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Philosophy of mathematics

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Philosophy of mathematics is an active discipline of inquiry and the resulting subject matter. As a branch of philosophy, it addresses questions about the character of mathematics, the conduct of mathematical inquiry, and the role of mathematical objects in describing empirical phenomena. As a form of philosophical inquiry, it examines the record of mathematical inquiry and poses questions regarding its aims, its conduct, and its results. Although the questions are diverse and never-ending, a number of recurrent themes can be recognized:

  1. What are the sources of mathematical subject matter?
  2. What does it mean to refer to a mathematical object?
  3. What is the character of a mathematical proposition?
  4. What kinds of inquiry play a role in mathematics?
  5. What are the objectives of mathematical inquiry?
  6. What gives mathematics its grip on experience?
  7. What is the bearing of beauty on mathematics?

A quick run through this slate of questions, touching on a sample of the answers that have been given so far in human history, makes for a ready introduction to the philosophy of mathematics.

Many thinkers down through the ages have contributed their ideas concerning the conduct of mathematical inquiry and the nature of mathematical objects and knowledge. Today, many 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.

Relation to mathematics proper

Observers from other fields ? anthropologists, biologists, linguists, logicians, philosophers, psychologists, and sociologists to name a few ? have put forward proposals as to where mathematics comes from, what it's about, and how it ought to be done. Mathematicians vary in their responses to these suggestions, and the search for true accounts will no doubt continue. Above and beyond these questions, however, there remains the question of where the philosophy of mathematics comes from.

From the writings that have come down to us, it appears that the philosophy of mathematics and the practice of mathematics went hand in hand for most of human history, with the same people engaged in both aspects of a single activity, the natural emphasis being on the practical side, but by its very nature demanding considerable examination of alternative ways that it might be done better. What little external critique there was appears to have come mainly from the great comic writers like Aristophanes. That makes for a likely story, as seen from a distance, but it is more likely an artifact of the sample of works that are extant.

The record grows clearer and more detailed with the advents of the Renaissance and the Enlightenment that the ways of mathematics and science in general were beginning to attract the attention of artisans and astute thinkers from areas beyond the practice of mathematics proper.

It is necessary at the outset to distinguish the philosophy of mathematics from the philosophy of mathematicians. Philosophy of mathematics has its source in any moment that a person reflects on mathematical practice, whether it is another person's practice or that person's own. Having made that distinction between the more generic reflection on mathematics and its more specialized reflexive application, it is possible to see yet another distinction, analogous to what medieval logicians call logica docens, logic as taught, and logica utens, logic as used. C.S. Peirce, as a logician, mathematician, and philosopher who found it useful to study the past on the way to creating the future of mathematics, is a useful source here:

But mathematics performs its reasoning by a logica utens which it develops for itself, and has no need of any appeal to a logica docens; for no disputes about reasoning arise in mathematics which need to be submitted to the principles of the philosophy of thought for decision. (C.S. Peirce, CP 1.417).

Peirce is here signing on to a declaration of independence for mathematics that he knows is nothing new, that many others have signed on before, but it remains a steadfast position that he affirms on many occasions and in many different ways all throughout his work on the foundations of mathematics.

Relation to philosophy proper

The terms philosophy of mathematics and mathematical philosophy are not synonyms. The latter is more often used to mean at least three distinct things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologicians, 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. Apart from that, some philosophers understand the term mathematical philosophy as an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy.

Some philosophers of mathematics view their task as giving an account of mathematics and mathematical practice as it stands, as interpretation rather than criticism. Criticisms can, however, have important ramifications for mathematical practice, so the philosophy of mathematics can be of direct interest to working mathematicians, particularly in new fields where the process of peer review of mathematical proofs is not firmly established, raising the 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.

Perennial questions

Three features of mathematical reasoning ? its abstract, hypothetical, and necessary qualities ? are so inseparable that their logical linkage is already a commonplace paradigm of Classical philosophy. The need to understand this complex of features leads to some of the initial encounters between mathematics and philosophy in general. For example, Plato bases one of his gnomic parables, the analogy of the divided line, on the way that students of mathematics use visible forms as images or simulacra of formal realities:

The very things which they mould and draw, which have shadows and images of themselves in water, these things they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind. (Plato, Republic 510E).

Plato is not engaged in the philosophy of mathematics, since mathematics is not his main object either here or elsewhere, and he is not proposing the type of mathematical philosophy that aims to reduce philosophy in general to mathematics. Plato's point is a wider one, having to do with the teaching that is now called Platonic realism. But the form of analogy that maps reality into representation is a familiar theme in mathematics, and so it serves as the analogical image of a further analogy that Plato uses to illustrate his broader philosophy. Plato's reasoning in this part of the Republic is Plato at his subtlest, but it lays bare many of the founding metaphors of the Western tradition and will repay further consideration below.

This same form of argument, that Stanislaw Ulam (1990) would later dub analogies between analogies, brings our story right up to the present time frame, as mathematical category theory, a formalism that many mathematicians regard as the natural language of contemporary mathematics, is nothing more in the first instance than a formalization of mathematical metaphors.

Aristotle routinely derives his initial philosophical impulses from the parables of his predecessors, especially Plato, but his natural attraction to earthly topics just as dependably brings him back to empirical grounds. For a topical example, he starts from Plato's treatment of analogy as the mathematical form of a logos, a proportion, or a ratio but he goes on to analyze the pattern of reasoning by analogy or example ? the Greek word he uses is παραδειγμα, the root of paradigms both grammatical and philosophical ? as a mixed syllogism, in particular, a two-stage inference that follows a step of inductive reasoning with a step of deductive reasoning.

A point of departure for the question of "Where mathematics comes from" can be taken from the following narrative, chosen for its typicality more than its novelty, of how abstractions are derived from the matrix of experience:

Our concept of physical space is the result of a desire to order our experiences of the external world. This ordering process is accompanied by successive approximations and abstractions which lead to our concept of mathematical space. For the physicist the correspondence between the data of experience and his concept of physical space is all important. As the abstraction process continues, this correspondence becomes less significant, so that the mathematician feels free to concentrate upon the logical relations involved. (G. de B. Robinson, 5).

The author describes a process of abstraction that produces empirically bound concepts and formally free concepts in tandem, and that brings about a threefold relation among contingent experiences, concepts of physical space, and concepts of mathematical space. Achieving a more thorough understanding of this process, by which mathematical patterns are abstracted from concrete experience, developed as quasi-autonomous forms, and then applied back to experience in far-reaching and surprising ways, is one of the essential services that philosophical examination can perform for the benefit of mathematical thought.

Mathematical propositions, at least at first sight, appear to differ from other sorts of propositions, but in ways that have, historically speaking, been difficult to define precisely. One distinctive feature of mathematical propositions is, as Hilary Putnam sketches a common view of it, "the very wide variety of equivalent formulations that they possess", by which he does not mean the sheer number of ways of saying the same thing but "rather that in mathematics the number of ways of expressing what is in some sense the same fact (if the proposition is true) while apparently not talking about the same objects is especially striking" (Putnam, 170).

Another characteristic of mathematical propositions, the recognition of which is drilled into the character of every student, is epitomized in the precept: "What's true is what you can prove". W.W. Tait (1986) takes up the relation between truth and proof in the process of examining the role of Platonism in mathematics.

Placed within a broader context, proof may be seen as a form of inquiry, being one of many proceedings that reduce the amount of uncertainty a reasoner has about a given question. Viewing proof in this light leads to the further question: What other forms of inquiry are involved in the actual practice of mathematics? In particular, what are the roles of analogy, beauty, conjecture, and various types of experiential reasoning, from empirical induction to chance inspiration to concrete intuition, in the actual life of mathematical inquiry?

Philosophical inquiry into the grounds of mathematics sooner or later comes to a question about its relation to logic. The answers that suggest themselves naturally depend on the definitions of logic and mathematics that are taken to be in force at the moment in question, or the basic intuitions that about them that a given inquirer takes for granted if real definitions have yet to be found.

One answer is that logic and mathematics, taken at the full, are identical subjects. A second answer is that mathematics depends on logic more than the reverse. A third answer is that logic depends on mathematics more than the reverse. Either one of the first two answers is given from the philosophical stance known as logicism. The last answer is put forth in the pragmatic philosophy of Charles Sanders Peirce. Of course, it is not necessarily the case that all respondents are using the same definitions of either logic or mathematics to argue for their answers.

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 is characterized by a predominant interest in formal logic, set theory, and foundational issues.

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 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 as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories lead to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55).

At the midpoint 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:

When philosophy discovers something wrong with science, sometimes science has to be changed ? Russell's paradox comes to mind, as does 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).

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

Mathematical realism, like 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 one sort of mathematics that can be discovered: Triangles, 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 G?. G? 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. G? 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 naive 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", an unchanging ultimate reality that the everyday 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.

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?

G?'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 synthetic a priori.) Davis and 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 formalism (see below).

Some mathematicians hold opinions that amount to more nuanced versions of Platonism. These ideas are sometimes described as Neo-Platonism.

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, 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:

1. The concepts of mathematics can be derived from logical concepts through explicit definitions.
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.

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 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, 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 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 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 Quine and 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 electrons 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. 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 G?'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).

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.) 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.

A major early proponent of formalism was David Hilbert, whose program 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 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 G?'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, G?'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 G? 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 axiom systems. Mathematical logicians study formal systems but are just as often realists 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 critique of 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 effort required in space and time would be prohibitive (witness Principia Mathematica.) In addition, 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.

Intuitionism

Main article : Mathematical intuitionism

In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (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 the excluded middle and therefore frowns upon 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 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. Not surprisingly, then, constructive mathematics is sometimes associated with theoretical computer science.

Constructivism

Main article : Mathematical constructivism

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 was introduced 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 true, Field suggested that mathematics was dispensable, and therefore should be rejected as false. 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" axioms of Hilbert geometry to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

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.

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. Although intriguing, Field's approach has not been very influential.

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. N?]. In addition, mathemetician Keith Devlin has investigated similar concepts with his book The Math Instinct. For more on the science 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.

The social nature of mathematics is highlighted in its 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. Social constructivists argue 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. 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 mathemetician's 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. 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 [1], similar to but not quite the same as that associated with Alvin White [2]; one of Hersh's co-authors, Philip J. Davis, has expressed sympathy for the social view as well.

Beyond the traditional 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 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-empricism 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 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).

Action

Some practitioners and scholars who are not engaged primarily in proof-oriented approaches have suggested an interesting and important theory about the nature of mathematics. For example, 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 knowledge relates to action. The most important output of this was new theories of truth, notably those appropriate to activism and grounding empirical methods.

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 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 Gottlob Frege and Alfred 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.

Aesthetics

Many practicising mathemeticians have been drawn to their subject because of a sense of beauty they perceive in it. One sometimes hears the sentiment that mathemeticians 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 exhileration 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 sculpture. Indeed, one can study mathematical and scientific writings as literature.

Philip 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 √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 mathemeticians 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. Gregory Chaitin rejected Erdős's book. By way of example, he provided three separate proofs of the infinitude of primes. The first was Euclid's, the second was based on the Euler zeta function, and the third was Chaitin's own, derived from algorithmic information theory. Chaitin then argued that each one was as beautiful as the others, because all three reveal different aspects of the same problem.

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 desriable 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 mathemeticians have characterized Hardy's views as mildly datedTemplate:Fact, with the applicablility 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.Template:Fact

Note Bene. Older Version To Be Merged

Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: "why is mathematics useful in describing nature?", "in which sense, if any, do mathematical entities such as numbers exist?" and "why and how are mathematical statements true?". Some philosophers of mathematics view their task as being to give an account of mathematics as it stands, as interpretation rather than criticism. However, others' conclusions can have important ramifications for mathematical practice and so the philosophy of mathematics can be of very direct interest to working mathematicians.

The philosophy of mathematics has seen several different schools which will be presented in this article. Three of these, intuitionism, logicism and formalism, emerged around the start of the 20th century in response to the increasingly widespread realisation that (as it stood) mathematics, and analysis in particular, did not live up to the standards of certainty and rigour with which it was credited. Each school addresses the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge.

Mathematical Realism, or Platonism

Mathematical Realism holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. The term Platonism is used because such a view is seen to parallel Plato's belief in a "heaven of ideas", an unchanging ultimate reality that the everday world can only imperfectly approximate. Plato's view probably derives from Pythagoras, and his followers the pythagoreans, who believed that the world was, quite literally, built up by the numbers. This idea may have even older origins that are unknown to us.

Many working mathematicians are mathematical realists; they see themselves as discoverers. Examples are [[Paul Erd? and Kurt G?. Psychological reasons have been given for this preference: it appears to be very hard to preoccupy oneself over long periods of time with the investigation of an entity in whose existence one doesn't firmly believe. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (eg. For any 2 mathematical objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis, might prove not be decidable just on the basis of such principles. Gödel suggested 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.

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?'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.

See also: Mathematical constructivism, Mathematical intuitionism

Embodied mind theories

These theories hold that mathematical thought is a natural outgrowth of the human cognitive apparatus which finds itself in our physical universe. For example, the abstract concept of number springs from the experience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.

The physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation.

The effectiveness of mathematics is thus easily explained: mathematics was constructed by the brain in order to be effective in this universe.

The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. N? (Since this book was first published in the year 2000, it may still be one of the only treatments of this perspective.) For more on the science that inspired this perspective, see cognitive science of mathematics.

Social constructivism

This theory sees mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly compared to reality and may be discarded if they don't agree with observation or prove pointless. The direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it.

Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko.

References

  • Audi, Robert (ed., 1999), The Cambridge Dictionary of Philosophy, Cambridge University Press, Cambridge, UK, 1995. 2nd edition, 1999. Cited as CDP.
  • Benacerraf, Paul, and 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.
  • 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.), Eprint.
  • Bourbaki, N. (1994), Elements of the History of Mathematics, John Meldrum (trans.), Springer-Verlag, Berlin, Germany.
  • 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).
  • 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.
  • Hardy, G.H. (1940), A Mathematician's Apology, 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.
  • Hart, W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
  • Huntley, H.E. (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover Publications, New York, NY.
  • Kleene, S.C. (1971), Introduction to Metamathematics, North?Holland Publishing Company, Amsterdam, Netherlands.
  • 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.
  • 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.
  • K?, Julius (Gyula) (1905), "ܢer 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).
  • Maddy, Penelope (1990), Realism in Mathematics, Oxford University Press, Oxford, UK.
  • Maddy, Penelope (1997), Naturalism in Mathematics, , Oxford University Press, Oxford, UK.
  • Maziarz, Edward A., and Greenwood, Thomas (1995), Greek Mathematical Philosophy, Barnes and Noble Books.
  • Peirce, Benjamin (1870), "Linear Associative Algebra", ? 1. See American Journal of Mathematics 4 (1881).
  • Peirce, C.S., Writings of Charles S. Peirce, A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN, 1981?. Cited as CE (volume), (page).
  • Peirce, C.S. (1870), "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", Memoirs of the American Academy of Arts and Sciences 9 (1870), 317?378. Reprinted (CP 3.45?149), (CE 2, 359?429).
  • Peirce, C.S. (c. 1896), "The Logic of Mathematics; An Attempt to Develop My Categories from Within", 1st published as CP 1.417?519 in Collected Papers.
  • Peirce, C.S. (1902), "The Simplest Mathematics", MS dated January?February 1902, intended as Chapter 3 of the "Minute Logic", CP 4.227?323 in Collected Papers.
  • 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.
  • Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5?22. Reprinted, pp. 168?184 in W.D. Hart (ed., 1996).
  • 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.
  • Smullyan, Raymond M. (1993), Recursion Theory for Metamathematics, Oxford University Press, Oxford, UK.
  • Steiner, Mark (1998), The Applicability of Mathematics as a Philosophical Problem, Harvard University Press, Cambridge, MA.
  • Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA.
  • Styazhkin, N.I. (1969), History of Mathematical Logic from Leibniz to Peano, MIT Press, Cambridge, MA.
  • Tait, W.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.
  • Ulam, S.M. (1990), Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators, A.R. Bednarek and Fran篩se Ulam (eds.), University of California Press, Berkeley, CA.
  • van Heijenoort, Jean (ed. 1967), From Frege To G?: A Source Book in Mathematical Logic, 1879?1931, Harvard University Press, Cambridge, MA.
  • Wilder, Raymond L. (1952), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY.

Further reading

  • Colyvan, Mark (2004), "Indispensability Arguments in the Philosophy of Mathematics", Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), Eprint.
  • 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.
  • 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.
  • Ernest, Paul (1998), Social Constructivism as a Philosophy of Mathematics, State University of New York Press, Albany, NY.
  • George, Alexandre (ed., 1994), Mathematics and Mind, Oxford University Press, Oxford, UK.
  • Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, NY.
  • Raymond, Eric S. (1993), "The Utility of Mathematics", Eprint.
  • Shapiro, Stewart (2000), Thinking About Mathematics: The Philosophy of Mathematics, Oxford University Press, Oxford, UK.

Journals

See also

Related topics

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Related works

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Historical topics

External links

  • Tymoczko, Thomas (1998), New Directions in the Philosophy of Mathematics, Princeton University Press, Princeton, NJ. Link.

Document history

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