Philosophy of mathematics
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The '''philosophy of mathematics''' is the branch of [[philosophy]] that studies the philosophical assumptions, foundations, and implications of [[mathematics]].
Recurrent themes include:
:* What are the sources of mathematical subject matter?
:* What is the [[ontology|ontological]] status of mathematical entities?
:* What does it mean to refer to a mathematical object?
:* What is the character of a mathematical proposition?
:* What is the relation between [[logic]] and mathematics?
:* What is the role of [[hermeneutics]] in mathematics?
:* What kinds of inquiry play a role in mathematics?
:* What are the objectives of mathematical inquiry?
:* What gives mathematics its hold on [[experience]]?
:* What are the [[human trait]]s behind mathematics?
:* What is [[mathematical beauty]]?
:* What is the source and nature of mathematical truth?
:* What is the relationship between the abstract world of mathematics and the material universe?
The terms ''philosophy of mathematics'' and ''mathematical philosophy'' are frequently used as synonyms.<ref>{{cite journal |first=Edward A. |last=Maziars |title=Problems in the Philosophy of Mathematics (Book Review) | journal=Philosophy of Science | volume=36 | issue=3 |pages=p. 325 |year=1969}}. For example, when Edward Maziars proposes in a 1969 book review ''"to distinguish philosophical mathematics (which is primarily a specialized task for a mathematician) from mathematical philosophy (which ordinarily may be the philosopher's metier)"'', he uses the term ''mathematical philosophy'' as being synonymous with ''philosophy of mathematics''.</ref>
<!-- LONG REF ENDS HERE!-->
The latter, however, may be used to mean at least three other things. One sense refers to a project of formalizing a philosophical subject matter, say, [[aesthetics]], [[ethics]], [[logic]], [[metaphysics]], or [[theology]], in a purportedly more exact and rigorous form, as for example the labors of [[Scholasticism|Scholastic]] theologians, or the systematic aims of [[Leibniz]] and [[Spinoza]]. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by [[Bertrand Russell]] in his book ''[[Introduction to Mathematical Philosophy]]''.
== Historical overview ==
Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.
There are traditions of mathematical philosophy in both [[History of Western philosophy|Western philosophy]] and [[Eastern philosophy]]. Western philosophies of mathematics go as far back as [[Plato]], who studied the [[ontology|ontological status]] of mathematical objects, and [[Aristotle]], who studied [[logic]] and issues related to [[infinity]] (actual versus potential). [[Ancient Greece|Greek]] philosophy on mathematics was strongly influenced by their study of [[geometry]]. At one time, the Greeks held the opinion that 1 (one) was not a [[number]], but rather a unit of arbitrary length. A number was defined as a multitude. Therefore 3, for example, represented a certain multitude of units, and truly was ''not'' a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportional to the arbitrary first "number" or "one." These earlier Greek ideas of number were later upended by the discovery of the [[Irrational number|irrationality]] of the square root of two. Hippasus, a disciple of Pythagoras, showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea.
Beginning with [[Leibniz]], the focus shifted strongly to the relationship between mathematics and logic. This view dominated the philosophy of mathematics through the time of [[Gottlob Frege|Frege]] and of Russell, but was brought into question by developments in the late 19th and early 20th century.
==Philosophy of mathematics in the 20th century==
A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in [[formal logic]], [[set theory]], and foundational issues.
It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the [[foundations of mathematics]] program.
At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical [[epistemology]] and [[ontology]]. Three schools, [[formalism]], [[intuitionism]], and [[logicism]], emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and [[mathematical analysis|analysis]] in particular, did not live up to the standards of [[certainty]] and [[rigor]] that had been taken for granted. Each school addressed 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.
Surprising and counterintuitive developments in [[formal logic]] and [[set theory]] early in the 20th century led to new questions concerning what was traditionally called the ''[[foundations of mathematics]]''. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of [[Euclid]] around 300 BCE as the natural basis for mathematics. Notions of [[axiom]], [[proposition]] and [[proof]], as well as the notion of a proposition being true of a mathematical object (see [[Assignment (mathematical logic)]]), were formalized, allowing them to be treated mathematically. The [[Zermelo-Fraenkel set theory|Zermelo-Fraenkel]] axioms for [[set|set theory]] were formulated which provided a conceptual framework in which much mathematical discourse would be interpreted. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. With [[Gödel numbering]], propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the [[consistency proof|consistency]] of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led [[Hilbert]] to call such study ''[[metamathematics]]'' or ''[[proof theory]]''.<ref name="Kleene 1971">
{{cite book
|last=Kleene
|first=Stephen
|authorlink = Stephen Cole Kleene
|title=Introduction to Metamathematics
|pages=p. 5
|year=1971
|publisher=North-Holland Publishing Company
|location=Amsterdam, Netherlands
|language =
}}
</ref>
At the middle of the century, a new mathematical theory known as [[category theory]] arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. [[Hilary Putnam]] summed up one common view of the situation in the last third of the century by saying:
<blockquote>
When philosophy discovers something wrong with science, sometimes science has to be changed — [[Russell's paradox]] comes to mind, as does [[George Berkeley|Berkeley]]'s attack on the actual [[infinitesimal]] — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need. (Putnam, 169-170).
</blockquote>
Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.
==Contemporary schools of thought==
===Mathematical realism===
{{Refimprove|date=February 2007}}
''Mathematical realism'', like [[Philosophical realism|realism]] in general, 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. In this point of view, there is really one sort of mathematics that can be discovered: [[Triangle]]s, for example, are real entities, not the creations of the human mind.
Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include [[Paul Erdős]] and [[Kurt Gödel]]. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two 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 that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.
Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.
====Platonism====
''Platonism'' is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. The term ''Platonism'' is used because such a view is seen to parallel [[Plato]]'s belief in a "World of Ideas" (typified by [[Allegory of the cave|Plato's cave]]): the everyday world can only imperfectly approximate of an unchanging, ultimate reality. Both ''Plato's cave'' and ''Platonism'' have meaningful, not just a superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular ''[[Pythagoreans]]'' of ancient Greece, who believed that the world was, quite literally, generated by [[number]]s.
The major problem of mathematical platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? 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? One answer might be [[Ultimate ensemble]], which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.
Gödel's platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things [[Husserl]] said about mathematics, and supports [[Kant]]'s idea that mathematics is [[Analytic-synthetic distinction#Conceptual containment|synthetic]] [[A priori and a posteriori (philosophy)|a priori]].) [[Philip J. Davis|Davis]] and [[Reuben Hersh|Hersh]] have suggested in their book ''The Mathematical Experience'' that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to [[Philosophy_of_mathematics#Formalism|formalism]] (see below).
Some mathematicians hold opinions that amount to more nuanced versions of Platonism. These ideas are sometimes described as [[neoplatonism|Neo-Platonism]].
====Logicism====
[[Image:Gottlob Frege.jpg|right|thumb|[[Gottlob Frege]], founder of logicism]]
''Logicism'' is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists hold that mathematics can be known ''[[a priori]]'', but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus [[analytic proposition|analytic]], not requiring any special faculty of mathematical intuition. In this view, [[logic]] is the proper foundation of mathematics, and all mathematical statements are necessary logical truths.
[[Rudolf Carnap]] (1931) presents the logicist thesis in two parts:
:{|
|- valign="top"
| 1. || The ''concepts'' of mathematics can be derived from logical concepts through explicit definitions.
|- valign="top"
| 2. || The ''theorems'' of mathematics can be derived from logical axioms through purely logical deduction.
|}
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 a general principle of comprehension, which he called "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.
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 [[Alfred North Whitehead|Whitehead]]. They attributed the paradox to "vicious circularity" and built up what they called [[ramified type theory]] 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, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of mathematics, such as an "[[axiom of reducibility]]". Even Russell said that this axiom did not really belong to logic.
Modern logicists (like [[Bob Hale (philosopher)|Bob Hale]], [[Crispin Wright]], and perhaps others) 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 [[bijection|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 possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.
If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense, logicism can be seen as shifting questions about the philosophy of mathematics to questions about logic without fully answering them.
====Empiricism====
''Empiricism'' is a form of realism that denies that mathematics can be known [[A priori and a posteriori (philosophy)|a priori]] at all. It says that we discover mathematical facts by [[empirical]] research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th century, but primarily arose in the middle of the century. However, an important early proponent of a view like this was [[John Stuart Mill]]. Mill's view was widely criticized, because it makes statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.
Contemporary mathematical empiricism, formulated by [[W.V. Quine|Quine]] and [[Hilary Putnam|Putnam]], is primarily supported by 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. That is, since physics needs to talk about [[electron]]s to say why light bulbs behave as they do, then electrons must exist. Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist. 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 distinctness from the other sciences.
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]]. Putnam was involved in coining the term "pure realism" (see below).
The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e. [[consilience]] after [[E O Wilson]]. Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is incredibly central, and that it would be extremely difficult for us to revise it, though not impossible.
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 [[Penelope Maddy]]'s ''Realism in Mathematics''. Another example of a realist theory is the [[embodied mind theory]] (see below).
For experimental evidence suggesting that one-day-old babies can do elementary arithmetic, see [[Brian Butterworth]].
====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). Mathematical truths are not about numbers and sets and triangles and the like — in fact, they aren't "about" anything at all!
Another 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. (Compare this position to [[Structuralism#Structuralism in the philosophy of mathematics|structuralism]].) But it does allow the working mathematician to continue in his or her 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.
[[Image:Hilbert1912.jpg|right|thumb|[[David Hilbert]]]]
A major early proponent of formalism was [[David Hilbert]], whose [[Hilbert's program|program]] was intended to be a [[Gödel's completeness theorem|complete]] and [[consistency proof|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 [[integers]], chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent was dealt a fatal blow by the second of [[Gödel's incompleteness theorem]]s, 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). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.
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.
Other formalists, such as [[Rudolf Carnap]], [[Alfred Tarski]] and [[Haskell Curry]], considered mathematics to be the investigation of [[formal system|formal axiom systems]]. [[Mathematical logic]]ians study formal systems but are just as often realists as they are formalists.
Formalists are relatively 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 usually not arbitrary.
The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent to the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.
Recently, some formalist mathematicians have proposed that all of our ''formal'' mathematical knowledge should be systematically encoded in [[computer-readable]] formats, so as to facilitate [[proof checking|automated proof checking]] of mathematical proofs and the use of [[proof assistant|interactive theorem proving]] in the development of mathematical theories and computer software. Because of their close connection with [[computer science]], this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition (see below). See [[QED project]] for a general overview.
===Intuitionism===
{{main|Mathematical intuitionism}}
In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" ([[Luitzen Egbertus Jan Brouwer|L.E.J. Brouwer]]). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the ''a priori'' forms of the volitions that inform the perception of empirical objects. (CDP, 542)
[[Leopold Kronecker]] said: "The natural numbers come from God, everything else is man's work." A major force behind Intuitionism was [[L.E.J. Brouwer]], who rejected the usefulness of formalized logic of any sort for mathematics. His student [[Arend Heyting]] postulated an [[intuitionistic logic]], different from the classical [[Aristotelian logic]]; this logic does not contain the [[Law of excluded middle|law of the excluded middle]] and therefore frowns upon [[Reductio ad absurdum|proofs by contradiction]]. The [[axiom of choice]] is also rejected in most intuitionistic set theories, though in some versions it is accepted. 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 led to criticisms. Attempts have been made to use the concepts of [[Turing machine]] or [[computable function]] to fill this gap, leading to the claim that only questions regarding the behavior of finite [[algorithm]]s are meaningful and should be investigated in mathematics. This has led to the study of the [[computable number]]s, first introduced by [[Alan Turing]]. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical [[computer science]].
===Constructivism===
{{main|Mathematical constructivism}}
{{Sectstub|date=May 2008}}
Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.
===Fictionalism===
[[Fictionalism]] in mathematics was brought to fame in 1980 when [[Hartry Field]] published ''Science Without Numbers'', which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of [[Newtonian mechanics]] that didn't reference numbers or functions at all. He started with the "betweenness" of [[Hilbert's axioms]] to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by [[vector field]]s. Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.
Having shown how to do science without using mathematics, he proceeded to rehabilitate mathematics as a kind of [[useful fiction]]. He showed that mathematical physics is a [[conservative extension]] of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from his system), so that the mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as false as "[[Sherlock Holmes]] lived at 221B Baker Street" — but both are true according to the relevant fictions.
By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about [[fiction]] in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of [[second-order logic]] to carry out his reduction, and because the statement of conservativity seems to require [[quantification]] over abstract models or deductions. Another objection is that it is not clear how one could have certain results in science, such as quantum theory or the periodic table, without mathematics. If what distinguishes one element from another is ''precisely'' the number of electrons, neutrons and protons, how does one distinguish between elements without a concept of number?{{Or|date=April 2008}}
===Embodied mind theories===
''Embodied mind 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.
With this view, 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]].
Embodied mind theorists thus explain the effectiveness of mathematics — 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]]. In addition, mathematician [[Keith Devlin]] has investigated similar concepts with his book ''[[The Math Instinct]]''. For more on the philosophical ideas that inspired this perspective, see [[cognitive science of mathematics]].
===Social constructivism or social realism===
''Social constructivism'' or ''social realism'' theories see 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 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 [[folk mathematics]] not enough, due to an over-emphasis on axiomatic proof and peer review as practices. However, this might be seen as merely saying that rigorously proven results are overemphasized, and then "look how chaotic and uncertain the rest of it all is!"
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. Some social scientists also argue that mathematics is not real or objective at all, but is affected by [[racism]] and [[ethnocentrism]]. Some of these ideas are close to [[postmodernism]].
Contributions to this school have been made by [[Imre Lakatos]] and [[Thomas Tymoczko]], although it is not clear that either would endorse the title. More recently [[Paul Ernest]] has explicitly formulated a social constructivist philosophy of mathematics. [http://www.people.ex.ac.uk/PErnest/pome12/article2.htm] 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 [http://edge.org/documents/archive/edge5.html], similar to but not quite the same as that associated with Alvin White [http://mathforum.org/mathed/humanistic.math.html]; one of Hersh's co-authors, [[Philip J. Davis]], has expressed sympathy for the social view as well.
A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity. To observe that rigorous proof comes only after unrigorous conjecture, experimentation and speculation is true, but it is trivial and no-one would deny this. So it's a bit of a stretch to characterize a philosophy of mathematics in this way, on something trivially true. The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrass in order to rigorously prove the theorems thereof. There is nothing special or interesting about this, as it fits in with the more general trend of unrigorous ideas which are later made rigorous. There needs to be a clear distinction between the objects of study of mathematics and the study of the objects of study of mathematics. The former doesn't seem to change a great deal; the latter is forever in flux. The latter is what the Social theory is about, and the former is what Platonism et al. are about.
However, this criticism is rejected by supporters of the [[social constructivist]] perspective because it misses the point that the very objects of mathematics are social constructs. These objects, it asserts, are primarily [[semiotic]] objects existing in the sphere of human culture, sustained by social practices (after [[Wittgenstein]]) that utilize physically embodied [[signs]] and give rise to intrapersonal (mental) constructs. Social constructivists view the [[reification]] of the sphere of human culture into a [[Platonic]] realm, or some other heaven-like domain of existence beyond the physical world, a long standing [[category error]].
=== Beyond the traditional schools ===
{{Refimprove|date=February 2007}}
Rather than focus on narrow debates about the true nature of mathematical [[truth]], or even on practices unique to mathematicians such as the [[mathematical proof|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 that the happy coincidence of mathematics and physics being 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 instead 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. Quasi-empiricism argues that in doing their research, mathematicians test hypotheses as well as proving theorems. A mathematical argument can transmit falsity from the conclusion to the premises just as well as it can transmit truth from the premises to the conclusion. [[Quasi-empiricism]] was developed by [[Imre Lakatos]], inspired by the philosophy of science of [[Karl Popper]].
[[Imre Lakatos|Lakatos]]' philosophy of mathematics is sometimes regarded as a kind of social constructivism, but this was not his intention.
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]] has 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 gave a detailed argument for this in ''New Directions'' (ed. Tymockzo, 1998).
==== Unification ====
Few philosophers are able to penetrate mathematical notations and culture to relate conventional notions of [[metaphysics]] to the more specialized metaphysical notions of the schools above. This may lead to a disconnection in which some mathematicians continue to profess 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]].
==== Language ====
Innovations in the [[philosophy of language]] during the 20th century renewed interest in the question as to whether mathematics is, as if often said, the ''language'' of science. Although most mathematicians and physicists (and many philosophers) would accept the statement "[[mathematics as a language|mathematics is a language]]", linguists believe that the implications of such a statement must be considered. For example, the tools of [[linguistics]] are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way than other languages. If mathematics is a language, it is a different type of language than [[natural languages]]. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student [[Richard Montague]] and other linguists working in [[formal semantics]] to show that the distinction between mathematical language and natural language may not be as great as it seems.
See also [[philosophy of language]].
== Aesthetics ==
Many practising mathematicians have been drawn to their subject because of a sense of [[mathematical beauty|beauty]] they perceive in it. One sometimes hears the sentiment that mathematicians would like to leave philosophy to the philosophers and get back to mathematics — where, presumably, the beauty lies.
In his work on the [[divine proportion]], H. E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art — the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor. Indeed, one can study mathematical and scientific writings as [[literature]].
[[Philip J. Davis]] and [[Reuben Hersh]] have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of the [[Square root of 2|√2]]. The first is the traditional proof by [[contradiction]], ascribed to [[Euclid]]; the second is a more direct proof involving the [[fundamental theorem of arithmetic]] that, they argue, gets to the heart of the issue. Davis and Hersh argue that mathematicians find the second proof more aesthetically appealing because it gets closer to the nature of the problem.
[[Paul Erdős]] was well-known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs. There is not universal agreement that a result has one "most elegant" proof; [[Gregory Chaitin]] has argued against this idea.
Philosophers have sometimes criticized mathematicians' sense of beauty or elegance as being, at best, vaguely stated. By the same token, however, philosophers of mathematics have sought to characterize what makes one proof more desirable than another when both are logically sound.
Another aspect of aesthetics concerning mathematics is mathematicians' views towards the possible uses of mathematics for purposes deemed unethical or inappropriate. The best-known exposition of this view occurs in [[G.H. Hardy]]'s book [[A Mathematician's Apology]], in which Hardy argues that pure mathematics is superior in beauty to [[applied mathematics]] precisely because it cannot be used for war and similar ends. Some later mathematicians have characterized Hardy's views as mildly dated{{Fact|date=February 2007}}, with the applicability of number theory to modern-day [[cryptography]]. While this would force Hardy to change his primary example if he were writing today, many practicing mathematicians still subscribe to Hardy's general sentiments.{{Fact|date=February 2007}}
==Mathematics of philosophy==
Mathematics of philosophy is the branch of mathematics which, with mathematic methods, attempts to approach philosophic matters.
For instance, in [[utilitarism]], the [[units of measurement]]s called [[hedons and dolor]]s may be used in formulas of various complexity in order to get to what is the best action to do in different situations.
==See also==
===Related topics===
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{{col-break}}
* [[Axiomatic set theory]]
* [[Axiomatic system]]
* [[Category theory]]
* [[Formal language]]
* [[Formal system]]
* [[Foundations of mathematics]]
* [[Golden ratio]]
* [[History of mathematics]]
* [[Intuitionistic logic]]
* [[Logic]]
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* [[Mathematical beauty]]
* [[Mathematical constructivism]]
* [[Mathematical logic]]
* [[Mathematical proof]]
* [[Metamathematics]]
* [[Model theory]]
* [[Naive set theory]]
* [[Non-standard analysis]]
* [[Philosophy of language]]
* [[Philosophy of science]]
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* [[Philosophy of probability]]
* [[Proof theory]]
* [[Rule of inference]]
* [[Science studies]]
* [[Scientific method]]
* [[Set theory]]
* [[Truth]]
* ''[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]''
{{col-end}}
===Related works===
{{col-begin}}
{{col-break}}
* ''[[The Analyst]]''
* [[Euclid's Elements|Euclid's ''Elements'']]
* [[Original proof of Gödel's completeness theorem|Gödel's completeness theorem]]
* ''[[Introduction to Mathematical Philosophy]]''
{{col-break}}
* [[Kaina Stoicheia]]
* ''[[New Foundations]]''
* ''[[Principia Mathematica]]''
* ''[[The Simplest Mathematics]]''
{{col-end}}
===Historical topics===
* [[History and philosophy of science]]
* [[History of mathematics]]
* [[History of philosophy]]
==Further reading==
*The [http://www.ucl.ac.uk/philosophy/LPSG/ London Philosophy Study Guide] offers many suggestions on what to read, depending on the student's familiarity with the subject:
**[http://www.ucl.ac.uk/philosophy/LPSG/PhilMath.htm Philosophy of Mathematics]
**[http://www.ucl.ac.uk/philosophy/LPSG/MathLogic.htm Mathematical Logic]
**[http://www.ucl.ac.uk/philosophy/LPSG/SetTheory.htm Set Theory & Further Logic]
* Colyvan, Mark (2004), "Indispensability Arguments in the Philosophy of Mathematics", ''Stanford Encyclopedia of Philosophy'', [[Edward N. Zalta]] (ed.), [http://plato.stanford.edu/entries/mathphil-indis/ Eprint].
* [[Philip J. Davis|Davis, Philip J.]] and [[Reuben Hersh|Hersh, Reuben]] (1981), ''[[The Mathematical Experience]]'', Mariner Books, New York, NY.
* [[Keith Devlin|Devlin, Keith]] (2005), ''The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs)'', Thunder's Mouth Press, New York, NY.
* [[Michael Dummett|Dummett, Michael]] (1991 a), ''Frege, Philosophy of Mathematics'', Harvard University Press, Cambridge, MA.
* Dummett, Michael (1991 b), ''Frege and Other Philosophers'', Oxford University Press, Oxford, UK.
* Dummett, Michael (1993), ''Origins of Analytical Philosophy'', Harvard University Press, Cambridge, MA.
* [[Paul Ernest|Ernest, Paul]] (1998), ''Social Constructivism as a Philosophy of Mathematics'', State University of New York Press, Albany, NY.
* [[Alexandre George|George, Alexandre]] (ed., 1994), ''Mathematics and Mind'', Oxford University Press, Oxford, UK.
* [[Morris Kline|Kline, Morris]] (1972), ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, New York, NY.
* [[George Lakoff|Lakoff, George]], and [[Rafael E. Núñez|Núñez, Rafael E.]] (2000), ''[[Where Mathematics Comes From]]: How the Embodied Mind Brings Mathematics into Being'', Basic Books, New York, NY.
* [[Charles Peirce (Bibliography)|Peirce, C.S., Bibliography]].
* Raymond, Eric S. (1993), "The Utility of Mathematics", [http://catb.org/~esr/writings/utility-of-math/ Eprint].
* [[Stewart Shapiro|Shapiro, Stewart]] (2000), ''Thinking About Mathematics: The Philosophy of Mathematics'', Oxford University Press, Oxford, UK.
==External links==
*[http://www.rbjones.com/rbjpub/philos/maths/index.htm R.B. Jones' philosophy of mathematics page]
*{{dmoz|Society/Philosophy/Philosophy_of_Science/Mathematics/}}
*[http://www.dcorfield.pwp.blueyonder.co.uk/blog.html The Philosophy of Real Mathematics Blog]
*{{sep entry|philosophy-mathematics|Philosophy of Mathematics|Leon Horsten}}
===Journals===
* [http://philmat.oxfordjournals.org/ Philosophia Mathematica journal]
* [http://www.ex.ac.uk/~PErnest/ The Philosophy of Mathematics Education Journal homepage]
==References==
* [[Aristotle]], "[[Prior Analytics]]", [[Hugh Tredennick]] (trans.), pp. 181-531 in ''Aristotle, Volume 1'', [[Loeb Classical Library]], William Heinemann, London, UK, 1938.
* [[Robert Audi|Audi, Robert]] (ed., 1999), ''The Cambridge Dictionary of Philosophy'', Cambridge University Press, Cambridge, UK, 1995. 2nd edition, 1999. Cited as CDP.
* [[Paul Benacerraf|Benacerraf, Paul]], and [[Hilary Putnam|Putnam, Hilary]] (eds., 1983), ''Philosophy of Mathematics, Selected Readings'', 1st edition, Prentice-Hall, Englewood Cliffs, NJ, 1964. 2nd edition, Cambridge University Press, Cambridge, UK, 1983.
* [[George Berkeley|Berkeley, George]] (1734), ''[[The Analyst]]; or, a Discourse Addressed to an Infidel Mathematician. Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith'', London & Dublin. Online text, [[David R. Wilkins]] (ed.), [http://www.maths.tcd.ie/pub/HistMath/People/Berkeley/Analyst/Analyst.html Eprint].
* [[Nicolas Bourbaki|Bourbaki, N.]] (1994), ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany.
* [[Rudolf Carnap|Carnap, Rudolf]] (1931), "Die logizistische Grundlegung der Mathematik", ''Erkenntnis'' 2, 91-121. Republished, "The Logicist Foundations of Mathematics", E. Putnam and G.J. Massey (trans.), in Benacerraf and Putnam (1964). Reprinted, pp. 41-52 in Benacerraf and Putnam (1983).
* [[Subrahmanyan Chandrasekhar|Chandrasekhar, Subrahmanyan]] (1987), ''Truth and Beauty. Aesthetics and Motivations in Science'', University of Chicago Press, Chicago, IL.
* [[Jacques Hadamard|Hadamard, Jacques]] (1949), ''The Psychology of Invention in the Mathematical Field'', 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954.
* [[G.H. Hardy|Hardy, G.H.]] (1940), ''A Mathematician's Apology'', 1st published, 1940. Reprinted, [[C.P. Snow]] (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.
* [[Wilbur Dyre Hart|Hart, W.D.]] (ed., 1996), ''The Philosophy of Mathematics'', Oxford University Press, Oxford, UK.
* [[Vincent F. Hendricks|Hendricks, Vincent F.]] and Hannes Leitgeb (eds.). ''Philosophy of Mathematics: 5 Questions'', New York: Automatic Press / VIP, 2006. [http://www.phil-math.org]
* [[H.E. Huntley|Huntley, H.E.]] (1970), ''The Divine Proportion: A Study in Mathematical Beauty'', Dover Publications, New York, NY.
* [[Jacob Klein|Klein, Jacob]] (1968), ''Greek Mathematical Thought and the Origin of Algebra'', [[Eva Brann]] (trans.), MIT Press, Cambridge, MA, 1968. Reprinted, Dover Publications, Mineola, NY, 1992.
* [[Morris Kline|Kline, Morris]] (1959), ''Mathematics and the Physical World'', Thomas Y. Crowell Company, New York, NY, 1959. Reprinted, Dover Publications, Mineola, NY, 1981.
* Kline, Morris (1972), ''Mathematical Thought from Ancient to Modern Times'', Oxford University Press, New York, NY.
* [[Gyula König|König, Julius (Gyula)]] (1905), "Über die Grundlagen der Mengenlehre und das Kontinuumproblem", ''Mathematische Annalen'' 61, 156-160. Reprinted, "On the Foundations of Set Theory and the Continuum Problem", [[Stefan Bauer-Mengelberg]] (trans.), pp. 145-149 in Jean van Heijenoort (ed., 1967).
*Lakatos, Imre 1976 ''Proofs and Refutations:The Logic of Mathematical Discovery'' (Eds) J. Worrall & E. Zahar Cambridge University Press
* Lakatos, Imre 1978 ''Mathematics, Science and Epistemology: Philosophical Papers'' Volume 2 (Eds) J.Worrall & G.Currie Cambridge University Press
* Lakatos, Imre 1968 ''Problems in the Philosophy of Mathematics'' North Holland
* [[Gottfried Wilhelm Leibniz|Leibniz, G.W.]], ''Logical Papers'' (1666-1690), [[G.H.R. Parkinson]] (ed., trans.), Oxford University Press, London, UK, 1966.
* [[Saunders Mac Lane|Mac Lane, Saunders]] (1998), ''[[Categories for the Working Mathematician]]'', 1st edition, Springer-Verlag, New York, NY, 1971, 2nd edition, Springer-Verlag, New York, NY.
* [[Penelope Maddy|Maddy, Penelope]] (1990), ''Realism in Mathematics'', Oxford University Press, Oxford, UK.
* Maddy, Penelope (1997), ''Naturalism in Mathematics'', Oxford University Press, Oxford, UK.
* [[Edward A. Maziarz|Maziarz, Edward A.]], and [[Thomas Greenwood|Greenwood, Thomas]] (1995), ''Greek Mathematical Philosophy'', Barnes and Noble Books.
* [[Matthew Mount|Mount, Matthew]], ''Classical Greek Mathematical Philosophy'', {{Fact|date=February 2007}}.
* [[Benjamin Peirce|Peirce, Benjamin]] (1870), "Linear Associative Algebra", § 1. See ''American Journal of Mathematics'' 4 (1881).
* [[Charles Peirce|Peirce, C.S.]], ''Collected Papers of Charles Sanders Peirce'', vols. 1-6, [[Charles Hartshorne]] and [[Paul Weiss (philosopher)|Paul Weiss]] (eds.), vols. 7-8, [[Arthur W. Burks]] (ed.), Harvard University Press, Cambridge, MA, 1931 – 1935, 1958. Cited as CP (volume).(paragraph).
* Plato, "The Republic, Volume 1", [[Paul Shorey]] (trans.), pp. 1-535 in ''Plato, Volume 5'', Loeb Classical Library, William Heinemann, London, UK, 1930.
* Plato, "The Republic, Volume 2", [[Paul Shorey]] (trans.), pp. 1-521 in ''Plato, Volume 6'', Loeb Classical Library, William Heinemann, London, UK, 1935.
* [[Hilary Putnam|Putnam, Hilary]] (1967), "Mathematics Without Foundations", ''Journal of Philosophy'' 64/1, 5-22. Reprinted, pp. 168-184 in W.D. Hart (ed., 1996).
* [[Gilbert de Beauregard Robinson|Robinson, Gilbert de B.]] (1959), ''The Foundations of Geometry'', University of Toronto Press, Toronto, Canada, 1940, 1946, 1952, 4th edition 1959.
* Russell, Bertrand (1919), ''Introduction to Mathematical Philosophy'', George Allen and Unwin, London, UK. Reprinted, [[John G. Slater]] (intro.), Routledge, London, UK, 1993.
* [[Raymond M. Smullyan|Smullyan, Raymond M.]] (1993), ''Recursion Theory for Metamathematics'', Oxford University Press, Oxford, UK.
* Strohmeier, John, and Westbrook, Peter (1999), ''Divine Harmony, The Life and Teachings of Pythagoras'', Berkeley Hills Books, Berkeley, CA.
* [[N.I. Styazhkin|Styazhkin, N.I.]] (1969), ''History of Mathematical Logic from Leibniz to Peano'', MIT Press, Cambridge, MA.
* [[William W. Tait|Tait, William W.]] (1986), "Truth and Proof: The Platonism of Mathematics", ''Synthese'' 69 (1986), 341-370. Reprinted, pp. 142-167 in W.D. Hart (ed., 1996).
* Tarski, A. (1983), ''Logic, Semantics, Metamathematics: Papers from 1923 to 1938'', [[J.H. Woodger]] (trans.), Oxford University Press, Oxford, UK, 1956. 2nd edition, John Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.
* [[Thomas Tymoczko|Tymoczko, Thomas]] (1998), ''New Directions in the Philosophy of Mathematics'', [http://pup.princeton.edu/titles/6172.html Catalog entry?]
* [[Stanisław Marcin Ulam|Ulam, S.M.]] (1990), ''Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators'', A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.
* [[Jean van Heijenoort|van Heijenoort, Jean]] (ed. 1967), ''From Frege To Gödel: A Source Book in Mathematical Logic, 1879-1931'', Harvard University Press, Cambridge, MA.
* [[Eugene Wigner|Wigner, Eugene]] (1960), "[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]", ''[[Communications on Pure and Applied Mathematics]]'' '''13'''(1): 1-14. [http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html Eprint]
==References==
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