not allow us to view mathematics as a body of certain knowledge: on this view, mathematics is dependent upon science for validation.
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 antirealist philosophy of mathematics.
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, you can generate the string corresponding to the Pythagorean theorem).
According to some versions of formalism, the subject matter of mathematics is then literally the written symbols themselves. Then any game is equally good, and one can only play the games, not prove things about them. Unfortunately, this does not solve the epistemic problems (what are symbols? do they exist in an eternal, unchanging realm?), does not explain the usefulness of mathematics, and renders mathematics an utterly spurious activity. This version of formalism is not widely accepted.
A second version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem, is not an absolute truth, but a relative one: if you assign meaning to the strings in such a way that the rules of the game become true (ie. True statements are assigned to the axioms and the rules of inference are truth preserving), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. But it does allow the working mathematician to continue in his work and leave such problems to the philosopher or scientist. Many formalists would say that in practice the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics.
A major early proponent of Formalism was David Hilbert, whose goal was a complete and consistent axiomatization of all of mathematics. ("Consistent" here means that no contradictions can be derived from the system.) Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive whole numbers, chosen to be philosophically uncontroversial) was consistent. Hilbert's program was dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible).
Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.
Modern Formalists, such as Rudolf Carnap, Alfred Tarski and Haskell Curry, continue to maintain that mathematics is the investigation of formal axiom systems. mathematical logicians study formal systems but are just as often platonists as they are formalists.
Formalists are usually very tolerant and inviting to new approaches to logic, nonstandard number systems, new set theories etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The 'games' are never arbitrarily chosen.
The main problem with Formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the minute string manipulation games mentioned above. While published proofs (if correct) could in principle be formulated in terms of these games, the rules are certainly not substantial to the initial creation of those proofs. Formalism is also silent to the question of which axiom systems ought to be studied.
Logicism holds that logic is the proper foundation of mathematics, and that all mathematical statements are necessary logical truths. For instance, the statement "If Aristotle is a human, and every human is mortal, then Aristotle is mortal" is a necessary logical truth. To the Logicist, all mathematical statements are precisely of the same type; they are analytic truths, or tautologies.
Gottlob Frege was the founder of logicism. In his seminal Grundlagen Der Aritmertik (Foundations of arithmetic) he built up arithmetic from a system of logic with Basic Law V (for concepts F and G, the extension of F equals the extension of G if and only if for all objects a, Fa if and only if Ga), a principle that he took to be acceptable as part of logic.
But Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent (this is Russell's Paradox). Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead. They attributed the paradox to "vicious circularity" and built up an elaborate theory of ramified types to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex, form (for example, the numbers were different in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of maths, such as an "axiom of reducibility". Even Russell said that this axiom did not really belong to logic.
Modern logicists, have returned to a program closer to Frege's. They have abandoned Basic Law V in favour of abstraction principles such as Hume's Principle (the number of objects falling under the concept F equals the number of objects falling under the concept G if and only if the extension of F and the extension of G can be put into onetoone correspondence). . Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's Principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possiblility that Julius Caesar=2.
Constructivism and Intuitionism
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 intuistic 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 lead 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.
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. Núńez. (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.
Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko[?].
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