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?".

Relation to philosophy proper

Some philosophers of mathematics view their task as being to give 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 claims for finished mathematics 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.

Why does it work?

The philosophy of mathematics has seen several different schools or strains, which primarily focus on metaphysics questions, ie, "Why does it work?", "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 metamathematics 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, 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 Erd�s and Kurt G�del. 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 hypothesis, 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 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 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.

Formalism

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 G�del'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, non-standard 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

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 Die Grundgesetze der Arithmetik (Basic Laws 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 one-to-one 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 led 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. 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. N��ez. (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 or Social Realism

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.

This theory seems intuitively wrong given the 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.

Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko. Some consider the work of Paul Erd�s 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.

Beyond the "schools"

Rather than focus on narrow debates about the "true nature" of mathematical truth, or even on practices unique to mathematicians such as the proof, a growing movement from the 1960s to the 1990s began to question the idea of seeking "foundations" or finding any one "right answer" to why mathematics works. The starting point for this was Eugene Wigner's famous 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences, in which he argued the happy coincidence that mathematics and physics were so well matched, seemed to be "unreasonable" and hard to explain.

The embodied-mind or "cognitive" school and the "social" school were responses to this challenge. But the debates raised were difficult to confine to those:

Quasi-empiricism

One parallel concern that does not actually challenge the schools directly but questions their focus is the notion of quasi-empiricism in mathematics. This grew from the increasingly popular assertion in the late 20th century that no one foundation of mathematics could be ever proven to exist. It is also sometimes called 'postmodernism in mathematics' although that term is considered overloaded by some and insulting by others. It is a very minimal form of social realism/constructivism that accepts that quasi-empirical methods and even sometimes empirical methods can be part of modern mathematical practice.

Such methods have always been part of folk mathematics by which great feats of calculation and measurement are sometimes achieved. Indeed, such methods may be the only notion of "proof" a culture has.

Hilary Putnam argued that any theory of mathematical realism would include quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic "proofs", and still be "doing mathematics" - at perhaps a somewhat greater risk of failure of their calculations. He laid out a quite detailed argument for this in New Directions (ed. Tymockzo, 1998).

Action

Many practitioners and scholars who are not engaged primarily in proofs have made interesting and important observations about the nature of mathematics:

Judea Pearl claimed that all of mathematics as presently understood was based on an algebra of seeing - and proposed an algebra of doing to complement it - this is a central concern of the philosophy of action and other studies of how "knowing" relates to "doing", or knowledge to action. The most important output of this was new theories of truth, notably those appropriate to activism and grounding empirical methods.

Unification

The notion of a philosophy of mathematics separate from philosophy as such has been criticized as leading to "good mathematicians doing bad philosophy" - few philosophers being able to penetrate mathematical notations and culture to actually relate conventional notions of metaphysics to the more specialized metaphysical notions of the 'schools' above. This may lead to a disconnection in which the mathematicians continue to spout bad and discredited philosophy as a justification for their continued belief in a world-view promoting their work.

Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human perception and everyday understandings of knowledge.

Ethics

As well, there is little or no consideration given to the ethics of doing mathematics, it being seen in a technological culture as an absolute necessity whose value cannot be questioned and whose implications cannot be avoided - even if particular branches have no known purpose, or are considered useful primarily or only to enable conflict, e.g. cryptography, steganography, which are about keeping secrets, or the mathematics involved in optimizing nuclear fission reactions in bombs. While most would accept that physicists bear some moral responsibility for these activities, few have been willing to also so criticize mathematicians.

Some of these criticisms have been explored in the sociology of knowledge, but in general mathematics itself has evaded the scrutiny often applied to the sciences of genetics, physics, economics or medicine. Which is interesting in itself, as mathematics is necessary to enable those and other sciences.

Evolutionary psychology for instance has embraced the idea that "the mind is a computer" in the sense of a Turing Machine. What are the implications of adopting an abstraction originating to explain computers formally, to explain the mind?

Aesthetics

Another criticism is that mathematics can be seen very narrowly as the science of measurement and as a vast number of trustworthy shortcuts to reduce the need to measure directly, and simplify calculation. Some of the schools have assigned rather more significance to mathematics than this mere utility -- even seeking sometimes moral guidance, or aesthetics of truth and beauty, in its abstractions. Some consider this a symptom of scientism. Keeping the philosophy of mathematics as a subfield that asks only or primarily 'why does it work?' assuming that it in fact does work in a social or biological sense, as opposed to the narrow sense of physics. It is as inappropriate in this view as having, say, a philosophy of weapons or of war, separate from that of the larger social and species and planetary context of it.

This question is usually rejected by working mathematicians as "irrelevant", but of course they are exactly those people whose aesthetics of proof and of rigour have been already accepted -- they may thus be practicing self-selection of a particular aesthetic, and propagating it with few constraints, especially in those fields where mathematics is not immediately applied to life.

Language

Finally, although many or most mathematicians or philosophers would accept the statement " mathematics is a language", there is little attention paid to the implications of that statement. Linguistics is not applied to discourses or symbol systems of mathematics, that is, mathematics is studied in a markedly different way than other languages. The capacity to acquire mathematics, and competence in it, called numeracy, is seen as separate from literacy and the acquisition of language.

Some argue that this is due to failures not of the philosophy of mathematics, but of linguistics and the study of natural grammar. These fields, they say, are not rigorous enough, and that linguistics needs to "catch up". But this implies that mathematics is inherently superior to all other knowledge, e.g. ecological wisdom accrued by a culture of people living on the land. Standards of rigour vary in language, but "more" may not be "better".

Others argue that computer science is the proper study of these more "linguistic" questions, and that its analysis of programming languages is also often just as applicable to mathematics or at least some metamathematics.