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Philosophy of mathematics is the branch ofphilosophy that deals with the nature ofmathematics and its relationship to other areas of philosophy, particularlyepistemology andmetaphysics. Central questions posed include whether or not mathematical objects are purely abstract entities or are in some way concrete, and in what the relationship such objects have with physical reality consists.[1]
Major themes that are dealt with in philosophy of mathematics include:
Reality: The question is whether mathematics is a pure product of human mind or whether it has some reality by itself.
Logic and rigor
Relationship with physical reality
Relationship with science
Relationship with applications
Mathematical truth
Nature as human activity (science,art,game, or all together)
The connection between mathematics and material reality has led to philosophical debates since at least the time ofPythagoras. The ancient philosopherPlato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to asPlatonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects.[2]
Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.[4] Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with afeeling of an objective existence, of a reality of mathematics ...
Nevertheless, Platonism and the concurrent views on abstraction do not explain theunreasonable effectiveness of mathematics (as Platonism assumes mathematics exists independently, but does not explain why it matches reality).[5]
Mathematical reasoning requiresrigor. This means that the definitions must be absolutely unambiguous and theproofs must be reducible to a succession of applications ofsyllogisms orinference rules,[a] without any use of empirical evidence andintuition.[b][6]
The rules of rigorous reasoning have been established by theancient Greek philosophers under the name oflogic. Logic is not specific to mathematics, but, in mathematics, the standard of rigor is much higher than elsewhere.
For many centuries, logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.[7] Circa the end of the 19th century, severalparadoxes made questionable the logical foundation of mathematics, and consequently the validity of the whole of mathematics. This has been called thefoundational crisis of mathematics. Some of these paradoxes consist of results that seem to contradict the common intuition, such as the possibility to construct validnon-Euclidean geometries in which theparallel postulate is wrong, theWeierstrass function that iscontinuous but nowheredifferentiable, and the study byGeorg Cantor ofinfinite sets, which led to consider several sizes of infinity (infinitecardinals). Even more striking,Russell's paradox shows that the phrase "the set of all sets" is self contradictory.
Several methods have been proposed to solve the problem by changing of logical framework, such asconstructive mathematics andintuitionistic logic. Roughly speaking, the first one consists of requiring that every existence theorem must provide an explicit example, and the second one excludes from mathematical reasoning thelaw of excluded middle anddouble negation elimination.
These logics have less inference rules than classical logic. On the other hand classical logic was afirst-order logic, which means roughly thatquantifiers cannot be applied to infinite sets. This means, for example that the sentence "every set ofnatural numbers has a least element" is nonsensical in any formalization of classical logic. This led to the introduction ofhigher-order logics, which are presently used commonly in mathematics.
The problems offoundation of mathematics has been eventually resolved with the rise ofmathematical logic as a new area of mathematics. In this framework, a mathematical orlogical theory consists of aformal language that defines thewell-formed of assertions, a set of basic assertions calledaxioms and a set ofinference rules that allow producing new assertions from one or several known assertions. Atheorem of such a theory is either an axiom or an assertion that can be obtained from previously known theorems by the application of an inference rule. TheZermelo–Fraenkel set theory with theaxiom of choice, generally calledZFC, is a higher-order logic in which all mathematics have been restated; it is used implicitely in all mathematics texts that do not specify explicitly on which foundations they are based. Moreover, the other proposed foundations can be modeled and studied inside ZFC.
It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply apleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof. In particular, proofs are rarely written in full details, and some steps of a proof are generally considered astrivial,easy, orstraightforward, and therefore left to the reader. As most proof errors occur in these skipped steps, a new proof requires to be verified by other specialists of the subject, and can be considered as reliable only after having been accepted by the community of the specialists, which may need several years.[8]
Also, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.[9]
Mathematics is used in mostsciences formodeling phenomena, which then allows predictions to be made from experimental laws.[10] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model.[11] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used.[12] For example, theperihelion precession of Mercury could only be explained after the emergence ofEinstein'sgeneral relativity, which replacedNewton's law of gravitation as a better mathematical model.[13]
There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it isfalsifiable, which means in mathematics that if a result or a theory is wrong, this can be proved by providing acounterexample. Similarly as in science,theories and results (theorems) are often obtained fromexperimentation.[14] In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation).[15] However, some authors emphasize that mathematics differs from the modern notion of science by notrelying on empirical evidence.[16][17][18][19]
Theunreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicistEugene Wigner.[20] It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.[21] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three andmanifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century,Albert Einstein developed thetheory of relativity that uses fundamentally these concepts. In particular,spacetime ofspecial relativity is a non-Euclidean space of dimension four, and spacetime ofgeneral relativity is a (curved) manifold of dimension four.[24][25]
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of thepositron and thebaryon In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknownparticle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.[26][27][28]
Pythagoras is considered the father of mathematics and geometry as he set the foundation forEuclid andEuclidean geometry. Pythagoras was the founder ofPythagoreanism: a mathematical and philosophical model to map the universe.
The origin of mathematics is of arguments and disagreements. Whether the birth of mathematics was by chance or induced by necessity during the development of similar subjects, such as physics, remains an area of contention.[29][30]
Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some[who?] 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 bothWestern philosophy andEastern philosophy. Western philosophies of mathematics go as far back asPythagoras, who described the theory "everything is mathematics" (mathematicism),Plato, who paraphrased Pythagoras, and studied theontological status of mathematical objects, andAristotle, who studiedlogic and issues related toinfinity (actual versus potential).
Greek philosophy on mathematics was strongly influenced by their study ofgeometry. For example, at one time, the Greeks held the opinion that 1 (one) was not anumber, but rather a unit of arbitrary length. A number was defined as a multitude. Therefore, 3, for example, represented a certain multitude of units, and was thus "truly" a number. At another point, a similar argument was made that 2 was not a number but a fundamental notion of a pair. These views come from the heavily geometric straight-edge-and-compass viewpoint of the Greeks: just as lines drawn in a geometric problem are measured in proportion to the first arbitrarily drawn line, so too are the numbers on a number line measured in proportion to the arbitrary first "number" or "one".[citation needed]
These earlier Greek ideas of numbers were later upended by the discovery of theirrationality of the square root of two.Hippasus, a disciple ofPythagoras, 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.[31]Simon Stevin was one of the first in Europe to challenge Greek ideas in the 16th century. Beginning withLeibniz, the focus shifted strongly to the relationship between mathematics and logic. This perspective dominated the philosophy of mathematics through the time ofBoole,Frege andRussell, but was brought into question by developments in the late 19th and early 20th centuries.
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 informal logic,set theory (bothnaive set theory andaxiomatic 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 thefoundations of mathematics 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 mathematicalepistemology andontology. Three schools,formalism,intuitionism, andlogicism, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, andanalysis in particular, did not live up to the standards ofcertainty andrigor 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 thefoundations 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 ofEuclid around 300 BCE as the natural basis for mathematics. Notions ofaxiom,proposition andproof, as well as the notion of a proposition being true of a mathematical object(seeAssignment), were formalized, allowing them to be treated mathematically. TheZermelo–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. WithGödel numbering, propositions could be interpreted as referring to themselves or other propositions, enabling inquiry into theconsistency of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" ledHilbert to call such studymetamathematics orproof theory.[32]
At the middle of the century, a new mathematical theory was created bySamuel Eilenberg andSaunders Mac Lane, known ascategory theory, and it became a new contender for the natural language of mathematical thinking.[33] 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 comes to mind, as doesBerkeley's attack on the actualinfinitesimal—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.[34]: 169–170
Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.
Contemporary schools of thought in the philosophy of mathematics include: artistic, Platonism, mathematicism, logicism, formalism, conventionalism, intuitionism, constructivism, finitism, structuralism, embodied mind theories (Aristotelian realism, psychologism, empiricism), fictionalism, social constructivism, and non-traditional schools.
However, many of these schools of thought are mutually compatible. For example, most living mathematicians are together Platonists and formalists, give a great importance toaesthetic, and consider that axioms should be chosen for the results they produce, not for their coherence with human intuition of reality (conventionalism).[35]
The view that claims thatmathematics is the aesthetic combination of assumptions, and then also claims that mathematics is anart. A famousmathematician who claims that is the BritishG. H. Hardy.[36] For Hardy, in his book,A Mathematician's Apology, the definition of mathematics was more like the aesthetic combination of concepts.[37]
Mathematical Platonism is the form ofrealism that suggests thatmathematical 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.
Max Tegmark'smathematical universe hypothesis (ormathematicism) 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 [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world".[38][39]
Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic.[40]: 41 Logicists hold that mathematics can be knowna priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thusanalytic, not requiring any special faculty of mathematical intuition. In this view,logic is the proper foundation of mathematics, and all mathematical statements are necessarylogical truths.
Theconcepts of mathematics can be derived from logical concepts through explicit definitions.
Thetheorems of mathematics can be derived from logical axioms through purely logical deduction.
Gottlob Frege was the founder of logicism. In his seminalDie Grundgesetze der Arithmetik (Basic Laws of Arithmetic) he built uparithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for conceptsF andG, the extension ofF equals the extension ofG if and only if for all objectsa,Fa equalsGa), a principle that he took to be acceptable as part of logic.
Frege's construction was flawed.Bertrand Russell discovered that Basic Law V is inconsistent (this isRussell's paradox). Frege abandoned his logicist program soon after this, but it was continued by Russell andWhitehead. They attributed the paradox to "vicious circularity" and built up what they calledramified 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 (likeBob 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 asHume's principle (the number of objects falling under the conceptF equals the number of objects falling under the conceptG if and only if the extension ofF and the extension ofG can be put intoone-to-one correspondence). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" ofEuclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that thePythagorean 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 known asdeductivism.[41] In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one, if it follows deductively from the appropriate axioms. The same is held to be true for all other mathematical statements.
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 tostructuralism.) 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 wasDavid Hilbert, whoseprogram was intended to be acomplete andconsistent axiomatization of all of mathematics.[42] Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usualarithmetic of the positiveintegers, 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 ofGödel'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 anyaxiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.
Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.
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[who?] formalist mathematicians have proposed that all of ourformal mathematical knowledge should be systematically encoded incomputer-readable formats, so as to facilitateautomated proof checking of mathematical proofs and the use ofinteractive theorem proving in the development of mathematical theories and computer software. Because of their close connection withcomputer science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition—seeQED project for a general overview.
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 thea priori forms of the volitions that inform the perception of empirical objects.[43]
In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts ofTuring machine orcomputable function to fill this gap, leading to the claim that only questions regarding the behavior of finitealgorithms are meaningful and should be investigated in mathematics. This has led to the study of thecomputable numbers, first introduced byAlan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoreticalcomputer science.
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 byErrett Bishop, who managed to prove versions of the most important theorems inreal analysis asconstructive analysis in his 1967Foundations of Constructive Analysis.[44]
Finitism is an extreme form ofconstructivism, according to which a mathematical object does not exist unless it can be constructed fromnatural numbers in afinite number of steps. In her bookPhilosophy of Set Theory,Mary Tiles characterized those who allowcountably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists.
God created the natural numbers, all else is the work of man.
Ultrafinitism 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 byJohn Penn Mayberry in his bookThe Foundations of Mathematics in the Theory of Sets.[46] 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 is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by theirplaces in such structures, consequently having nointrinsic 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, thenumber line. Other examples of mathematical objects might includelines andplanes in geometry, or elements and operations inabstract algebra.
Structuralism is anepistemologicallyrealistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to whatkind of entity a mathematical object is, not to what kind ofexistence mathematical objects or structures have (not, in other words, to theirontology). 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.[47]
Theante rem structuralism ("before the thing") has a similar ontology toPlatonism. 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(seeBenacerraf's identification problem).
Thein re structuralism ("in the thing") is the equivalent ofAristotelian 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.
Thepost rem structuralism ("after the thing") isanti-realist about structures in a way that parallelsnominalism. Like nominalism, thepost rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view mathematicalsystems 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 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 ofnumber 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 toneuroscience; if mathematics is considered to be relevant to a natural world (such as fromrealism or a degree of it, as opposed to puresolipsism).
Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested theevolution 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 asillusion, assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, andhallucinations). As such, this may also raise questions for the modernscientific method for its compatibility with general mathematics; as while relatively reliable, it is still limited by what can be measured byempiricism which may not be as reliable as previously assumed (see also: 'counterintuitive' concepts in such asquantum nonlocality, andaction at a distance).
Another issue is that onenumeral system may not necessarily be applicable to problem solving. Subjects such ascomplex numbers orimaginary numbers require specific changes to more commonly used axioms of mathematics; otherwise they cannot be adequately understood.
Alternatively, computer programmers may usehexadecimal for its 'human-friendly' representation ofbinary-coded values, rather thandecimal (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 ofzero).
Asperceptions from the human brain are subject toillusions, assumptions, deceptions, (induced)hallucinations, 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 ofempiricism 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 asEuler's identity are true then they are true as a map of the human mind andcognition.
Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this universe.
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.[48][49] Aristotelian realism is defended byJames Franklin and theSydney School in the philosophy of mathematics and is close to the view ofPenelope 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).[50] 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 byJohn Penn Mayberry in his bookThe Foundations of Mathematics in the Theory of Sets[46] 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 in the philosophy of mathematics is the position thatmathematicalconcepts 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 asSigwart andErdmann as well as a number ofpsychologists, past and present: for example,Gustave Le Bon. Psychologism was famously criticized byFrege in hisThe Foundations of Arithmetic, and many of his works and essays, including his review ofHusserl'sPhilosophy of Arithmetic. Edmund Husserl, in the first volume of hisLogical 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 byCharles Sanders Peirce andMaurice Merleau-Ponty.
Mathematical empiricism is a form of realism that denies that mathematics can be knowna priori at all. It says that we discover mathematical facts byempirical 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 wasJohn Stuart Mill. Mill's view was widely criticized, because, according to critics, such as A.J. Ayer,[51] it makes statements like"2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.
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."[52] Popper also noted he would "admit a system as empirical or scientific only if it is capable of being tested by experience."[53]
Contemporary mathematical empiricism, formulated byW. V. O. Quine andHilary Putnam, is primarily supported by theindispensability 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 aboutelectrons to say why light bulbs behave as they do, then electrons mustexist. 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" as implying an over-specificontology that was not necessary tomathematical practice in any real sense. He advocated a form of "pure realism" that rejected mystical notions oftruth and accepted muchquasi-empiricism in mathematics. This grew from the increasingly popular assertion in the late 20th century that no onefoundation 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 inNew Directions.[54] Quasi-empiricism was also developed byImre Lakatos.
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 theempirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e.consilience afterE.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 seePenelope Maddy'sRealism in Mathematics. Another example of a realist theory is theembodied mind theory.
For experimental evidence suggesting that human infants can do elementary arithmetic, seeBrian Butterworth.
Mathematical fictionalism was brought to fame in 1980 whenHartry Field publishedScience Without Numbers,[55] 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 ofNewtonian mechanics with no reference to numbers or functions at all. He started with the "betweenness" ofHilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done byvector 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 ofuseful fiction. He showed that mathematical physics is aconservative extension 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"2 + 2 = 4" 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.[56]
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 aboutfiction in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments ofsecond-order logic to carry out his reduction, and because the statement of conservativity seems to requirequantification over abstract models or deductions.[citation needed]
Social constructivism sees mathematics primarily as asocial 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: asmathematical practice evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, andfolk mathematics not enough, due to an overemphasis on axiomatic proof and peer review as practices.
The social nature of mathematics is highlighted in itssubcultures. 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 ownepistemic community and often has great difficulty communicating, or motivating the investigation ofunifying conjectures that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human'scognitive 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 byImre Lakatos andThomas Tymoczko, although it is not clear that either would endorse the title.[clarification needed] More recentlyPaul Ernest has explicitly formulated a social constructivist philosophy of mathematics.[57] Some consider the work ofPaul 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 theErdős number.Reuben Hersh has also promoted the social view of mathematics, calling it a "humanistic" approach,[58] similar to but not quite the same as that associated with Alvin White;[59] one of Hersh's co-authors,Philip J. Davis, has expressed sympathy for the social view as well.
Rather than focus on narrow debates about the true nature of mathematicaltruth, or even on practices unique to mathematicians such as theproof, 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 wasEugene Wigner's famous 1960 paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.
Realist and constructivist theories are normally taken to be contraries. However,Karl Popper[60] argued that a number statement such as"2 apples + 2 apples = 4 apples" 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.[61]
Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said,[citation needed] thelanguage of science. Although some[who?] mathematicians and philosophers would accept the statement "mathematics is a language" (most consider that thelanguage of mathematics is a part of mathematics to which mathematics cannot be reduced),[citation needed] linguists[who?] believe that the implications of such a statement must be considered. For example, the tools oflinguistics 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 fromnatural 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 studentRichard Montague and other linguists working informal 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.[62] Ganesalingam notes that some features of natural language are not necessary when analysing mathematical language (such astense), but many of the same analytical tools can be used (such ascontext-free grammars). One important difference is that mathematical objects have clearly definedtypes, which can be explicitly defined in a text: "Effectively, we are allowed to introduce a word in one part of a sentence, and declare itspart of speech in another; and this operation has no analogue in natural language."[62]: 251
This argument, associated withWillard Quine andHilary Putnam, is considered byStephen 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.[63] The form of the argument is as follows.
One must haveontological commitments toall entities that are indispensable to the best scientific theories, and to those entitiesonly (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.[64]
The justification for the first premise is the most controversial. Both Putnam and Quine invokenaturalism 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 byconfirmation 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 thenominalist who wishes to exclude the existence ofsets andnon-Euclidean geometry, but to include the existence ofquarks and other undetectable entities of physics, for example, in a difficult position.[64]
Theanti-realist "epistemic argument" against Platonism has been made byPaul Benacerraf andHartry Field. Platonism posits that mathematical objects areabstract entities. By general agreement, abstract entities cannot interactcausally 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"[65]). Whilst our knowledge of concrete, physical objects is based on our ability toperceive them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects.[66][67][68] 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 generateproofs, etc., which is already fully accountable in terms of physical processes in their brains.
Field developed his views intofictionalism. Benacerraf also developed the philosophy ofmathematical 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 satisfactorynaturalistic 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 specialintuition that involves contact with the Platonic realm. A modern form of this argument is given bySir Roger Penrose.[69]
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 byJerrold Katz in his 2000 bookRealistic Rationalism.
A more radical defense is denial of physical reality, i.e. themathematical universe hypothesis. In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.
Many practicing mathematicians have been drawn to their subject because of a sense ofbeauty 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 thedivine 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 asliterature.
Philip J. Davis andReuben Hersh have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of√2. The first is the traditional proof bycontradiction, ascribed toEuclid; the second is a more direct proof involving thefundamental theorem of arithmetic 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 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 inG. H. Hardy's bookA Mathematician's Apology, in which Hardy argues that pure mathematics is superior in beauty toapplied mathematics precisely because it cannot be used for war and similar ends.
^This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, withoutcomputers andproof assistants. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.
^This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.
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^SeeWhite, L. (1947). "The locus of mathematical reality: An anthropological footnote".Philosophy of Science.14 (4):289–303.doi:10.1086/286957.S2CID119887253. 189303; also inNewman, J. R. (1956).The World of Mathematics. Vol. 4. New York: Simon and Schuster. pp. 2348–2364.
^Dorato, Mauro (2005)."Why are laws mathematical?"(PDF).The Software of the Universe, An Introduction to the History and Philosophy of Laws of Nature. Ashgate. pp. 31–66.ISBN978-0-7546-3994-7.Archived(PDF) from the original on August 17, 2023. RetrievedDecember 5, 2022.
^Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs".Philosophy of Mathematics.42 (167 (4)). Revue Internationale de Philosophie:500–508.
^Davis, Jon D.; McDuffie, Amy Roth; Drake, Corey; Seiwell, Amanda L. (2019). "Teachers' perceptions of the official curriculum: Problem solving and rigor".International Journal of Educational Research.93:91–100.doi:10.1016/j.ijer.2018.10.002.S2CID149753721.
^Nickles, Thomas (2013). "The Problem of Demarcation".Philosophy of Pseudoscience: Reconsidering the Demarcation Problem. Chicago: The University of Chicago Press. p. 104.ISBN978-0-226-05182-6.
^Sarukkai, Sundar (February 10, 2005). "Revisiting the 'unreasonable effectiveness' of mathematics".Current Science.88 (3):415–423.JSTOR24110208.
^Wagstaff, Samuel S. Jr. (2021)."History of Integer Factoring"(PDF). In Bos, Joppe W.; Stam, Martijn (eds.).Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp. 41–77.Archived(PDF) from the original on November 20, 2022. RetrievedNovember 20, 2022.
^"Curves: Ellipse".MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland.Archived from the original on October 14, 2022. RetrievedNovember 20, 2022.
^Wilson, Edwin B.; Lewis, Gilbert N. (November 1912). "The Space-Time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics".Proceedings of the American Academy of Arts and Sciences.48 (11):389–507.doi:10.2307/20022840.JSTOR20022840.
^Ginammi, Michele (February 2016). "Avoiding reification: Heuristic effectiveness of mathematics and the prediction of the Ω– particle".Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics.53:20–27.Bibcode:2016SHPMP..53...20G.doi:10.1016/j.shpsb.2015.12.001.
^abCarnap, 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).
^Zach, Richard (2019),"Hilbert's Program", in Zalta, Edward N. (ed.),The Stanford Encyclopedia of Philosophy (Summer 2019 ed.), Metaphysics Research Lab, Stanford University,archived from the original on 2022-02-08, retrieved2019-05-25
^Audi, Robert (1999),The Cambridge Dictionary of Philosophy, Cambridge University Press, Cambridge, UK, 1995. 2nd edition. Page 542.
^From an 1886 lecture at the 'Berliner Naturforscher-Versammlung', according toH. M. Weber's memorial article, as quoted and translated inGonzalez Cabillon, Julio (2000-02-03)."FOM: What were Kronecker's f.o.m.?".Archived from the original on 2007-10-09. Retrieved2008-07-19.Gonzalez gives as the sources for the memorial article, the following: Weber, H: "Leopold Kronecker",Jahresberichte der Deutschen Mathematiker Vereinigung, vol ii (1893), pp. 5-31. Cf. page 19. See alsoMathematische Annalen vol. xliii (1893), pp. 1-25.
^Hersh, Reuben (February 10, 1997)."What Kind of a Thing is a Number?" (Interview). Interviewed by John Brockman. Edge Foundation. Archived fromthe original on May 16, 2008. Retrieved2008-12-26.
^abPutnam, H.Mathematics, Matter and Method. Philosophical Papers, vol. 1. Cambridge: Cambridge University Press, 1975. 2nd. ed., 1985.
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^"Since abstract objects are outside the nexus of causes and effects, and thus perceptually inaccessible, they cannot be known through their effects on us" — Katz, J.Realistic Rationalism, 2000, p. 15
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