Philosophy of mathematics

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?".

The philosophy of mathematics has seen several different schools which will be presented in this article.

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.

The main argument for Mathematical Realism, formulated by Quine and Putnam, 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.

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.

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?

Formalism

Formalism holds that mathematical statements are statements about the consequences of certain string manipulation rules. In the game of chess, for instance, one can prove that a king and a rook suffice to mate the lone king. Similarly, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some rules to generate new strings from given ones), one can prove that the Pythagorean theorem holds. The Pythagorean theorem, therefore, is not an absolute truth, but a relative one: if you accept the rules of the game, then you have to accept the theorem. The same is held to be true for all other mathematical statements.

Formalism need not (and is not necessarily intended to) mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some ontology 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 may 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) 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.

Modern Formalists, such as Rudolf Carnap, Alfred Tarski and Haskell Curry, continue to maintain that mathematics is the investigation of formal axiom systems. Most mathematical logicians nowadays describe themselves as Formalists.

Formalists are usually very tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc. The more games we study, the better.

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

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.

This view was defended by Gottlob Frege, and also by Russell and Whitehead with their Principia Mathematica.

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.

Embodied mind theories

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. 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.

Social Constructivism

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.