HOME

TheInfoList



OR:

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The philosophy of mathematics has two major themes: mathematical realism and mathematical anti-realism.


History

The origin of mathematics is subject to arguments and disagreements. Whether the birth of mathematics was a random happening or induced by necessity during the development of other subjects, like physics, is still a matter of prolific debates. 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 Western philosophy encompasses the philosophical thought and work of the Western world. Historically, the term refers to the philosophical thinking of Western culture, beginning with the ancient Greek philosophy of the pre-Socratics. The word ' ...
and Eastern philosophy. Western philosophies of mathematics go as far back as
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...
, who described the theory "everything is mathematics" ( mathematicism),
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, who paraphrased Pythagoras, and studied the ontological status of mathematical objects, and
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
, who studied
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
and issues related to infinity (actual versus potential). Greek philosophy on mathematics was strongly influenced by their study of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
. For example, at one time, the Greeks held the opinion that 1 (one) was not a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
, 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 not "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". These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the square root of two.
Hippasus Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes c ...
, a disciple of
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...
, 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.
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning with Leibniz, the focus shifted strongly to the relationship between mathematics and logic. This perspective 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 centuries.


Contemporary philosophy

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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
, set theory (both naive set theory and axiomatic set theory), and foundational issues. It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive. Investigations into this issue are known as the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
program. 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
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Epis ...
and
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
. Three schools, formalism, intuitionism, and
logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...
, 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 Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and o ...
and
rigor Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as ma ...
that had been taken for granted. Each school addressed the issues that came to the fore at that time, either attempting to resolve them or claiming that mathematics is not entitled to its status as our most trusted knowledge. Surprising and counter-intuitive 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 Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
around 300 BCE as the natural basis for mathematics. Notions of axiom, proposition and proof, as well as the notion of a proposition being true of a mathematical object (see
Assignment Assignment, assign or The Assignment may refer to: * Homework * Sex assignment * The process of sending National Basketball Association players to its development league; see Computing * Assignment (computer science), a type of modification to ...
), were formalized, allowing them to be treated mathematically. The Zermelo–Fraenkel 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 Gödel numbering, propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into the
consistency In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
to call such study '' metamathematics'' or '' proof theory''. At the middle of the century, a new mathematical theory was created by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
and
Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...
, known as category theory, 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. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:
When philosophy discovers something wrong with science, sometimes science has to be changed—
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
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.
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.


Major themes


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;
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s, for example, are real entities, not the creations of the human mind. Many working mathematicians have been mathematical realists; they see themselves as discoverers of naturally occurring objects. Examples include Paul Erdős and Kurt Gödel. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but the
continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
conjecture might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture. Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them. Major forms of mathematical realism include
Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...
and
Aristotelianism Aristotelianism ( ) is a philosophical tradition inspired by the work of Aristotle, usually characterized by deductive logic and an analytic inductive method in the study of natural philosophy and metaphysics. It covers the treatment of the so ...
.


Mathematical anti-realism

Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities. Major forms of mathematical anti-realism include formalism and
fictionalism Fictionalism is the view in philosophy according to which statements that appear to be descriptions of the world should not be construed as such, but should instead be understood as cases of "make believe", of pretending to treat something as liter ...
.


Contemporary schools of thought


Artistic

The view that claims that mathematics is the aesthetic combination of assumptions, and then also claims that mathematics is an
art Art is a diverse range of human activity, and resulting product, that involves creative or imaginative talent expressive of technical proficiency, beauty, emotional power, or conceptual ideas. There is no generally agreed definition of wha ...
. A famous
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
who claims that is the British G. H. Hardy. For Hardy, in his book, '' A Mathematician's Apology'', the definition of mathematics was more like the aesthetic combination of concepts.


Platonism

Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. The term ''Platonism'' is used because such a view is seen to parallel
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
's
Theory of Forms The theory of Forms or theory of Ideas is a philosophical theory, fuzzy concept, or world-view, attributed to Plato, that the physical world is not as real or true as timeless, absolute, unchangeable ideas. According to this theory, ideas in th ...
and a "World of Ideas" (Greek: ''eidos'' (εἶδος)) described in Plato's
allegory of the cave The Allegory of the Cave, or Plato's Cave, is an allegory presented by the Greek philosopher Plato in his work ''Republic'' (514a–520a) to compare "the effect of education ( παιδεία) and the lack of it on our nature". It is written as ...
: 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 ''
Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
'' of ancient Greece, who believed that the world was, quite literally, generated by
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s. 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 world and discover truths about the entities? One proposed answer is the
Ultimate Ensemble In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory and struogony (from mathematical structure, Latin: struō), is a speculative "theory of everything" (TOE) proposed by cosmologist Max ...
, a theory that postulates that all structures that exist mathematically also exist physically in their own universe. Kurt 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 , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
said about mathematics, and supports
Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and aest ...
's idea that mathematics is synthetic '' a priori''.) Davis and Hersh have suggested in their 1999 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. Full-blooded Platonism 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
excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
, and the axiom of choice). 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. Set-theoretic realism (also set-theoretic Platonism) a position defended by Penelope Maddy, is the view that set theory is about a single universe of sets. This position (which is also known as naturalized Platonism because it is a naturalized version of mathematical Platonism) has been criticized by Mark Balaguer on the basis of
Paul Benacerraf Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. He ...
's epistemological problem. A similar view, termed Platonized naturalism, was later defended by the Stanford–Edmonton School: according to this view, a more traditional kind of Platonism is consistent with naturalism; the more traditional kind of Platonism they defend is distinguished by general principles that assert the existence of abstract objects.


Mathematicism

Max Tegmark Max Erik Tegmark (born 5 May 1967) is a Swedish-American physicist, cosmologist and machine learning researcher. He is a professor at the Massachusetts Institute of Technology and the president of the Future of Life Institute. He is also a scienti ...
's mathematical universe hypothesis (or mathematicism) goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does. Tegmark's sole postulate is: ''All structures that exist mathematically also exist physically''. That is, in the sense that "in those orldscomplex enough to contain self-aware substructures heywill subjectively perceive themselves as existing in a physically 'real' world".


Logicism

Logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...
is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic. Carnap, Rudolf (1931), "Die logizistische Grundlegung der Mathematik", ''Erkenntnis'' 2, 91-121. Republished, "The Logicist Foundations of Mathematics", E. Putnam and G.J. Massey (trans.), in Benacerraf and Putnam (1964). Reprinted, pp. 41–52 in Benacerraf and Putnam (1983). 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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
is the proper foundation of mathematics, and all mathematical statements are necessary
logical truth Logical truth is one of the most fundamental concepts in logic. Broadly speaking, a logical truth is a statement which is true regardless of the truth or falsity of its constituent propositions. In other words, a logical truth is a statement whic ...
s. Rudolf Carnap (1931) presents the logicist thesis in two parts: #The ''concepts'' of mathematics can be derived from logical concepts through explicit definitions. #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'' equals ''Ga''), a principle that he took to be acceptable as part of logic. Frege's construction was flawed.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
discovered that Basic Law V is inconsistent (this is
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
). 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 much of mathematics, such as the " axiom of reducibility". Even Russell said that this axiom did not really belong to logic. Modern logicists (like Bob Hale, Crispin Wright, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favor of abstraction principles such as
Hume's principle Hume's principle or HP says that the number of ''F''s is equal to the number of ''G''s if and only if there is a one-to-one correspondence (a bijection) between the ''F''s and the ''G''s. HP can be stated formally in systems of second-order logic. ...
(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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
). 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.


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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
(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, 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. Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: ''if'' one assigns meaning to the strings in such a way that the rules of the game become true (i.e., true statements are assigned to the axioms and the rules of inference are truth-preserving), ''then'' one must accept the theorem, or, rather, the interpretation one has given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism.) But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics. A major early proponent of formalism was David Hilbert, whose program was intended to be a complete and
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
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 arithmetic of the positive
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, 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 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 In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...
of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent. Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation. Other formalists, such as Rudolf Carnap,
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
, and
Haskell Curry Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic. While the initial concept of combinatory logic was based on a single paper by ...
, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists. Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories etc. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns. The "games" are usually not arbitrary. The main critique of formalism is that the actual mathematical ideas that occupy mathematicians are far removed from the string manipulation games mentioned above. Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view. Recently, some formalist mathematicians have proposed that all of our ''formal'' mathematical knowledge should be systematically encoded in computer-readable formats, so as to facilitate automated proof checking of mathematical proofs and the use of interactive theorem proving in the development of mathematical theories and computer software. Because of their close connection with
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition—see
QED project The QED manifesto was a proposal for a computer-based database of all mathematical knowledge, strictly formalized and with all proofs having been checked automatically. (Q.E.D. means in Latin, meaning "which was to be demonstrated.") Overview Th ...
for a general overview.


Conventionalism

The French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
Henri Poincaré was among the first to articulate a conventionalist view. Poincaré's use of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
in his work on
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s convinced him that
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
should not be regarded as ''a priori'' truth. He held that
axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.


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. 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 Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
, different from the classical Aristotelian logic; this logic does not contain the
law of the excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
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. 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 A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
or
computable function Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable if there exists an algorithm that can do ...
to fill this gap, leading to the claim that only questions regarding the behavior of finite
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
s are meaningful and should be investigated in mathematics. This has led to the study of the
computable number In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive ...
s, first introduced by
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...
. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
.


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 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 Errett Bishop, who managed to prove versions of the most important theorems in
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
as constructive analysis in his 1967 ''Foundations of Constructive Analysis.''


Finitism

Finitism Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are ...
is an extreme form of
constructivism Constructivism may refer to: Art and architecture * Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes * Constructivist architecture, an architectural movement in Russia in the 1920s a ...
, according to which a mathematical object does not exist unless it can be constructed from
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s in a
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
number of steps. In her book ''Philosophy of Set Theory'',
Mary Tiles Mary Tiles (born 1946) is a philosopher and historian of mathematics and science. From 2006 until 2009, she served as chair of the philosophy department of the University of Hawaii at Manoa. She retired in 2009. Life At Bristol University, Tiles ...
characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists. The most famous proponent of finitism was
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
, who said:
Ultrafinitism In the philosophy of mathematics, ultrafinitism (also known as ultraintuitionism,International Workshop on Logic and Computational Complexity, ''Logic and Computational Complexity'', Springer, 1995, p. 31. strict formalism,St. Iwan (2000),On the U ...
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 John Penn Mayberry in his book ''The Foundations of Mathematics in the Theory of Sets''. 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.


Structuralism

Structuralism is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their ''places'' in such structures, consequently having no intrinsic properties. 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
number line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra. Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what ''kind'' of entity a mathematical object is, not to what kind of ''existence'' mathematical objects or structures have (not, in other words, to their
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
). 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. The ''ante rem'' structuralism ("before the thing") has a similar ontology to
Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...
. 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 Benacerraf's identification problem). The ''in re'' structuralism ("in the thing") is the equivalent of Aristotelian realism. 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. The ''post rem'' structuralism ("after the thing") is anti-realist about structures in a way that parallels nominalism. Like nominalism, the ''post rem'' approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view mathematical ''systems'' 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.


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 A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
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 not discover, mathematics. The cognitive processes of pattern-finding and distinguishing objects are also subject to
neuroscience Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, developme ...
; if mathematics is considered to be relevant to a natural world (such as from realism or a degree of it, as opposed to pure
solipsism Solipsism (; ) is the philosophical idea that only one's mind is sure to exist. As an epistemological position, solipsism holds that knowledge of anything outside one's own mind is unsure; the external world and other minds cannot be known a ...
). Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested the
evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
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
illusion An illusion is a distortion of the senses, which can reveal how the mind normally organizes and interprets sensory stimulation. Although illusions distort the human perception of reality, they are generally shared by most people. Illusions may oc ...
, assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, and
hallucination A hallucination is a perception in the absence of an external stimulus that has the qualities of a real perception. Hallucinations are vivid, substantial, and are perceived to be located in external objective space. Hallucination is a combinati ...
s). As such, this may also raise questions for the modern
scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific ...
for its compatibility with general mathematics; as while relatively reliable, it is still limited by what can be measured by empiricism which may not be as reliable as previously assumed (see also: 'counterintuitive' concepts ins science such as quantum nonlocality, and
action at a distance In physics, action at a distance is the concept that an object can be affected without being physically touched (as in mechanical contact) by another object. That is, it is the non-local interaction of objects that are separated in space. Non- ...
). Another issue is that one
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbo ...
may not necessarily be applicable to problem solving. Subjects such as
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s or imaginary numbers require specific changes to more commonly used axioms of mathematics; otherwise they cannot be adequately understood. Alternatively, computer programmers may use hexadecimal for its 'human-friendly' representation of binary-coded values, rather than decimal (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
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
). As
perception Perception () is the organization, identification, and interpretation of sensory information in order to represent and understand the presented information or environment. All perception involves signals that go through the nervous system ...
s from the human brain are subject to
illusion An illusion is a distortion of the senses, which can reveal how the mind normally organizes and interprets sensory stimulation. Although illusions distort the human perception of reality, they are generally shared by most people. Illusions may oc ...
s, assumptions, deceptions, (induced)
hallucination A hallucination is a perception in the absence of an external stimulus that has the qualities of a real perception. Hallucinations are vivid, substantial, and are perceived to be located in external objective space. Hallucination is a combinati ...
s, cognitive errors or assumptions in a general context, it can be questioned whether they are accurate or strictly indicative of truth (see also: philosophy of being), and the nature of empiricism itself in relation to the universe and whether it is independent to the senses and the universe. The human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition. Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this universe. The most accessible, famous, and infamous treatment of this perspective is '' Where Mathematics Comes From'', by George Lakoff and Rafael E. Núñez. In addition, mathematician
Keith Devlin Keith J. Devlin (born 16 March 1947) is a British mathematician and popular science writer. Since 1987 he has lived in the United States. He has dual British-American citizenship.
has investigated similar concepts with his book '' The Math Instinct'', as has neuroscientist Stanislas Dehaene with his book ''The Number Sense''. For more on the philosophical ideas that inspired this perspective, see cognitive science of mathematics.


Aristotelian realism

Aristotelian realism 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 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 James Franklin and th
Sydney School
in the philosophy of mathematics and is close to the view of Penelope Maddy that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world). Maddy, Penelope (1990), ''Realism in Mathematics'', Oxford University Press, Oxford, UK. A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world. The Euclidean arithmetic developed by John Penn Mayberry in his book ''The Foundations of Mathematics in the Theory of Sets'' 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 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.


Psychologism

Psychologism Psychologism is a family of philosophical positions, according to which certain psychological facts, laws, or entities play a central role in grounding or explaining certain non-psychological facts, laws, or entities. The word was coined by Johan ...
in the philosophy of mathematics is the position that
mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws). John Stuart Mill seems to have been an advocate of a type of logical psychologism, as were many 19th-century German logicians such as Sigwart and Erdmann as well as a number of
psychologists A psychologist is a professional who practices psychology and studies mental states, perceptual, cognitive, emotional, and social processes and behavior. Their work often involves the experimentation, observation, and interpretation of how indi ...
, past and present: for example,
Gustave Le Bon Charles-Marie Gustave Le Bon (; 7 May 1841 – 13 December 1931) was a leading French polymath whose areas of interest included anthropology, psychology, sociology, medicine, invention, and physics. He is best known for his 1895 work '' The Crowd ...
. Psychologism was famously criticized by Frege in his '' The Foundations of Arithmetic'', and many of his works and essays, including his review of
Husserl , thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations) , thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view , thesis1_year = 1883 , thesis2_title ...
's '' Philosophy of Arithmetic''. Edmund Husserl, in the first volume of his '' Logical Investigations'', 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 Charles Sanders Peirce and
Maurice Merleau-Ponty Maurice Jean Jacques Merleau-Ponty. (; 14 March 1908 – 3 May 1961) was a French phenomenological philosopher, strongly influenced by Edmund Husserl and Martin Heidegger. The constitution of meaning in human experience was his main interest an ...
.


Empiricism

Mathematical 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, according to critics, such as A.J. Ayer, it makes statements like come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet. Karl Popper 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 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." Contemporary mathematical empiricism, formulated by W. V. O. Quine and Hilary Putnam, is primarily supported by the indispensability argument: mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. That is, since physics needs to talk about
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s to say why light bulbs behave as they do, then electrons must exist. Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of being distinct from the other sciences. Putnam strongly rejected the term "
Platonist Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at l ...
" as implying an over-specific
ontology In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
that was not necessary to
mathematical practice Mathematical practice comprises the working practices of professional mathematicians: selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking peer re ...
in any real sense. He advocated a form of "pure realism" that rejected mystical notions of
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belie ...
and accepted much 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 prove 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. 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''. Quasi-empiricism was also developed by
Imre Lakatos Imre Lakatos (, ; hu, Lakatos Imre ; 9 November 1922 – 2 February 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pr ...
. The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e.
consilience In science and history, consilience (also convergence of evidence or concordance of evidence) is the principle that evidence from independent, unrelated sources can "converge" on strong conclusions. That is, when multiple sources of evidence are ...
after E.O. Wilson. 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. 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. For experimental evidence suggesting that human infants can do elementary arithmetic, see Brian Butterworth.


Fictionalism

Mathematical fictionalism was brought to fame in 1980 when Hartry Field published ''Science Without Numbers'', which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics with no reference to numbers or functions at all. He started with the "betweenness" of
Hilbert's axioms Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book ''Grundlagen der Geometrie'' (tr. ''The Foundations of Geometry'') as the foundation for a modern treatment of Euclidean geometry. Other well-known modern ax ...
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. Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of useful fiction. He showed that mathematical physics is a
conservative extension In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superthe ...
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 see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like is just as fictitious as " Sherlock Holmes lived at 221B Baker Street"—but both are true according to the relevant fictions. Another fictionalist, Mary Leng, 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. 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 In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
to carry out his reduction, and because the statement of conservativity seems to require quantification over abstract models or deductions.


Social constructivism

Social constructivism sees mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints—the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated—that work to conserve the historically-defined discipline. This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as
mathematical practice Mathematical practice comprises the working practices of professional mathematicians: selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking peer re ...
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 overemphasis on axiomatic proof and peer review as practices. The social nature of mathematics is highlighted in its subcultures. Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Social constructivists argue each speciality forms its own
epistemic community An epistemic community is a network of knowledge-based experts who help decision-makers to define the problems they face, identify various policy solutions and assess the policy outcomes. The definitive conceptual framework of an epistemic communit ...
and often has great difficulty communicating, or motivating the investigation of
unifying conjecture There have been several attempts in history to reach a unified theory of mathematics. Some of the most respected mathematicians in the academia have expressed views that the whole subject should be fitted into one theory. Historical perspective ...
s that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias, or of mathematicians' collective intelligence as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless. Contributions to this school have been made by
Imre Lakatos Imre Lakatos (, ; hu, Lakatos Imre ; 9 November 1922 – 2 February 1974) was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its "methodology of proofs and refutations" in its pr ...
and Thomas Tymoczko, although it is not clear that either would endorse the title. More recently Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics. Some consider the work of Paul Erdős as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g., via the
Erdős number The Erdős number () describes the "collaborative distance" between mathematician Paul Erdős and another person, as measured by authorship of mathematical papers. The same principle has been applied in other fields where a particular individual ...
.
Reuben Hersh Reuben Hersh (December 9, 1927 – January 3, 2020) was an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. Although he was generally known as Reuben Hersh, late in life h ...
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, Philip J. Davis, has expressed sympathy for the social view as well.


Beyond the traditional schools


Unreasonable effectiveness

Rather than focus on narrow debates about the true nature of mathematical
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belie ...
, 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 Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his co ...
'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.


Popper's two senses of number statements

Realist and constructivist theories are normally taken to be contraries. However, Karl Popper argued that a number statement such as 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 on realist lines.


Philosophy of language

Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said, the ''language'' of science. Although some mathematicians and philosophers would accept the statement " mathematics is a language", linguists believe that the implications of such a statement must be considered. For example, the tools of
linguistics Linguistics is the science, scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure ...
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 natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague and other linguists working in formal semantics to show that the distinction between mathematical language and natural language may not be as great as it seems. 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 tense), but many of the same analytical tools can be used (such as context-free grammars). One important difference is that mathematical objects have clearly defined
types Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type In computer science and computer programming, a data type (or simply type) is a set of possible values and a set of allo ...
, 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 part of speech in another; and this operation has no analogue in natural language."


Arguments


Indispensability argument for realism

This argument, associated with
Willard Quine Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...
and Hilary Putnam, is considered by Stephen Yablo 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. #One must have
ontological In metaphysics, ontology is the philosophical study of being, as well as related concepts such as existence, becoming, and reality. Ontology addresses questions like how entities are grouped into categories and which of these entities exi ...
commitments to ''all'' entities that are indispensable to the best scientific theories, and to those entities ''only'' (commonly referred to as "all and only"). #Mathematical entities are indispensable to the best scientific theories. Therefore, #One must have ontological commitments to mathematical entities.Putnam, H. ''Mathematics, Matter and Method. Philosophical Papers, vol. 1''. Cambridge: Cambridge University Press, 1975. 2nd. ed., 1985. The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism 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 confirmation holism. 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
nominalist In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are at least two main versions of nominalism. One version denies the existence of universalsthings th ...
who wishes to exclude the existence of sets and
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
, but to include the existence of quarks and other undetectable entities of physics, for example, in a difficult position.


Epistemic argument against realism

The anti-realist " epistemic argument" against Platonism has been made by
Paul Benacerraf Paul Joseph Salomon Benacerraf (; born 26 March 1931) is a French-born American philosopher working in the field of the philosophy of mathematics who taught at Princeton University his entire career, from 1960 until his retirement in 2007. He ...
and Hartry Field. Platonism posits that mathematical objects are '' abstract'' entities. By general agreement, abstract entities cannot interact
causal Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the ca ...
ly 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
perceive Perception () is the organization, identification, and interpretation of sensory information in order to represent and understand the presented information or environment. All perception involves signals that go through the nervous system ...
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 proofs, etc., which is already fully accountable in terms of physical processes in their brains. Field developed his views into
fictionalism Fictionalism is the view in philosophy according to which statements that appear to be descriptions of the world should not be construed as such, but should instead be understood as cases of "make believe", of pretending to treat something as liter ...
. Benacerraf also developed the philosophy of mathematical structuralism, according to which there are no mathematical objects. Nonetheless, some versions of structuralism are compatible with some versions of realism. The argument hinges on the idea that a satisfactory naturalistic 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
intuition Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
that involves contact with the Platonic realm. A modern form of this argument is given by Sir Roger Penrose.Review
of The Emperor's New Mind.
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 Jerrold Katz in his 2000 book ''
Realistic Rationalism In analytic philosophy, anti-realism is a position which encompasses many varieties such as metaphysical, mathematical, semantic, scientific, moral and epistemic. The term was first articulated by British philosopher Michael Dummett in an argument ...
''. A more radical defense is denial of physical reality, i.e. the mathematical universe hypothesis. In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.


Aesthetics

Many practicing mathematicians have been drawn to their subject because of a sense of
beauty Beauty is commonly described as a feature of objects that makes these objects pleasurable to perceive. Such objects include landscapes, sunsets, humans and works of art. Beauty, together with art and taste, is the main subject of aesthetics, o ...
they perceive in it. One sometimes hears the sentiment that mathematicians would like to leave philosophy to the philosophers and get back to mathematics—where, presumably, the beauty lies. In his work on the divine proportion, H.E. Huntley relates the feeling of reading and understanding someone else's proof of a theorem of mathematics to that of a viewer of a masterpiece of art—the reader of a proof has a similar sense of exhilaration at understanding as the original author of the proof, much as, he argues, the viewer of a masterpiece has a sense of exhilaration similar to the original painter or sculptor. Indeed, one can study mathematical and scientific writings as
literature Literature is any collection of written work, but it is also used more narrowly for writings specifically considered to be an art form, especially prose fiction, drama, and poetry. In recent centuries, the definition has expanded to include ...
. Philip J. Davis and
Reuben Hersh Reuben Hersh (December 9, 1927 – January 3, 2020) was an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. Although he was generally known as Reuben Hersh, late in life h ...
have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of . The first is the traditional proof by
contradiction In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
, ascribed to
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
; the second is a more direct proof involving the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
that, they argue, gets to the heart of the issue. Davis and Hersh argue that mathematicians find the second proof more aesthetically appealing because it gets closer to the nature of the problem. Paul Erdős was well known for his notion of a hypothetical "Book" containing the most elegant or beautiful mathematical proofs. There is not universal agreement that a result has one "most elegant" proof;
Gregory Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-t ...
has argued against this idea. Philosophers have sometimes criticized mathematicians' sense of beauty or elegance as being, at best, vaguely stated. By the same token, however, philosophers of mathematics have sought to characterize what makes one proof more desirable than another when both are logically sound. Another aspect of aesthetics concerning mathematics is mathematicians' views towards the possible uses of mathematics for purposes deemed unethical or inappropriate. The best-known exposition of this view occurs in G. H. Hardy's book '' A Mathematician's Apology'', in which Hardy argues that pure mathematics is superior in beauty to
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
precisely because it cannot be used for war and similar ends.


Journals


''Philosophia Mathematica'' journal''The Philosophy of Mathematics Education Journal'' homepage


See also

*
Definitions of mathematics Mathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions. All proposed definitions are controversial in their own ways. Early definitions Pythag ...
*
Formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sy ...
*
Foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
*
Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
* Model theory *
Non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
* Philosophy of language *
Philosophy of logic Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application ...
*
Philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ult ...
*
Philosophy of physics In philosophy, philosophy of physics deals with conceptual and interpretational issues in modern physics, many of which overlap with research done by certain kinds of theoretical physicists. Philosophy of physics can be broadly divided into thr ...
* Philosophy of probability * Rule of inference *
Science studies Science studies is an interdisciplinary research area that seeks to situate scientific expertise in broad social, historical, and philosophical contexts. It uses various methods to analyze the production, representation and reception of scient ...
*
Scientific method The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientific ...


Related works

*'' The Analyst'' *Euclid's ''Elements'' *"
On Formally Undecidable Propositions of Principia Mathematica and Related Systems "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical logic by Kurt Gödel. Submitted November ...
" *" On Computable Numbers, with an Application to the Entscheidungsproblem" *'' Introduction to Mathematical Philosophy'' *" New Foundations for Mathematical Logic" *''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' *'' The Simplest Mathematics''


Historical topics

*
History and philosophy of science The history and philosophy of science (HPS) is an academic discipline that encompasses the philosophy of science and the history of science. Although many scholars in the field are trained primarily as either historians or as philosophers, there ...
*
History of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
* History of philosophy


Notes


Further reading

*
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
, " Prior Analytics", Hugh Tredennick (trans.), pp. 181–531 in ''Aristotle, Volume 1'', Loeb Classical Library, William Heinemann, London, UK, 1938. * * Berkeley, George (1734), '' The Analyst; or, a Discourse Addressed to an Infidel Mathematician. Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith'', London & Dublin. Online text, David R. Wilkins (ed.)
Eprint
* * *Colyvan, Mark (2004), "Indispensability Arguments in the Philosophy of Mathematics", ''Stanford Encyclopedia of Philosophy'', Edward N. Zalta (ed.)
Eprint
* * * * * * * * * * * / VIP,
キャッシング対策局【審査・在籍確認・増額・おまとめ・借り換え】
* * * * * * Reprinted, "On the Foundations of Set Theory and the Continuum Problem",
Stefan Bauer-Mengelberg Stefan may refer to: * Stefan (given name) * Stefan (surname) * Ștefan, a Romanian given name and a surname * Štefan, a Slavic given name and surname * Stefan (footballer) (born 1988), Brazilian footballer * Stefan Heym, pseudonym of German writ ...
(trans.), pp. 145–149 in Jean van Heijenoort (ed., 1967). * * * * * * * * * Mount, Matthew, ''Classical Greek Mathematical Philosophy'', . * * Peirce, Benjamin (1870), "Linear Associative Algebra", § 1. See * Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1-6,
Charles Hartshorne Charles Hartshorne (; June 5, 1897 – October 9, 2000) was an American philosopher who concentrated primarily on the philosophy of religion and metaphysics, but also contributed to ornithology. He developed the neoclassical idea of God and ...
and Paul Weiss (eds.), vols. 7-8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931 – 1935, 1958. Cited as CP (volume).(paragraph). *Peirce, C.S., various pieces on mathematics and logic, many readable online through links at the
Charles Sanders Peirce bibliography This Charles Sanders Peirce bibliography consolidates numerous references to the writings of Charles Sanders Peirce, including letters, manuscripts, publications, and . For an extensive chronological list of Peirce's works (titled in English), s ...
, especially under Books authored or edited by Peirce, published in his lifetime and the two sections following it. *Plato, "The Republic, Volume 1", Paul Shorey (trans.), pp. 1–535 in ''Plato, Volume 5'', Loeb Classical Library, William Heinemann, London, UK, 1930. *Plato, "The Republic, Volume 2", Paul Shorey (trans.), pp. 1–521 in ''Plato, Volume 6'', Loeb Classical Library, William Heinemann, London, UK, 1935. * * * * * * * * * * Reprinted in * * * * * *


External links

* * * * *
''Mathematical Structuralism'', Internet Encyclopaedia of Philosophy
*
''Abstractionism'', Internet Encyclopaedia of Philosophy
** *Th
London Philosophy Study Guide
offers many suggestions on what to read, depending on the student's familiarity with the subject:







* * * {{DEFAULTSORT:Philosophy Of Mathematics