Philosophy of mathematics

2949:, 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. 1289:, 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 was thus "truly" 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 proportion to the arbitrary first "number" or "one". 1212: 11156: 9360: 3534: 504: 2803:

argument can transmit falsity from the conclusion to the premises just as well as it can transmit truth from the premises to the conclusion. 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

1304:, 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. 1902:). 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. 1837: 9372: 8336: 1073:. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from the modern notion of science by not 2874:, 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 3430:, 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 225: 3367: 76: 3079: 35: 9396: 1981:, 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 9384: 1093:. It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. 286: 178: 2700:, called "The Prolegomena of Pure Logic", criticized psychologism thoroughly and sought to distance himself from it. The "Prolegomena" is considered a more concise, fair, and thorough refutation of psychologism than the criticisms made by Frege, and also it is considered today by many as being a memorable refutation for its decisive blow to psychologism. Psychologism was also criticized by 8763: 8323: 3548: 1481:—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. 2775:. 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 being distinct from the other sciences. 1944: 2221: 2191:

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 using proof by contradiction when showing the existence of an object or when trying to establish the truth of some proposition. Important work was done by

950:. Since the existence of such a monster seemed impossible, people had two choices: either they accept such unrealistic facts, which implies that mathematics does not need to reflect the physical reality; or they changes the logical rules for excluding such monsters. The first choice led to the philosophical school of 2913:, expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence". This rejection separates fictionalism from other forms of anti-realism, which see mathematics itself as artificial but still bounded or fitted to reality in some way. 2373:. Structures are held to exist inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise legitimate structures. 1033:

phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical

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them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects. Another way of making the point is that if the Platonic world were to disappear, it would make no difference to the ability of mathematicians to generate

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can be taken in two senses. In one sense it is irrefutable and logically true. In the second sense it is factually true and falsifiable. Another way of putting this is to say that a single number statement can express two propositions: one of which can be explained on constructivist lines; the other

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also falls into the Aristotelian realist tradition. Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". Whether or not there are definite multitudes of units

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springs from the experience of counting discrete objects (requiring the human senses such as sight for detecting the objects, touch; and signalling from the brain). 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

2284:. Mayberry's system is Aristotelian in general inspiration and, despite his strong rejection of any role for operationalism or feasibility in the foundations of mathematics, comes to somewhat similar conclusions, such as, for instance, that super-exponentiation is not a legitimate finitary function. 2190:

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

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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

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of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from Field's system), so that mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can

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points and lines does not matter at all for the validity of the theory. This means that one can verify the correctness of a proof without referring to any figure, and that a proved theorem remains true independently of any interpretation of the entities involved in the axioms. For example, in plane

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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

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holds that mathematics studies properties such as symmetry, continuity and order that can be literally realized in the physical world (or in any other world there might be). It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world

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can be axiomatized with two sorts of objects, the points and the lines, and a relation "belonging to" or "passing through" that relates points and lines. One of the axioms is "there is exactly one line that passes through two points". The interpretation of points and lines (of the theory) as usual

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Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together. Because the language of mathematics is so precise, it is ideally suited to defining

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was another philosopher to point out empirical aspects of mathematics, observing that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it

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A major question considered in mathematical Platonism is: 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, that is occupied by the mathematical entities? How can we gain access to this separate

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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 on the question of which axiom systems ought to be studied, as none is more meaningful than another from a

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As a consequence of this closeness to physical reality, mathematicians were very cautious when problems they want to solve led them to introduce new concepts that are not directly related the real world. These precautions are still reflected in modern terminology, where the numbers that are not

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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

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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 prove theorems. A mathematical

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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

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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

3152:. 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 856:, and therefore left to the reader. As most proof errors occur in these skipped steps, a new proof requires to be verified by other specialists of the subject, and can be considered as reliable only after having been accepted by the community of the specialists, which may need several years. 1988:

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

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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.

2114:). 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 2814:

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

1940:.) 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. 2396:

exist, and have structural features in common. If something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence.

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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.

994:). Nevertheless the interpretation of the objects of a theory in terms of physical reality (when possible) or of previously studied abstractions remains fundamental for guiding the choice of the axioms, understanding the subject of the theory, and follow the steps of a long proof. 2767:: 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 1461:, and it became a new contender for the natural language of mathematical thinking. As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at the century's beginning. 2609:

that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world). A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world.

837:, is such a theory in which all mathematics have been restated; it is used implicitely in all mathematics texts that do not specify explicitly on which foundations they are based. Moreover, the other proposed foundations can be modeled and studied inside ZFC. 3296:

with concrete, physical entities ("the truth-values of our mathematical assertions depend on facts involving Platonic entities that reside in a realm outside of space-time"). Whilst our knowledge of concrete, physical objects is based on our ability to

1569:: the everyday world can only imperfectly approximate an unchanging, ultimate reality. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular 2737:, 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 630:

argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as

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axioms for 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

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to justify the exclusion of all non-scientific entities, and hence to defend the "only" part of "all and only". The assertion that "all" entities postulated in scientific theories, including numbers, should be accepted as real is justified by

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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

2969:. 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 2275:

is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources. Another variant of finitism is Euclidean arithmetic, a system developed by

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for which Euclid's Common Notion 5 (the whole is greater than the part) fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.

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holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all.

2358:. Structures are held to have a real but abstract and immaterial existence. As such, it faces the standard epistemological problem of explaining the interaction between such abstract structures and flesh-and-blood mathematicians (see 1548:

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

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account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else. One line of defense is to maintain that this is false, so that mathematical reasoning uses some special

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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

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As a consequence, the relationship between mathematics and physical reality is no more a mathematical question, but the nature of this relationship remains a philosophical question that does not have any uncontroversial answer.

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but can be physically realized. For example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots. Aristotelian realism is defended by

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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.

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The origin of mathematics is of arguments and disagreements. Whether the birth of mathematics was by chance or induced by necessity during the development of similar subjects, such as physics, remains an area of contention.

2347:). The kind of existence mathematical objects have would clearly be dependent on that of the structures in which they are embedded; different sub-varieties of structuralism make different ontological claims in this regard. 1644:

is a modern variation of Platonism, which is in reaction to the fact that different sets of mathematical entities can be proven to exist depending on the axioms and inference rules employed (for instance, the law of the

2827:. Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is extraordinarily central, and that it would be extremely difficult for us to revise it, though not impossible. 3147:

are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way from other languages. If mathematics is a language, it is a different type of language from

2309:. For instance, it would maintain that all that needs to be known about the number 1 is that it is the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the 3484:

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

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Mohan Ganesalingam has analysed mathematical language using tools from formal linguistics. Ganesalingam notes that some features of natural language are not necessary when analysing mathematical language (such as

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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

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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

2886:. 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. 1932:. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one, if it follows deductively from the appropriate axioms. The same is held to be true for all other mathematical statements. 3006:

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

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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.

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At the start of the 20th 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

990:, one may interpret points as lines and vice versa. This implies that for every theorem relating points and lines, one gets immediately a new theorem by exchanging the role of the points and the lines (see 844:. Where a special concept of rigor comes into play is in the socialized aspects of a proof. In particular, proofs are rarely written in full details, and some steps of a proof are generally considered as 1397:. 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 1057:

debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is

635:. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects (see 3467:

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.

1977:, chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent were seriously undermined by the second of 5166:

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).

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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;

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Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is non-causal, and not analogous to perception. This argument is developed by

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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.

3246:. Since theories are not confirmed in a piecemeal fashion, but as a whole, there is no justification for excluding any of the entities referred to in well-confirmed theories. This puts the 3211:

to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets. The form of the argument is as follows.

3045: 3040:, 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 1086: 3015:

has also promoted the social view of mathematics, calling it a "humanistic" approach, similar to but not quite the same as that associated with Alvin White; one of Hersh's co-authors,

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For many centuries, logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. Circa the end of the 19th century, several

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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

4741: 1591: 6202: 782:. Roughly speaking, the first one consists of requiring that every existence theorem must provide an explicit example, and the second one excludes from mathematical reasoning the 2408:

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

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axiomatization of all of mathematics. Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual

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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.

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In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and

1013:. The whole mathematics has been rebuilt inside this theory. Except if the contrary is explicitly stated, all modern mathematical texts use it as a foundation of mathematics. 1393:

Surprising and counter-intuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the

5697: 4603: 4011: 1732:. That is, in the sense that "in those complex enough to contain self-aware substructures will subjectively perceive themselves as existing in a physically 'real' world". 3143:

is a part of mathematics to which mathematics cannot be reduced), linguists believe that the implications of such a statement must be considered. For example, the tools of

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During the 19th century, there were an active research for giving more precise definitions to the basic concepts resulting of abstraction from the real world; for example

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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

2489:(convenient for counting because humans have ten fingers). The axioms or logical rules behind mathematics also vary through time (such as the adaption and invention of 1916:

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

5809:"Since abstract objects are outside the nexus of causes and effects, and thus perceptually inaccessible, they cannot be known through their effects on us" — Katz, J. 2392:

approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view mathematical

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of perceptions, the body, and the senses may have been necessary for survival) is not necessarily accurate to a full realism (and is still subject to flaws such as

1603:'s Platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things 1162:

A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the

199: 1920:(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 1312:, the focus shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time of 297:

about the point of view of notable mathematicians and many important philosophical problems, such as the relationship between mathematics and other sciences.

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Wilson, Edwin B.; Lewis, Gilbert N. (November 1912). "The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics".

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Davis, Jon D.; McDuffie, Amy Roth; Drake, Corey; Seiwell, Amanda L. (2019). "Teachers' perceptions of the official curriculum: Problem solving and rigor".

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Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this universe.

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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

954:; in its strong form this school may be understood as the fact that mathematicians must not take care of the physical reality. The second choice led to 4862: 825:

of such a theory is either an axiom or an assertion that can be obtained from previously known theorems by the application of an inference rule. The

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It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a

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seemed even recently." Popper also noted he would "admit a system as empirical or scientific only if it is capable of being tested by experience."

1653:). It holds that all mathematical entities exist. They may be provable, even if they cannot all be derived from a single consistent set of axioms. 883:(used for measurements), etc. were abstractions from the physical world, and it was commonly considered that it was sufficient for defining them. 5202: 4422: 3317:, according to which there are no mathematical objects. Nonetheless, some versions of structuralism are compatible with some versions of realism. 2920:

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

2508:, cognitive errors or assumptions in a general context, it can be questioned whether they are accurate or strictly indicative of truth (see also: 3048:", in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain. 2830:

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

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After strong debates, axiomatic approach became eventually a de facto norm in mathematics. This mean that mathematical theories must be based on

10210: 1728:) goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does. Tegmark's sole postulate is: 8820: 4453: 2593: 2578: 1878:, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favor of abstraction principles such as 5853: 2745:

come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.

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in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.

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of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led

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In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (

4486: 2439:, assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, and 1749:

is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. Logicists hold that mathematics can be known

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that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to

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made questionable the logical foundation of mathematics, and consequently the validity of the whole mathematics. This has been called the

6277: 3176:, which can be explicitly defined in a text: "Effectively, we are allowed to introduce a word in one part of a sentence, and declare its 3139:

of science. Although some mathematicians and philosophers would accept the statement "mathematics is a language" (most consider that the

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Ginammi, Michele (February 2016). "Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω particle".

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is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their

1612: 735:. Some of these paradoxes consist of results that seem to contradict the common intuition, such as the possibility to construct valid 186: 112: 7398: 1937: 17: 5612: 3135:

Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said, the

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view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what

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This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.

3671: 1684: 119: 6175: 1203:, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments. 1139:. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, 6151: 6138: 2882:

to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by

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In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown

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This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without

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for its compatibility with general mathematics; as while relatively reliable, it is still limited by what can be measured by

1911: 981:) involved in the axioms are considered as defined by the axioms, and nothing else is supposed on their nature. For example, 951: 9553: 6163: 3692: 10620: 9943: 9183: 4934: 4286: 2819:

comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e.

467: 126: 5634: 1978: 8419: 6124: 6094: 6080: 6064: 1699:; the more traditional kind of Platonism they defend is distinguished by general principles that assert the existence of 3964: 11180: 10625: 10615: 10352: 10205: 9558: 9328: 8813: 8548: 6604: 5876: 5759: 5591: 5553: 935: 732: 9549: 7661: 2474:

require specific changes to more commonly used axioms of mathematics; otherwise they cannot be adequately understood.

1962: 1587:, a theory that postulates that all structures that exist mathematically also exist physically in their own universe. 108: 10761: 9400: 8697: 7686: 6246: 6035: 6003: 5959: 5938: 5919: 5366: 3706: 3464: 3406: 3122: 2658:

seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as

2043:, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition—see 524: 325: 267: 159: 62: 10103: 3104: 622:

The connection between mathematics and material reality has led to philosophical debates since at least the time of

10858: 10602: 9427: 8864: 8781: 8218: 3353:. In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another. 3350: 2696: 2566: 1721: 1035: 724:. Logic is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere. 5700:". City University of Hong Kong. Republished in Philosophy of Mathematics Education Journal No. 26 (December 2011) 1377:, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and 10163: 9856: 9278: 8373: 8248: 6270: 3863: 3761:

Even if some people could consider that more accurates definitions were needed, they were unable to provide them.

3743:. Even with this modern technology, it may take years of human work for writing down a completely detailed proof. 3160:

to show that the distinction between mathematical language and natural language may not be as great as it seems.

2795: 2717: 1418: 1414: 826: 234: 9597: 7856: 2741:. Mill's view was widely criticized, because, according to critics, such as A.J. Ayer, it makes statements like 1426:, propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the 977:) that can be deduced (proved) from the axiom by using inference rules, and inference rules only. The entities ( 11119: 10821: 10584: 10579: 10404: 9825: 9509: 9376: 8702: 8274: 7434: 3388: 3157: 3089: 2889:

Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of

1753:, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus 1047: 565:: The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. 300: 97: 54: 5823: 1805:

from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts

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The human mind has no special claim on reality or approaches to it built out of math. If such constructs as

2431:

Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested the

934:. These formal definitions allowed to prove counterintuitive results, which are a part of the origin of the 11039: 10865: 10551: 10185: 9784: 9388: 8253: 8158: 7806: 6523: 6208: 3491: 2598: 2535: 2111: 1692: 1521: 688: 517: 4858: 2039:

in the development of mathematical theories and computer software. Because of their close connection with

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Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics

2257:

objects as classical finitists, and those who deny even countably infinite objects as strict finitists.

10846: 10436: 9830: 9798: 9489: 8874: 8728: 7924: 7593: 6899: 6675: 6263: 5358:

An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure

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has explicitly formulated a social constructivist philosophy of mathematics. Some consider the work of

2298: 2293: 2138: 1350: 1061:, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a 859:

Also, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.

717: 133: 6309: 5287:

Gonzalez gives as the sources for the memorial article, the following: Weber, H: "Leopold Kronecker",

11085: 10982: 10480: 10441: 9918: 9563: 9288: 9260: 8707: 8632: 8414: 8130: 6332: 5884: 3612: 2898:

see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like

2799: 2764: 2451:

which may not be as reliable as previously assumed (see also: 'counterintuitive' concepts in such as

2105: 1616: 1309: 794: 472: 462: 9592: 5384:"Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics" 4538:. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp. 41–77. 4500:

Sarukkai, Sundar (February 10, 2005). "Revisiting the 'unreasonable effectiveness' of mathematics".

10977: 10907: 10446: 10298: 10281: 10004: 9484: 9333: 8605: 8233: 7796: 7046: 6886: 6883: 6609: 6443: 6428: 3096: 2690: 2077: 775: 774:

Several methods have been proposed to solve the problem by changing of logical framework, such as

736: 666:

concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a

6145:

offers many suggestions on what to read, depending on the student's familiarity with the subject:

5845: 2516:

itself in relation to the universe and whether it is independent to the senses and the universe.

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was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning with

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The justification for the first premise is the most controversial. Both Putnam and Quine invoke

2172:. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical 1320:, but was brought into question by developments in the late 19th and early 20th centuries. 11099: 10826: 10804: 10771: 10664: 10510: 10495: 10468: 10419: 10303: 10238: 10063: 10029: 10024: 9898: 9729: 9706: 9316: 9213: 9193: 9188: 9117: 8842: 7944: 7876: 6956: 6794: 6372: 6347: 6337: 4660: 3711: 3592: 3255: 2946: 2925: 2894: 2721: 2701: 2355: 1871: 1855: 1700: 943: 783: 477: 452: 355: 238: 6234: 5177: 4212: 2149:

is also rejected in most intuitionistic set theories, though in some versions it is accepted.

1100:

of natural numbers that was discovered more than 2,000 years before its common use for secure

11029: 10882: 10674: 10392: 10128: 10034: 9893: 9878: 9759: 9734: 9343: 9273: 9150: 9074: 9013: 8998: 8993: 8970: 8852: 8743: 8181: 8120: 8100: 7934: 7846: 7826: 7816: 7449: 7298: 6931: 6863: 6771: 6738: 6566: 6546: 6352: 6071: 5949: 5453: 5446: 5000: 4983: 4320: 3716: 3685: 3607: 3602: 3587: 3342: 3326: 2959: 2954: 2942: 2937: 2787: 2705: 2663: 2659: 2421: 2200: 2081: 1864: 1854:). Frege abandoned his logicist program soon after this, but it was continued by Russell and 1378: 487: 401: 6025: 5993: 5909: 4955: 2237:, according to which a mathematical object does not exist unless it can be constructed from 2160:

to fill this gap, leading to the claim that only questions regarding the behavior of finite

1469:

When philosophy discovers something wrong with science, sometimes science has to be changed—

245: 11002: 10964: 10841: 10645: 10485: 10409: 10387: 10215: 10173: 10072: 10039: 9903: 9691: 9602: 9323: 9203: 9198: 9122: 9023: 8662: 8508: 8198: 8153: 8115: 8062: 7991: 7747: 7543: 7454: 7277: 7247: 6988: 6516: 6362: 6357: 5489: 5114: 4892: 4707: 4462: 4355: 3649: 3306:, etc., which is already fully accountable in terms of physical processes in their brains. 3169: 2790:

in any real sense. He advocated a form of "pure realism" that rejected mystical notions of

2652:

and/or truths are grounded in, derived from or explained by psychological facts (or laws).

2544: 2456: 2332: 2164:

are meaningful and should be investigated in mathematics. This has led to the study of the

2142: 2130: 2125:, who rejected the usefulness of formalized logic of any sort for mathematics. His student 1958: 1859: 1851: 1754: 1566: 1470: 1342: 1211: 1144: 939: 915: 779: 768: 744: 447: 406: 375: 6051: 3329:

that involves contact with the Platonic realm. A modern form of this argument is given by

2845:

For experimental evidence suggesting that human infants can do elementary arithmetic, see

2416:

The cognitive processes of pattern-finding and distinguishing objects are also subject to

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to carry out his reduction, and because the statement of conservativity seems to require

2879: 2875: 2798:. This grew from the increasingly popular assertion in the late 20th century that no one 2584: 2520: 2452: 2157: 1921: 1879: 1097: 1066: 987: 978: 919: 911: 810: 748: 698: 636: 7051: 5493: 5118: 4896: 4711: 4564: 4466: 1679:

version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of

299:

Please expand the article to include this information. Further details may exist on the

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entities that are indispensable to the best scientific theories, and to those entities

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with no reference to numbers or functions at all. He started with the "betweenness" of

2816: 2734: 2614: 2277: 2254: 2134: 2085: 2009: 1917: 1761:

is the proper foundation of mathematics, and all mathematical statements are necessary

1465:

summed up one common view of the situation in the last third of the century by saying:

1410: 1244: 1223: 1156: 1152: 1043: 1030: 798: 740: 694: 507: 411: 10113: 3419:

Many practicing mathematicians have been drawn to their subject because of a sense of

1423: 11155: 11095: 10902: 10712: 10702: 10594: 10475: 10310: 10286: 10067: 10051: 9956: 9933: 9810: 9779: 9744: 9639: 9474: 9359: 9079: 9064: 9008: 8955: 8767: 8738: 8733: 8723: 8657: 8585: 8470: 8327: 8228: 8223: 8208: 8148: 8110: 8095: 8052: 7623: 7583: 7501: 7429: 7412: 7390: 7018: 6993: 6804: 6624: 6367: 6242: 6230: 6172: 6160: 6031: 6009: 5999: 5977: 5955: 5934: 5915: 5726: 5587: 5549: 5522: 5497: 5482: 5457: 5407: 5362: 5335: 5246: 4826: 4408: 4388: 4363: 4328: 4292: 4228: 4125: 4097: 4076: 3934: 3884: 3842: 3627: 3617: 3553: 3533: 3427: 3330: 3251: 3165: 3149: 2846: 2637: 2558: 2444: 2329: 2261: 2224: 2165: 1966: 1858:. They attributed the paradox to "vicious circularity" and built up what they called 1584: 1525:, the definition of mathematics was more like the aesthetic combination of concepts. 1454: 1427: 1293: 1113: 892: 503: 6148: 5969: 5740: 4217:

The Map and the Territory: Exploring the Foundations of Science, Thought and Reality

4007: 3861:

White, L. (1947). "The locus of mathematical reality: An anthropological footnote".

2962:

not enough, due to an overemphasis on axiomatic proof and peer review as practices.

2602: 2069: 654: 11109: 11104: 10997: 10954: 10776: 10737: 10732: 10717: 10543: 10500: 10397: 10195: 10145: 9719: 9681: 9293: 9268: 9140: 8988: 8925: 8672: 8398: 8393: 8279: 8213: 8203: 7904: 7851: 7801: 7781: 7742: 7737: 7578: 7506: 7224: 7135: 7026: 6998: 6983: 6946: 6652: 6632: 6599: 6504: 6466: 5989: 5718: 5659: 5395: 5134: 5122: 5069: 5038: 4960:"Platonism in the Philosophy of Mathematics", (Stanford Encyclopedia of Philosophy) 4912: 4900: 4715: 4672: 4634: 4528: 4482: 4470: 4220: 4178: 4174: 4149: 4137: 4064: 3987: 3979: 3922: 3872: 3832: 3446: 3230:

Mathematical entities are indispensable to the best scientific theories. Therefore,

3153: 2738: 2655: 2588: 2471: 2322: 2318: 2173: 2122: 2040: 1982: 1847: 1840: 1786:

of mathematics can be derived from logical axioms through purely logical deduction.

1540: 1450: 1338: 1317: 1278: 1105: 1006: 907: 900: 868: 645:

summarized this view of mathematics reality as follows, and provided quotations of

4255: 3262:

and other undetectable entities of physics, for example, in a difficult position.

3056:

Realist and constructivist theories are normally taken to be contraries. However,

2906:

lived at 221B Baker Street"—but both are true according to the relevant fictions.

1695:: according to this view, a more traditional kind of Platonism is consistent with 1575:

of ancient Greece, who believed that the world was, quite literally, generated by

1413:, as well as the notion of a proposition being true of a mathematical object (see 1218:

is considered the father of mathematics and geometry as he set the foundation for

11090: 11080: 11034: 11017: 10972: 10934: 10836: 10756: 10563: 10490: 10463: 10451: 10357: 10271: 10245: 10200: 10168: 9969: 9771: 9714: 9664: 9629: 9587: 9233: 9160: 9089: 8882: 8518: 8460: 7985: 7954: 7919: 7884: 7762: 7613: 7511: 7469: 7380: 7368: 7353: 7328: 7303: 7073: 6941: 6936: 6853: 6838: 6511: 6395: 6197: 6179: 6167: 6155: 6142: 6075: 6021: 5901: 5880: 5541: 5057: 5005:

Naturalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy)

4988:

Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences

4536:

Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL

4224: 4091: 3740: 3678: 3658:

On Formally Undecidable Propositions of Principia Mathematica and Related Systems

3622: 3567: 3474: 3437: 3288: 3279: 3016: 2995: 2973:

and often has great difficulty communicating, or motivating the investigation of

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

1680: 1650: 1646: 1621: 1558: 1474: 1458: 1436: 1140: 1125: 1070: 1034:

concepts, imply the need to change the mathematical model used. For example, the

1010: 970: 872: 830: 806: 658: 650: 429: 7125: 5873: 4926: 4719: 3008: 1292:

These earlier Greek ideas of numbers were later upended by the discovery of the

11075: 11054: 11012: 10992: 10887: 10742: 10340: 10330: 10320: 10315: 10249: 10123: 9999: 9888: 9883: 9861: 9462: 9311: 9238: 8945: 8772: 8465: 8444: 8359: 7894: 7889: 7757: 7722: 7654: 7628: 7464: 7313: 7252: 7165: 7008: 6904: 6647: 6327: 6218: 5638: 5427: 5399: 5264:

From an 1886 lecture at the 'Berliner Naturforscher-Versammlung', according to

4417: 4068: 3177: 2979: 2890: 2831: 2756: 2671: 2606: 2467: 2463: 2420:; if mathematics is considered to be relevant to a natural world (such as from 2238: 2220: 2153: 2032: 1875: 1664: 1285:. For example, at one time, the Greeks held the opinion that 1 (one) was not a 1227: 1062: 982: 947: 888: 876: 818: 802: 706: 684: 457: 7335: 7110: 6089: 6013: 5981: 5722: 5126: 4676: 4141: 3926: 3172:). One important difference is that mathematical objects have clearly defined 2998:, although it is not clear that either would endorse the title. More recently 11174: 11049: 10727: 10234: 10019: 10009: 9979: 9964: 9634: 9099: 9031: 8983: 8622: 8563: 7949: 7861: 7791: 7533: 7323: 7257: 7214: 7083: 6708: 6494: 6476: 6193: 6060: 5905: 5755: 5238: 5159: 5042: 4444: 4360:

Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists

3846: 3456: 3208: 3204: 3200: 3041: 2760: 2675: 2667: 2540: 2505: 2440: 2405: 2272: 2196: 2192: 2126: 2066: 2005: 2001: 1997: 1993: 1954: 1947: 1790: 1768: 1762: 1757:, not requiring any special faculty of mathematical intuition. In this view, 1725: 1712: 1608: 1512: 1478: 1462: 1431: 1313: 1252: 1117: 1090: 760: 382: 4571:. School of Mathematics and Statistics, University of St Andrews, Scotland. 3470: 3003: 2012:

study formal systems but are just as often realists as they are formalists.

1600: 1594: 821:

that allow producing new assertions from one or several known assertions. A

10949: 10796: 10697: 10689: 10569: 10517: 10426: 10362: 10345: 10276: 10135: 9994: 9696: 9479: 9041: 9036: 8940: 8612: 8434: 7980: 7970: 7929: 7909: 7681: 7644: 7603: 7489: 7439: 7100: 7078: 7056: 7003: 6971: 6843: 6703: 6614: 6410: 5565: 4792: 4474: 4325:

Mathematical Enculturation: A Cultural Perspective on Mathematics Education

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The most accessible, famous, and infamous treatment of this perspective is

2417: 2381: 1625: 1571: 1442: 1370: 1358: 1330: 1305: 1167: 1129: 997:

This axiomatic approach has been applied to the whole mathematics, through

955: 756: 642: 8798: 5637:(Interview). Interviewed by John Brockman. Edge Foundation. Archived from 3032:

Rather than focus on narrow debates about the true nature of mathematical

2565:. For more on the philosophical ideas that inspired this perspective, see 1671:

is about a single universe of sets. This position (which is also known as

11059: 10939: 10118: 10108: 10055: 9739: 9659: 9644: 9524: 9469: 9243: 8907: 8830: 8647: 8642: 8595: 7975: 7914: 7786: 7766: 7671: 7608: 7568: 7548: 7474: 7444: 7105: 7041: 6733: 6718: 6594: 6584: 6533: 6499: 6438: 5715:

The Language of Mathematics: A Linguistic and Philosophical Investigation

5687:

Popper, Karl Raimund (1946) Aristotelian Society Supplementary Volume XX.

5477: 5265: 5228:, Cambridge University Press, Cambridge, UK, 1995. 2nd edition. Page 542. 5221: 4170: 3642: 3486: 3216: 3144: 3057: 2999: 2824: 2820: 2748: 2482: 2478: 2310: 2169: 2044: 1929: 1717: 1583:

world and discover truths about the entities? One proposed answer is the

1516: 1500: 1406: 1058: 927: 896: 880: 764: 697:. This means that the definitions must be absolutely unambiguous and the 646: 552: 420: 346: 6135: 5831: 4646: 4513: 2118:

forms of the volitions that inform the perception of empirical objects.

1836: 1507:

combination of assumptions, and then also claims that mathematics is an

9989: 9844: 9815: 9621: 9228: 9107: 8902: 8590: 8558: 8523: 7821: 7649: 7598: 7588: 7459: 7363: 7308: 7115: 7095: 6961: 6728: 6642: 6471: 6418: 6382: 6286: 6241:. Vol. 2 (1893–1913). Indiana University Press. pp. 300–324. 6222: 6055: 5201:(Summer 2019 ed.), Metaphysics Research Lab, Stanford University, 5081: 4596:"Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry" 4449:"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" 3999: 3991: 3790:(Spring 2016 ed.). Metaphysics Research Lab, Stanford University. 3495:, in which Hardy argues that pure mathematics is superior in beauty to 3431: 3391: in this section. Unsourced material may be challenged and removed. 3298: 3247: 2966: 2513: 2497: 2466:

may not necessarily be applicable to problem solving. Subjects such as

2448: 2385: 2343:

mathematical objects or structures have (not, in other words, to their

2250: 2242: 1970: 1802: 1668: 1449:

At the middle of the century, a new mathematical theory was created by

1334: 1301: 1248: 1215: 623: 558:

Major themes that are dealt with in philosophy of mathematics include:

548: 425: 415: 4816:. Amsterdam, Netherlands: North-Holland Publishing Company. p. 5. 4684: 4327:. Norwell, Massachusetts: Kluwer Academic Publishers. pp. 20–59. 4036:

Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs".

3665:

On Computable Numbers, with an Application to the Entscheidungsproblem

2023:

Recently, some formalist mathematicians have proposed that all of our

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in another; and this operation has no analogue in natural language."

3046:

The Unreasonable Effectiveness of Mathematics in the Natural Sciences

2910: 2779: 2772: 2524: 2432: 2425: 2161: 1899: 1867:". Even Russell said that this axiom did not really belong to logic. 1534: 1504: 1401:

around 300 BCE as the natural basis for mathematics. Notions of

1382: 1264: 1148: 710: 702: 7061: 5073: 3983: 3366: 3107:. Statements consisting only of original research should be removed. 1417:), were formalized, allowing them to be treated mathematically. The 1089:

is a phenomenon that was named and first made explicit by physicist

75: 11126: 10924: 10372: 10077: 9671: 9132: 9051: 8978: 8573: 7666: 7573: 7538: 7496: 7484: 7272: 7066: 6966: 6909: 6713: 6667: 6551: 6110: 6102: 3876: 3736: 3508: 2783: 2768: 2509: 2501: 2436: 2344: 2230: 2215: 1974: 1746: 1741: 1374: 1362: 1297: 1282: 1272: 1260: 1200: 1163: 1136: 1101: 1039: 969:(basic assumptions that are considered as true) and a fixed set of 931: 841: 801:

as a new area of mathematics. In this framework, a mathematical or

771:

shows that the phrase "the set of all sets" is self contradictory.

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In Search of a Better World: Lectures and Essays from Thirty Years

5109: 5029:

Balaguer, Mark (1994). "Against (Maddian) naturalized Platonism".

4385:

Philosophy of Pseudoscience: Reconsidering the Demarcation Problem

3292:

entities. By general agreement, abstract entities cannot interact

2195:, who managed to prove versions of the most important theorems in 1590: 1239:

analysis. There are traditions of mathematical philosophy in both

895:, originally meaning that reason cannot conceive them. Similarly, 867:

Before the 19th century, the basic mathematical concepts, such as

10722: 9514: 8917: 8637: 8568: 7293: 7267: 7262: 7204: 7199: 7031: 6919: 6914: 6873: 6695: 6541: 6423: 5954:. Handbook of the Philosophy of Science. North-Holland Elsevier. 4848:

64/1, 5-22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).

2917: 2685: 2649: 2486: 1604: 1109: 1026: 1002: 974: 728: 597: 392: 3895:. Vol. 4. New York: Simon and Schuster. pp. 2348–2364. 3516: 285: 177: 8475: 7558: 7479: 7209: 6868: 6858: 6556: 6458: 6212: 5717:. Lecture Notes in Computer Science. Vol. 7805. Springer. 4126:"Mathematical Models and Reality: A Constructivist Perspective" 3460: 3349:

A more radical defense is denial of physical reality, i.e. the

3293: 3233:

One must have ontological commitments to mathematical entities.

2729:

is a form of realism that denies that mathematics can be known

2409: 2093: 1833:), a principle that he took to be acceptable as part of logic. 1576: 1398: 1286: 1219: 923: 814: 5787:. Cambridge: Cambridge University Press, 1975. 2nd. ed., 1985. 4931:

Platonism in Metaphysics (Stanford Encyclopedia of Philosophy)

3499:

precisely because it cannot be used for war and similar ends.

3168:), but many of the same analytical tools can be used (such as 2771:

to say why light bulbs behave as they do, then electrons must

1730:

All structures that exist mathematically also exist physically

10266: 9612: 9457: 8667: 8382: 7374: 7036: 6322: 6255: 3259: 3033: 2791: 2605:

in the philosophy of mathematics and is close to the view of

2354:

structuralism ("before the thing") has a similar ontology to

2268:

God created the natural numbers, all else is the work of man.

1758: 1554: 1402: 1386: 1268: 1256: 1230:: a mathematical and philosophical model to map the universe. 1155:

is a non-Euclidean space of dimension four, and spacetime of

966: 716:

The rules of rigorous reasoning have been established by the

680: 627: 5785:

Mathematics, Matter and Method. Philosophical Papers, vol. 1

4096:. Mathematical Modeling. Vol. 1. CRC Press. p. 1. 3036:, or even on practices unique to mathematicians such as the 2684:, and many of his works and essays, including his review of 763:, which led to consider several sizes of infinity (infinite 8627: 5095:

Tegmark, Max (February 2008). "The Mathematical Universe".

5060:(1995). "Naturalized Platonism vs. Platonized Naturalism". 4844:*Putnam, Hilary (1967), "Mathematics Without Foundations", 3051: 2523:

are true then they are true as a map of the human mind and

2490: 2027:

mathematical knowledge should be systematically encoded in

1121: 605: 1943: 670:

of an objective existence, of a reality of mathematics ...

7219: 6027:

Thinking About Mathematics: The Philosophy of Mathematics

5175: 4387:. Chicago: The University of Chicago Press. p. 104. 4093:

Modelling Mathematical Methods and Scientific Computation

4054: 2443:). As such, this may also raise questions for the modern 1508: 1281:

on mathematics was strongly influenced by their study of

998: 8351: 5383: 5269: 5162:(1931), "Die logizistische Grundlegung der Mathematik", 4627:

Proceedings of the American Academy of Arts and Sciences

4219:. The Frontiers Collection. Springer. pp. 345–356. 4215:. In Wuppuluri, Shyam; Doria, Francisco Antonio (eds.). 4044:(167 (4)). Revue Internationale de Philosophie: 500–508. 3188: 2339:

of entity a mathematical object is, not to what kind of

1251:, who described the theory "everything is mathematics" ( 1247:. Western philosophies of mathematics go as far back as 899:

are the numbers that can be used for measurement, while

5852:. Metaphysics Research Lab, Stanford University. 2018. 5698:

Arithmetic and Reality: A Development of Popper's Ideas

5007:. Metaphysics Research Lab, Stanford University. 2016. 4962:. Metaphysics Research Lab, Stanford University. 2018. 4933:. Metaphysics Research Lab, Stanford University. 2016. 2965:

The social nature of mathematics is highlighted in its

2733:

at all. It says that we discover mathematical facts by

2313:. Other examples of mathematical objects might include 1886:

equals the number of objects falling under the concept

1147:

that uses fundamentally these concepts. In particular,

6120:"Ludwig Wittgenstein: Later Philosophy of Mathematics" 4383:

Nickles, Thomas (2013). "The Problem of Demarcation".

4090:

Bellomo, Nicola; Preziosi, Luigi (December 22, 1994).

3019:, has expressed sympathy for the social view as well. 2645:

in the philosophy of mathematics is the position that

6239:

The Essential Peirce, Selected Philosophical Writings

5289:

Jahresberichte der Deutschen Mathematiker Vereinigung

3265: 1850:

discovered that Basic Law V is inconsistent (this is

1175: 862: 701:

must be reducible to a succession of applications of

5313:

The Foundations of Mathematics in the Theory of Sets

3529: 3320:

The argument hinges on the idea that a satisfactory

2755:

Contemporary mathematical empiricism, formulated by

2619:

The Foundations of Mathematics in the Theory of Sets

2369:

structuralism ("in the thing") is the equivalent of

2282:

The Foundations of Mathematics in the Theory of Sets

2004:, considered mathematics to be the investigation of 689:

Mathematics § Mathematical logic and set theory

4883:S, F. (January 1941). "A Mathematician's Apology". 4321:"Environmental activities and mathematical culture" 2958:mathematics is often accorded too much status, and 1381:in particular, did not live up to the standards of 100:. Unsourced material may be challenged and removed. 5481: 5445: 3463:; the second is a more direct proof involving the 1771:(1931) presents the logicist thesis in two parts: 1489: 1191: 1124:with planes). It is almost 2,000 years later that 918:(infinite sums that may have a finite value), and 555:and its relationship with other human activities. 5291:, vol ii (1893), pp. 5-31. Cf. page 19. See also 5268:'s memorial article, as quoted and translated in 4797:Mathematical Thought from Ancient to Modern Times 4665:The British Journal for the Philosophy of Science 4663:(November 1961). "Discovering the Positron (I)". 3286:. Platonism posits that mathematical objects are 3022: 1882:(the number of objects falling under the concept 11172: 5900: 5303: 5301: 5182:. Metaphysics Research Lab, Stanford University. 4859:"A Mathematician's Apology Quotes by G.H. Hardy" 4671:(47). The University of Chicago Press: 194–214. 2551:has investigated similar concepts with his book 1553:is used because such a view is seen to parallel 5708: 5706: 5546:New Directions in the Philosophy of Mathematics 4089: 3518:The Philosophy of Mathematics Education Journal 3027: 2990:Contributions to this school have been made by 1108:. A second historical example is the theory of 1069:and results (theorems) are often obtained from 1038:could only be explained after the emergence of 930:, the definition of real numbers by Cauchy and 910:for natural numbers, the formal definitions of 5974:The Philosophy of Mathematics, An Introduction 4454:Communications on Pure and Applied Mathematics 3313:. Benacerraf also developed the philosophy of 2579:Aristotelian realist philosophy of mathematics 1259:, who paraphrased Pythagoras, and studied the 797:has been eventually resolved with the rise of 9428: 8814: 8367: 6271: 5521:. Abingdon-on-Thames: Routledge. p. 18. 5452:. New York: Dover Publications, Inc. p. 5298: 5001:"Naturalism in the Philosophy of Mathematics" 4835:, 2nd edition, Springer-Verlag, New York, NY. 4401: 4291:. London: Taylor & Francis. p. 100. 4169: 4057:International Journal of Educational Research 3278:argument" against Platonism has been made by 2838:. Another example of a realist theory is the 2694:. Edmund Husserl, in the first volume of his 1863:to develop much of mathematics, such as the " 1080: 1020: 525: 5929:Hart, W. D. (1996). Wilbur Dyre Hart (ed.). 5914:(2nd ed.). Cambridge University Press. 5911:Philosophy of Mathematics, Selected Readings 5846:"Platonism in the Philosophy of Mathematics" 5712: 5703: 5245:(Paperback ed.), New York: Ishi Press, 5179:Deductivism in the Philosophy of Mathematics 4956:"Platonism in the Philosophy of Mathematics" 4354: 4256:"The science checklist applied: Mathematics" 3965:"The Road to Modern Logic—An Interpretation" 3775: 2477:Alternatively, computer programmers may use 1793:was the founder of logicism. In his seminal 709:, without any use of empirical evidence and 8828: 6223:"The Philosophy of Real Mathematics – Blog" 5088: 5055: 4802: 4624: 4376: 2893:. He showed that mathematical physics is a 2839: 2481:for its 'human-friendly' representation of 2321:in geometry, or elements and operations in 2305:in such structures, consequently having no 1329:characterized by a predominant interest in 63:Learn how and when to remove these messages 9620: 9435: 9421: 8821: 8807: 8374: 8360: 6278: 6264: 6187:R.B. Jones' philosophy of mathematics page 5666:. Humanistic Mathematics Network Journal. 4534:. In Bos, Joppe W.; Stam, Martijn (eds.). 4437: 4312: 2674:. Psychologism was famously criticized by 2260:The most famous proponent of finitism was 2186:Constructivism (philosophy of mathematics) 1989:whatsoever, regardless of interpretation. 1323: 1159:is a (curved) manifold of dimension four. 946:that is everywhere continuous and nowhere 532: 518: 248:. Please do not remove this message until 5753: 5155: 5153: 5108: 4594:Mukunth, Vasudevan (September 10, 2015). 4407: 3962: 3836: 3407:Learn how and when to remove this message 3227:(commonly referred to as "all and only"). 3123:Learn how and when to remove this message 3068: 2807:. Quasi-empiricism was also developed by 2400: 2384:about structures in a way that parallels 2370: 1953:A major early proponent of formalism was 1928:Another version of formalism is known as 1087:unreasonable effectiveness of mathematics 326:Learn how and when to remove this message 268:Learn how and when to remove this message 160:Learn how and when to remove this message 5609:"Is Mathematics Discovered or Invented?" 5586:. Oxford University Press. p. 239. 5381: 5354: 5307: 5028: 4832:Categories for the Working Mathematician 4734:"Is mathematics discovered or invented?" 4526: 4499: 4035: 3781: 3455:. The first is the traditional proof by 3052:Popper's two senses of number statements 2931: 2219: 1942: 1835: 1589: 1210: 244:Relevant discussion may be found on the 202:of all important aspects of the article. 6020: 5995:Introduction to Mathematical Philosophy 5988: 5348: 5270:Gonzalez Cabillon, Julio (2000-02-03). 5199:The Stanford Encyclopedia of Philosophy 5094: 4697: 4593: 4489:from the original on February 28, 2011. 4382: 4207: 3811: 3809: 3788:The Stanford Encyclopedia of Philosophy 3672:Introduction to Mathematical Philosophy 3250:who wishes to exclude the existence of 3060:argued that a number statement such as 2572: 14: 11173: 9442: 6229: 6114:, Internet Encyclopaedia of Philosophy 6106:, Internet Encyclopaedia of Philosophy 5968: 5947: 5828:: "Mathematical Knowledge: A dilemma"" 5779: 5777: 5516: 5476: 5434:, Oxford University Press, Oxford, UK. 5237: 5226:The Cambridge Dictionary of Philosophy 5150: 4808: 4764:"Math: Discovered, Invented, or Both?" 4659: 4606:from the original on November 20, 2022 4545:from the original on November 20, 2022 4443: 4318: 4284: 4262:. University of California, Berkeley. 4189:from the original on November 17, 2022 4123: 3905: 3890: 3679:New Foundations for Mathematical Logic 3195:Quine–Putnam indispensability argument 2613:The Euclidean arithmetic developed by 2424:or a degree of it, as opposed to pure 2121:A major force behind intuitionism was 2035:of mathematical proofs and the use of 973:; the theory consists of the results ( 198:Please consider expanding the lead to 9416: 8802: 8355: 8017: 6759: 6297: 6259: 5632: 5421: 5329: 4919: 4575:from the original on October 14, 2022 4266:from the original on October 27, 2019 4017:from the original on February 2, 2023 3944:from the original on December 5, 2022 3860: 3815: 3794:from the original on January 30, 2022 3189:Indispensability argument for realism 2504:, assumptions, deceptions, (induced) 2380:structuralism ("after the thing") is 2205:Foundations of Constructive Analysis. 1912:Formalism (philosophy of mathematics) 1607:said about mathematics, and supports 9383: 6217: 5928: 5850:Standard Encyclopaedia of Philosophy 5762:from the original on January 7, 2020 5660:"Humanism and Mathematics Education" 5581: 5575: 5443: 5272:"FOM: What were Kronecker's f.o.m.?" 5243:Foundations of Constructive Analysis 5192: 4413:"Are There 'Other' Ways of Knowing?" 3806: 3700: 3389:adding citations to reliable sources 3360: 3072: 2928:over abstract models or deductions. 2500:from the human brain are subject to 2072:was among the first to articulate a 279: 218: 171: 98:adding citations to reliable sources 69: 28: 9395: 8549:Analytic and synthetic propositions 8420:Formal semantics (natural language) 6237:. In Peirce Edition Project (ed.). 6235:"22. New Elements (Καινα Στοιχεία)" 6125:Internet Encyclopedia of Philosophy 6095:Internet Encyclopedia of Philosophy 6081:Stanford Encyclopedia of Philosophy 6069: 6065:Indiana Philosophy Ontology Project 5774: 5635:"What Kind of a Thing is a Number?" 5633:Hersh, Reuben (February 10, 1997). 5488:. New York: Routledge. p. 56. 4799:, page 32. Oxford University Press. 4288:Dictionary of Scientific Quotations 4183:Stanford Encyclopedia of Philosophy 2778:Putnam strongly rejected the term " 2360:Benacerraf's identification problem 813:, a set of basic assertions called 24: 5894: 5798:Realism, Mathematics, and Modality 5747: 5606: 4774:from the original on 28 March 2018 3266:Epistemic argument against realism 3258:, but to include the existence of 2137:; this logic does not contain the 2050: 1628:have suggested in their 1999 book 1177: 936:foundational crisis of mathematics 863:Relationship with physical reality 733:foundational crisis of mathematics 674: 574:Relationship with physical reality 25: 11192: 6045: 5519:The Logic of Scientific Discovery 5176:Alexander Paseau; Fabian Pregel. 4882: 4744:from the original on 27 July 2018 4425:from the original on May 13, 2020 3707:History and philosophy of science 3465:fundamental theorem of arithmetic 3310: 2866:was brought to fame in 1980 when 2670:, past and present: for example, 2179: 1846:Frege's construction was flawed. 44:This article has multiple issues. 11154: 9394: 9382: 9371: 9370: 9358: 8761: 8335: 8334: 8321: 4527:Wagstaff, Samuel S. Jr. (2021). 3634: 3546: 3532: 3365: 3351:mathematical universe hypothesis 3077: 3062:"2 apples + 2 apples = 4 apples" 2945:sees mathematics primarily as a 2763:, is primarily supported by the 2567:cognitive science of mathematics 2287: 1890:if and only if the extension of 1722:mathematical universe hypothesis 1706: 1226:. Pythagoras was the founder of 1050:as a better mathematical model. 1036:perihelion precession of Mercury 693:Mathematical reasoning requires 502: 284: 223: 176: 74: 33: 9279:Computational complexity theory 5867: 5856:from the original on 2010-12-04 5838: 5816: 5803: 5790: 5690: 5681: 5670:from the original on 2008-07-24 5652: 5626: 5615:from the original on 2008-04-05 5600: 5559: 5535: 5510: 5470: 5437: 5375: 5323: 5278:from the original on 2007-10-09 5258: 5231: 5215: 5205:from the original on 2022-02-08 5186: 5169: 5141: 5049: 5022: 5011:from the original on 2018-06-11 4993: 4977: 4966:from the original on 2018-11-25 4948: 4937:from the original on 2019-04-28 4876: 4865:from the original on 2021-05-08 4851: 4838: 4820: 4814:Introduction to Metamathematics 4786: 4756: 4726: 4691: 4653: 4618: 4587: 4557: 4520: 4493: 4348: 4278: 4248: 4213:"Mathematics, Maps, and Models" 4201: 4163: 4117: 4083: 3755: 3376:needs additional citations for 3309:Field developed his views into 3199:This argument, associated with 3156:and other linguists working in 2852: 2796:quasi-empiricism in mathematics 2782:" as implying an over-specific 2718:Quasi-empiricism in mathematics 2625: 2133:, different from the classical 2099: 1979:Gödel's incompleteness theorems 1821:if and only if for all objects 1795:Die Grundgesetze der Arithmetik 1561:and a "World of Ideas" (Greek: 1515:who claims that is the British 1490:Contemporary schools of thought 612: 190:may be too short to adequately 85:needs additional citations for 52:or discuss these issues on the 6285: 6173:Set Theory & Further Logic 5754:Yablo, S. (November 8, 1998). 5696:Gregory, Frank Hutson (1996) " 4529:"History of Integer Factoring" 4048: 4029: 3956: 3908:"Mathematical Rigor and Proof" 3899: 3853: 3825:The Mathematical Intelligencer 3821:"Mathematics: Art and Science" 3746: 3729: 3477:has argued against this idea. 3023:Beyond the traditional schools 2253:characterized those who allow 1635: 1565:(εἶδος)) described in Plato's 584:Relationship with applications 551:that deals with the nature of 200:provide an accessible overview 13: 1: 11115:History of mathematical logic 8018: 6136:London Philosophy Study Guide 5951:The Philosophy of Mathematics 5931:The Philosophy of Mathematics 5197:, in Zalta, Edward N. (ed.), 3786:. In Zalta, Edward N. (ed.). 3768: 3356: 2711: 2681:The Foundations of Arithmetic 2547:. In addition, mathematician 2245:number of steps. In her book 2031:formats, so as to facilitate 1263:of mathematical objects, and 1132:of the planets are ellipses. 11040:Primitive recursive function 7807:Ordinary language philosophy 6298: 5713:Ganesalingam, Mohan (2013). 5448:Language, Truth, & Logic 5295:vol. xliii (1893), pp. 1-25. 4225:10.1007/978-3-319-72478-2_18 3915:The Review of Symbolic Logic 3183: 2536:Where Mathematics Comes From 1905: 1691:, was later defended by the 1611:'s idea that mathematics is 1528: 1345:), and foundational issues. 1192:{\displaystyle \Omega ^{-}.} 1114:ancient Greek mathematicians 1025:Mathematics is used in most 7: 7857:Contemporary utilitarianism 7772:Internalism and externalism 6090:"Philosophy of mathematics" 6072:"Philosophy of Mathematics" 6030:. Oxford University Press. 5933:. Oxford University Press. 5444:Ayer, Alfred Jules (1952). 4720:10.1016/j.shpsb.2015.12.001 4358:; Lazere, Cathy A. (1998). 3906:Hamami, Yacin (June 2022). 3525: 3502: 3103:the claims made and adding 2413:not discover, mathematics. 2209: 2037:interactive theorem proving 2020:formalistic point of view. 1735: 1631:The Mathematical Experience 1494: 1296:of the square root of two. 1275:(actual versus potential). 1120:(that is, intersections of 1112:. They were studied by the 1104:communications through the 1065:. Similarly as in science, 1048:Newton's law of gravitation 827:Zermelo–Fraenkel set theory 788:double negation elimination 250:conditions to do so are met 109:"Philosophy of mathematics" 10: 11197: 10104:Schröder–Bernstein theorem 9831:Monadic predicate calculus 9490:Foundations of mathematics 9329:Films about mathematicians 7121:Svatantrika and Prasangika 6760: 6104:Mathematical Structuralism 5800:, Oxford: Blackwell, p. 68 5400:10.1007/s10699-021-09786-1 5317:Cambridge University Press 4990:, Routledge, 2002, p. 681. 4927:"Platonism in Metaphysics" 4124:Hennig, Christian (2010). 4069:10.1016/j.ijer.2018.10.002 3972:Bulletin of Symbolic Logic 3784:"Platonism in Metaphysics" 3573:Foundations of mathematics 3563:Definitions of mathematics 3315:mathematical structuralism 3192: 3028:Unreasonable effectiveness 2935: 2902:is just as fictitious as " 2856: 2786:that was not necessary to 2715: 2635: 2629: 2582: 2576: 2462:Another issue is that one 2294:Mathematical structuralism 2291: 2213: 2183: 2141:and therefore frowns upon 2139:law of the excluded middle 2103: 2088:should not be regarded as 2054: 1992:Other formalists, such as 1909: 1739: 1710: 1687:. A similar view, termed 1538: 1532: 1519:. For Hardy, in his book, 1499:The view that claims that 1395:foundations of mathematics 1351:foundations of mathematics 1206: 1081:Unreasonable effectiveness 1021:Relationship with sciences 718:ancient Greek philosophers 678: 626:. The ancient philosopher 617: 442:Relationship with sciences 11181:Philosophy of mathematics 11150: 11137:Philosophy of mathematics 11086:Automated theorem proving 11068: 10963: 10795: 10688: 10540: 10257: 10233: 10211:Von Neumann–Bernays–Gödel 10156: 10050: 9954: 9852: 9843: 9770: 9705: 9611: 9533: 9450: 9352: 9302: 9259: 9169: 9131: 9098: 9050: 9022: 8969: 8916: 8898:Philosophy of mathematics 8873: 8838: 8756: 8716: 8688: 8681: 8633:Necessity and sufficiency 8536: 8501: 8453: 8407: 8389: 8381: 8315: 8267: 8167: 8129: 8076: 8043: 8034: 8030: 8013: 7963: 7875: 7713: 7704: 7637: 7420: 7411: 7389: 7344: 7286: 7238: 7192: 7183: 7146: 7017: 6882: 6829: 6820: 6770: 6766: 6755: 6694: 6666: 6623: 6575: 6532: 6485: 6457: 6409: 6381: 6343:Philosophy of mathematics 6333:Philosophy of information 6308: 6304: 6293: 6209:Philosophy of mathematics 6149:Philosophy of Mathematics 6061:Philosophy of mathematics 6052:Philosophy of mathematics 5723:10.1007/978-3-642-37012-0 5332:Philosophy of Mathematics 5127:10.1007/s10701-007-9186-9 5062:The Journal of Philosophy 4362:. Springer. p. 228. 4142:10.1007/s10699-009-9167-x 4038:Philosophy of Mathematics 3927:10.1017/S1755020319000443 3613:Philosophy of probability 3492:A Mathematician's Apology 2864:Mathematical fictionalism 2800:foundation of mathematics 2765:indispensability argument 2106:Mathematical intuitionism 1900:one-to-one correspondence 1663:) a position defended by 1522:A Mathematician's Apology 1096:A notable example is the 811:well-formed of assertions 795:foundation of mathematics 579:Relationship with science 545:Philosophy of mathematics 18:Mathematical fictionalism 9334:Recreational mathematics 5948:Irvine, A., ed. (2009). 5756:"A Paradox of Existence" 5611:. University of Exeter. 5480:(1995). "On knowledge". 5382:Franklin, James (2022). 5355:Franklin, James (2014). 3963:Ferreirós, José (2001). 3893:The World of Mathematics 3722: 3693:The Simplest Mathematics 2982:, or of mathematicians' 2691:Philosophy of Arithmetic 2557:, as has neuroscientist 2247:Philosophy of Set Theory 2078:non-Euclidean geometries 2076:view. Poincaré's use of 2047:for a general overview. 2033:automated proof checking 1817:equals the extension of 1799:Basic Laws of Arithmetic 1693:Stanford–Edmonton School 1477:'s attack on the actual 776:constructive mathematics 737:non-Euclidean geometries 685:Mathematics § Rigor 661:that support his views. 594:Nature as human activity 10787:Self-verifying theories 10608:Tarski's axiomatization 9559:Tarski's undefinability 9554:incompleteness theorems 9219:Mathematical statistics 9209:Mathematical psychology 9179:Engineering mathematics 9113:Algebraic number theory 7812:Postanalytic philosophy 7753:Experimental philosophy 6203:considered for deletion 5584:Mathematics and Reality 5570:Science Without Numbers 5334:. New York: Routledge. 5031:Philosophia Mathematica 4677:10.1093/bjps/xiii.49.54 4661:Hanson, Norwood Russell 3782:Balaguer, Mark (2016). 3510:Philosophia Mathematica 3141:language of mathematics 2984:collective intelligence 2872:Science Without Numbers 2817:empirical justification 2727:Mathematical empiricism 2666:as well as a number of 2388:. Like nominalism, the 2143:proofs by contradiction 2016:usually not arbitrary. 1870:Modern logicists (like 1685:epistemological problem 1661:set-theoretic Platonism 1473:comes to mind, as does 1324:Contemporary philosophy 1077:on empirical evidence. 767:). Even more striking, 11161:Mathematics portal 10772:Proof of impossibility 10420:propositional variable 9730:Propositional calculus 9365:Mathematics portal 9214:Mathematical sociology 9194:Mathematical economics 9189:Mathematical chemistry 9118:Analytic number theory 8999:Differential equations 7945:Social constructionism 6957:Hellenistic philosophy 6373:Theoretical philosophy 6348:Philosophy of religion 6338:Philosophy of language 5885:The Emperor's New Mind 5517:Popper, Karl (2002) . 5432:Realism in Mathematics 5388:Foundations of Science 5361:. Palgrave Macmillan. 5193:Zach, Richard (2019), 5097:Foundations of Physics 5043:10.1093/philmat/2.2.97 4475:10.1002/cpa.3160130102 4285:Mackay, A. L. (1991). 4130:Foundations of Science 3891:Newman, J. R. (1956). 3712:History of mathematics 3593:Philosophy of language 3256:non-Euclidean geometry 3069:Philosophy of language 3044:'s famous 1960 paper " 2909:Another fictionalist, 2895:conservative extension 2836:Realism in Mathematics 2722:Postmodern mathematics 2702:Charles Sanders Peirce 2697:Logical Investigations 2401:Embodied mind theories 2270: 2233:is an extreme form of 2227: 2168:, first introduced by 2082:differential equations 2010:Mathematical logicians 1950: 1843: 1642:Full-blooded Platonism 1597: 1546:Mathematical Platonism 1483: 1271:and issues related to 1231: 1193: 784:law of excluded middle 672: 295:is missing information 11030:Kolmogorov complexity 10983:Computably enumerable 10883:Model complete theory 10675:Principia Mathematica 9735:Propositional formula 9564:Banach–Tarski paradox 9344:Mathematics education 9274:Theory of computation 8994:Hypercomplex analysis 8768:Philosophy portal 8328:Philosophy portal 7847:Scientific skepticism 7827:Reformed epistemology 6353:Philosophy of science 5811:Realistic Rationalism 5796:Field, Hartry, 1989, 5330:Brown, James (2008). 5293:Mathematische Annalen 5147:Tegmark (1998), p. 1. 4984:Ivor Grattan-Guinness 4846:Journal of Philosophy 4356:Shasha, Dennis Elliot 4319:Bishop, Alan (1991). 4260:Understanding Science 3864:Philosophy of Science 3831:(4). Springer: 9–17. 3717:History of philosophy 3686:Principia Mathematica 3608:Philosophy of physics 3603:Philosophy of science 3588:Non-standard analysis 3343:Realistic Rationalism 3170:context-free grammars 2955:mathematical practice 2943:Social constructivism 2938:Social constructivism 2932:Social constructivism 2788:mathematical practice 2706:Maurice Merleau-Ponty 2512:), and the nature of 2266: 2223: 2201:constructive analysis 1961:was intended to be a 1946: 1894:and the extension of 1865:axiom of reducibility 1839: 1689:Platonized naturalism 1673:naturalized Platonism 1657:Set-theoretic realism 1593: 1467: 1214: 1194: 663: 10978:Church–Turing thesis 10965:Computability theory 10174:continuum hypothesis 9692:Square of opposition 9550:Gödel's completeness 9324:Informal mathematics 9204:Mathematical physics 9199:Mathematical finance 9184:Mathematical biology 9123:Diophantine geometry 7748:Critical rationalism 7455:Edo neo-Confucianism 7299:Acintya bheda abheda 7278:Renaissance humanism 6989:School of the Sextii 6363:Practical philosophy 6358:Political philosophy 5834:on February 7, 2011. 4738:University of Exeter 4177:(February 4, 2020). 3385:improve this article 2975:unifying conjectures 2840:embodied mind theory 2594:Aristotelian realism 2573:Aristotelian realism 2485:values, rather than 2457:action at a distance 2371:Aristotelian realism 2328:Structuralism is an 2307:intrinsic properties 2131:intuitionistic logic 2092:truth. He held that 2006:formal axiom systems 1860:ramified type theory 1567:allegory of the cave 1343:axiomatic set theory 1173: 1145:theory of relativity 1128:discovered that the 1009:set theory with the 979:mathematical objects 940:Weierstrass function 780:intuitionistic logic 745:Weierstrass function 407:Discrete mathematics 94:improve this article 11132:Mathematical object 11023:P versus NP problem 10988:Computable function 10782:Reverse mathematics 10708:Logical consequence 10585:primitive recursive 10580:elementary function 10353:Free/bound variable 10206:Tarski–Grothendieck 9725:Logical connectives 9655:Logical equivalence 9505:Logical consequence 9339:Mathematics and art 9249:Operations research 9004:Functional analysis 8430:Philosophy of logic 7319:Nimbarka Sampradaya 7230:Korean Confucianism 6977:Academic Skepticism 5582:Leng, Mary (2010). 5494:1992sbwl.book.....P 5195:"Hilbert's Program" 5119:2008FoPh...38..101T 4897:1941Natur.147....3S 4712:2016SHPMP..53...20G 4467:1960CPAM...13....1W 4179:"Models in Science" 3598:Philosophy of logic 3497:applied mathematics 3426:In his work on the 3244:confirmation holism 3207:, is considered by 2971:epistemic community 2876:Newtonian mechanics 2585:In re structuralism 2510:philosophy of being 2453:quantum nonlocality 2158:computable function 2084:convinced him that 1922:Pythagorean theorem 1813:, the extension of 1667:, is the view that 1434:to call such study 1098:prime factorization 988:projective geometry 833:, generally called 755:, and the study by 637:Mathematical object 341:Part of a series on 237:of this article is 10930:Transfer principle 10893:Semantics of logic 10878:Categorical theory 10854:Non-standard model 10368:Logical connective 9495:Information theory 9444:Mathematical logic 9284:Numerical analysis 8893:Mathematical logic 8888:Information theory 8729:Rules of inference 8698:Mathematical logic 8440:Semantics of logic 7940:Post-structuralism 7842:Scientific realism 7797:Quinean naturalism 7777:Logical positivism 7733:Analytical Marxism 6952:Peripatetic school 6864:Chinese naturalism 6391:Aesthetic response 6318:Applied philosophy 6178:2009-02-27 at the 6166:2009-01-25 at the 6161:Mathematical Logic 6154:2009-06-20 at the 6141:2009-09-23 at the 5879:2011-05-14 at the 5572:, Blackwell, 1980. 4827:Mac Lane, Saunders 4409:Pigliucci, Massimo 3838:10.4171/news/103/8 3540:Mathematics portal 3088:possibly contains 3065:on realist lines. 2922:second-order logic 2794:and accepted much 2735:empirical research 2615:John Penn Mayberry 2278:John Penn Mayberry 2255:countably infinite 2228: 2166:computable numbers 2135:Aristotelian logic 2086:Euclidean geometry 1951: 1918:Euclidean geometry 1844: 1598: 1261:ontological status 1245:Eastern philosophy 1241:Western philosophy 1232: 1224:Euclidean geometry 1189: 1157:general relativity 1153:special relativity 1044:general relativity 893:irrational numbers 799:mathematical logic 741:parallel postulate 720:under the name of 608:, or all together) 589:Mathematical truth 508:Mathematics Portal 11168: 11167: 11100:Abstract category 10903:Theories of truth 10713:Rule of inference 10703:Natural deduction 10684: 10683: 10229: 10228: 9934:Cartesian product 9839: 9838: 9745:Many-valued logic 9720:Boolean functions 9603:Russell's paradox 9578:diagonal argument 9475:First-order logic 9410: 9409: 9009:Harmonic analysis 8796: 8795: 8752: 8751: 8586:Deductive closure 8532: 8531: 8471:Critical thinking 8349: 8348: 8311: 8310: 8307: 8306: 8303: 8302: 8009: 8008: 8005: 8004: 8001: 8000: 7728:Analytic feminism 7700: 7699: 7662:Kierkegaardianism 7624:Transcendentalism 7584:Neo-scholasticism 7430:Classical Realism 7407: 7406: 7179: 7178: 6994:Neopythagoreanism 6751: 6750: 6747: 6746: 6368:Social philosophy 5990:Russell, Bertrand 5732:978-3-642-37011-3 5528:978-0-415-27843-0 5503:978-0-415-13548-1 5463:978-0-486-20010-1 5341:978-0-415-96047-2 5252:978-4-87187-714-5 4770:. 13 April 2015. 4565:"Curves: Ellipse" 4394:978-0-226-05182-6 4369:978-0-387-98269-4 4334:978-0-7923-1270-3 4298:978-0-7503-0106-0 4234:978-3-319-72478-2 4175:Hartmann, Stephan 4103:978-0-8493-8331-1 3701:Historical topics 3628:Scientific method 3618:Rule of inference 3554:Philosophy portal 3428:divine proportion 3417: 3416: 3409: 3340:in his 2000 book 3331:Sir Roger Penrose 3150:natural languages 3133: 3132: 3125: 3090:original research 2847:Brian Butterworth 2638:Anti-psychologism 2559:Stanislas Dehaene 2554:The Math Instinct 2472:imaginary numbers 2445:scientific method 2330:epistemologically 2262:Leopold Kronecker 2225:Leopold Kronecker 2029:computer-readable 1852:Russell's paradox 1585:Ultimate Ensemble 1471:Russell's paradox 1455:Saunders Mac Lane 1365:. Three schools, 1053:There is still a 1046:, which replaced 901:imaginary numbers 809:that defines the 769:Russell's paradox 547:is the branch of 542: 541: 497: 496: 336: 335: 328: 318: 317: 278: 277: 270: 217: 216: 170: 169: 162: 144: 67: 16:(Redirected from 11188: 11159: 11158: 11110:History of logic 11105:Category of sets 10998:Decision problem 10777:Ordinal analysis 10718:Sequent calculus 10616:Boolean algebras 10556: 10555: 10530: 10501:logical/constant 10255: 10254: 10241: 10164:Zermelo–Fraenkel 9915:Set operations: 9850: 9849: 9787: 9618: 9617: 9598:Löwenheim–Skolem 9485:Formal semantics 9437: 9430: 9423: 9414: 9413: 9398: 9397: 9386: 9385: 9374: 9373: 9363: 9362: 9294:Computer algebra 9269:Computer science 8989:Complex analysis 8823: 8816: 8809: 8800: 8799: 8766: 8765: 8764: 8686: 8685: 8451: 8450: 8415:Computer science 8376: 8369: 8362: 8353: 8352: 8338: 8337: 8326: 8325: 8324: 8041: 8040: 8032: 8031: 8015: 8014: 7905:Frankfurt School 7852:Transactionalism 7802:Normative ethics 7782:Legal positivism 7758:Falsificationism 7743:Consequentialism 7738:Communitarianism 7711: 7710: 7579:New Confucianism 7418: 7417: 7225:Neo-Confucianism 7190: 7189: 6999:Second Sophistic 6984:Middle Platonism 6827: 6826: 6768: 6767: 6757: 6756: 6600:Epiphenomenalism 6467:Consequentialism 6401:Institutionalism 6306: 6305: 6295: 6294: 6280: 6273: 6266: 6257: 6256: 6252: 6226: 6206: 6129: 6099: 6085: 6076:Zalta, Edward N. 6041: 6022:Shapiro, Stewart 6017: 5985: 5976:. 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E. J. Brouwer 2112:L. E. J. Brouwer 2041:computer science 1983:axiomatic system 1973:of the positive 1898:can be put into 1880:Hume's principle 1848:Bertrand Russell 1841:Bertrand Russell 1701:abstract objects 1675:because it is a 1541:Modern Platonism 1451:Samuel Eilenberg 1419:Zermelo–Fraenkel 1339:naive set theory 1300:, a disciple of 1279:Greek philosophy 1198: 1196: 1195: 1190: 1185: 1184: 1106:RSA cryptosystem 908:Peano arithmetic 793:The problems of 534: 527: 520: 506: 370: 369: 338: 337: 331: 324: 313: 310: 304: 288: 280: 273: 266: 262: 259: 253: 227: 226: 219: 212: 209: 203: 180: 172: 165: 158: 154: 151: 145: 143: 102: 78: 70: 59: 37: 36: 29: 21: 11196: 11195: 11191: 11190: 11189: 11187: 11186: 11185: 11171: 11170: 11169: 11164: 11153: 11146: 11091:Category theory 11081:Algebraic logic 11064: 11035:Lambda calculus 10973:Church encoding 10959: 10935:Truth predicate 10791: 10757:Complete theory 10680: 10549: 10545: 10541: 10536: 10528: 10248: and 10244: 10239: 10225: 10201:New Foundations 10169:axiom of choice 10152: 10114:Gödel numbering 10054: and 10046: 9950: 9835: 9785: 9766: 9715:Boolean algebra 9701: 9665:Equiconsistency 9630:Classical logic 9607: 9588:Halting problem 9576: and 9552: and 9540: and 9539: 9534:Theorems ( 9529: 9446: 9441: 9411: 9406: 9357: 9348: 9298: 9255: 9234:Systems science 9165: 9161:Homotopy theory 9127: 9094: 9046: 9018: 8965: 8912: 8883:Category theory 8869: 8834: 8827: 8797: 8792: 8762: 8760: 8748: 8712: 8703:Boolean algebra 8677: 8528: 8519:Metamathematics 8497: 8449: 8403: 8385: 8380: 8350: 8345: 8322: 8320: 8299: 8263: 8163: 8125: 8072: 8026: 8025: 7997: 7986:Russian cosmism 7959: 7955:Western Marxism 7920:New Historicism 7885:Critical theory 7871: 7867:Wittgensteinian 7763:Foundationalism 7696: 7633: 7614:Social contract 7470:Foundationalism 7403: 7385: 7369:Illuminationism 7354:Aristotelianism 7340: 7329:Vishishtadvaita 7282: 7234: 7175: 7142: 7013: 6942:Megarian school 6937:Eretrian school 6878: 6839:Agriculturalism 6816: 6762: 6743: 6690: 6662: 6619: 6571: 6528: 6512:Incompatibilism 6481: 6453: 6405: 6377: 6300: 6289: 6284: 6249: 6219:Corfield, David 6191: 6180:Wayback Machine 6168:Wayback Machine 6156:Wayback Machine 6143:Wayback Machine 6118: 6088: 6070:Horsten, Leon. 6048: 6038: 6006: 5970:Körner, Stephan 5962: 5941: 5922: 5908:, eds. (1983). 5897: 5895:Further reading 5892: 5891: 5881:Wayback Machine 5872: 5868: 5859: 5857: 5844: 5843: 5839: 5822: 5821: 5817: 5808: 5804: 5795: 5791: 5782: 5775: 5765: 5763: 5752: 5748: 5733: 5711: 5704: 5695: 5691: 5686: 5682: 5673: 5671: 5658: 5657: 5653: 5644: 5642: 5641:on May 16, 2008 5631: 5627: 5618: 5616: 5605: 5601: 5594: 5580: 5576: 5564: 5560: 5540: 5536: 5529: 5515: 5511: 5504: 5478:Popper, Karl R. 5475: 5471: 5464: 5442: 5438: 5428:Maddy, Penelope 5426: 5422: 5412: 5410: 5380: 5376: 5369: 5353: 5349: 5342: 5328: 5324: 5306: 5299: 5281: 5279: 5263: 5259: 5253: 5236: 5232: 5220: 5216: 5208: 5206: 5191: 5187: 5174: 5170: 5158: 5151: 5146: 5142: 5093: 5089: 5074:10.2307/2940786 5068:(10): 525–555. 5054: 5050: 5027: 5023: 5014: 5012: 4999: 4998: 4994: 4982: 4978: 4969: 4967: 4954: 4953: 4949: 4940: 4938: 4925: 4924: 4920: 4881: 4877: 4868: 4866: 4857: 4856: 4852: 4843: 4839: 4825: 4821: 4810:Kleene, Stephen 4807: 4803: 4791: 4787: 4777: 4775: 4762: 4761: 4757: 4747: 4745: 4732: 4731: 4727: 4696: 4692: 4658: 4654: 4633:(11): 389–507. 4623: 4619: 4609: 4607: 4592: 4588: 4578: 4576: 4563: 4562: 4558: 4548: 4546: 4542: 4531: 4525: 4521: 4502:Current Science 4498: 4494: 4442: 4438: 4428: 4426: 4406: 4402: 4395: 4381: 4377: 4370: 4353: 4349: 4339: 4337: 4335: 4317: 4313: 4303: 4301: 4299: 4283: 4279: 4269: 4267: 4254: 4253: 4249: 4239: 4237: 4235: 4206: 4202: 4192: 4190: 4168: 4164: 4154: 4152: 4122: 4118: 4108: 4106: 4104: 4088: 4084: 4053: 4049: 4034: 4030: 4020: 4018: 4014: 3984:10.2307/2687794 3967: 3961: 3957: 3947: 3945: 3941: 3910: 3904: 3900: 3858: 3854: 3814: 3807: 3797: 3795: 3780: 3776: 3771: 3766: 3765: 3760: 3756: 3751: 3747: 3734: 3730: 3725: 3703: 3698: 3637: 3632: 3623:Science studies 3568:Formal language 3552: 3547: 3545: 3538: 3531: 3528: 3505: 3475:Gregory Chaitin 3449: 3447: 3445: 3438:Philip J. 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Núñez 2468:complex numbers 2403: 2296: 2290: 2239:natural numbers 2218: 2212: 2188: 2182: 2147:axiom of choice 2108: 2102: 2080:in his work on 2074:conventionalist 2063: 2061:Preintuitionism 2057:Conventionalism 2055:Main articles: 2053: 2051:Conventionalism 1914: 1908: 1744: 1738: 1715: 1709: 1681:Paul Benacerraf 1651:axiom of choice 1647:excluded middle 1559:Theory of Forms 1543: 1537: 1531: 1497: 1492: 1459:category theory 1437:metamathematics 1424:Gödel numbering 1326: 1209: 1180: 1176: 1174: 1171: 1170: 1141:Albert Einstein 1126:Johannes Kepler 1083: 1071:experimentation 1023: 1011:axiom of choice 971:inference rules 938:. For example, 889:natural numbers 877:natural numbers 865: 854:straightforward 831:axiom of choice 819:inference rules 807:formal language 707:inference rules 691: 677: 675:Logic and rigor 659:Albert Einstein 651:Charles Hermite 620: 615: 569:Logic and rigor 538: 493: 492: 443: 435: 434: 430:Decision theory 378: 332: 321: 320: 319: 314: 308: 305: 298: 289: 274: 263: 257: 254: 243: 228: 224: 213: 207: 204: 197: 185:This article's 181: 166: 155: 149: 146: 103: 101: 91: 79: 38: 34: 23: 22: 15: 12: 11: 5: 11194: 11184: 11183: 11166: 11165: 11151: 11148: 11147: 11145: 11144: 11139: 11134: 11129: 11124: 11123: 11122: 11112: 11107: 11102: 11093: 11088: 11083: 11078: 11076:Abstract logic 11072: 11070: 11066: 11065: 11063: 11062: 11057: 11055:Turing machine 11052: 11047: 11042: 11037: 11032: 11027: 11026: 11025: 11020: 11015: 11010: 11005: 10995: 10993:Computable set 10990: 10985: 10980: 10975: 10969: 10967: 10961: 10960: 10958: 10957: 10952: 10947: 10942: 10937: 10932: 10927: 10922: 10921: 10920: 10915: 10910: 10900: 10895: 10890: 10888:Satisfiability 10885: 10880: 10875: 10874: 10873: 10863: 10862: 10861: 10851: 10850: 10849: 10844: 10839: 10834: 10829: 10819: 10818: 10817: 10812: 10805:Interpretation 10801: 10799: 10793: 10792: 10790: 10789: 10784: 10779: 10774: 10769: 10759: 10754: 10753: 10752: 10751: 10750: 10740: 10735: 10725: 10720: 10715: 10710: 10705: 10700: 10694: 10692: 10686: 10685: 10682: 10681: 10679: 10678: 10670: 10669: 10668: 10667: 10662: 10661: 10660: 10655: 10650: 10630: 10629: 10628: 10626:minimal axioms 10623: 10612: 10611: 10610: 10599: 10598: 10597: 10592: 10587: 10582: 10577: 10572: 10559: 10557: 10538: 10537: 10535: 10534: 10533: 10532: 10520: 10515: 10514: 10513: 10508: 10503: 10498: 10488: 10483: 10478: 10473: 10472: 10471: 10466: 10456: 10455: 10454: 10449: 10444: 10439: 10429: 10424: 10423: 10422: 10417: 10412: 10402: 10401: 10400: 10395: 10390: 10385: 10380: 10375: 10365: 10360: 10355: 10350: 10349: 10348: 10343: 10338: 10333: 10323: 10318: 10316:Formation rule 10313: 10308: 10307: 10306: 10301: 10291: 10290: 10289: 10279: 10274: 10269: 10264: 10258: 10252: 10235:Formal systems 10231: 10230: 10227: 10226: 10224: 10223: 10218: 10213: 10208: 10203: 10198: 10193: 10188: 10183: 10178: 10177: 10176: 10171: 10160: 10158: 10154: 10153: 10151: 10150: 10149: 10148: 10138: 10133: 10132: 10131: 10124:Large cardinal 10121: 10116: 10111: 10106: 10101: 10087: 10086: 10085: 10080: 10075: 10060: 10058: 10048: 10047: 10045: 10044: 10043: 10042: 10037: 10032: 10022: 10017: 10012: 10007: 10002: 9997: 9992: 9987: 9982: 9977: 9972: 9967: 9961: 9959: 9952: 9951: 9949: 9948: 9947: 9946: 9941: 9936: 9931: 9926: 9921: 9913: 9912: 9911: 9906: 9896: 9891: 9889:Extensionality 9886: 9884:Ordinal number 9881: 9871: 9866: 9865: 9864: 9853: 9847: 9841: 9840: 9837: 9836: 9834: 9833: 9828: 9823: 9818: 9813: 9808: 9803: 9802: 9801: 9791: 9790: 9789: 9776: 9774: 9768: 9767: 9765: 9764: 9763: 9762: 9757: 9752: 9742: 9737: 9732: 9727: 9722: 9717: 9711: 9709: 9703: 9702: 9700: 9699: 9694: 9689: 9684: 9679: 9674: 9669: 9668: 9667: 9657: 9652: 9647: 9642: 9637: 9632: 9626: 9624: 9615: 9609: 9608: 9606: 9605: 9600: 9595: 9590: 9585: 9580: 9568:Cantor's 9566: 9561: 9556: 9546: 9544: 9531: 9530: 9528: 9527: 9522: 9517: 9512: 9507: 9502: 9497: 9492: 9487: 9482: 9477: 9472: 9467: 9466: 9465: 9454: 9452: 9448: 9447: 9440: 9439: 9432: 9425: 9417: 9408: 9407: 9405: 9404: 9392: 9380: 9368: 9353: 9350: 9349: 9347: 9346: 9341: 9336: 9331: 9326: 9321: 9320: 9319: 9312:Mathematicians 9308: 9306: 9304:Related topics 9300: 9299: 9297: 9296: 9291: 9286: 9281: 9276: 9271: 9265: 9263: 9257: 9256: 9254: 9253: 9252: 9251: 9246: 9241: 9239:Control theory 9231: 9226: 9221: 9216: 9211: 9206: 9201: 9196: 9191: 9186: 9181: 9175: 9173: 9167: 9166: 9164: 9163: 9158: 9153: 9148: 9143: 9137: 9135: 9129: 9128: 9126: 9125: 9120: 9115: 9110: 9104: 9102: 9096: 9095: 9093: 9092: 9087: 9082: 9077: 9072: 9067: 9062: 9056: 9054: 9048: 9047: 9045: 9044: 9039: 9034: 9028: 9026: 9020: 9019: 9017: 9016: 9014:Measure theory 9011: 9006: 9001: 8996: 8991: 8986: 8981: 8975: 8973: 8967: 8966: 8964: 8963: 8958: 8953: 8948: 8943: 8938: 8933: 8928: 8922: 8920: 8914: 8913: 8911: 8910: 8905: 8900: 8895: 8890: 8885: 8879: 8877: 8871: 8870: 8868: 8867: 8862: 8857: 8856: 8855: 8850: 8839: 8836: 8835: 8826: 8825: 8818: 8811: 8803: 8794: 8793: 8791: 8790: 8785: 8775: 8770: 8757: 8754: 8753: 8750: 8749: 8747: 8746: 8741: 8736: 8731: 8726: 8720: 8718: 8714: 8713: 8711: 8710: 8705: 8700: 8694: 8692: 8683: 8679: 8678: 8676: 8675: 8670: 8665: 8660: 8655: 8650: 8645: 8640: 8635: 8630: 8625: 8620: 8615: 8610: 8609: 8608: 8598: 8593: 8588: 8583: 8578: 8577: 8576: 8571: 8561: 8556: 8551: 8546: 8540: 8538: 8534: 8533: 8530: 8529: 8527: 8526: 8521: 8516: 8511: 8505: 8503: 8499: 8498: 8496: 8495: 8490: 8485: 8480: 8479: 8478: 8473: 8463: 8457: 8455: 8448: 8447: 8442: 8437: 8432: 8427: 8422: 8417: 8411: 8409: 8405: 8404: 8402: 8401: 8396: 8390: 8387: 8386: 8379: 8378: 8371: 8364: 8356: 8347: 8346: 8344: 8343: 8331: 8316: 8313: 8312: 8309: 8308: 8305: 8304: 8301: 8300: 8298: 8297: 8292: 8287: 8282: 8277: 8271: 8269: 8265: 8264: 8262: 8261: 8256: 8251: 8246: 8241: 8236: 8231: 8226: 8221: 8216: 8211: 8206: 8201: 8196: 8195: 8194: 8184: 8179: 8173: 8171: 8165: 8164: 8162: 8161: 8156: 8151: 8146: 8141: 8135: 8133: 8131:Middle Eastern 8127: 8126: 8124: 8123: 8118: 8113: 8108: 8103: 8098: 8093: 8088: 8082: 8080: 8074: 8073: 8071: 8070: 8065: 8060: 8055: 8049: 8047: 8038: 8028: 8027: 8024: 8023: 8019: 8011: 8010: 8007: 8006: 8003: 8002: 7999: 7998: 7996: 7995: 7988: 7983: 7978: 7973: 7967: 7965: 7961: 7960: 7958: 7957: 7952: 7947: 7942: 7937: 7932: 7927: 7922: 7917: 7912: 7907: 7902: 7897: 7895:Existentialism 7892: 7890:Deconstruction 7887: 7881: 7879: 7873: 7872: 7870: 7869: 7864: 7859: 7854: 7849: 7844: 7839: 7834: 7829: 7824: 7819: 7814: 7809: 7804: 7799: 7794: 7789: 7784: 7779: 7774: 7769: 7760: 7755: 7750: 7745: 7740: 7735: 7730: 7725: 7723:Applied ethics 7719: 7717: 7708: 7702: 7701: 7698: 7697: 7695: 7694: 7689: 7687:Nietzscheanism 7684: 7679: 7674: 7669: 7664: 7659: 7658: 7657: 7647: 7641: 7639: 7635: 7634: 7632: 7631: 7629:Utilitarianism 7626: 7621: 7616: 7611: 7606: 7601: 7596: 7591: 7586: 7581: 7576: 7571: 7566: 7561: 7556: 7551: 7546: 7541: 7536: 7531: 7530: 7529: 7527:Transcendental 7524: 7519: 7514: 7509: 7504: 7494: 7493: 7492: 7482: 7477: 7472: 7467: 7465:Existentialism 7462: 7457: 7452: 7447: 7442: 7437: 7432: 7427: 7421: 7415: 7409: 7408: 7405: 7404: 7402: 7401: 7395: 7393: 7387: 7386: 7384: 7383: 7378: 7371: 7366: 7361: 7356: 7350: 7348: 7342: 7341: 7339: 7338: 7333: 7332: 7331: 7326: 7321: 7316: 7311: 7306: 7301: 7290: 7288: 7284: 7283: 7281: 7280: 7275: 7270: 7265: 7260: 7255: 7253:Augustinianism 7250: 7244: 7242: 7236: 7235: 7233: 7232: 7227: 7222: 7217: 7212: 7207: 7202: 7196: 7194: 7187: 7181: 7180: 7177: 7176: 7174: 7173: 7168: 7166:Zoroastrianism 7163: 7158: 7152: 7150: 7144: 7143: 7141: 7140: 7139: 7138: 7133: 7128: 7123: 7118: 7113: 7108: 7103: 7098: 7088: 7087: 7086: 7081: 7071: 7070: 7069: 7064: 7059: 7054: 7049: 7044: 7039: 7034: 7023: 7021: 7015: 7014: 7012: 7011: 7009:Church Fathers 7006: 7001: 6996: 6991: 6986: 6981: 6980: 6979: 6974: 6969: 6964: 6954: 6949: 6944: 6939: 6934: 6929: 6924: 6923: 6922: 6917: 6912: 6907: 6902: 6891: 6889: 6880: 6879: 6877: 6876: 6871: 6866: 6861: 6856: 6851: 6846: 6841: 6835: 6833: 6824: 6818: 6817: 6815: 6814: 6813: 6812: 6807: 6802: 6797: 6792: 6782: 6776: 6774: 6764: 6763: 6753: 6752: 6749: 6748: 6745: 6744: 6742: 6741: 6736: 6731: 6726: 6721: 6716: 6711: 6706: 6700: 6698: 6692: 6691: 6689: 6688: 6683: 6678: 6672: 6670: 6664: 6663: 6661: 6660: 6655: 6650: 6645: 6640: 6635: 6629: 6627: 6621: 6620: 6618: 6617: 6612: 6607: 6602: 6597: 6592: 6587: 6581: 6579: 6573: 6572: 6570: 6569: 6564: 6559: 6554: 6549: 6544: 6538: 6536: 6530: 6529: 6527: 6526: 6524:Libertarianism 6521: 6520: 6519: 6509: 6508: 6507: 6497: 6491: 6489: 6483: 6482: 6480: 6479: 6474: 6469: 6463: 6461: 6455: 6454: 6452: 6451: 6446: 6441: 6436: 6431: 6426: 6421: 6415: 6413: 6407: 6406: 6404: 6403: 6398: 6393: 6387: 6385: 6379: 6378: 6376: 6375: 6370: 6365: 6360: 6355: 6350: 6345: 6340: 6335: 6330: 6328:Metaphilosophy 6325: 6320: 6314: 6312: 6302: 6301: 6291: 6290: 6283: 6282: 6275: 6268: 6260: 6254: 6253: 6247: 6227: 6215: 6189: 6184: 6183: 6182: 6170: 6158: 6132: 6131: 6130: 6116: 6112:Abstractionism 6108: 6086: 6067: 6058: 6047: 6046:External links 6044: 6043: 6042: 6036: 6018: 6004: 5986: 5966: 5960: 5945: 5939: 5926: 5920: 5906:Putnam, Hilary 5896: 5893: 5890: 5889: 5866: 5837: 5826:Philosophy Now 5815: 5802: 5789: 5773: 5746: 5731: 5702: 5689: 5680: 5651: 5625: 5607:Ernest, Paul. 5599: 5593:978-0199280797 5592: 5574: 5558: 5554:978-0691034980 5534: 5527: 5509: 5502: 5469: 5462: 5436: 5420: 5394:(2): 327–344. 5374: 5367: 5347: 5340: 5322: 5309:Mayberry, J.P. 5297: 5257: 5251: 5239:Bishop, Errett 5230: 5214: 5185: 5168: 5160:Carnap, Rudolf 5149: 5140: 5103:(2): 101–150. 5087: 5048: 5021: 4992: 4976: 4947: 4918: 4875: 4850: 4837: 4819: 4801: 4785: 4755: 4725: 4690: 4652: 4617: 4586: 4556: 4519: 4508:(3): 415–423. 4492: 4445:Wigner, Eugene 4436: 4418:Philosophy Now 4400: 4393: 4375: 4368: 4347: 4333: 4311: 4297: 4277: 4247: 4233: 4200: 4162: 4116: 4102: 4082: 4047: 4028: 3978:(4): 441–484. 3955: 3921:(2): 409–449. 3898: 3877:10.1086/286957 3871:(4): 289–303. 3852: 3805: 3773: 3772: 3770: 3767: 3764: 3763: 3754: 3745: 3727: 3726: 3724: 3721: 3720: 3719: 3714: 3709: 3702: 3699: 3697: 3696: 3689: 3682: 3675: 3668: 3661: 3654: 3646: 3638: 3636: 3633: 3631: 3630: 3625: 3620: 3615: 3610: 3605: 3600: 3595: 3590: 3585: 3580: 3575: 3570: 3565: 3559: 3558: 3557: 3543: 3527: 3524: 3523: 3522: 3514: 3504: 3501: 3459:, ascribed to 3415: 3414: 3373: 3371: 3364: 3358: 3355: 3267: 3264: 3235: 3234: 3231: 3228: 3215:One must have 3193:Main article: 3190: 3187: 3185: 3182: 3178:part of speech 3131: 3130: 3085: 3083: 3076: 3070: 3067: 3053: 3050: 3029: 3026: 3024: 3021: 2980:cognitive bias 2936:Main article: 2933: 2930: 2926:quantification 2891:useful fiction 2854: 2851: 2832:Penelope Maddy 2805:New Directions 2757:W. V. O. Quine 2713: 2710: 2672:Gustave Le Bon 2630:Main article: 2627: 2624: 2607:Penelope Maddy 2599:James Franklin 2577:Main article: 2574: 2571: 2561:with his book 2506:hallucinations 2464:numeral system 2441:hallucinations 2402: 2399: 2292:Main article: 2289: 2286: 2235:constructivism 2214:Main article: 2211: 2208: 2184:Main article: 2181: 2180:Constructivism 2178: 2154:Turing machine 2129:postulated an 2104:Main article: 2101: 2098: 2070:Henri Poincaré 2052: 2049: 1910:Main article: 1907: 1904: 1876:Crispin Wright 1801:) he built up 1788: 1787: 1780: 1763:logical truths 1740:Main article: 1737: 1734: 1711:Main article: 1708: 1705: 1665:Penelope Maddy 1533:Main article: 1530: 1527: 1496: 1493: 1491: 1488: 1325: 1322: 1267:, who studied 1228:Pythagoreanism 1208: 1205: 1188: 1183: 1179: 1143:developed the 1118:conic sections 1082: 1079: 1076: 1063:counterexample 1022: 1019: 983:plane geometry 960:constructivism 948:differentiable 864: 861: 805:consists of a 803:logical theory 753:differentiable 743:is wrong, the 676: 673: 669: 655:Henri Poincaré 619: 616: 614: 611: 610: 609: 591: 586: 581: 576: 571: 566: 540: 539: 537: 536: 529: 522: 514: 511: 510: 499: 498: 495: 494: 491: 490: 485: 480: 475: 470: 465: 460: 455: 450: 444: 441: 440: 437: 436: 433: 432: 423: 418: 409: 404: 395: 390: 385: 379: 374: 373: 366: 365: 364: 363: 358: 350: 349: 343: 342: 334: 333: 316: 315: 292: 290: 283: 276: 275: 231: 229: 222: 215: 214: 194:the key points 184: 182: 175: 168: 167: 82: 80: 73: 68: 42: 41: 39: 32: 9: 6: 4: 3: 2: 11193: 11182: 11179: 11178: 11176: 11163: 11162: 11157: 11149: 11143: 11140: 11138: 11135: 11133: 11130: 11128: 11125: 11121: 11118: 11117: 11116: 11113: 11111: 11108: 11106: 11103: 11101: 11097: 11094: 11092: 11089: 11087: 11084: 11082: 11079: 11077: 11074: 11073: 11071: 11067: 11061: 11058: 11056: 11053: 11051: 11050:Recursive set 11048: 11046: 11043: 11041: 11038: 11036: 11033: 11031: 11028: 11024: 11021: 11019: 11016: 11014: 11011: 11009: 11006: 11004: 11001: 11000: 10999: 10996: 10994: 10991: 10989: 10986: 10984: 10981: 10979: 10976: 10974: 10971: 10970: 10968: 10966: 10962: 10956: 10953: 10951: 10948: 10946: 10943: 10941: 10938: 10936: 10933: 10931: 10928: 10926: 10923: 10919: 10916: 10914: 10911: 10909: 10906: 10905: 10904: 10901: 10899: 10896: 10894: 10891: 10889: 10886: 10884: 10881: 10879: 10876: 10872: 10869: 10868: 10867: 10864: 10860: 10859:of arithmetic 10857: 10856: 10855: 10852: 10848: 10845: 10843: 10840: 10838: 10835: 10833: 10830: 10828: 10825: 10824: 10823: 10820: 10816: 10813: 10811: 10808: 10807: 10806: 10803: 10802: 10800: 10798: 10794: 10788: 10785: 10783: 10780: 10778: 10775: 10773: 10770: 10767: 10766:from ZFC 10763: 10760: 10758: 10755: 10749: 10746: 10745: 10744: 10741: 10739: 10736: 10734: 10731: 10730: 10729: 10726: 10724: 10721: 10719: 10716: 10714: 10711: 10709: 10706: 10704: 10701: 10699: 10696: 10695: 10693: 10691: 10687: 10677: 10676: 10672: 10671: 10666: 10665:non-Euclidean 10663: 10659: 10656: 10654: 10651: 10649: 10648: 10644: 10643: 10641: 10638: 10637: 10635: 10631: 10627: 10624: 10622: 10619: 10618: 10617: 10613: 10609: 10606: 10605: 10604: 10600: 10596: 10593: 10591: 10588: 10586: 10583: 10581: 10578: 10576: 10573: 10571: 10568: 10567: 10565: 10561: 10560: 10558: 10553: 10547: 10542:Example 10539: 10531: 10526: 10525: 10524: 10521: 10519: 10516: 10512: 10509: 10507: 10504: 10502: 10499: 10497: 10494: 10493: 10492: 10489: 10487: 10484: 10482: 10479: 10477: 10474: 10470: 10467: 10465: 10462: 10461: 10460: 10457: 10453: 10450: 10448: 10445: 10443: 10440: 10438: 10435: 10434: 10433: 10430: 10428: 10425: 10421: 10418: 10416: 10413: 10411: 10408: 10407: 10406: 10403: 10399: 10396: 10394: 10391: 10389: 10386: 10384: 10381: 10379: 10376: 10374: 10371: 10370: 10369: 10366: 10364: 10361: 10359: 10356: 10354: 10351: 10347: 10344: 10342: 10339: 10337: 10334: 10332: 10329: 10328: 10327: 10324: 10322: 10319: 10317: 10314: 10312: 10309: 10305: 10302: 10300: 10299:by definition 10297: 10296: 10295: 10292: 10288: 10285: 10284: 10283: 10280: 10278: 10275: 10273: 10270: 10268: 10265: 10263: 10260: 10259: 10256: 10253: 10251: 10247: 10242: 10236: 10232: 10222: 10219: 10217: 10214: 10212: 10209: 10207: 10204: 10202: 10199: 10197: 10194: 10192: 10189: 10187: 10186:Kripke–Platek 10184: 10182: 10179: 10175: 10172: 10170: 10167: 10166: 10165: 10162: 10161: 10159: 10155: 10147: 10144: 10143: 10142: 10139: 10137: 10134: 10130: 10127: 10126: 10125: 10122: 10120: 10117: 10115: 10112: 10110: 10107: 10105: 10102: 10099: 10095: 10091: 10088: 10084: 10081: 10079: 10076: 10074: 10071: 10070: 10069: 10065: 10062: 10061: 10059: 10057: 10053: 10049: 10041: 10038: 10036: 10033: 10031: 10030:constructible 10028: 10027: 10026: 10023: 10021: 10018: 10016: 10013: 10011: 10008: 10006: 10003: 10001: 9998: 9996: 9993: 9991: 9988: 9986: 9983: 9981: 9978: 9976: 9973: 9971: 9968: 9966: 9963: 9962: 9960: 9958: 9953: 9945: 9942: 9940: 9937: 9935: 9932: 9930: 9927: 9925: 9922: 9920: 9917: 9916: 9914: 9910: 9907: 9905: 9902: 9901: 9900: 9897: 9895: 9892: 9890: 9887: 9885: 9882: 9880: 9876: 9872: 9870: 9867: 9863: 9860: 9859: 9858: 9855: 9854: 9851: 9848: 9846: 9842: 9832: 9829: 9827: 9824: 9822: 9819: 9817: 9814: 9812: 9809: 9807: 9804: 9800: 9797: 9796: 9795: 9792: 9788: 9783: 9782: 9781: 9778: 9777: 9775: 9773: 9769: 9761: 9758: 9756: 9753: 9751: 9748: 9747: 9746: 9743: 9741: 9738: 9736: 9733: 9731: 9728: 9726: 9723: 9721: 9718: 9716: 9713: 9712: 9710: 9708: 9707:Propositional 9704: 9698: 9695: 9693: 9690: 9688: 9685: 9683: 9680: 9678: 9675: 9673: 9670: 9666: 9663: 9662: 9661: 9658: 9656: 9653: 9651: 9648: 9646: 9643: 9641: 9638: 9636: 9635:Logical truth 9633: 9631: 9628: 9627: 9625: 9623: 9619: 9616: 9614: 9610: 9604: 9601: 9599: 9596: 9594: 9591: 9589: 9586: 9584: 9581: 9579: 9575: 9571: 9567: 9565: 9562: 9560: 9557: 9555: 9551: 9548: 9547: 9545: 9543: 9537: 9532: 9526: 9523: 9521: 9518: 9516: 9513: 9511: 9508: 9506: 9503: 9501: 9498: 9496: 9493: 9491: 9488: 9486: 9483: 9481: 9478: 9476: 9473: 9471: 9468: 9464: 9461: 9460: 9459: 9456: 9455: 9453: 9449: 9445: 9438: 9433: 9431: 9426: 9424: 9419: 9418: 9415: 9403: 9402: 9393: 9391: 9390: 9381: 9379: 9378: 9369: 9367: 9366: 9361: 9355: 9354: 9351: 9345: 9342: 9340: 9337: 9335: 9332: 9330: 9327: 9325: 9322: 9318: 9315: 9314: 9313: 9310: 9309: 9307: 9305: 9301: 9295: 9292: 9290: 9287: 9285: 9282: 9280: 9277: 9275: 9272: 9270: 9267: 9266: 9264: 9262: 9261:Computational 9258: 9250: 9247: 9245: 9242: 9240: 9237: 9236: 9235: 9232: 9230: 9227: 9225: 9222: 9220: 9217: 9215: 9212: 9210: 9207: 9205: 9202: 9200: 9197: 9195: 9192: 9190: 9187: 9185: 9182: 9180: 9177: 9176: 9174: 9172: 9168: 9162: 9159: 9157: 9154: 9152: 9149: 9147: 9144: 9142: 9139: 9138: 9136: 9134: 9130: 9124: 9121: 9119: 9116: 9114: 9111: 9109: 9106: 9105: 9103: 9101: 9100:Number theory 9097: 9091: 9088: 9086: 9083: 9081: 9078: 9076: 9073: 9071: 9068: 9066: 9063: 9061: 9058: 9057: 9055: 9053: 9049: 9043: 9040: 9038: 9035: 9033: 9032:Combinatorics 9030: 9029: 9027: 9025: 9021: 9015: 9012: 9010: 9007: 9005: 9002: 9000: 8997: 8995: 8992: 8990: 8987: 8985: 8984:Real analysis 8982: 8980: 8977: 8976: 8974: 8972: 8968: 8962: 8959: 8957: 8954: 8952: 8949: 8947: 8944: 8942: 8939: 8937: 8934: 8932: 8929: 8927: 8924: 8923: 8921: 8919: 8915: 8909: 8906: 8904: 8901: 8899: 8896: 8894: 8891: 8889: 8886: 8884: 8881: 8880: 8878: 8876: 8872: 8866: 8863: 8861: 8858: 8854: 8851: 8849: 8846: 8845: 8844: 8841: 8840: 8837: 8832: 8824: 8819: 8817: 8812: 8810: 8805: 8804: 8801: 8789: 8786: 8783: 8779: 8776: 8774: 8771: 8769: 8759: 8758: 8755: 8745: 8744:Logic symbols 8742: 8740: 8737: 8735: 8732: 8730: 8727: 8725: 8722: 8721: 8719: 8715: 8709: 8706: 8704: 8701: 8699: 8696: 8695: 8693: 8691: 8687: 8684: 8680: 8674: 8671: 8669: 8666: 8664: 8661: 8659: 8656: 8654: 8651: 8649: 8646: 8644: 8641: 8639: 8636: 8634: 8631: 8629: 8626: 8624: 8623:Logical truth 8621: 8619: 8616: 8614: 8611: 8607: 8604: 8603: 8602: 8599: 8597: 8594: 8592: 8589: 8587: 8584: 8582: 8579: 8575: 8572: 8570: 8567: 8566: 8565: 8564:Contradiction 8562: 8560: 8557: 8555: 8552: 8550: 8547: 8545: 8542: 8541: 8539: 8535: 8525: 8522: 8520: 8517: 8515: 8512: 8510: 8509:Argumentation 8507: 8506: 8504: 8500: 8494: 8493:Philosophical 8491: 8489: 8488:Non-classical 8486: 8484: 8481: 8477: 8474: 8472: 8469: 8468: 8467: 8464: 8462: 8459: 8458: 8456: 8452: 8446: 8443: 8441: 8438: 8436: 8433: 8431: 8428: 8426: 8423: 8421: 8418: 8416: 8413: 8412: 8410: 8406: 8400: 8397: 8395: 8392: 8391: 8388: 8384: 8377: 8372: 8370: 8365: 8363: 8358: 8357: 8354: 8342: 8341: 8332: 8330: 8329: 8318: 8317: 8314: 8296: 8293: 8291: 8288: 8286: 8283: 8281: 8278: 8276: 8273: 8272: 8270: 8268:Miscellaneous 8266: 8260: 8257: 8255: 8252: 8250: 8247: 8245: 8242: 8240: 8237: 8235: 8232: 8230: 8227: 8225: 8222: 8220: 8217: 8215: 8212: 8210: 8207: 8205: 8202: 8200: 8197: 8193: 8190: 8189: 8188: 8185: 8183: 8180: 8178: 8175: 8174: 8172: 8170: 8166: 8160: 8157: 8155: 8152: 8150: 8147: 8145: 8142: 8140: 8137: 8136: 8134: 8132: 8128: 8122: 8119: 8117: 8114: 8112: 8109: 8107: 8104: 8102: 8099: 8097: 8094: 8092: 8089: 8087: 8084: 8083: 8081: 8079: 8075: 8069: 8066: 8064: 8061: 8059: 8056: 8054: 8051: 8050: 8048: 8046: 8042: 8039: 8037: 8033: 8029: 8021: 8020: 8016: 8012: 7994: 7993: 7989: 7987: 7984: 7982: 7979: 7977: 7974: 7972: 7969: 7968: 7966: 7964:Miscellaneous 7962: 7956: 7953: 7951: 7950:Structuralism 7948: 7946: 7943: 7941: 7938: 7936: 7935:Postmodernism 7933: 7931: 7928: 7926: 7925:Phenomenology 7923: 7921: 7918: 7916: 7913: 7911: 7908: 7906: 7903: 7901: 7898: 7896: 7893: 7891: 7888: 7886: 7883: 7882: 7880: 7878: 7874: 7868: 7865: 7863: 7862:Vienna Circle 7860: 7858: 7855: 7853: 7850: 7848: 7845: 7843: 7840: 7838: 7835: 7833: 7830: 7828: 7825: 7823: 7820: 7818: 7815: 7813: 7810: 7808: 7805: 7803: 7800: 7798: 7795: 7793: 7792:Moral realism 7790: 7788: 7785: 7783: 7780: 7778: 7775: 7773: 7770: 7768: 7764: 7761: 7759: 7756: 7754: 7751: 7749: 7746: 7744: 7741: 7739: 7736: 7734: 7731: 7729: 7726: 7724: 7721: 7720: 7718: 7716: 7712: 7709: 7707: 7703: 7693: 7690: 7688: 7685: 7683: 7680: 7678: 7675: 7673: 7670: 7668: 7665: 7663: 7660: 7656: 7653: 7652: 7651: 7648: 7646: 7643: 7642: 7640: 7636: 7630: 7627: 7625: 7622: 7620: 7617: 7615: 7612: 7610: 7607: 7605: 7602: 7600: 7597: 7595: 7594:Phenomenology 7592: 7590: 7587: 7585: 7582: 7580: 7577: 7575: 7572: 7570: 7567: 7565: 7562: 7560: 7557: 7555: 7552: 7550: 7547: 7545: 7542: 7540: 7537: 7535: 7534:Individualism 7532: 7528: 7525: 7523: 7520: 7518: 7515: 7513: 7510: 7508: 7505: 7503: 7500: 7499: 7498: 7495: 7491: 7488: 7487: 7486: 7483: 7481: 7478: 7476: 7473: 7471: 7468: 7466: 7463: 7461: 7458: 7456: 7453: 7451: 7448: 7446: 7443: 7441: 7438: 7436: 7433: 7431: 7428: 7426: 7423: 7422: 7419: 7416: 7414: 7410: 7400: 7399:Judeo-Islamic 7397: 7396: 7394: 7392: 7388: 7382: 7379: 7377: 7376: 7375:ʿIlm al-Kalām 7372: 7370: 7367: 7365: 7362: 7360: 7357: 7355: 7352: 7351: 7349: 7347: 7343: 7337: 7334: 7330: 7327: 7325: 7324:Shuddhadvaita 7322: 7320: 7317: 7315: 7312: 7310: 7307: 7305: 7302: 7300: 7297: 7296: 7295: 7292: 7291: 7289: 7285: 7279: 7276: 7274: 7271: 7269: 7266: 7264: 7261: 7259: 7258:Scholasticism 7256: 7254: 7251: 7249: 7246: 7245: 7243: 7241: 7237: 7231: 7228: 7226: 7223: 7221: 7218: 7216: 7213: 7211: 7208: 7206: 7203: 7201: 7198: 7197: 7195: 7191: 7188: 7186: 7182: 7172: 7169: 7167: 7164: 7162: 7159: 7157: 7154: 7153: 7151: 7149: 7145: 7137: 7134: 7132: 7129: 7127: 7124: 7122: 7119: 7117: 7114: 7112: 7109: 7107: 7104: 7102: 7099: 7097: 7094: 7093: 7092: 7089: 7085: 7082: 7080: 7077: 7076: 7075: 7072: 7068: 7065: 7063: 7060: 7058: 7055: 7053: 7050: 7048: 7045: 7043: 7040: 7038: 7035: 7033: 7030: 7029: 7028: 7025: 7024: 7022: 7020: 7016: 7010: 7007: 7005: 7002: 7000: 6997: 6995: 6992: 6990: 6987: 6985: 6982: 6978: 6975: 6973: 6970: 6968: 6965: 6963: 6960: 6959: 6958: 6955: 6953: 6950: 6948: 6945: 6943: 6940: 6938: 6935: 6933: 6930: 6928: 6925: 6921: 6918: 6916: 6913: 6911: 6908: 6906: 6903: 6901: 6898: 6897: 6896: 6893: 6892: 6890: 6888: 6885: 6881: 6875: 6872: 6870: 6867: 6865: 6862: 6860: 6857: 6855: 6852: 6850: 6847: 6845: 6842: 6840: 6837: 6836: 6834: 6832: 6828: 6825: 6823: 6819: 6811: 6808: 6806: 6803: 6801: 6798: 6796: 6793: 6791: 6788: 6787: 6786: 6783: 6781: 6778: 6777: 6775: 6773: 6769: 6765: 6758: 6754: 6740: 6737: 6735: 6732: 6730: 6727: 6725: 6722: 6720: 6717: 6715: 6712: 6710: 6709:Conceptualism 6707: 6705: 6702: 6701: 6699: 6697: 6693: 6687: 6684: 6682: 6679: 6677: 6674: 6673: 6671: 6669: 6665: 6659: 6656: 6654: 6651: 6649: 6646: 6644: 6641: 6639: 6638:Particularism 6636: 6634: 6631: 6630: 6628: 6626: 6622: 6616: 6613: 6611: 6608: 6606: 6605:Functionalism 6603: 6601: 6598: 6596: 6593: 6591: 6590:Eliminativism 6588: 6586: 6583: 6582: 6580: 6578: 6574: 6568: 6565: 6563: 6560: 6558: 6555: 6553: 6550: 6548: 6545: 6543: 6540: 6539: 6537: 6535: 6531: 6525: 6522: 6518: 6515: 6514: 6513: 6510: 6506: 6503: 6502: 6501: 6498: 6496: 6495:Compatibilism 6493: 6492: 6490: 6488: 6484: 6478: 6475: 6473: 6470: 6468: 6465: 6464: 6462: 6460: 6456: 6450: 6447: 6445: 6442: 6440: 6437: 6435: 6434:Particularism 6432: 6430: 6427: 6425: 6422: 6420: 6417: 6416: 6414: 6412: 6408: 6402: 6399: 6397: 6394: 6392: 6389: 6388: 6386: 6384: 6380: 6374: 6371: 6369: 6366: 6364: 6361: 6359: 6356: 6354: 6351: 6349: 6346: 6344: 6341: 6339: 6336: 6334: 6331: 6329: 6326: 6324: 6321: 6319: 6316: 6315: 6313: 6311: 6307: 6303: 6296: 6292: 6288: 6281: 6276: 6274: 6269: 6267: 6262: 6261: 6258: 6250: 6248:9780253007810 6244: 6240: 6236: 6232: 6228: 6224: 6220: 6216: 6214: 6210: 6204: 6200: 6199: 6195: 6190: 6188: 6185: 6181: 6177: 6174: 6171: 6169: 6165: 6162: 6159: 6157: 6153: 6150: 6147: 6146: 6144: 6140: 6137: 6133: 6127: 6126: 6121: 6117: 6115: 6113: 6109: 6107: 6105: 6101: 6100: 6097: 6096: 6091: 6087: 6083: 6082: 6077: 6073: 6068: 6066: 6062: 6059: 6057: 6053: 6050: 6049: 6039: 6037:9780192893062 6033: 6029: 6028: 6023: 6019: 6015: 6011: 6007: 6005:9780486277240 6001: 5998:. Routledge. 5997: 5996: 5991: 5987: 5983: 5979: 5975: 5971: 5967: 5963: 5961:9780080930589 5957: 5953: 5952: 5946: 5942: 5940:9780198751199 5936: 5932: 5927: 5923: 5921:9781107268135 5917: 5913: 5912: 5907: 5903: 5899: 5898: 5886: 5882: 5878: 5875: 5870: 5855: 5851: 5847: 5841: 5833: 5829: 5827: 5819: 5813:, 2000, p. 15 5812: 5806: 5799: 5793: 5786: 5780: 5778: 5761: 5757: 5750: 5742: 5738: 5734: 5728: 5724: 5720: 5716: 5709: 5707: 5699: 5693: 5684: 5669: 5665: 5661: 5655: 5640: 5636: 5629: 5614: 5610: 5603: 5595: 5589: 5585: 5578: 5571: 5567: 5566:Field, Hartry 5562: 5555: 5551: 5547: 5543: 5538: 5530: 5524: 5520: 5513: 5505: 5499: 5495: 5491: 5486: 5485: 5479: 5473: 5465: 5459: 5455: 5450: 5449: 5440: 5433: 5429: 5424: 5409: 5405: 5401: 5397: 5393: 5389: 5385: 5378: 5370: 5368:9781137400727 5364: 5360: 5359: 5351: 5343: 5337: 5333: 5326: 5318: 5314: 5310: 5304: 5302: 5294: 5290: 5277: 5273: 5267: 5261: 5254: 5248: 5244: 5240: 5234: 5227: 5223: 5218: 5204: 5200: 5196: 5189: 5181: 5180: 5172: 5165: 5161: 5156: 5154: 5144: 5136: 5132: 5128: 5124: 5120: 5116: 5111: 5106: 5102: 5098: 5091: 5083: 5079: 5075: 5071: 5067: 5063: 5059: 5052: 5044: 5040: 5037:(2): 97–108. 5036: 5032: 5025: 5010: 5006: 5002: 4996: 4989: 4985: 4980: 4965: 4961: 4957: 4951: 4936: 4932: 4928: 4922: 4914: 4910: 4906: 4902: 4898: 4894: 4891:(3714): 3–5. 4890: 4886: 4879: 4864: 4860: 4854: 4847: 4841: 4834: 4833: 4828: 4823: 4815: 4811: 4805: 4798: 4794: 4789: 4773: 4769: 4765: 4759: 4743: 4739: 4735: 4729: 4721: 4717: 4713: 4709: 4705: 4701: 4694: 4686: 4682: 4678: 4674: 4670: 4666: 4662: 4656: 4648: 4644: 4640: 4636: 4632: 4628: 4621: 4605: 4601: 4597: 4590: 4574: 4570: 4566: 4560: 4541: 4537: 4530: 4523: 4515: 4511: 4507: 4503: 4496: 4488: 4484: 4480: 4476: 4472: 4468: 4464: 4460: 4456: 4455: 4450: 4446: 4440: 4424: 4420: 4419: 4414: 4410: 4404: 4396: 4390: 4386: 4379: 4371: 4365: 4361: 4357: 4351: 4336: 4330: 4326: 4322: 4315: 4300: 4294: 4290: 4289: 4281: 4265: 4261: 4257: 4251: 4236: 4230: 4226: 4222: 4218: 4214: 4210: 4204: 4188: 4184: 4180: 4176: 4172: 4166: 4151: 4147: 4143: 4139: 4135: 4131: 4127: 4120: 4105: 4099: 4095: 4094: 4086: 4078: 4074: 4070: 4066: 4062: 4058: 4051: 4043: 4039: 4032: 4013: 4009: 4005: 4001: 3997: 3993: 3989: 3985: 3981: 3977: 3973: 3966: 3959: 3940: 3936: 3932: 3928: 3924: 3920: 3916: 3909: 3902: 3894: 3886: 3882: 3878: 3874: 3870: 3866: 3865: 3856: 3848: 3844: 3839: 3834: 3830: 3826: 3822: 3818: 3817:Borel, Armand 3812: 3810: 3793: 3789: 3785: 3778: 3774: 3758: 3749: 3742: 3738: 3732: 3728: 3718: 3715: 3713: 3710: 3708: 3705: 3704: 3695: 3694: 3690: 3688: 3687: 3683: 3680: 3676: 3674: 3673: 3669: 3666: 3662: 3659: 3655: 3653: 3652: 3647: 3645: 3644: 3640: 3639: 3635:Related works 3629: 3626: 3624: 3621: 3619: 3616: 3614: 3611: 3609: 3606: 3604: 3601: 3599: 3596: 3594: 3591: 3589: 3586: 3584: 3581: 3579: 3576: 3574: 3571: 3569: 3566: 3564: 3561: 3560: 3555: 3544: 3541: 3535: 3530: 3521: 3519: 3515: 3513: 3511: 3507: 3506: 3500: 3498: 3494: 3493: 3488: 3482: 3478: 3476: 3472: 3468: 3466: 3462: 3458: 3457:contradiction 3453: 3443: 3439: 3435: 3433: 3429: 3424: 3422: 3411: 3408: 3400: 3397:November 2015 3390: 3386: 3380: 3379: 3374:This section 3372: 3368: 3363: 3362: 3354: 3352: 3347: 3345: 3344: 3339: 3334: 3332: 3328: 3323: 3318: 3316: 3312: 3307: 3305: 3300: 3295: 3291: 3290: 3285: 3281: 3277: 3273: 3263: 3261: 3257: 3253: 3249: 3245: 3240: 3232: 3229: 3226: 3222: 3218: 3214: 3213: 3212: 3210: 3209:Stephen Yablo 3206: 3205:Hilary Putnam 3202: 3201:Willard Quine 3196: 3181: 3179: 3175: 3171: 3167: 3161: 3159: 3155: 3151: 3146: 3142: 3138: 3127: 3124: 3116: 3113:February 2023 3106: 3102: 3098: 3092: 3091: 3086:This section 3084: 3075: 3074: 3066: 3059: 3049: 3047: 3043: 3042:Eugene Wigner 3039: 3035: 3020: 3018: 3014: 3010: 3005: 3001: 2997: 2993: 2988: 2985: 2981: 2976: 2972: 2968: 2963: 2961: 2956: 2950: 2948: 2944: 2939: 2929: 2927: 2923: 2919: 2914: 2912: 2907: 2905: 2896: 2892: 2887: 2885: 2884:vector fields 2881: 2877: 2873: 2869: 2865: 2860: 2850: 2848: 2843: 2841: 2837: 2833: 2828: 2826: 2822: 2818: 2812: 2810: 2806: 2801: 2797: 2793: 2789: 2785: 2781: 2776: 2774: 2770: 2766: 2762: 2761:Hilary Putnam 2758: 2753: 2750: 2746: 2740: 2736: 2732: 2728: 2723: 2719: 2709: 2707: 2703: 2699: 2698: 2693: 2692: 2687: 2683: 2682: 2677: 2673: 2669: 2668:psychologists 2665: 2661: 2657: 2653: 2651: 2648: 2644: 2639: 2633: 2623: 2620: 2616: 2611: 2608: 2604: 2603:Sydney School 2600: 2595: 2590: 2586: 2580: 2570: 2568: 2564: 2560: 2556: 2555: 2550: 2546: 2542: 2541:George Lakoff 2538: 2537: 2531: 2528: 2526: 2522: 2517: 2515: 2511: 2507: 2503: 2499: 2494: 2492: 2488: 2484: 2480: 2475: 2473: 2469: 2465: 2460: 2458: 2454: 2450: 2446: 2442: 2438: 2434: 2429: 2427: 2423: 2419: 2414: 2411: 2407: 2406:Embodied mind 2398: 2395: 2391: 2387: 2383: 2379: 2374: 2372: 2368: 2363: 2361: 2357: 2353: 2348: 2346: 2342: 2338: 2334: 2331: 2326: 2324: 2320: 2316: 2312: 2308: 2304: 2300: 2299:Structuralism 2295: 2288:Structuralism 2285: 2283: 2279: 2274: 2273:Ultrafinitism 2269: 2265: 2263: 2258: 2256: 2252: 2248: 2244: 2240: 2236: 2232: 2226: 2222: 2217: 2207: 2206: 2202: 2198: 2197:real analysis 2194: 2193:Errett Bishop 2187: 2177: 2175: 2171: 2167: 2163: 2159: 2155: 2150: 2148: 2144: 2140: 2136: 2132: 2128: 2127:Arend Heyting 2124: 2119: 2117: 2113: 2107: 2097: 2095: 2091: 2087: 2083: 2079: 2075: 2071: 2068: 2067:mathematician 2062: 2058: 2048: 2046: 2042: 2038: 2034: 2030: 2026: 2021: 2017: 2013: 2011: 2007: 2003: 2002:Haskell Curry 1999: 1998:Alfred Tarski 1995: 1994:Rudolf Carnap 1990: 1986: 1984: 1980: 1976: 1972: 1968: 1964: 1960: 1956: 1955:David Hilbert 1949: 1948:David Hilbert 1945: 1941: 1939: 1938:structuralism 1933: 1931: 1926: 1923: 1919: 1913: 1903: 1901: 1897: 1893: 1889: 1885: 1881: 1877: 1873: 1868: 1866: 1861: 1857: 1853: 1849: 1842: 1838: 1834: 1832: 1828: 1824: 1820: 1816: 1812: 1808: 1804: 1800: 1796: 1792: 1791:Gottlob Frege 1785: 1781: 1778: 1774: 1773: 1772: 1770: 1769:Rudolf Carnap 1766: 1764: 1760: 1756: 1752: 1748: 1743: 1733: 1731: 1727: 1726:mathematicism 1723: 1719: 1714: 1713:Mathematicism 1707:Mathematicism 1704: 1702: 1698: 1694: 1690: 1686: 1682: 1678: 1674: 1670: 1666: 1662: 1658: 1654: 1652: 1648: 1643: 1639: 1637: 1633: 1632: 1627: 1623: 1619: 1618: 1614: 1610: 1606: 1602: 1596: 1592: 1588: 1586: 1580: 1578: 1574: 1573: 1568: 1564: 1560: 1556: 1552: 1547: 1542: 1536: 1526: 1524: 1523: 1518: 1514: 1513:mathematician 1510: 1506: 1502: 1487: 1482: 1480: 1479:infinitesimal 1476: 1472: 1466: 1464: 1463:Hilary Putnam 1460: 1456: 1452: 1447: 1445: 1444: 1439: 1438: 1433: 1429: 1425: 1420: 1416: 1412: 1408: 1404: 1400: 1396: 1391: 1388: 1384: 1380: 1376: 1372: 1368: 1364: 1360: 1354: 1352: 1346: 1344: 1340: 1336: 1332: 1321: 1319: 1315: 1311: 1307: 1303: 1299: 1295: 1294:irrationality 1290: 1288: 1284: 1280: 1276: 1274: 1270: 1266: 1262: 1258: 1254: 1253:mathematicism 1250: 1246: 1242: 1236: 1229: 1225: 1221: 1217: 1213: 1204: 1202: 1186: 1181: 1169: 1165: 1160: 1158: 1154: 1150: 1146: 1142: 1138: 1133: 1131: 1127: 1123: 1119: 1115: 1111: 1107: 1103: 1099: 1094: 1092: 1091:Eugene Wigner 1088: 1078: 1074: 1072: 1068: 1064: 1060: 1056: 1055:philosophical 1051: 1049: 1045: 1041: 1037: 1032: 1028: 1018: 1014: 1012: 1008: 1004: 1000: 995: 993: 989: 984: 980: 976: 972: 968: 963: 961: 957: 953: 949: 945: 941: 937: 933: 929: 925: 921: 917: 913: 909: 904: 902: 898: 894: 890: 884: 882: 878: 874: 870: 860: 857: 855: 851: 847: 843: 838: 836: 832: 828: 824: 820: 817:and a set of 816: 812: 808: 804: 800: 796: 791: 789: 785: 781: 777: 772: 770: 766: 762: 761:infinite sets 758: 754: 751:but nowhere 750: 746: 742: 739:in which the 738: 734: 730: 725: 723: 719: 714: 712: 708: 704: 700: 696: 690: 686: 682: 671: 667: 662: 660: 656: 652: 648: 644: 640: 638: 634: 629: 625: 607: 603: 599: 595: 592: 590: 587: 585: 582: 580: 577: 575: 572: 570: 567: 564: 561: 560: 559: 556: 554: 550: 546: 535: 530: 528: 523: 521: 516: 515: 513: 512: 509: 505: 501: 500: 489: 486: 484: 481: 479: 476: 474: 471: 469: 466: 464: 461: 459: 456: 454: 451: 449: 446: 445: 439: 438: 431: 427: 424: 422: 419: 417: 413: 410: 408: 405: 403: 399: 396: 394: 391: 389: 386: 384: 383:Number theory 381: 380: 377: 372: 371: 368: 367: 362: 359: 357: 354: 353: 352: 351: 348: 345: 344: 340: 339: 330: 327: 312: 309:November 2022 302: 296: 293:This article 291: 287: 282: 281: 272: 269: 261: 258:November 2022 251: 247: 241: 240: 236: 230: 221: 220: 211: 201: 195: 193: 188: 183: 179: 174: 173: 164: 161: 153: 142: 139: 135: 132: 128: 125: 121: 118: 114: 111: – 110: 106: 105:Find sources: 99: 95: 89: 88: 83:This article 81: 77: 72: 71: 66: 64: 57: 56: 51: 50: 45: 40: 31: 30: 27: 19: 11152: 11136: 10950:Ultraproduct 10797:Model theory 10762:Independence 10698:Formal proof 10690:Proof theory 10673: 10646: 10603:real numbers 10575:second-order 10486:Substitution 10363:Metalanguage 10304:conservative 10277:Axiom schema 10221:Constructive 10191:Morse–Kelley 10157:Set theories 10136:Aleph number 10129:inaccessible 10035:Grothendieck 9919:intersection 9806:Higher-order 9794:Second-order 9740:Truth tables 9697:Venn diagram 9480:Formal proof 9399: 9387: 9375: 9356: 9289:Optimization 9151:Differential 9075:Differential 9042:Order theory 9037:Graph theory 8941:Group theory 8897: 8663:Substitution 8483:Mathematical 8408:Major fields 8333: 8319: 7990: 7981:Postcritique 7971:Kyoto School 7930:Posthumanism 7910:Hermeneutics 7765: / 7706:Contemporary 7682:Newtonianism 7645:Cartesianism 7604:Reductionism 7440:Conservatism 7435:Collectivism 7373: 7101:Sarvāstivadā 7079:Anekantavada 7004:Neoplatonism 6972:Epicureanism 6905:Pythagoreans 6844:Confucianism 6810:Contemporary 6800:Early modern 6704:Anti-realism 6658:Universalism 6615:Subjectivism 6411:Epistemology 6342: 6238: 6231:Peirce, C.S. 6196: 6123: 6111: 6103: 6093: 6079: 6026: 5994: 5973: 5950: 5930: 5910: 5869: 5858:. 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26, 5674:2008-12-26 5664:Math Forum 5645:2008-12-26 5619:2008-12-26 5282:2008-07-19 5209:2019-05-25 5164:Erkenntnis 5015:2018-08-18 4970:2018-08-17 4941:2018-08-24 4869:2020-07-20 4063:: 91–100. 3769:References 3471:Paul Erdős 3432:literature 3357:Aesthetics 3248:nominalist 3239:naturalism 3097:improve it 3004:Paul Erdős 2870:published 2857:See also: 2712:Empiricism 2636:See also: 2583:See also: 2514:empiricism 2449:empiricism 2386:nominalism 2251:Mary Tiles 2162:algorithms 1971:arithmetic 1967:consistent 1803:arithmetic 1697:naturalism 1669:set theory 1649:, and the 1601:Kurt Gödel 1595:Kurt Gödel 1539:See also: 1415:Assignment 1335:set theory 1302:Pythagoras 1249:Pythagoras 1216:Pythagoras 920:continuity 749:continuous 703:syllogisms 679:See also: 624:Pythagoras 549:philosophy 483:Philosophy 426:Statistics 416:Set theory 235:neutrality 150:April 2022 120:newspapers 49:improve it 11142:Supertask 11045:Recursion 11003:decidable 10837:saturated 10815:of models 10738:deductive 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10810:function 10658:Tarski's 10647:Elements 10634:geometry 10590:Robinson 10511:variable 10496:function 10469:spectrum 10459:Sentence 10415:variable 10358:Language 10311:Relation 10272:Automata 10262:Alphabet 10246:language 10100:-jection 10078:codomain 10064:Function 10025:Universe 9995:Infinite 9899:Relation 9682:Validity 9672:Argument 9570:theorem, 9377:Category 9133:Topology 9080:Discrete 9065:Analytic 9052:Geometry 9024:Discrete 8979:Calculus 8971:Analysis 8926:Abstract 8865:Glossary 8848:Timeline 8773:Category 8673:Validity 8574:Antinomy 8502:Theories 8466:Informal 8340:Category 8295:Yugoslav 8285:Romanian 8192:Scottish 8177:American 8106:Japanese 8086:Buddhist 8068:Africana 8058:Egyptian 7900:Feminist 7822:Rawlsian 7817:Quietism 7715:Analytic 7667:Krausism 7574:Nihilism 7539:Kokugaku 7502:Absolute 7497:Idealism 7485:Humanism 7273:Occamism 7240:European 7185:Medieval 7131:Yogacara 7091:Buddhist 7084:Syādvāda 6967:Stoicism 6932:Cynicism 6920:Sophists 6915:Atomists 6910:Eleatics 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