Philosophy of mathematics(philosophymathematics,philosophy mathematics)
Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: "why is mathematics useful indescribing nature?", "in which sense, if any, do mathematical entities such as numbers exist?" and "why and how are mathematicalstatements true?". The various approaches to answering these questions will be presented in this article.
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 thancriticism. Criticisms can however have important ramifications for mathematical practice and so the philosophy of mathematics canbe of very direct interest to working mathematicians, particularly in new fields where the process of peer review of mathematical proofs isnot firmly established, raising probability of an undetected error. Such errors can thus only be reduced by knowing where theyare 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?". And, the related but logicallyseparate, "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, eitherattempting 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 whichothers 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 aboutmathematics".
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 notinvent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. Theterm Platonism is used because such a view is seen to parallel Plato 's belief ina "heaven of ideas", an unchanging ultimate reality that the everyday world can only imperfectly approximate. Plato's viewprobably 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 unknownto 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 oftime with the investigation of an entity in whose existence one doesn't firmly believe. Gödel believed in an objectivemathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (eg, for any twomathematical objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to betrue, 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 toprovide 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 aworld, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to thisseparate world and discover truths about the entities? Gödel's and Plato's answers to each of these questions are muchcriticised.
An important argument for mathematical realism, formulated by Quine and Putnam , is the Indispensability Argument: mathematics isindispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, weought also believe in the reality of those entities required for this description. In keeping with Quine and Putnam's overallphilosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation forexperience, thus stripping mathematics of some of its epistemic status.
Most forms of logicism (see below) are forms of mathematical realism. For a philosophy of mathematics that attempts toovercome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Maddy's Realism inMathematics. 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 rejectedmystical notions of truth and accepted much quasi-empiricism in mathematics - a term that he was involved in coining (see below). Anexample of a theory that both embraces realism and rejects Platonism is the embodied mind theory - see below.
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain stringmanipulation rules. For example, in the "game" of Euclideangeometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new stringsfrom 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 solvethe epistemic problems (What are symbols? Do they exist in an eternal, unchanging realm?), does not explain the usefulness ofmathematics, 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 absolutetruth, 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 toaccept the theorem, or, rather, the interpretation you have given it must be a true statement. The same is held to be true forall other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolicgame. It is usually hoped that there exists some interpretation in which the rules of the game hold. But it does allow theworking mathematician to continue in his work and leave such problems to the philosopher or scientist. Many formalists would saythat 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 wasa complete and consistent axiomatization of all of mathematics. ("Consistent" here means that no contradictions can be derivedfrom the system.) Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitaryarithmetic" (a subsystem of the usual arithmetic of the positive whole numbers, chosen to be philosophically uncontroversial) wasconsistent. Hilbert's program was dealt a fatal blow by the second of Gödel's incompleteness theorems , which states that sufficiently expressive consistent axiomsystems 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 proveits 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 yieldintrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that therewas no other meaningful mathematics whatsoever, regardless of interpretation.
Modern formalists, such as Rudolf Carnap , Alfred Tarski and HaskellCurry , considered mathematics to be 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 theoriesetc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematicalor 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 theminute string manipulation games mentioned above. While published proofs (if correct) could in principle be formulated in termsof these games, the rules are certainly not substantial to the initial creation of those proofs. Formalism is also silent to thequestion of which axiom systems ought to be studied.
Logicism holds that logic isthe proper foundation of mathematics, and that all mathematical statements are necessary logical truths. For instance, thestatement "If Socrates is a human, and every human is mortal, then Socrates 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 DieGrundgesetze 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 theextension 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 oflogic.
But Frege's construction was flawed. Russell discovered thatBasic Law V is inconsistent (this is Russell's Paradox ). Fregeabandoned 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 oframified types to deal with it. In this system, they were eventually able to build up much of modern mathematics but in analtered, and excessively complex, form (for example, the numbers were different in each type, and there were infinitely manytypes). 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 abstractionprinciples such as Hume's Principle (the number of objectsfalling under the concept F equals the number of objects falling under the concept G if and only if the extension of F and theextension of G can be put into one-to-one correspondence). Frege required Basic Law V to be able to give an explicit definitionof the numbers, but all the properties of numbers can be derived from Hume's Principle. This would not have been enough for Fregebecause (to paraphrase him) it does not exclude the possibility that Julius Caesar=2.
Constructivism and intuitionism
These schools maintain that only mathematical entities which can be explicitly constructed have a claim to existence andshould be admitted in mathematical discourse.
A typical quote comes from Leopold Kronecker : "The naturalnumbers 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; thisintuitionistic logic does not contain the law ofthe excluded middle and therefore frowns upon proofs bycontradiction . The axiom of choice is also rejected. Importantwork was later done by Errett Bishop , who managed to prove versions ofthe 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 beenmade to use the concepts of Turing machine or recursive function to fill this gap, leading to the claim that onlyquestions regarding the behavior of finite algorithms are meaningful and shouldbe investigated in mathematics. This has led to the study of the computable numbers , first introduced by AlanTuring .
Embodied mind theories
These theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself inour physical universe. For example, the abstract concept of number springs from theexperience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, otherthan 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 andlater 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 effectivein 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 thatinspired this perspective, see cognitive science of mathematics .
Social constructivism or social realism
This theory sees mathematics primarily as a social construct , asa product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavorwhose 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 thesociety financing it.
To some mathematicians, this theory seems intuitively wrong given their seeming permanence of mathematics. But this permanenceis in fact grounded by much uncertainty: as mathematicalpractice evolves, the status of previous finished mathematics is cast into doubt, and is re-examined and corrected only to the degree itis 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 toomuch status, and folk mathematics not enough, due to an over-beliefin axiomatic proof and peer review as practices.
Mathematics also has subcultures . Major discoveries can be made in onebranch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact betweenmathematicians. Each speciality forms its own epistemiccommunity and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas ofmathematics.
If the social process of 'doing mathematics' is seen as actually creating the meaning, the term social constructivism is moreappropriate. If deficiency of human capacity to abstract, human cognitivebias , or lack of sufficient collectiveintelligence is seen as preventing the comprehension of a 'real' universe of 'mathematical objects', the term social realismis more appropriate.
Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or evenmeaningless. Some social scientists also argue that mathematics is not real or objective at all, but is affected by racism and ethnocentrism . Some of theseideas are close to postmodernism .
Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko . Some consider thework of Paul Erdös as a whole to have advanced this view (although hepersonally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as asocial activity", e.g. via the Erdos number . This strongly influenced workon measuringreputation 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 , oreven 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" orfinding 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 theNatural Sciences , in which he argued the happy coincidence that mathematics and physics were so well matched, seemed tobe "unreasonable" and hard to explain.
The embodied-mind or "cognitive" school and the "social" school were responses to this challenge. But the debates raised weredifficult to confine to those:
One parallel concern that does not actually challenge the schools directly but questions their focus is the notion of quasi-empiricism in mathematics . This grewfrom 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 minimalform 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 whichgreat feats of calculation and measurement are sometimes achieved. Indeed, such methods may be the only notion of "proof" aculture has.
Hilary Putnam argued that any theory of mathematical realism wouldinclude quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methodsprimarily, being willing often to forgo rigorous and axiomatic "proofs", and still be "doing mathematics" - at perhaps a somewhatgreater risk of failure of their calculations. He laid out a quite detailed argument for this in New Directions (ed.Tymockzo, 1998).
Many practitioners and scholars who are not engaged primarily in proofs have made interesting and important observations aboutthe nature of mathematics:
Judea Pearl claimed that all of mathematics as presently understood wasbased 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 . Themost important output of this was new theories of truth , notably those appropriate to activism and grounding empirical methods .
The notion of a philosophy of mathematics separate from philosophy as suchhas been criticized as leading to "good mathematicians doing bad philosophy" - few philosophers being able to penetratemathematical notations and culture to actually relate conventional notions of metaphysics to the more specialized metaphysical notions of the 'schools' above. This may lead to adisconnection in which the mathematicians continue to spout bad and discredited philosophy as a justification for their continuedbelief 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 shortof actually relating this to ordinary human perception and everydayunderstandings of knowledge .
As well, there is little or no consideration given to the ethics of doingmathematics, it being seen in a technological culture as an absolute necessity whose value cannot be questioned and whoseimplications cannot be avoided - even if particular branches have no known purpose, or are considered useful primarily or only toenable 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 toalso 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 tothe sciences of genetics , physics , economics or medicine . Which isinteresting in itself, as mathematics is necessary to enable those and other sciences.
Evolutionary psychology for instance has embracedthe idea that "the mind is a computer" in the sense of a TuringMachine . What are the implications of adopting an abstraction originating to explain computers formally, to explain themind?
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 tomathematics 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 itwork?' 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 ofwar, 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 whoseaesthetics 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 thosefields where mathematics is not immediately applied to life.
Finally, although many or most mathematicians or philosophers would accept the statement " mathematics is a language ", there is little attentionpaid to the implications of that statement. Linguistics is not applied todiscourses 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 seenas 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 needsto "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 thesemore "linguistic" questions, and that its analysis of programming languages is also often just as applicable to mathematics or at least some metamathematics .
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