Philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The philosopher Socrates about to take poison hemlock as ordered by the court. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

Recurrent themes include:

• What are the sources of mathematical subject matter?
• What is the ontological status of mathematical entities?
• What does it mean to refer to a mathematical object?
• What is the character of a mathematical proposition?
• What is the relation between logic and mathematics?
• What is the role of hermeneutics in mathematics?
• What kinds of inquiry play a role in mathematics?
• What are the objectives of mathematical inquiry?
• What gives mathematics its hold on experience?
• What are the human traits behind mathematics?
• What is mathematical beauty?
• What is the source and nature of mathematical truth?
• What is the relationship between the abstract world of mathematics and the material universe?

The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms.^[1] The latter, however, may be used to mean at least three other things. One sense refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors of Scholastic theologians, or the systematic aims of Leibniz and Spinoza. Another sense refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term mathematical philosophy to be an allusion to the approach taken by Bertrand Russell in his book Introduction to Mathematical Philosophy. In philosophy, ontology (from the Greek , genitive : of being (part. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Hermeneutics may be described as the development and study of theories of the interpretation and understanding of texts. ... Look up Experience in Wiktionary, the free dictionary This article discusses the general concept of experience. ... Psychology (from Greek: ψυχή, psukhÄ“, spirit, soul; and λόγος, logos, knowledge) is an academic or applied discipline involving the scientific study of mental processes and behavior. ... Most mathematicians derive aesthetic pleasure from their work, and from mathematics in general. ... The Parthenons facade showing an interpretation of golden rectangles in its proportions. ... For other uses, see Ethics (disambiguation). ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Plato (Left) and Aristotle (right), by Raphael (Stanza della Segnatura, Rome) Metaphysics is the branch of philosophy concerned with explaining the ultimate nature of reality, being, and the world. ... Theology finds its scholars pursuing the understanding of and providing reasoned discourse of religion, spirituality and God or the gods. ... Scholastic is the official student publication of the University of Notre Dame. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Baruch Spinoza Benedictus de Spinoza (November 24, 1632 - February 21, 1677), named Baruch Spinoza by his synagogue elders and known as Bento de Spinoza or Bento dEspiñoza in the community in which he grew up. ... Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... Introduction to Mathematical Philosophy is a book by Bertrand Russell, published in 1919, written in part to exposit in a less technical way the main ideas of his and Whiteheads Principia Mathematica (1910–1913). ...

Historical overview

Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis. There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Western philosophies of mathematics go as far back as Plato, who studied the ontological status of mathematical objects, and Aristotle, who studied logic and issues related to infinity (actual versus potential). Greek philosophy on mathematics was strongly influenced by their study of geometry. At one time, the Greeks held the opinion that 1 (one) was not a number, but rather a unit of arbitrary length. A number was defined as a multitude. Therefore 3, for example, represented a certain multitude of units, and truly was a number). At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportional to the arbitrary first "number" or "one." These earlier Greek ideas of number were later upended by the discovery of the irrationality of the square root of two. Hippasus, a disciple of Pythagoras, showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. For the book by Bertrand Russell, see History of Western Philosophy (Russell) Philosophy has a long history conventionally divided into three large eras: the Ancient, Medieval and Modern. ... This article needs additional references or sources for verification. ... PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... Aristotle (Greek: Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... The infinity symbol ∞ in several typefaces. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... For other uses, see Number (disambiguation). ... In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ...

Beginning with Leibniz, the focus shifted strongly to the relationship between mathematics and logic. This view dominated the philosophy of mathematics through the time of Frege and of Russell, but was brought into question by developments in the late 19th and early 20th century. Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Russell is a Scottish or French name derived from the colour red or from the fox animal: // Spanish - Quesada French - Roussel Italian - Rufino Latin - Rufus American - Rusty Notable people with the surname Russell include: William Russell (disambiguation page) Members of this family have held the title of Earl of Bedford...

Philosophy of mathematics in the 20th century

A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in formal logic, set theory, and foundational issues. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s As a means of recording the passage of time, the 20th century was that century which lasted from 1901–2000 in the sense of the Gregorian calendar (1900–1999... Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

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 'truthfullness' remains elusive. Investigations into this issue are known as the foundations of mathematics program. Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...

At the start of the century, philosophers of mathematics were already beginning to divide into various schools of thought about all these questions, broadly distinguished by their pictures of mathematical epistemology and ontology. Three schools, formalism, intuitionism, and logicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis in particular, did not live up to the standards of certainty and rigor that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge. It has been suggested that Meta-epistemology be merged into this article or section. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... The term formalism describes an emphasis on form over content or meaning in the arts, literature, or philosophy. ... In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. ... Analysis has its beginnings in the rigorous formulation of calculus. ... A related article is titled uncertainty. ... For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. ...

Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of Euclid as the natural basis for mathematics. Core concepts such as axiom, order, and set received fresh emphasis. In mathematics as in physics, new and unexpected ideas had arisen and significant changes were coming. Inquiries into the consistency of mathematical theories led to the development of a new level of study, a reflective critique in which the theory under review "becomes itself the object of a mathematical study", what Hilbert called metamathematics or proof theory (Kleene, 55). Logic (from ancient Greek λόγος (logos), meaning reason) is the study of arguments. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ... For other uses, see Euclid (disambiguation). ... This article is about a logical statement. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematical logic, a formal system is consistent if it does not contain a contradiction, or, more precisely, for no proposition φ are both φ and ¬φ provable. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... In general, metamathematics or meta-mathematics is reflection about mathematics seen as an entity/object in human consciousness and culture. ... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Stephen Cole Kleene (January 5, 1909 – January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ...

At the midpoint of the century, a new mathematical theory known as category theory arose as a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam summed up one common view of the situation in the last third of the century by saying: In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ...

When philosophy discovers something wrong with science, sometimes science has to be changed — Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal — but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that 'philosophical interpretation' is just what mathematics doesn't need. (Putnam, 169–170). Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... George Berkeley (IPA: , Bark-Lee) (12 March 1685 – 14 January 1753), also known as Bishop Berkeley, was an influential Irish philosopher whose primary philosophical achievement is the advancement of a theory he called immaterialism (later referred to as subjective idealism by others). ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.

Contemporary schools of thought

Mathematical realism

Mathematical realism, like realism in general, holds that mathematical entities exist independently of the human mind. Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same. In this point of view, there is really one sort of mathematics that can be discovered: Triangles, for example, are real entities, not the creations of the human mind. Contemporary philosophical realism, also referred to as metaphysical realism, is the belief in a reality that is completely ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc. ... For other uses, see Mind (disambiguation). ... A triangle. ...

Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis, might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture. Paul ErdÅ‘s, also ErdÅ‘s Pál, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... Kurt Gödel (IPA: ) (April 28, 1906 Brünn, Austria-Hungary (now Brno, Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.

Platonism

Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the naive view most people have of numbers. The term Platonism is used because such a view is seen to parallel Plato's belief in a "World of Ideas", an unchanging ultimate reality that the everyday world can only imperfectly approximate. The two ideas have a meaningful, not just a superficial connection, because Plato probably derived his understanding from the Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers. PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ... The Pythagoreans were a Hellenic organization of astronomers, musicians, mathematicians, and philosophers who believed that all things are, essentially, numeric. ... For other uses, see Number (disambiguation). ...

The major problem of mathematical platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be Ultimate ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe. The Ultimate Ensemble is a speculative possible feature of theories of everything (TOEs), suggested by Max Tegmark. ...

Gödel's platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Husserl said about mathematics, and supports Kant's idea that mathematics is synthetic a priori.) Davis and Hersh have suggested in their book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism (see below). Edmund Husserl Edmund Gustav Albrecht Husserl (April 8, 1859 - April 26, 1938), philosopher, was born into a Jewish family in Prossnitz, Moravia (Prostejov, Czech Republic), Empire of Austria-Hungary. ... Immanuel Kant Immanuel Kant (April 22, 1724 – February 12, 1804) was a Prussian philosopher, generally regarded as one of Europes most influential thinkers and the last major philosopher of the Enlightenment. ... Th analytic-synthetic distinction (or dichotomy) is a conceptual distinction, used primarily in philosophy to distinguish propositions into two types: analytic propositions and synthetic propositions. ... The terms a priori and a posteriori are used in philosophy to distinguish between two different types of propositional knowledge. ... Philip J. Davis is an American applied mathematician. ... Reuben Hersh (December 9, 1927 - ) is an American mathematician, now an emeritus professor of the University of New Mexico. ...

Some mathematicians hold opinions that amount to more nuanced versions of Platonism. These ideas are sometimes described as Neo-Platonism. Neoplatonism (also Neo-Platonism) is the modern term for a school of religious and mystical philosophy that took shape in the 3rd century AD, based on the teachings of Plato and earlier Platonists. ...

Logicism

Gottlob Frege, founder of logicism

Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists hold that mathematics can be known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths. Image File history File links Gottlob_Frege. ... Image File history File links Gottlob_Frege. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... The terms analytic and synthetic are philosophical terms, used by philosophers to divide propositions into two types: analytic propositions and synthetic propositions. ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ...

Rudolf Carnap (1931) presents the logicist thesis in two parts: Rudolf Carnap (May 18, 1891, Ronsdorf, Germany – September 14, 1970, Santa Monica, California) was an influential philosopher who was active in central Europe before 1935 and in the United States thereafter. ...

1.	The concepts of mathematics can be derived from logical concepts through explicit definitions.
2.	The theorems of mathematics can be derived from logical axioms through purely logical deduction.

Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic) he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G, the extension of F equals the extension of G if and only if for all objects a, Fa if and only if Ga), a principle that he took to be acceptable as part of logic. Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ...

But Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent (this is Russell's paradox). Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead. They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex, form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of mathematics, such as an "axiom of reducibility". Even Russell said that this axiom did not really belong to logic. Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 – 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ... Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... Alfred North Whitehead, OM (February 15, 1861 Ramsgate, Kent, England – December 30, 1947 Cambridge, Massachusetts, USA) was an English-born mathematician who became a philosopher. ... The title page of the shortened version of the work, Principia Mathematica to *56. ... The axiom of reducibility was introduced by Bertrand Russel as part of his ramified theory of types, an attempt to ground mathematics in first-order logic. ...

Modern logicists (like Bob Hale, Crispin Wright, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favour of abstraction principles such as Hume's principle (the number of objects falling under the concept F equals the number of objects falling under the concept G if and only if the extension of F and the extension of G can be put into one-to-one correspondence). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical. Crispin Wright (born 1942) is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgensteins later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity. ... Humes principle is a standard for comparing any two sets of objects as to size. ... A bijective function. ...

If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense, logicism can be seen as shifting questions about the philosophy of mathematics to questions about logic without fully answering them.

Empiricism

Empiricism is a form of realism that denies that mathematics can be known a priori at all. It says that we discover mathematical facts by empirical research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th century, but primarily arose in the middle of the century. However, an important early proponent of a view like this was John Stuart Mill. Mill's view was widely criticized, because it makes statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet. The terms a priori and a posteriori are used in philosophy to distinguish between two different types of propositional knowledge. ... A central concept in science and the scientific method is that all evidence must be empirical, or empirically based, that is, dependent on evidence or consequences that are observable by the senses. ... John Stuart Mill (20 May 1806 – 8 May 1873), British philosopher, political economist civil servant, and Member of Parliament, was an influential liberal thinker of the 19th century. ...

Contemporary mathematical empiricism, formulated by Quine and Putnam, is primarily supported by the indispensability argument: mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. That is, since physics needs to talk about electrons to say why light bulbs behave as they do, then electrons must exist. Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of some of its distinctness from the other sciences. W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ... For other uses, see Electron (disambiguation). ...

Putnam strongly rejected the term "Platonist" as implying an overly-specific ontology that was not necessary to mathematical practice in any real sense. He advocated a form of "pure realism" that rejected mystical notions of truth and accepted much quasi-empiricism in mathematics. Putnam was involved in coining the term "pure realism" (see below). Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... In the philosophy of mathematics, mathematical practice is used to distinguish the working practices of professional mathematicians (eg. ... A common dictionary definition of truth is agreement with fact or reality.[1] There is no single definition of truth about which the majority of philosophers agree. ... Quasi-empiricism in mathematics is the movement in the philosophy of mathematics to direct philosophers attention to mathematical practice, in particular, relations with physics and social sciences, rather than the foundations problem in mathematics. ...

The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole. Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is incredibly central, and that it would be extremely difficult for us to revise it, though not impossible.

For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy's Realism in Mathematics. Another example of a realist theory is the embodied mind theory (see below). Penelope Maddy is Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. ...

For experimental evidence suggesting that one-day-old babies can do elementary arithmetic, see Brian Butterworth.

Formalism

Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, you can generate the string corresponding to the Pythagorean theorem). Mathematical truths are not about numbers and sets and triangles and the like — in fact, they aren't "about" anything at all! Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: if you assign meaning to the strings in such a way that the rules of the game become true (ie, true statements are assigned to the axioms and the rules of inference are truth-preserving), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism.) But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics. Structuralism as a term refers to various theories across the humanities, social sciences and economics many of which share the assumption that structural relationships between concepts vary between different cultures/languages and that these relationships can be usefully exposed and explored. ...

David Hilbert

A major early proponent of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics. ("Consistent" here means that no contradictions can be derived from the system.) Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent was dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent. David Hilbert, 1912 photograph This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... David Hilbert, 1912 photograph This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Hilberts program, formulated by German mathematician David Hilbert in the 1920s, was to formalize all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. ... Gödels completeness theorem is an important theorem in mathematical logic which was first proved by Kurt Gödel in 1929. ... In mathematical logic, a formal system is consistent if it does not contain a contradiction, or, more precisely, for no proposition φ are both φ and ¬φ provable. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...

Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.

Other formalists, such as Rudolf Carnap, Alfred Tarski and Haskell Curry, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists. Rudolf Carnap (May 18, 1891, Ronsdorf, Germany – September 14, 1970, Santa Monica, California) was an influential philosopher who was active in central Europe before 1935 and in the United States thereafter. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ... Haskell Brooks Curry (September 12, 1900, Millis, Massachusetts - September 1, 1982, State College, Pennsylvania) was an American mathematician and logician. ... Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ...

Formalists are usually very tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary.

The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the minute string manipulation games mentioned above. While published proofs (if correct) could in principle be formulated in terms of these games, the effort required in space and time would be prohibitive (witness Principia Mathematica.) In addition, the rules are certainly not substantial to the initial creation of those proofs. Formalism is also silent to the question of which axiom systems ought to be studied. The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ...

Intuitionism

Main article: Mathematical intuitionism

In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L.E.J. Brouwer). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. (CDP, 542) In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ...

Leopold Kronecker said: "The natural numbers come from God, everything else is man's work." A major force behind Intuitionism was L.E.J. Brouwer, who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting postulated an intuitionistic logic, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle and therefore frowns upon proofs by contradiction. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. Important work was later done by Errett Bishop, who managed to prove versions of the most important theorems in real analysis within this framework. Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ... Arend Heyting (May 9, 1898 – July 9, 1980) was a Dutch mathematician and logician. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ... “Excluded middle” redirects here. ... Reductio ad absurdum (Latin: reduction to the absurd) also known as an apagogical argument, reductio ad impossibile, or proof by contradiction, is a type of logical argument where one assumes a claim for the sake of argument, derives an absurd or ridiculous outcome, and then concludes that the original assumption... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... Errett Albert Bishop (1928-1983) was an American mathematician known for is work on analysis. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...

In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science. An artistic representation of a Turing Machine . ... Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. ... Alan Mathison Turing, OBE, FRS (23 June 1912 – 7 June 1954) was an English mathematician, logician, and cryptographer. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

Constructivism

Main article: Mathematical constructivism

Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction. In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...

Fictionalism

Fictionalism was introduced in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics that didn't reference numbers or functions at all. He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields. 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. Hartry H. Field (born 1946) is a philosopher working at New York University (NYU). ... It has been suggested that this article or section be merged with Classical mechanics. ... Hilberts axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

Having shown how to do science without using mathematics, he proceeded to rehabilitate mathematics as a kind of useful fiction. He showed that mathematical physics is a conservative extension of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from his system), so that the mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as false as "Sherlock Holmes lived at 221B Baker Street" — but both are true according to the relevant fictions. // Definition A logical theory T2 is a conservative extension of theory T1 if any consequence of T2 involving symbols of T1 only is already a consequence of T1. ... A portrait of Sherlock Holmes by Sidney Paget from the Strand Magazine, 1891 Sherlock Holmes is a fictional detective of the late 19th and early 20th centuries, who first appeared in publication in 1887. ...

By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions. Another objection is that it is not clear how one could have certain results in science, such as quantum theory or the periodic table, without mathematics. If what distinguishes one element from another is *precisely* the number of electrons, neutrons and protons, how does one distinguish between elements without a concept of number? For other uses, see Fiction (disambiguation). ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...

Embodied mind theories

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

With this view, the physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation. However, the human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition. For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... Look up Cognition in Wiktionary, the free dictionary. ...

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

The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From, by George Lakoff and Rafael E. Núñez. In addition, mathematician Keith Devlin has investigated similar concepts with his book The Math Instinct. For more on the science that inspired this perspective, see cognitive science of mathematics. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. ... This article or section does not cite any references or sources. ... Rafael E. Núñez is a professor of Cognitive science at the University of California, San Diego and is well known for promoting the idea of embodied cognition. ... Keith Devlin is an English mathematician and writer. ... The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science. ...

Social constructivism or social realism

Social constructivism or social realism theories see mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with 'reality', social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints- the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated- that work to conserve the historically defined discipline. Social scientists and literary scholars have claimed that many things are social constructions or social constructs, or that they have been socially constructed. ...

This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an over-emphasis on axiomatic proof and peer review as practices. However, this might be seen as merely saying that rigorously proven results are overemphasized, and then "look how chaotic and uncertain the rest of it all is!" In the philosophy of mathematics, mathematical practice is used to distinguish the working practices of professional mathematicians (eg. ... As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. ...

The social nature of mathematics is highlighted in its subcultures. Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Social constructivists argue each speciality forms its own epistemic community and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics. Social constructivists see the process of 'doing mathematics' as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias, or of mathematicians' collective intelligence as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless. Some social scientists also argue that mathematics is not real or objective at all, but is affected by racism and ethnocentrism. Some of these ideas are close to postmodernism. In sociology, anthropology and cultural studies, a subculture is a set of people with a set of behaviors and beliefs, culture, which could be distinct or hidden, that differentiate them from the larger culture to which they belong. ... An epistemic community may consist of those who accept one version of a story, or one version of validating a story. ... In mathematics, there have been many attempts down the centuries to unify the whole subject. ... This article or section does not cite its references or sources. ... It has been suggested that symbiotic intelligence be merged into this article or section. ... Racism is the prejudice that members of one race are intrinsically superior or inferior to members of other races. ... Ethnocentrism is the tendency to look at the world primarily from the perspective of ones own culture. ... Postmodernism is a term applied to a wide-ranging set of developments in critical theory, philosophy, architecture, art, literature, and culture, which are generally characterized as either emerging from, in reaction to, or superseding, modernism. ...

Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko, although it is not clear that either would endorse the title. More recently Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics. [1] Some consider the work of Paul Erdős as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g. via the Erdős number. Reuben Hersh has also promoted the social view of mathematics, calling it a 'humanistic' approach [2], similar to but not quite the same as that associated with Alvin White [3]; one of Hersh's co-authors, Philip J. Davis, has expressed sympathy for the social view as well. Imre Lakatos (November 9, 1922 – February 2, 1974) was a philosopher of mathematics and science. ... A. Thomas Tymoczko (1943-1996) was a philosopher specializing in logic and the philosophy of mathematics. ... Paul Ernest He is a recent contributor to the social constructivist philosophy of mathematics. ... Paul ErdÅ‘s, also ErdÅ‘s Pál, in English Paul Erdos or Paul Erdös (March 26, 1913 – September 20, 1996), was an immensely prolific (and famously eccentric) Hungarian-born mathematician. ... The ErdÅ‘s number, honouring the late Hungarian mathematician Paul ErdÅ‘s, one of the most prolific writers of mathematical papers, is a way of describing the collaborative distance, in regard to mathematical papers, between an author and ErdÅ‘s. ... Reuben Hersh (December 9, 1927 - ) is an American mathematician, now an emeritus professor of the University of New Mexico. ... Philip J. Davis is an American applied mathematician. ...

A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity. To observe that rigorous proof comes only after unrigorous conjecture, experimentation and speculation is true, but it is trivial and noone would deny this. So it's a bit of a stretch to characterize a philosophy of mathematics in this way, on something trivially true. The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrauss in order to rigorously prove the theorems thereof. There is nothing special or interesting about this, as it fits in with the more general trend of unrigorous ideas which are later made rigorous. There needs to be a clear distinction between the objects of study of mathematics and the study of the objects of study of mathematics. The former doesn't seem to change a great deal; the latter is forever in flux. The latter is what the Social theory is about, and the former is what Platonism et al. are about.

Beyond the traditional schools

Rather than focus on narrow debates about the true nature of mathematical truth, or even on practices unique to mathematicians such as the proof, a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner's famous 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences, in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain. A common dictionary definition of truth is agreement with fact or reality.[1] There is no single definition of truth about which the majority of philosophers agree. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... The 1960s decade refers to the years from January 1, 1960 to December 31, 1969, inclusive. ... For the band, see 1990s (band). ... Eugene Wigner Eugene Paul Wigner (Hungarian Wigner Pál Jenő) (November 17, 1902 – January 1, 1995) was a Hungarian physicist and mathematician who received the Nobel Prize in Physics in 1963 for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and... Year 1960 (MCMLX) was a leap year starting on Friday (link will display full calendar) of the Gregorian calendar. ... In 1960, the physicist Eugene Wigner published an article titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences, arguing that the way in which the mathematical structure of a physical theory often points the way to further advances in that theory and even to empirical predictions, is not a...

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

Quasi-empiricism

One parallel concern that does not actually challenge the schools directly but instead questions their focus is the notion of quasi-empiricism in mathematics. This grew from the increasingly popular assertion in the late 20th century that no one foundation of mathematics could be ever proven to exist. It is also sometimes called 'postmodernism in mathematics' although that term is considered overloaded by some and insulting by others. Quasi-empiricism argues that in doing their research, mathematicians test hypotheses as well as proving theorems. A mathematical argument can transmit falsity from the conclusion to the premises just as well as it can transmit truth from the premises to the conclusion. Quasi-empiricism was developed by Imre Lakatos, inspired by the philosophy of science of Karl Popper. Quasi-empiricism in mathematics is the movement in the philosophy of mathematics to direct philosophers attention to mathematical practice, in particular, relations with physics and social sciences, rather than the foundations problem in mathematics. ... The foundations problem in mathematics was the late 19th century and early 20th century term for the search for the simplest metamathematics. ... Quasi-empiricism refers to applying quasi-empirical methods and accepting their results as valid or true, eg, as in quasi-empiricism in mathematics. ... Imre Lakatos (November 9, 1922 – February 2, 1974) was a philosopher of mathematics and science. ... Sir Karl Raimund Popper, CH, FRS, FBA, (July 28, 1902 – September 17, 1994), was an Austrian and British[1] philosopher and a professor at the London School of Economics. ...

Lakatos's philosophy of mathematics is sometimes regarded as a kind of social constructivism, but this was not his intention. Imre Lakatos (November 9, 1922 – February 2, 1974) was a philosopher of mathematics and science. ...

Such methods have always been part of folk mathematics by which great feats of calculation and measurement are sometimes achieved. Indeed, such methods may be the only notion of proof a culture has. As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. ...

Hilary Putnam has argued that any theory of mathematical realism would include quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics - at perhaps a somewhat greater risk of failure of their calculations. He gave a detailed argument for this in New Directions (ed. Tymockzo, 1998). Hilary Whitehall Putnam (born July 31, 1926) is an American philosopher who has been a central figure in Western philosophy since the 1960s, especially in philosophy of mind, philosophy of language, and philosophy of science. ...

Action

Some practitioners and scholars who are not engaged primarily in proof-oriented approaches have suggested an interesting and important theory about the nature of mathematics. For example, Judea Pearl claimed that all of mathematics as presently understood was based on an algebra of seeing - and proposed an algebra of doing to complement it - this is a central concern of the philosophy of action and other studies of how knowledge relates to action. The most important output of this was new theories of truth, notably those appropriate to activism and grounding empirical methods. Judea Pearl is a computer science professor at UCLA. He was one of the pioneers of Bayesian networks and the probabilistic approach to artificial intelligence. ... Philosophy of action is chiefly concerned with human action, intending to distinguish between activity and passivity, voluntary, intentional, culpable and involuntary actions, and related question. ... This article needs additional references or sources for verification. ... Action, as a concept in philosophy, is what an agent can do, as for instance humans as agents can do. ... A common dictionary definition of truth is agreement with fact or reality.[1] There is no single definition of truth about which the majority of philosophers agree. ... Activism, in a general sense, can be described as intentional action to bring about social or political change. ... Empirical methods are the means by which scientists gather information about the world in order to develop theories. ...

Unification

Few philosophers are able to penetrate mathematical notations and culture to relate conventional notions of metaphysics to the more specialized metaphysical notions of the schools above. This may lead to a disconnection in which some mathematicians continue to profess discredited philosophy as a justification for their continued belief in a world-view promoting their work. Plato (Left) and Aristotle (right), by Raphael (Stanza della Segnatura, Rome) Metaphysics is the branch of philosophy concerned with explaining the ultimate nature of reality, being, and the world. ...

Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human perception and everyday understandings of knowledge. It has been suggested that Meta-epistemology be merged into this article or section. ... In psychology and the cognitive sciences, perception is the process of acquiring, interpreting, selecting, and organizing sensory information. ...<