Philosophy of mathematics
Encyclopedia
The philosophy of mathematics is the branch of philosophy
that studies the philosophical assumptions, foundations, and implications of mathematics
. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand 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.
Recurrent themes include:
The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms.
The latter, however, may be used to refer to several other areas of study. One refers to a project of formalising a philosophical subject matter, say, aesthetics
, ethics
, logic, metaphysics
, or theology
, in a purportedly more exact and rigorous form, as for example the labours of Scholastic
theologians, or the systematic aims of Leibniz and Spinoza. Another refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term "mathematical philosophy" to be an allusion to the approach taken by Bertrand Russell
in his books The Principles of Mathematics
and Introduction to Mathematical Philosophy
.
Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand, while others emphasize a role for themselves that goes beyond simple interpretation to critical analysis.
There are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy
. Western philosophies of mathematics go as far back as Plato
, who studied the ontological status
of mathematical objects, and Aristotle
, who studied logic
and issues related to infinity
(actual versus potential).
Greek
philosophy on mathematics was strongly influenced by their study of geometry
. For example, at one time, the Greeks held the opinion that 1 (one) was not a number
, but rather a unit of arbitrary length. A number was defined as a multitude. Therefore 3, for example, represented a certain multitude of units, and 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 proportional 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
, a disciple of Pythagoras
, showed that the diagonal of a unit square was incommensurable with its (unit-length) edge: in other words he proved there was no existing (rational) number that accurately depicts the proportion of the diagonal of the unit square to its edge. This caused a significant re-evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatised by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. Simon Stevin
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 century.
, 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
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 and ontology
. Three schools, formalism
, intuitionism
, and logicism
, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis
in particular, did not live up to the standards of certainty
and rigour
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
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 (mathematical logic)), were formalised, 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
of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert
to call such study metamathematics
or proof theory
.
At the middle of the century, a new mathematical theory was created by Samuel Eilenberg
and Saunders Mac Lane
, known as category theory
, and it became a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam
summed up one common view of the situation in the last third of the century by saying:
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.
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
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 some conjectures, like the continuum hypothesis
, might prove undecidable just on the basis of such principles. Gödel suggested that quasi-empirical methodology could be used to provide sufficient evidence to be able to reasonably assume such a conjecture.
Within realism, there are distinctions depending on what sort of existence one takes mathematical entities to have, and how we know about them.
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
's Theory of Forms
and a "World of Ideas" (Greek: Eidos (εἶδος))
described in Plato's Allegory of the cave
: 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 of ancient Greece, who believed that the world was, quite literally, generated by number
s.
The major problem of mathematical platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be Ultimate ensemble
, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.
Plato spoke of mathematics by:
In context, chapter 8, H.D.P. Lee translation, reports the education of a philosopher containing five mathematical disciplines:
1. arithmetic, written in unit fraction 'parts' using theoretical unities and abstract numbers.
2. plane geometry and solid geometry also considered the line to be segmented into rational and irrational unit 'parts',
3. astronomy
4. harmonics
Translators of the works of Plato rebelled against practical versions of his culture's practical mathematics. However, Plato himself and Greeks had copied 1,500 older Egyptian fraction abstract unities, one being a hekat unity scaled to (64/64) in the Akhmim Wooden Tablet
, thereby not getting lost in fractions.
Gödel's platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Husserl said about mathematics, and supports Kant
's idea that mathematics is synthetic a priori
.) Davis
and Hersh
have suggested in their book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism (see below).
Some mathematicians hold opinions that amount to more nuanced versions of Platonism.
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, and the axiom of choice). It holds that all mathematical entities exist, however they may be provable, even if they cannot all be derived from a single consistent set of axioms.
at all. It says that we discover mathematical facts by empirical
research, just like facts in any of the other sciences. It is not one of the classical three positions advocated in the early 20th century, but primarily arose in the middle of the century. However, an important early proponent of a view like this was John Stuart Mill
. Mill's view was widely criticized, because it makes statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.
Contemporary mathematical empiricism, formulated by Quine and Putnam
, is primarily supported by the indispensability argument: mathematics is indispensable to all empirical sciences, and if we want to believe in the reality of the phenomena described by the sciences, we ought also believe in the reality of those entities required for this description. That is, since physics needs to talk about electron
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 some of its distinctness from the other sciences.
Putnam strongly rejected the term "Platonist" as implying an over-specific ontology
that was not necessary to mathematical practice
in any real sense. He advocated a form of "pure realism" that rejected mystical notions of truth
and accepted much quasi-empiricism in mathematics
. Putnam was involved in coining the term "pure realism" (see below).
The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e. consilience
after E O Wilson. Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is incredibly central, and that it would be extremely difficult for us to revise it, though not impossible.
For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy
's Realism in Mathematics. Another example of a realist theory is the embodied mind theory (below). For a modern revision of mathematical empiricism see New Empiricism (below).
For experimental evidence suggesting that one-day-old babies can do elementary arithmetic, see Brian Butterworth
.
's Mathematical universe hypothesis goes further than full-blooded 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".
is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists hold that mathematics can be known a priori
, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic
is the proper foundation of mathematics, and all mathematical statements are necessary logical truth
s.
Rudolf Carnap
(1931) presents the logicist thesis in two parts:
Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic) he built up arithmetic
from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G, the extension of F equals the extension of G if and only if for all objects a, Fa if and only if Ga), a principle that he took to be acceptable as part of logic.
Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent. (This is Russell's paradox
) Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead
. They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex, form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of mathematics, such as an "axiom of reducibility
". Even Russell said that this axiom did not really belong to logic.
Modern logicists (like Bob Hale
, Crispin Wright
, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favour of abstraction principles such as Hume's principle
(the number of objects falling under the concept F equals the number of objects falling under the concept G if and only if the extension of F and the extension of G can be put into one-to-one correspondence
). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.
If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense, logicism can be seen as shifting questions about the philosophy of mathematics to questions about logic without fully answering them.
(which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem
holds (that is, you can generate the string corresponding to the Pythagorean theorem). According to Formalism, mathematical truths are not about numbers and sets and triangles and the like — in fact, they aren't "about" anything at all.
Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: if you assign meaning to the strings in such a way that the rules of the game become true (i.e., true statements are assigned to the axioms and the rules of inference are truth-preserving), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism
.) But it does allow the working mathematician to continue in his or her work and leave such problems to the philosopher or scientist. Many formalists would say that in practice, the axiom systems to be studied will be suggested by the demands of science or other areas of mathematics.
A major early proponent of formalism was David Hilbert
, whose program
was intended to be a complete
and consistent
axiomatization of all of mathematics. ("Consistent" here means that no contradictions can be derived from the system.) Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic
of the positive integers, chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent were dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.
Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.
Other formalists, such as Rudolf Carnap
, Alfred Tarski
and Haskell Curry
, considered mathematics to be the investigation of formal axiom systems
. Mathematical logic
ians 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
, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition (see below). See QED project
for a general overview.
Henri Poincaré
was among the first to articulate a conventionalist
view. Poincaré's use of non-Euclidean geometries in his work on differential equations convinced him that Euclidean geometry
should not be regarded as a priori
truth. He held that axioms in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.
in the philosophy of mathematics is the position that mathematical
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 nineteenth-century German logicians such as Sigwart
and Erdmann
as well as a number of psychologists, past and present: for example, Gustave Le Bon
. Psychologism was famously criticized by Frege
in his The Foundations of Arithmetic, and many of his works and essays, including his review of Husserl'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
.
). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. (CDP, 542)
A major force behind Intuitionism was L.E.J. Brouwer, who rejected the usefulness of formalized logic of any sort for mathematics. His student Arend Heyting
postulated an intuitionistic logic
, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle
and therefore frowns upon proofs by contradiction
. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. Important work was later done by Errett Bishop
, who managed to prove versions of the most important theorems in real analysis
within this framework.
In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine
or computable function
to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithm
s are meaningful and should be investigated in mathematics. This has led to the study of the computable number
s, first introduced by Alan Turing
. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science
is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural number
s in a finite number of steps. In her book Philosophy of Set Theory, Mary 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
, who said:
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.
. 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
. Other examples of mathematical objects might include lines
and planes in geometry, or elements and operations in abstract algebra
.
Structuralism is a 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
). 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, or fully realist, variation of structuralism has a similar ontology to Platonism in that structures are held to have a real but abstract and immaterial existence. As such, it faces the usual problems of explaining the interaction between such abstract structures and flesh-and-blood mathematicians.
In Re, or moderately realistic, structuralism is the equivalent of Aristotelean 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 Res or eliminative variant of structuralism is anti-realist
about structures in a way that parallels nominalism
. 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 convenient to talk of structures being "held in common" between systems: they in fact have no independent existence.
springs from the experience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.
With this view, the physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation. However, the human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition
.
Embodied mind theorists thus explain the effectiveness of mathematics — mathematics was constructed by the brain in order to be effective in this universe.
The most accessible, famous, and infamous treatment of this perspective is Where Mathematics Comes From
, by George Lakoff
and Rafael E. Núñez
. In addition, mathematician Keith Devlin
has investigated similar concepts with his book The Math Instinct. For more on the philosophical ideas that inspired this perspective, see cognitive science of mathematics.
To this principle it adds a materialist connection: All the processes of logic which interpret, organize and abstract observations, are physical phenomena which take place in real time and physical space: namely, in the brains of human beings. Abstract objects, such as mathematical objects, are ideas, which in turn exist as electrical and chemical states of the billions of neurons in the human brain.
This second concept is reminiscent of the social constructivist approach, which holds that mathematics is produced by humans rather than being “discovered” from abstract, a priori truths. However, it differs sharply from the constructivist implication that humans arbitrarily construct mathematical principles that have no inherent truth but which instead are created on a conveniency basis. On the contrary, new empiricism shows how mathematics, although constructed by humans, follows rules and principles that will be agreed on by all who participate in the process, with the result that everyone practicing mathematics comes up with the same answer — except in those areas where there is philosophical disagreement on the meaning of fundamental concepts. This is because the new empiricism perceives this agreement as being a physical phenomenon. One which is observed by other humans in the same way that other physical phenomena, like the motions of inanimate bodies, or the chemical interaction of various elements, are observed.
A difficulty lies in the observation that mathematical truths based on logical deduction appear to be more certainly true than knowledge of the physical world itself. (The physical world in this case is taken to mean the portion of it lying outside the human brain.)
Kant argued that the structures of logic which organize, interpret and abstract observations were built into the human mind and were true and valid a priori. Mill, on the contrary, said that we believe them to be true because we have enough individual instances of their truth to generalize: in his words, "From instances we have observed, we feel warranted in concluding that what we found true in those instances holds in all similar ones, past, present and future, however numerous they may be." Although the psychological or epistemological specifics given by Mill through which we build our logical apparatus may not be completely warranted, his explanation still nonetheless manages to demonstrate that there is no way around Kant’s a priori logic. To recant Mill's original idea in an empiricist twist: “Indeed, the very principles of logical deduction are true because we observe that using them leads to true conclusions.”, which is itself an a priori pressuposition.
For most mathematicians the empiricist principle that all knowledge comes from the senses contradicts a more basic principle: that mathematical propositions are true independent of the physical world. Everything about a mathematical proposition is independent of what appears to be the physical world. It all takes place in the mind. And the mind operates on infallible principles of deductive logic. It is not influenced by exterior inputs from the physical world, distorted by having to pass through the tentative, contingent universe of the senses. It all happens internally, so to say. This in turn may be the answer to what brings about Gödel's special kind of mathematical intuition, which was mentioned earlier in the article.
If all this is true, then where do the world senses come in? The early empiricists all stumbled over this point. Hume asserted that all knowledge comes from the senses, and then gave away the ballgame by excepting abstract propositions, which he called “relations of ideas.” These, he said, were absolutely true (although the mathematicians who thought them up, being human, might get them wrong). Mill, on the other hand, tried to deny that abstract ideas exist outside the physical world: all numbers, he said, “must be numbers of something: there are no such things as numbers in the abstract.” When we count to eight or add five and three we are really counting spoons or bumblebees. “All things possess quantity,” he said, so that propositions concerning numbers are propositions concerning “all things whatever.” But then in almost a contradiction of himself he went on to acknowledge that numerical and algebraic expressions are not necessarily attached to real world objects: they “do not excite in our minds ideas of any things in particular.” Mill’s low reputation as a philosopher of logic, and the low estate of empiricism in the century and a half following him, derives from this failed attempt to link abstract thoughts to the physical world, when it is obvious that abstraction consists precisely of separating the thought from its physical foundations.
The conundrum created by our certainty that abstract deductive propositions, if valid (i.e., if we can “prove” them), are true, exclusive of observation and testing in the physical world, gives rise to a further reflection...What if thoughts themselves, and the minds that create them, are physical objects, existing only in the physical world?
This would not reconcile the contradiction between our belief in the certainty of abstract deductions and the empiricist principle that knowledge comes from observation of individual instances. We know that Euler’s equation is true because every time a human mind derives the equation, it gets the same result, unless it has made a mistake, which can be acknowledged and corrected. We observe this phenomenon, and we extrapolate to the general proposition that it is always true. However, based on this rationale, one would still not be warranted in concluding that mathematics are purely empirical in nature.
This applies not only to physical principles, like the law of gravity, but to abstract phenomena that we observe only in human brains: in ours and in those of others.
Aristotelian realism is defended by James Franklin
and the Sydney School in the philosophy of mathematics and is close to the view of Penelope Maddy
(1990) that when I open an egg carton I perceive a set of three eggs (that is, a mathematical entity realized in the physical world). A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world.
in mathematics 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 that didn't reference numbers or functions at all. He started with the "betweenness" of Hilbert's axioms
to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector field
s. 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
of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from Field's system), so that the mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as fictitious as "Sherlock Holmes
lived at 221B Baker Street" — but both are true according to the relevant fictions.
By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction
in general. Field's approach has been very influential, but is widely rejected. This is in part because of the requirement of strong fragments of second-order logic
to carry out his reduction, and because the statement of conservativity seems to require quantification
over abstract models or deductions.
This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice
evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an over-emphasis on axiomatic proof and peer review as practices. However, this might be seen as merely saying that rigorously proven results are overemphasized, and then "look how chaotic and uncertain the rest of it all is!"
The social nature of mathematics is highlighted in its subculture
s. Major discoveries can be made in one branch of mathematics and be relevant to another, yet the relationship goes undiscovered for lack of social contact between mathematicians. Social constructivists argue each speciality forms its own epistemic community
and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias
, or of mathematicians' collective intelligence
as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless. Some social scientists also argue that mathematics is not real or objective at all, but is affected by racism
and ethnocentrism
. Some of these ideas are close to postmodernism
.
Contributions to this school have been made by Imre Lakatos
and Thomas Tymoczko
, although it is not clear that either would endorse the title. More recently Paul Ernest
has explicitly formulated a social constructivist philosophy of mathematics. Some consider the work of Paul Erdős
as a whole to have advanced this view (although he personally rejected it) because of his uniquely broad collaborations, which prompted others to see and study "mathematics as a social activity", e.g., via the Erdős number
. Reuben Hersh
has also promoted the social view of mathematics, calling it a "humanistic" approach, 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.
A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity. To observe that rigorous proof comes only after unrigorous conjecture, experimentation and speculation is true, but it is trivial and no-one would deny this. So it's a bit of a stretch to characterize a philosophy of mathematics in this way, on something trivially true. The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrass in order to rigorously prove the theorems thereof. There is nothing special or interesting about this, as it fits in with the more general trend of unrigorous ideas which are later made rigorous. There needs to be a clear distinction between the objects of study of mathematics and the study of the objects of study of mathematics. The former doesn't seem to change a great deal; the latter is forever in flux. The latter is what the Social theory is about, and the former is what Platonism et al. are about.
However, this criticism is rejected by supporters of the social constructivist perspective because it misses the point that the very objects of mathematics are social constructs. These objects, it asserts, are primarily semiotic objects existing in the sphere of human culture, sustained by social practices (after Wittgenstein) that utilize physically embodied signs and give rise to intrapersonal (mental) constructs. Social constructivists view the reification of the sphere of human culture into a Platonic
realm, or some other heaven-like domain of existence beyond the physical world, a long standing category error.
, or even on practices unique to mathematicians such as the proof
, a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner's famous 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences
, in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.
The embodied-mind or cognitive school and the social school were responses to this challenge, but the debates raised were difficult to confine to those.
. 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. Quasi-empiricism was developed by Imre Lakatos
, inspired by the philosophy of science of Karl Popper
.
Lakatos' philosophy of mathematics is sometimes regarded as a kind of social constructivism, but this was not his intention.
Such methods have always been part of folk mathematics by which great feats of calculation and measurement are sometimes achieved. Indeed, such methods may be the only notion of proof a culture has.
Hilary Putnam
has argued that any theory of mathematical realism would include quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics — at perhaps a somewhat greater risk of failure of their calculations. He gave a detailed argument for this in New Directions (ed. Tymockzo, 1998).
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.
to the more specialized metaphysical notions of the schools above. This may lead to a disconnection in which some mathematicians continue to profess discredited philosophy as a justification for their continued belief in a world-view promoting their work.
Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human perception
and everyday understandings of knowledge
.
", linguists believe that the implications of such a statement must be considered. For example, the tools of linguistics
are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way than other languages. If mathematics is a language, it is a different type of language than natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by 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.
, 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.
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
who wishes to exclude the existence of sets and non-Euclidean geometry
, but to include the existence of quark
s and other undetectable entities of physics, for example, in a difficult position.
and Hartry Field
. Platonism posits that mathematical objects are abstract
entities. By general agreement, abstract entities cannot
interact causally 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
them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects. ("An account of mathematical truth ..must be consistent with the possibility of mathematical knowledge"). 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 proof
s, etc., which is already fully accountable in terms of physical processes in their brains.
Field developed his views into fictionalism. 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 defence is to maintain that this is false, so that mathematical reasoning uses some special intuition
that involves contact with the Platonic realm. A modern form of this argument is given by Sir Roger Penrose.
Another line of defence 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 book Realistic Rationalism.
A more radical defense is denial of physical reality, i.e. the mathematical universe hypothesis. In that case, a mathematicians knowledge of mathematics is one mathematical object making contact with another.
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
.
Philip J. Davis
and Reuben Hersh
have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of the √2
. The first is the traditional proof by contradiction
, ascribed to Euclid
; the second is a more direct proof involving the fundamental theorem of arithmetic
that, they argue, gets to the heart of the issue. Davis and Hersh argue that 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 in G.H. Hardy's book A Mathematician's Apology
, in which Hardy argues that pure mathematics is superior in beauty to applied mathematics
precisely because it cannot be used for war and similar ends. Some later mathematicians have characterized Hardy's views as mildly dated, with the applicability of number theory to modern-day cryptography
.
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
that studies the philosophical assumptions, foundations, and implications of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand 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.
Recurrent themes include:
- What are the sources of mathematical subject matter?
- What is the ontologicalOntologyOntology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
status of mathematical entities? - What does it mean to refer to a mathematical objectMathematical objectIn mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
? - What is the character of a mathematical proposition?
- What is the relation between logicLogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and mathematics? - What is the role of hermeneutics in mathematics?
- What kinds of inquiry play a role in mathematics?
- What are the objectives of mathematical inquiry?
- What gives mathematics its hold on experienceExperienceExperience as a general concept comprises knowledge of or skill in or observation of some thing or some event gained through involvement in or exposure to that thing or event....
? - What are the human traits behind mathematics?
- What is mathematical beautyMathematical beautyMany mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity...
? - What is the source and nature of mathematical truth?
- What is the relationship between the abstract world of mathematics and the material universe?
The terms philosophy of mathematics and mathematical philosophy are frequently used as synonyms.
The latter, however, may be used to refer to several other areas of study. One refers to a project of formalising a philosophical subject matter, say, aesthetics
Aesthetics
Aesthetics is a branch of philosophy dealing with the nature of beauty, art, and taste, and with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgments of sentiment and taste...
, ethics
Ethics
Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...
, logic, metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
, or theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...
, in a purportedly more exact and rigorous form, as for example the labours of Scholastic
Scholasticism
Scholasticism is a method of critical thought which dominated teaching by the academics of medieval universities in Europe from about 1100–1500, and a program of employing that method in articulating and defending orthodoxy in an increasingly pluralistic context...
theologians, or the systematic aims of Leibniz and Spinoza. Another refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Additionally, some understand the term "mathematical philosophy" to be an allusion to the approach taken by Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
in his books The Principles of Mathematics
The Principles of Mathematics
The Principles of Mathematics is a book written by Bertrand Russell in 1903. In it he presented his famous paradox and argued his thesis that mathematics and logic are identical....
and Introduction to Mathematical Philosophy
Introduction to Mathematical Philosophy
Introduction to Mathematical Philosophy is a book by Bertrand Russell, published in 1919, written in part to exposit in a less technical way the main ideas of his and Whitehead's Principia Mathematica , including the theory of descriptions....
.
History
The origin of mathematics is subject to argument. Whether the birth of mathematics was a random happening or induced by necessity duly contingent of other subjects, say for 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 and Eastern philosophy
Eastern philosophy
Eastern philosophy includes the various philosophies of Asia, including Chinese philosophy, Iranian philosophy, Japanese philosophy, Indian philosophy and Korean philosophy...
. Western philosophies of mathematics go as far back as Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
, who studied the ontological status
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
of mathematical objects, and Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, who studied logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and issues related to infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
(actual versus potential).
Greek
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
philosophy on mathematics was strongly influenced by their study of geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
. For example, at one time, the Greeks held the opinion that 1 (one) was not a number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
, 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 proportional to the arbitrary first "number" or "one."
These earlier Greek ideas of numbers were later upended by the discovery of the irrationality
Irrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
of the square root of two. Hippasus
Hippasus
Hippasus of Metapontum in Magna Graecia, was a Pythagorean philosopher. Little is known about his life or his beliefs, but he is sometimes credited with the discovery of the existence of irrational numbers.-Life:...
, a disciple of Pythagoras
Pythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
, 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 traumatised by this discovery that they murdered Hippasus to stop him from spreading his heretical idea. Simon Stevin
Simon Stevin
Simon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...
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
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
and of Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
, but was brought into question by developments in the late 19th and early 20th century.
20th century
A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20th century philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterised by a predominant interest in formal logicFormal logic
Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
, set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, 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
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
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 and ontology
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
. Three schools, formalism
Formalism (mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....
, intuitionism
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied...
, and logicism
Logicism
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind...
, emerged at this time, partly in response to the increasingly widespread worry that mathematics as it stood, and analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
in particular, did not live up to the standards of certainty
Certainty
Certainty can be defined as either:# perfect knowledge that has total security from error, or# the mental state of being without doubtObjectively defined, certainty is total continuity and validity of all foundational inquiry, to the highest degree of precision. Something is certain only if no...
and rigour
Rigour
Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
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
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
around 300 BCE as the natural basis for mathematics. Notions of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
, proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
and proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
, as well as the notion of a proposition being true of a mathematical object (see Assignment (mathematical logic)), were formalised, 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
Consistency proof
In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all...
of mathematical theories. This reflective critique in which the theory under review "becomes itself the object of a mathematical study" led Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
to call such study metamathematics
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
or proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
.
At the middle of the century, a new mathematical theory was created by Samuel Eilenberg
Samuel Eilenberg
Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...
and Saunders Mac Lane
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
, known as category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, and it became a new contender for the natural language of mathematical thinking (Mac Lane 1998). As the 20th century progressed, however, philosophical opinions diverged as to just how well-founded were the questions about foundations that were raised at its opening. Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
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 paradoxRussell's paradoxIn the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
comes to mind, as does BerkeleyGeorge BerkeleyGeorge Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...
's attack on the actual infinitesimalInfinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
— but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need. (Putnam, 169-170).
Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.
Mathematical realism
Mathematical realism, like realismPhilosophical realism
Contemporary philosophical realism is the belief that our reality, or some aspect of it, is ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc....
in general, holds that mathematical entities exist independently of the human mind
Mind
The concept of mind is understood in many different ways by many different traditions, ranging from panpsychism and animism to traditional and organized religious views, as well as secular and materialist philosophies. Most agree that minds are constituted by conscious experience and intelligent...
. 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
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
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
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
and Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
. Gödel believed in an objective mathematical reality that could be perceived in a manner analogous to sense perception. Certain principles (e.g., for any two objects, there is a collection of objects consisting of precisely those two objects) could be directly seen to be true, but some conjectures, like the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
, 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.
Platonism
Mathematical PlatonismPlatonism
Platonism is the philosophy of Plato or the name of other philosophical systems considered closely derived from it. In a narrower sense the term might indicate the doctrine of Platonic realism...
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
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
's Theory of Forms
Theory of Forms
Plato's theory of Forms or theory of Ideas asserts that non-material abstract forms , and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form is often capitalized...
and a "World of Ideas" (Greek: Eidos (εἶδος))
Eidos (disambiguation)
Eidos is a Greek word meaning "image", "shape", "look", "kind" or "species". The term became significant in Greek philosophy when Plato used it to refer to the ideal Forms or Ideas in his Theory of Forms...
described in Plato's Allegory of the cave
Allegory of the cave
The Allegory of the Cave—also known as the Analogy of the Cave, Plato's Cave, or the Parable of the Cave—is an allegory used by the Greek philosopher Plato in his work The Republic to illustrate "our nature in its education and want of education"...
: 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 of ancient Greece, who believed that the world was, quite literally, generated by number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s.
The major problem of mathematical platonism is this: precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be Ultimate ensemble
Ultimate ensemble
In physics and cosmology, the mathematical universe hypothesis , also known as the Ultimate Ensemble, is a speculative "theory of everything" proposed by the theoretical physicist, Max Tegmark.-Description:...
, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe.
Plato spoke of mathematics by:
In context, chapter 8, H.D.P. Lee translation, reports the education of a philosopher containing five mathematical disciplines:
1. arithmetic, written in unit fraction 'parts' using theoretical unities and abstract numbers.
2. plane geometry and solid geometry also considered the line to be segmented into rational and irrational unit 'parts',
3. astronomy
4. harmonics
Translators of the works of Plato rebelled against practical versions of his culture's practical mathematics. However, Plato himself and Greeks had copied 1,500 older Egyptian fraction abstract unities, one being a hekat unity scaled to (64/64) in the Akhmim Wooden Tablet
Akhmim wooden tablet
The Akhmim wooden tablets or Cairo wooden tablets are two ancient Egyptian wooden writing tablets. They each measure about 18 by 10 inches and are covered with plaster. The tablets are inscribed on both sides. The inscriptions on the first tablet includes a list of servants, which is followed...
, thereby not getting lost in fractions.
Gödel's platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly. (This view bears resemblances to many things Husserl said about mathematics, and supports Kant
KANT
KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...
's idea that mathematics is synthetic a priori
A priori and a posteriori (philosophy)
The terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...
.) Davis
Philip J. Davis
Philip J. Davis is an American applied mathematician.Davis was born in Lawrence, Massachusetts. He is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics...
and Hersh
Reuben Hersh
Reuben Hersh is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. This work challenges and complements mainstream philosophy of mathematics.After receiving a B.A...
have suggested in their book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism (see below).
Some mathematicians hold opinions that amount to more nuanced versions of Platonism.
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, and the axiom of choice). It holds that all mathematical entities exist, however they may be provable, even if they cannot all be derived from a single consistent set of axioms.
Empiricism
Empiricism is a form of realism that denies that mathematics can be known a prioriA priori and a posteriori (philosophy)
The terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...
at all. It says that we discover mathematical facts by empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
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
John Stuart Mill
John Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...
. Mill's view was widely criticized, because it makes statements like "2 + 2 = 4" come out as uncertain, contingent truths, which we can only learn by observing instances of two pairs coming together and forming a quartet.
Contemporary mathematical empiricism, formulated by Quine and Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
, 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
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s to say why light bulbs behave as they do, then electrons must exist
Existence
In common usage, existence is the world we are aware of through our senses, and that persists independently without them. In academic philosophy the word has a more specialized meaning, being contrasted with essence, which specifies different forms of existence as well as different identity...
. Since physics needs to talk about numbers in offering any of its explanations, then numbers must exist. In keeping with Quine and Putnam's overall philosophies, this is a naturalistic argument. It argues for the existence of mathematical entities as the best explanation for experience, thus stripping mathematics of some of its distinctness from the other sciences.
Putnam strongly rejected the term "Platonist" as implying an over-specific ontology
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
that was not necessary to mathematical practice
Mathematical practice
Mathematical practice is used to distinguish the working practices of professional mathematicians from the end result of proven and published theorems.-Quasi-empiricism:This distinction is...
in any real sense. He advocated a form of "pure realism" that rejected mystical notions of truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
and accepted much quasi-empiricism in mathematics
Quasi-empiricism in mathematics
Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of mathematics...
. Putnam was involved in coining the term "pure realism" (see below).
The most important criticism of empirical views of mathematics is approximately the same as that raised against Mill. If mathematics is just as empirical as the other sciences, then this suggests that its results are just as fallible as theirs, and just as contingent. In Mill's case the empirical justification comes directly, while in Quine's case it comes indirectly, through the coherence of our scientific theory as a whole, i.e. consilience
Consilience
Consilience, or the unity of knowledge , has its roots in the ancient Greek concept of an intrinsic orderliness that governs our cosmos, inherently comprehensible by logical process, a vision at odds with mystical views in many cultures that surrounded the Hellenes...
after E O Wilson. Quine suggests that mathematics seems completely certain because the role it plays in our web of belief is incredibly central, and that it would be extremely difficult for us to revise it, though not impossible.
For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy
Penelope Maddy
Penelope Maddy is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on realism and naturalism.Maddy received her Ph.D. from...
's Realism in Mathematics. Another example of a realist theory is the embodied mind theory (below). For a modern revision of mathematical empiricism see New Empiricism (below).
For experimental evidence suggesting that one-day-old babies can do elementary arithmetic, see Brian Butterworth
Brian Butterworth
Brian Butterworth is a professor of cognitive neuropsychology in the Institute of Cognitive Neuroscience at University College London. His research has ranged from speech errors and pauses, short-term memory deficits, dyslexia, reading both in alphabetic scripts and logograms, and mathematics and...
.
Mathematical monism
Max TegmarkMax Tegmark
Max Tegmark is a Swedish-American cosmologist. Tegmark is a professor at the Massachusetts Institute of Technology and belongs to the scientific directorate of the Foundational Questions Institute.-Early life:...
's Mathematical universe hypothesis goes further than full-blooded 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".
Logicism
LogicismLogicism
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind...
is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists hold that mathematics can be known a priori
A priori and a posteriori (philosophy)
The terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...
, 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
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
is the proper foundation of mathematics, and all mathematical statements are necessary logical truth
Logical truth
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement.Logical...
s.
Rudolf Carnap
Rudolf Carnap
Rudolf Carnap was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
(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
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G, the extension of F equals the extension of G if and only if for all objects a, Fa if and only if Ga), a principle that he took to be acceptable as part of logic.
Frege's construction was flawed. Russell discovered that Basic Law V is inconsistent. (This is Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
) Frege abandoned his logicist program soon after this, but it was continued by Russell and Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
. They attributed the paradox to "vicious circularity" and built up what they called ramified type theory to deal with it. In this system, they were eventually able to build up much of modern mathematics but in an altered, and excessively complex, form (for example, there were different natural numbers in each type, and there were infinitely many types). They also had to make several compromises in order to develop so much of mathematics, such as an "axiom of reducibility
Axiom of reducibility
The axiom of reducibility was introduced by Bertrand Russell as part of his ramified theory of types, an attempt to ground mathematics in first-order logic.- History: the problem of impredicativity :...
". Even Russell said that this axiom did not really belong to logic.
Modern logicists (like Bob Hale
Bob Hale (philosopher)
Robert Hale FBA, FRSE is a British philosopher, who is well-known for his contributions to the development of the neo-Fregean philosophy of mathematics in collaboration with Crispin Wright, and for his works in modality and philosophy of language....
, Crispin Wright
Crispin Wright
Crispin Wright is a British philosopher, who has written on neo-Fregean philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity....
, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favour of abstraction principles such as Hume's principle
Hume's principle
Hume's Principle or HP—the terms were coined by George Boolos—says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence between the Fs and the Gs. 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
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.
If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects. But what, one might ask, are the objects of logical concepts? In this sense, logicism can be seen as shifting questions about the philosophy of mathematics to questions about logic without fully answering them.
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 geometryEuclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
(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
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
holds (that is, you 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 aren't "about" anything at all.
Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: if you assign meaning to the strings in such a way that the rules of the game become true (i.e., true statements are assigned to the axioms and the rules of inference are truth-preserving), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true statement. The same is held to be true for all other mathematical statements. Thus, formalism need not mean that mathematics is nothing more than a meaningless symbolic game. It is usually hoped that there exists some interpretation in which the rules of the game hold. (Compare this position to structuralism
Structuralism (philosophy of mathematics)
Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their place in such structures, consequently having no intrinsic properties. For instance, it would maintain that all that...
.) 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.
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, whose program
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
was intended to be a complete
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
and consistent
Consistency proof
In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all...
axiomatization of all of mathematics. ("Consistent" here means that no contradictions can be derived from the system.) Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
of the positive integers, chosen to be philosophically uncontroversial) was consistent. Hilbert's goals of creating a system of mathematics that is both complete and consistent were dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency. Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that (since it would then prove its own consistency, which Gödel had shown was impossible). Thus, in order to show that any axiomatic system of mathematics is in fact consistent, one needs to first assume the consistency of a system of mathematics that is in a sense stronger than the system to be proven consistent.
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
Rudolf Carnap
Rudolf Carnap was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
, Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
and Haskell Curry
Haskell Curry
Haskell Brooks Curry 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 Moses Schönfinkel, much of the development was done by Curry. Curry is also known for Curry's...
, considered mathematics to be the investigation of formal axiom systems
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
. Mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
ians 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
Proof checking
Automated proof checking is the process of using software for checking proofs for correctness. It is one of the most developed fields in automated reasoning....
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
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition (see below). See QED project
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. The idea for the project arose in 1993, mainly under the impetus of Robert Boyer...
for a general overview.
Conventionalism
The French mathematicianMathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
was among the first to articulate a conventionalist
Conventionalism
Conventionalism is the philosophical attitude that fundamental principles of a certain kind are grounded on agreements in society, rather than on external reality...
view. Poincaré's use of non-Euclidean geometries in his work on differential equations convinced him that Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
should not be regarded as a priori
A priori and a posteriori (philosophy)
The terms a priori and a posteriori are used in philosophy to distinguish two types of knowledge, justifications or arguments...
truth. He held that axioms in geometry should be chosen for the results they produce, not for their apparent coherence with human intuitions about the physical world.
Psychologism
PsychologismPsychologism
Psychologism is a generic type of position in philosophy according to which psychology plays a central role in grounding or explaining some other, non-psychological type of fact or law...
in the philosophy of mathematics is the position that mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
concepts and/or truths are grounded in, derived from or explained by psychological facts (or laws).
John Stuart Mill
John Stuart Mill
John Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...
seems to have been an advocate of a type of logical psychologism, as were many nineteenth-century German logicians such as Sigwart
Christoph von Sigwart
Christoph von Sigwart was a German philosopher and logician. He was the son of philosopher Heinrich Christoph Wilhelm Sigwart .-Life:...
and Erdmann
Johann Eduard Erdmann
Johann Eduard Erdmann was a German philosophical writer....
as well as a number of psychologists, past and present: for example, Gustave Le Bon
Gustave Le Bon
Gustave Le Bon was a French social psychologist, sociologist, and amateur physicist...
. Psychologism was famously criticized by Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
in his The Foundations of Arithmetic, and many of his works and essays, including his review of Husserl's Philosophy of Arithmetic
Philosophy of Arithmetic
The Philosophy of Arithmetic is the English language title of Edmund Husserl's first published book. It was first published, in the German language, under the full title, Philosophie der Arithmetik...
. Edmund Husserl, in the first volume of his Logical Investigations
Logical Investigations
Logical Investigations can refer to:*Logical Investigations, 1840 work by Friedrich Adolf Trendelenburg*Logical Investigations, 1900 work by Edmund Husserl...
, 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 Merleau-Ponty
Maurice Merleau-Ponty was a French phenomenological philosopher, strongly influenced by Karl Marx, Edmund Husserl and Martin Heidegger in addition to being closely associated with Jean-Paul Sartre and Simone de Beauvoir...
.
Intuitionism
In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no non-experienced mathematical truths" (L.E.J. BrouwerLuitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
). From this springboard, intuitionists seek to reconstruct what they consider to be the corrigible portion of mathematics in accordance with Kantian concepts of being, becoming, intuition, and knowledge. Brouwer, the founder of the movement, held that mathematical objects arise from the a priori forms of the volitions that inform the perception of empirical objects. (CDP, 542)
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
Arend Heyting
Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
postulated an intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
, different from the classical Aristotelian logic; this logic does not contain the law of the excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
and therefore frowns upon proofs by contradiction
Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
. The axiom of choice is also rejected in most intuitionistic set theories, though in some versions it is accepted. Important work was later done by Errett Bishop
Errett Bishop
Errett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructive analysis, because of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
, who managed to prove versions of the most important theorems in real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
within this framework.
In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
or computable function
Computable function
Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register...
to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s are meaningful and should be investigated in mathematics. This has led to the study of the computable number
Computable number
In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...
s, first introduced by Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
Constructivism
Like intuitionism, constructivism involves the regulative principle that only mathematical entities which can be explicitly constructed in a certain sense should be admitted to mathematical discourse. In this view, mathematics is an exercise of the human intuition, not a game played with meaningless symbols. Instead, it is about entities that we can create directly through mental activity. In addition, some adherents of these schools reject non-constructive proofs, such as a proof by contradiction.Finitism
FinitismFinitism
In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps...
is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s in a finite number of steps. In her book Philosophy of Set Theory, Mary Tiles
Mary Tiles
Mary Tiles is a philosopher and historian of mathematics and science. she is professor and chair in the philosophy department of the University of Hawaii at Manoa.-Life:...
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
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
, who said:
Ultrafinitism
Ultrafinitism
In the philosophy of mathematics, ultrafinitism, also known as ultraintuitionism, strict-finitism, actualism, and strong-finitism is a form of finitism. There are various philosophies of mathematics which are called 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.
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 propertiesIntrinsic and extrinsic properties (philosophy)
An intrinsic property is a property that an object or a thing has of itself, independently of other things, including its context. An extrinsic property is a property that depends on a thing's relationship with other things...
. 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
Number line
In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...
. Other examples of mathematical objects might include lines
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
and planes in geometry, or elements and operations in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
.
Structuralism is a 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
Ontology
Ontology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
). 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, or fully realist, variation of structuralism has a similar ontology to Platonism in that structures are held to have a real but abstract and immaterial existence. As such, it faces the usual problems of explaining the interaction between such abstract structures and flesh-and-blood mathematicians.
In Re, or moderately realistic, structuralism is the equivalent of Aristotelean 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 Res or eliminative variant of structuralism is anti-realist
Anti-realism
In analytic philosophy, the term anti-realism is used to describe any position involving either the denial of an objective reality of entities of a certain type or the denial that verification-transcendent statements about a type of entity are either true or false...
about structures in a way that parallels nominalism
Nominalism
Nominalism is a metaphysical view in philosophy according to which general or abstract terms and predicates exist, while universals or abstract objects, which are sometimes thought to correspond to these terms, do not exist. Thus, there are at least two main versions of nominalism...
. 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 convenient 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 numberNumber
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
springs from the experience of counting discrete objects. It is held that mathematics is not universal and does not exist in any real sense, other than in human brains. Humans construct, but do not discover, mathematics.
With this view, the physical universe can thus be seen as the ultimate foundation of mathematics: it guided the evolution of the brain and later determined which questions this brain would find worthy of investigation. However, the human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition
Cognition
In science, cognition refers to mental processes. These processes include attention, remembering, producing and understanding language, solving problems, and making decisions. Cognition is studied in various disciplines such as psychology, philosophy, linguistics, and computer science...
.
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
Where Mathematics Comes From
Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist...
, by George Lakoff
George Lakoff
George P. Lakoff is an American cognitive linguist and professor of linguistics at the University of California, Berkeley, where he has taught since 1972...
and Rafael E. Núñez
Rafael E. Núñez
Rafael E. Núñez is a professor of cognitive science at the University of California, San Diego and a proponent of embodied cognition. He co-authored Where Mathematics Comes From with George Lakoff.-External links:*...
. In addition, mathematician Keith Devlin
Keith Devlin
Keith J. Devlin is a British mathematician and popular science writer. He has lived in the USA since 1987 and has dual American-British citizenship.- Biography :...
has investigated similar concepts with his book The Math Instinct. For more on the philosophical ideas that inspired this perspective, see cognitive science of mathematics.
New Empiricism
A more recent empiricism returns to the principle of the English empiricists of the 18th and 19th Centuries, in particular John Stuart Mill, who asserted that all knowledge comes to us from observation through the senses. This applies not only to matters of fact, but also to "relations of ideas," as Hume called them: the structures of logic which interpret, organize and abstract observations.To this principle it adds a materialist connection: All the processes of logic which interpret, organize and abstract observations, are physical phenomena which take place in real time and physical space: namely, in the brains of human beings. Abstract objects, such as mathematical objects, are ideas, which in turn exist as electrical and chemical states of the billions of neurons in the human brain.
This second concept is reminiscent of the social constructivist approach, which holds that mathematics is produced by humans rather than being “discovered” from abstract, a priori truths. However, it differs sharply from the constructivist implication that humans arbitrarily construct mathematical principles that have no inherent truth but which instead are created on a conveniency basis. On the contrary, new empiricism shows how mathematics, although constructed by humans, follows rules and principles that will be agreed on by all who participate in the process, with the result that everyone practicing mathematics comes up with the same answer — except in those areas where there is philosophical disagreement on the meaning of fundamental concepts. This is because the new empiricism perceives this agreement as being a physical phenomenon. One which is observed by other humans in the same way that other physical phenomena, like the motions of inanimate bodies, or the chemical interaction of various elements, are observed.
A difficulty lies in the observation that mathematical truths based on logical deduction appear to be more certainly true than knowledge of the physical world itself. (The physical world in this case is taken to mean the portion of it lying outside the human brain.)
Kant argued that the structures of logic which organize, interpret and abstract observations were built into the human mind and were true and valid a priori. Mill, on the contrary, said that we believe them to be true because we have enough individual instances of their truth to generalize: in his words, "From instances we have observed, we feel warranted in concluding that what we found true in those instances holds in all similar ones, past, present and future, however numerous they may be." Although the psychological or epistemological specifics given by Mill through which we build our logical apparatus may not be completely warranted, his explanation still nonetheless manages to demonstrate that there is no way around Kant’s a priori logic. To recant Mill's original idea in an empiricist twist: “Indeed, the very principles of logical deduction are true because we observe that using them leads to true conclusions.”, which is itself an a priori pressuposition.
For most mathematicians the empiricist principle that all knowledge comes from the senses contradicts a more basic principle: that mathematical propositions are true independent of the physical world. Everything about a mathematical proposition is independent of what appears to be the physical world. It all takes place in the mind. And the mind operates on infallible principles of deductive logic. It is not influenced by exterior inputs from the physical world, distorted by having to pass through the tentative, contingent universe of the senses. It all happens internally, so to say. This in turn may be the answer to what brings about Gödel's special kind of mathematical intuition, which was mentioned earlier in the article.
If all this is true, then where do the world senses come in? The early empiricists all stumbled over this point. Hume asserted that all knowledge comes from the senses, and then gave away the ballgame by excepting abstract propositions, which he called “relations of ideas.” These, he said, were absolutely true (although the mathematicians who thought them up, being human, might get them wrong). Mill, on the other hand, tried to deny that abstract ideas exist outside the physical world: all numbers, he said, “must be numbers of something: there are no such things as numbers in the abstract.” When we count to eight or add five and three we are really counting spoons or bumblebees. “All things possess quantity,” he said, so that propositions concerning numbers are propositions concerning “all things whatever.” But then in almost a contradiction of himself he went on to acknowledge that numerical and algebraic expressions are not necessarily attached to real world objects: they “do not excite in our minds ideas of any things in particular.” Mill’s low reputation as a philosopher of logic, and the low estate of empiricism in the century and a half following him, derives from this failed attempt to link abstract thoughts to the physical world, when it is obvious that abstraction consists precisely of separating the thought from its physical foundations.
The conundrum created by our certainty that abstract deductive propositions, if valid (i.e., if we can “prove” them), are true, exclusive of observation and testing in the physical world, gives rise to a further reflection...What if thoughts themselves, and the minds that create them, are physical objects, existing only in the physical world?
This would not reconcile the contradiction between our belief in the certainty of abstract deductions and the empiricist principle that knowledge comes from observation of individual instances. We know that Euler’s equation is true because every time a human mind derives the equation, it gets the same result, unless it has made a mistake, which can be acknowledged and corrected. We observe this phenomenon, and we extrapolate to the general proposition that it is always true. However, based on this rationale, one would still not be warranted in concluding that mathematics are purely empirical in nature.
This applies not only to physical principles, like the law of gravity, but to abstract phenomena that we observe only in human brains: in ours and in those of others.
Aristotelian realism
Similar to empiricism in emphasizing the relation of mathematics to the real world, 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
James Franklin (philosopher)
James Franklin is an Australian philosopher, mathematician and historian of ideas. He was educated at St. Joseph's College, Hunters Hill, New South Wales. His undergraduate work was at the University of Sydney , where he attended St John's College and he was influenced by philosophers David Stove...
and the Sydney School in the philosophy of mathematics and is close to the view of Penelope Maddy
Penelope Maddy
Penelope Maddy is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on realism and naturalism.Maddy received her Ph.D. from...
(1990) that when I open an egg carton I perceive a set of three eggs (that is, a mathematical entity realized in the physical world). A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world.
Fictionalism
FictionalismFictionalism
Fictionalism is a methodological theory in philosophy that suggests that statements of a certain sort should not be taken to be literally true, but merely as a useful fiction...
in mathematics was brought to fame in 1980 when Hartry Field
Hartry Field
Hartry H. Field is a philosopher, the Silver Professor of Philosophy at New York University. He previously taught at the University of Southern California and The Graduate Center of the City University of New York. He earned his Ph.D...
published Science Without Numbers, which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics that didn't reference numbers or functions at all. He started with the "betweenness" of Hilbert's axioms
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie , as the foundation for a modern treatment of Euclidean geometry...
to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s. 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
Conservative extension
In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...
of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from Field's system), so that the mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking as if numbers existed. For Field, a statement like "2 + 2 = 4" is just as fictitious as "Sherlock Holmes
Sherlock Holmes
Sherlock Holmes is a fictional detective created by Scottish author and physician Sir Arthur Conan Doyle. The fantastic London-based "consulting detective", Holmes is famous for his astute logical reasoning, his ability to take almost any disguise, and his use of forensic science skills to solve...
lived at 221B Baker Street" — but both are true according to the relevant fictions.
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
Fiction
Fiction is the form of any narrative or informative work that deals, in part or in whole, with information or events that are not factual, but rather, imaginary—that is, invented by the author. Although fiction describes a major branch of literary work, it may also refer to theatrical,...
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
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....
to carry out his reduction, and because the statement of conservativity seems to require quantification
Quantification
Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...
over abstract models or deductions.
Social constructivism or social realism
Social constructivism or social realism theories see mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints — the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated — that work to conserve the historically defined discipline.This runs counter to the traditional beliefs of working mathematicians, that mathematics is somehow pure or objective. But social constructivists argue that mathematics is in fact grounded by much uncertainty: as mathematical practice
Mathematical practice
Mathematical practice is used to distinguish the working practices of professional mathematicians from the end result of proven and published theorems.-Quasi-empiricism:This distinction is...
evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an over-emphasis on axiomatic proof and peer review as practices. However, this might be seen as merely saying that rigorously proven results are overemphasized, and then "look how chaotic and uncertain the rest of it all is!"
The social nature of mathematics is highlighted in its subculture
Subculture
In sociology, anthropology and cultural studies, a subculture is a group of people with a culture which differentiates them from the larger culture to which they belong.- Definition :...
s. 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
Epistemic community
An epistemic community is a transnational 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 community is widely accepted as that of Peter...
and often has great difficulty communicating, or motivating the investigation of unifying conjectures that might relate different areas of mathematics. Social constructivists see the process of "doing mathematics" as actually creating the meaning, while social realists see a deficiency either of human capacity to abstractify, or of human's cognitive bias
Cognitive bias
A cognitive bias is a pattern of deviation in judgment that occurs in particular situations. Implicit in the concept of a "pattern of deviation" is a standard of comparison; this may be the judgment of people outside those particular situations, or may be a set of independently verifiable...
, or of mathematicians' collective intelligence
Collective intelligence
Collective intelligence is a shared or group intelligence that emerges from the collaboration and competition of many individuals and appears in consensus decision making in bacteria, animals, humans and computer networks....
as preventing the comprehension of a real universe of mathematical objects. Social constructivists sometimes reject the search for foundations of mathematics as bound to fail, as pointless or even meaningless. Some social scientists also argue that mathematics is not real or objective at all, but is affected by racism
Racism
Racism is the belief that inherent different traits in human racial groups justify discrimination. In the modern English language, the term "racism" is used predominantly as a pejorative epithet. It is applied especially to the practice or advocacy of racial discrimination of a pernicious nature...
and ethnocentrism
Ethnocentrism
Ethnocentrism is the tendency to believe that one's ethnic or cultural group is centrally important, and that all other groups are measured in relation to one's own. The ethnocentric individual will judge other groups relative to his or her own particular ethnic group or culture, especially with...
. Some of these ideas are close to postmodernism
Postmodernism
Postmodernism is a philosophical movement evolved in reaction to modernism, the tendency in contemporary culture to accept only objective truth and to be inherently suspicious towards a global cultural narrative or meta-narrative. Postmodernist thought is an intentional departure from the...
.
Contributions to this school have been made by Imre Lakatos
Imre Lakatos
Imre Lakatos 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 pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his...
and Thomas Tymoczko
Thomas Tymoczko
A. Thomas Tymoczko was a philosopher specializing in logic and the philosophy of mathematics. He taught at Smith College in Northampton, Massachusetts from 1971 until his untimely death....
, although it is not clear that either would endorse the title. More recently Paul Ernest
Paul Ernest
Paul Ernest is a recent contributor to the social constructivist philosophy of mathematics. He illustrates this position in his discussion of the issue of whether mathematics is discovered or invented...
has explicitly formulated a social constructivist philosophy of mathematics. Some consider the work of Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
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
Erdos number
The Erdős number describes the "collaborative distance" between a person and mathematician Paul Erdős, as measured by authorship of mathematical papers.The same principle has been proposed for other eminent persons in other fields.- Overview :...
. Reuben Hersh
Reuben Hersh
Reuben Hersh is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. This work challenges and complements mainstream philosophy of mathematics.After receiving a B.A...
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
Philip J. Davis
Philip J. Davis is an American applied mathematician.Davis was born in Lawrence, Massachusetts. He is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics...
, has expressed sympathy for the social view as well.
A criticism of this approach is that it is trivial, based on the trivial observation that mathematics is a human activity. To observe that rigorous proof comes only after unrigorous conjecture, experimentation and speculation is true, but it is trivial and no-one would deny this. So it's a bit of a stretch to characterize a philosophy of mathematics in this way, on something trivially true. The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrass in order to rigorously prove the theorems thereof. There is nothing special or interesting about this, as it fits in with the more general trend of unrigorous ideas which are later made rigorous. There needs to be a clear distinction between the objects of study of mathematics and the study of the objects of study of mathematics. The former doesn't seem to change a great deal; the latter is forever in flux. The latter is what the Social theory is about, and the former is what Platonism et al. are about.
However, this criticism is rejected by supporters of the social constructivist perspective because it misses the point that the very objects of mathematics are social constructs. These objects, it asserts, are primarily semiotic objects existing in the sphere of human culture, sustained by social practices (after Wittgenstein) that utilize physically embodied signs and give rise to intrapersonal (mental) constructs. Social constructivists view the reification of the sphere of human culture into a Platonic
Platonism
Platonism is the philosophy of Plato or the name of other philosophical systems considered closely derived from it. In a narrower sense the term might indicate the doctrine of Platonic realism...
realm, or some other heaven-like domain of existence beyond the physical world, a long standing category error.
Beyond the traditional schools
Rather than focus on narrow debates about the true nature of mathematical truthTruth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
, or even on practices unique to mathematicians such as the proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
, a growing movement from the 1960s to the 1990s began to question the idea of seeking foundations or finding any one right answer to why mathematics works. The starting point for this was Eugene Wigner's famous 1960 paper The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist Eugene Wigner...
, in which he argued that the happy coincidence of mathematics and physics being so well matched seemed to be unreasonable and hard to explain.
The embodied-mind or cognitive school and the social school were responses to this challenge, but the debates raised were difficult to confine to those.
Quasi-empiricism
One parallel concern that does not actually challenge the schools directly but instead questions their focus is the notion of quasi-empiricism in mathematicsQuasi-empiricism in mathematics
Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely to issues in the foundations of 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. Quasi-empiricism was developed by Imre Lakatos
Imre Lakatos
Imre Lakatos 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 pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his...
, inspired by the philosophy of science of Karl Popper
Karl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
.
Lakatos' philosophy of mathematics is sometimes regarded as a kind of social constructivism, but this was not his intention.
Such methods have always been part of folk mathematics by which great feats of calculation and measurement are sometimes achieved. Indeed, such methods may be the only notion of proof a culture has.
Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
has argued that any theory of mathematical realism would include quasi-empirical methods. He proposed that an alien species doing mathematics might well rely on quasi-empirical methods primarily, being willing often to forgo rigorous and axiomatic proofs, and still be doing mathematics — at perhaps a somewhat greater risk of failure of their calculations. He gave a detailed argument for this in New Directions (ed. Tymockzo, 1998).
Popper's "two senses" theory
Realist and constructivist theories are normally taken to be contraries. However, Karl PopperKarl Popper
Sir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
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.
Unification
Few philosophers are able to penetrate mathematical notations and culture to relate conventional notions of metaphysicsMetaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
to the more specialized metaphysical notions of the schools above. This may lead to a disconnection in which some mathematicians continue to profess discredited philosophy as a justification for their continued belief in a world-view promoting their work.
Although the social theories and quasi-empiricism, and especially the embodied mind theory, have focused more attention on the epistemology implied by current mathematical practices, they fall far short of actually relating this to ordinary human perception
Perception
Perception is the process of attaining awareness or understanding of the environment by organizing and interpreting sensory information. All perception involves signals in the nervous system, which in turn result from physical stimulation of the sense organs...
and everyday understandings of knowledge
Knowledge
Knowledge is a familiarity with someone or something unknown, which can include information, facts, descriptions, or skills acquired through experience or education. It can refer to the theoretical or practical understanding of a subject...
.
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 most mathematicians and physicists (and many philosophers) would accept the statement "mathematics is a languageMathematics as a language
The language of mathematics is the system used by mathematicians to communicate mathematical ideas among themselves. This language consists of a substrate of some natural language using technical terms and grammatical conventions that are peculiar to mathematical discourse , supplemented by a...
", linguists believe that the implications of such a statement must be considered. For example, the tools of linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
are not generally applied to the symbol systems of mathematics, that is, mathematics is studied in a markedly different way than other languages. If mathematics is a language, it is a different type of language than natural languages. Indeed, because of the need for clarity and specificity, the language of mathematics is far more constrained than natural languages studied by linguists. However, the methods developed by Frege and Tarski for the study of mathematical language have been extended greatly by Tarski's student Richard Montague
Richard Montague
Richard Merett Montague was an American mathematician and philosopher.-Career:At the University of California, Berkeley, Montague earned an B.A. in Philosophy in 1950, an M.A. in Mathematics in 1953, and a Ph.D. in Philosophy 1957, the latter under the direction of the mathematician and logician...
and other linguists working in formal semantics
Formal semantics (linguistics)
In linguistics, formal semantics seeks to understand linguistic meaning by constructing precise mathematical models of the principles that speakers use to define relations between expressions in a natural language and the world which supports meaningful discourse.The mathematical tools used are the...
to show that the distinction between mathematical language and natural language may not be as great as it seems.
The indispensability argument for realism
This argument, associated with Willard Quine and Hilary PutnamHilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
, is considered by Stephen Yablo
Stephen Yablo
Stephen Yablo is a philosopher at the Massachusetts Institute of Technology . He specializes in the philosophy of logic, philosophy of mind, metaphysics, and philosophy of language....
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 ontologicalOntologyOntology is the philosophical study of the nature of being, existence or reality as such, as well as the basic categories of being and their relations...
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.
The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism
Naturalism (philosophy)
Naturalism commonly refers to the philosophical viewpoint that the natural universe and its natural laws and forces operate in the universe, and that nothing exists beyond the natural universe or, if it does, it does not affect the natural universe that we know...
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
Confirmation holism
Confirmation holism, also called epistemological holism is the claim that a single scientific theory cannot be tested in isolation; a test of one theory always depends on other theories and hypotheses....
. 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
Nominalism
Nominalism is a metaphysical view in philosophy according to which general or abstract terms and predicates exist, while universals or abstract objects, which are sometimes thought to correspond to these terms, do not exist. Thus, there are at least two main versions of nominalism...
who wishes to exclude the existence of sets and non-Euclidean geometry
Non-Euclidean geometry
Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...
, but to include the existence of quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
s and other undetectable entities of physics, for example, in a difficult position.
The epistemic argument against realism
The anti-realist "epistemic argument" against Platonism has been made by Paul BenacerrafPaul Benacerraf
Paul Joseph Salomon Benacerraf is an American philosopher working in the field of the philosophy of mathematics who has been teaching at Princeton University since he joined the faculty in 1960. He was appointed Stuart Professor of Philosophy in 1974, and recently retired as the James S....
and Hartry Field
Hartry Field
Hartry H. Field is a philosopher, the Silver Professor of Philosophy at New York University. He previously taught at the University of Southern California and The Graduate Center of the City University of New York. He earned his Ph.D...
. Platonism posits that mathematical objects are abstract
Abstract object
An abstract object is an object which does not exist at any particular time or place, but rather exists as a type of thing . In philosophy, an important distinction is whether an object is considered abstract or concrete. Abstract objects are sometimes called abstracta An abstract object is an...
entities. By general agreement, abstract entities cannot
interact causally 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
Perception is the process of attaining awareness or understanding of the environment by organizing and interpreting sensory information. All perception involves signals in the nervous system, which in turn result from physical stimulation of the sense organs...
them, and therefore to causally interact with them, there is no parallel account of how mathematicians come to have knowledge of abstract objects. ("An account of mathematical truth ..must be consistent with the possibility of mathematical knowledge"). 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 proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
s, etc., which is already fully accountable in terms of physical processes in their brains.
Field developed his views into fictionalism. 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
Naturalism (philosophy)
Naturalism commonly refers to the philosophical viewpoint that the natural universe and its natural laws and forces operate in the universe, and that nothing exists beyond the natural universe or, if it does, it does not affect the natural universe that we know...
account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else. One line of defence is to maintain that this is false, so that mathematical reasoning uses some special intuition
Intuition (knowledge)
Intuition is the ability to acquire knowledge without inference or the use of reason. "The word 'intuition' comes from the Latin word 'intueri', which is often roughly translated as meaning 'to look inside'’ or 'to contemplate'." Intuition provides us with beliefs that we cannot necessarily justify...
that involves contact with the Platonic realm. A modern form of this argument is given by Sir Roger Penrose.
Another line of defence 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
Jerrold Katz
Jerrold J. Katz was an American philosopher and linguist.After receiving a PhD in philosophy from Princeton University in 1960, Katz became a Research Associate in Linguistics at the Massachusetts Institute of Technology in 1961. He was appointed Assistant Professor of Philosophy there in 1963,...
in his book Realistic Rationalism.
A more radical defense is denial of physical reality, i.e. the mathematical universe hypothesis. In that case, a mathematicians knowledge of mathematics is one mathematical object making contact with another.
Aesthetics
Many practising mathematicians have been drawn to their subject because of a sense of beautyMathematical beauty
Many mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity...
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
Literature is the art of written works, and is not bound to published sources...
.
Philip J. Davis
Philip J. Davis
Philip J. Davis is an American applied mathematician.Davis was born in Lawrence, Massachusetts. He is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics...
and Reuben Hersh
Reuben Hersh
Reuben Hersh is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. This work challenges and complements mainstream philosophy of mathematics.After receiving a B.A...
have commented that the sense of mathematical beauty is universal amongst practicing mathematicians. By way of example, they provide two proofs of the irrationality of the √2
Square root of 2
The square root of 2, often known as root 2, is the positive algebraic number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2, to distinguish it from the negative number with the same property.Geometrically the square root of 2 is the...
. The first is the traditional proof by contradiction
Contradiction
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
, ascribed to Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
; the second is a more direct proof involving the fundamental theorem of arithmetic
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
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
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
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 Chaitin
Gregory John Chaitin is an Argentine-American mathematician and computer scientist.-Mathematics and computer science:Beginning in 2009 Chaitin has worked on metabiology, a field parallel to biology dealing with the random evolution of artificial software instead of natural software .Beginning in...
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
A Mathematician's Apology
A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...
, in which Hardy argues that pure mathematics is superior in beauty to applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
precisely because it cannot be used for war and similar ends. Some later mathematicians have characterized Hardy's views as mildly dated, with the applicability of number theory to modern-day cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
.
See also
- Axiomatic set theory
- Axiomatic systemAxiomatic systemIn mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems...
- Category theoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
- Definitions of mathematicsDefinitions of mathematicsMathematics has no generally accepted definition. Different schools of thought, particularly in philosophy, have put forth radically different definitions...
- Formal languageFormal languageA formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
- Formal systemFormal systemIn formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
- Foundations of mathematicsFoundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
- Golden ratioGolden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
- History of mathematicsHistory of mathematicsThe area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
- Intuitionistic logicIntuitionistic logicIntuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
- LogicLogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
- Mathematical beautyMathematical beautyMany mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity...
- Mathematical constructivism
- Mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
- Mathematical proofMathematical proofIn mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
- MetamathematicsMetamathematicsMetamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
- Model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
- Naive set theoryNaive set theoryNaive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
- Non-standard analysisNon-standard analysisNon-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
- Philosophy of languagePhilosophy of languagePhilosophy of language is the reasoned inquiry into the nature, origins, and usage of language. As a topic, the philosophy of language for analytic philosophers is concerned with four central problems: the nature of meaning, language use, language cognition, and the relationship between language...
- Philosophy of sciencePhilosophy of scienceThe philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
- Philosophy of probability
- Proof theoryProof theoryProof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
- Rule of inferenceRule of inferenceIn logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...
- Science studiesScience studiesScience studies is an interdisciplinary research area that seeks to situate scientific expertise in a broad social, historical, and philosophical context. It is concerned with the history of academic disciplines, the interrelationships between science and society, and the alleged covert purposes...
- Scientific methodScientific methodScientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...
- Set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
- The Unreasonable Effectiveness of Mathematics in the Natural SciencesThe Unreasonable Effectiveness of Mathematics in the Natural SciencesThe Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist Eugene Wigner...
- TruthTruthTruth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
- Ultimate ensembleUltimate ensembleIn physics and cosmology, the mathematical universe hypothesis , also known as the Ultimate Ensemble, is a speculative "theory of everything" proposed by the theoretical physicist, Max Tegmark.-Description:...
Related works
- The AnalystThe AnalystThe Analyst, subtitled "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", is a book published by...
- Euclid's ElementsEuclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
- Gödel's completeness theoremOriginal proof of Gödel's completeness theoremThe proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 is not easy to read today; it uses concepts and formalism that are outdated and terminology that is often obscure...
- Introduction to Mathematical PhilosophyIntroduction to Mathematical PhilosophyIntroduction to Mathematical Philosophy is a book by Bertrand Russell, published in 1919, written in part to exposit in a less technical way the main ideas of his and Whitehead's Principia Mathematica , including the theory of descriptions....
- New FoundationsNew FoundationsIn mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
- Principia MathematicaPrincipia MathematicaThe Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
- The Simplest Mathematics
Historical topics
- History and philosophy of scienceHistory and philosophy of scienceThe history and philosophy of science 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 are degree-granting departments of HPS at several...
- History of mathematicsHistory of mathematicsThe area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past....
- History of philosophyHistory of philosophyThe history of philosophy is the study of philosophical ideas and concepts through time. Issues specifically related to history of philosophy might include : How can changes in philosophy be accounted for historically? What drives the development of thought in its historical context? To what...
Further reading
- AristotleAristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, "Prior AnalyticsPrior AnalyticsThe Prior Analytics is Aristotle's work on deductive reasoning, specifically the syllogism. It is also part of his Organon, which is the instrument or manual of logical and scientific methods....
", Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical LibraryLoeb Classical LibraryThe Loeb Classical Library is a series of books, today published by Harvard University Press, which presents important works of ancient Greek and Latin Literature in a way designed to make the text accessible to the broadest possible audience, by presenting the original Greek or Latin text on each...
, William Heinemann, London, UK, 1938. - Audi, RobertRobert AudiRobert Audi is an American philosopher whose major work has focused on epistemology, ethics—especially on ethical intuitionism-and the theory of action. He is O'Brien Professor of Philosophy at the University of Notre Dame, and previously held a Chair in the Business School there...
(ed., 1999), The Cambridge Dictionary of Philosophy, Cambridge University Press, Cambridge, UK, 1995. 2nd edition, 1999. Cited as CDP. - Benacerraf, PaulPaul BenacerrafPaul Joseph Salomon Benacerraf is an American philosopher working in the field of the philosophy of mathematics who has been teaching at Princeton University since he joined the faculty in 1960. He was appointed Stuart Professor of Philosophy in 1974, and recently retired as the James S....
, and Putnam, HilaryHilary PutnamHilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
(eds., 1983), Philosophy of Mathematics, Selected Readings, 1st edition, Prentice-Hall, Englewood Cliffs, NJ, 1964. 2nd edition, Cambridge University Press, Cambridge, UK, 1983. - Berkeley, GeorgeGeorge BerkeleyGeorge Berkeley , also known as Bishop Berkeley , was an Irish philosopher whose primary achievement was the advancement of a theory he called "immaterialism"...
(1734), The AnalystThe AnalystThe Analyst, subtitled "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", is a book published by...
; or, a Discourse Addressed to an Infidel Mathematician. Wherein It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith, London & Dublin. Online text, David R. Wilkins (ed.), Eprint. - Bourbaki, N.Nicolas BourbakiNicolas Bourbaki is the collective pseudonym under which a group of 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. With the goal of founding all of mathematics on set theory, the group strove for rigour and generality...
(1994), Elements of the History of Mathematics, John Meldrum (trans.), Springer-Verlag, Berlin, Germany. - Carnap, RudolfRudolf CarnapRudolf Carnap was an influential German-born philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism....
(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). - Chandrasekhar, SubrahmanyanSubrahmanyan ChandrasekharSubrahmanyan Chandrasekhar, FRS ) was an Indian origin American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize for Physics for key discoveries that led to the currently accepted theory on the later evolutionary stages of massive stars...
(1987), Truth and Beauty. Aesthetics and Motivations in Science, University of Chicago Press, Chicago, IL. - Colyvan, Mark (2004), "Indispensability Arguments in the Philosophy of Mathematics", Stanford Encyclopedia of Philosophy, Edward N. ZaltaEdward N. ZaltaEdward N. Zalta is a Senior research scholar at the Center for the Study of Language and Information. He received his Ph.D. in philosophy from the University of Massachusetts - Amherst in 1980. Zalta has taught courses at Stanford University, Rice University, the University of Salzburg, and the...
(ed.), Eprint. - Davis, Philip J.Philip J. DavisPhilip J. Davis is an American applied mathematician.Davis was born in Lawrence, Massachusetts. He is known for his work in numerical analysis and approximation theory, as well as his investigations in the history and philosophy of mathematics...
and Hersh, ReubenReuben HershReuben Hersh is an American mathematician and academic, best known for his writings on the nature, practice, and social impact of mathematics. This work challenges and complements mainstream philosophy of mathematics.After receiving a B.A...
(1981), The Mathematical ExperienceThe Mathematical ExperienceThe Mathematical Experience is a 1981 book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective...
, Mariner Books, New York, NY. - Devlin, KeithKeith DevlinKeith J. Devlin is a British mathematician and popular science writer. He has lived in the USA since 1987 and has dual American-British citizenship.- Biography :...
(2005), The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs), Thunder's Mouth Press, New York, NY. - Dummett, MichaelMichael DummettSir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...
(1991 a), Frege, Philosophy of Mathematics, Harvard University Press, Cambridge, MA. - Dummett, Michael (1991 b), Frege and Other Philosophers, Oxford University Press, Oxford, UK.
- Dummett, Michael (1993), Origins of Analytical Philosophy, Harvard University Press, Cambridge, MA.
- Ernest, PaulPaul ErnestPaul Ernest is a recent contributor to the social constructivist philosophy of mathematics. He illustrates this position in his discussion of the issue of whether mathematics is discovered or invented...
(1998), Social Constructivism as a Philosophy of Mathematics, State University of New York Press, Albany, NY. - George, Alexandre (ed., 1994), Mathematics and Mind, Oxford University Press, Oxford, UK.
- Hadamard, JacquesJacques HadamardJacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
(1949), The Psychology of Invention in the Mathematical Field, 1st edition, Princeton University Press, Princeton, NJ. 2nd edition, 1949. Reprinted, Dover Publications, New York, NY, 1954. - Hardy, G.H. (1940), A Mathematician's Apology, 1st published, 1940. Reprinted, C.P. Snow (foreword), 1967. Reprinted, Cambridge University Press, Cambridge, UK, 1992.
- Hart, W.D. (ed., 1996), The Philosophy of Mathematics, Oxford University Press, Oxford, UK.
- Hendricks, Vincent F.Vincent F. HendricksVincent Fella Hendricks , is a Danish philosopher and logician. He holds two doctoral degrees in philosophy and is Professor of Formal Philosophy at University of Copenhagen, Denmark. He was previously Professor of Formal Philosophy at Roskilde University, Denmark. He is member of IIP, the...
and Hannes Leitgeb (eds.). Philosophy of Mathematics: 5 Questions, New York: Automatic Press / VIP, 2006. http://www.phil-math.org - Huntley, H.E. (1970), The Divine Proportion: A Study in Mathematical Beauty, Dover Publications, New York, NY.
- Irvine, A., ed (2009), The Philosophy of Mathematics, in Handbook of the Philosophy of Science series, North-Holland Elsevier, Amsterdam.
- Klein, JacobJacob Klein (philosopher)Jacob Klein was a German-American philosopher and interpreter of Plato.-Biography:Klein was born in Liepāja, Latvia. He studied at Berlin and Marburg, where he received his Ph.D. in 1922. A student of Nicolai Hartmann, Martin Heidegger, and Edmund Husserl, he later taught at St. John's College in...
(1968), Greek Mathematical Thought and the Origin of Algebra, Eva BrannEva BrannEva Brann is a former dean and the longest-serving tutor at St. John's College, Annapolis, and a 2005 recipient of the National Humanities Medal....
(trans.), MIT Press, Cambridge, MA, 1968. Reprinted, Dover Publications, Mineola, NY, 1992. - Kline, MorrisMorris KlineMorris Kline was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.Kline grew up in Brooklyn and in Jamaica, Queens...
(1959), Mathematics and the Physical World, Thomas Y. Crowell Company, New York, NY, 1959. Reprinted, Dover Publications, Mineola, NY, 1981. - Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York, NY.
- König, Julius (Gyula) (1905), "Über die Grundlagen der Mengenlehre und das Kontinuumproblem", Mathematische Annalen 61, 156-160. Reprinted, "On the Foundations of Set Theory and the Continuum Problem", Stefan Bauer-Mengelberg (trans.), pp. 145–149 in Jean van Heijenoort (ed., 1967).
- Körner, StephanStephan KörnerStephan Körner, FBA was a British philosopher, who specialised in the work of Kant, the study of concepts, and in the philosophy of mathematics...
, The Philosophy of Mathematics, An Introduction. Harper Books, 1960. - Lakoff, GeorgeGeorge LakoffGeorge P. Lakoff is an American cognitive linguist and professor of linguistics at the University of California, Berkeley, where he has taught since 1972...
, and Núñez, Rafael E.Rafael E. NúñezRafael E. Núñez is a professor of cognitive science at the University of California, San Diego and a proponent of embodied cognition. He co-authored Where Mathematics Comes From with George Lakoff.-External links:*...
(2000), Where Mathematics Comes FromWhere Mathematics Comes FromWhere Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist...
: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York, NY. - Lakatos, Imre 1976 Proofs and Refutations:The Logic of Mathematical Discovery (Eds) J. Worrall & E. Zahar Cambridge University Press
- Lakatos, Imre 1978 Mathematics, Science and Epistemology: Philosophical Papers Volume 2 (Eds) J.Worrall & G.Currie Cambridge University Press
- Lakatos, Imre 1968 Problems in the Philosophy of Mathematics North Holland
- Leibniz, G.W., Logical Papers (1666–1690), G.H.R. Parkinson (ed., trans.), Oxford University Press, London, UK, 1966.
- Mac Lane, SaundersSaunders Mac LaneSaunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
(1998), Categories for the Working MathematicianCategories for the Working MathematicianCategories for the Working Mathematician is a textbook in category theory written by American mathematician Saunders Mac Lane, who cofounded the subject together with Samuel Eilenberg. It was first published in 1971, and is based on his lectures on the subject given at the University of Chicago,...
, 1st edition, Springer-Verlag, New York, NY, 1971, 2nd edition, Springer-Verlag, New York, NY. - Maddy, PenelopePenelope MaddyPenelope Maddy is a UCI Distinguished Professor of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the philosophy of mathematics, where she has worked on realism and naturalism.Maddy received her Ph.D. from...
(1990), Realism in Mathematics, Oxford University Press, Oxford, UK. - Maddy, Penelope (1997), Naturalism in Mathematics, Oxford University Press, Oxford, UK.
- Maziarz, Edward A., and Greenwood, ThomasThomas GreenwoodThomas J. Greenwood was born in 1908 and was an Illinois labor and Indian affairs activist, of Scottish and Cherokee descent. He worked as the manager of a shipyard during World War II and was noted for his hiring of Oklahoma Indians and women...
(1995), Greek Mathematical Philosophy, Barnes and Noble Books. - Mount, Matthew, Classical Greek Mathematical Philosophy, .
- Peirce, BenjaminBenjamin PeirceBenjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics....
(1870), "Linear Associative Algebra", § 1. See American Journal of Mathematics 4 (1881). - Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1-6, Charles HartshorneCharles HartshorneCharles Hartshorne was a prominent American philosopher who concentrated primarily on the philosophy of religion and metaphysics. He developed the neoclassical idea of God and produced a modal proof of the existence of God that was a development of St. Anselm's Ontological Argument...
and Paul WeissPaul Weiss (philosopher)Paul Weiss was an American philosopher.-Background:Paul Weiss grew up on the lower east side of New York City. His father, Samuel Weiss , was a Hungarian emigrant who moved from Europe in the 1890s. He worked as a tinsmith, a coppersmith, and a boilermaker. Paul Weiss's mother, Emma Rothschild ...
(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 bibliographyCharles Sanders Peirce bibliographyThis Charles Sanders Peirce bibliography consolidates numerous references to Charles Sanders Peirce's writings, including letters, manuscripts, publications, and Nachlass...
, especially under Books authored or edited by Peirce, published in his lifetime and the two sections following it. - Plato, "The Republic, Volume 1", Paul ShoreyPaul ShoreyPaul Shorey, Ph.D., LL.D., Litt.D. was an American classical scholar, born at Davenport, Iowa After graduating from Harvard in 1878 he studied in Europe at Leipzig, Bonn, Athens, and Munich . He was a professor at several institutions from 1885 onward...
(trans.), pp. 1–535 in Plato, Volume 5, Loeb Classical Library, William Heinemann, London, UK, 1930. - Plato, "The Republic, Volume 2", Paul Shorey (trans.), pp. 1–521 in Plato, Volume 6, Loeb Classical Library, William Heinemann, London, UK, 1935.
- Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 5-22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).
- Resnik, Michael D. Frege and the Philosophy of Mathematics, Cornell University, 1980.
- Resnik, MichaelMichael ResnikMichael David Resnik is a leading contemporary philosopher of mathematics. He obtained his B.A. in mathematics and philosophy at Yale University in 1960, and his Ph.D. in Philosophy at Harvard University in 1964. He wrote his thesis on Frege...
(1997), Mathematics as a Science of Patterns, Clarendon Press, Oxford, UK, ISBN 9780198250142 - Robinson, Gilbert de B.Gilbert de Beauregard RobinsonGilbert de Beauregard Robinson was a Canadian mathematician most famous for his work on combinatorics and representation theory of the symmetric groups, including the Robinson-Schensted algorithm.-Biography:...
(1959), The Foundations of Geometry, University of Toronto Press, Toronto, Canada, 1940, 1946, 1952, 4th edition 1959. - Raymond, Eric S. (1993), "The Utility of Mathematics", Eprint.
- Smullyan, Raymond M. (1993), Recursion Theory for Metamathematics, Oxford University Press, Oxford, UK.
- Russell, Bertrand (1919), Introduction to Mathematical Philosophy, George Allen and Unwin, London, UK. Reprinted, John G. Slater (intro.), Routledge, London, UK, 1993.
- Shapiro, StewartStewart ShapiroStewart Shapiro is O'Donnell Professor of Philosophy at the Ohio State University and a regular visiting professor at the University of St Andrews in Scotland. He is an important contemporary figure in the philosophy of mathematics where he defends a version of structuralism. He studied...
(2000), Thinking About Mathematics: The Philosophy of Mathematics, Oxford University Press, Oxford, UK - Strohmeier, John, and Westbrook, Peter (1999), Divine Harmony, The Life and Teachings of Pythagoras, Berkeley Hills Books, Berkeley, CA.
- Styazhkin, N.I. (1969), History of Mathematical Logic from Leibniz to Peano, MIT Press, Cambridge, MA.
- Tait, William W.William W. TaitWilliam Walker Tait is an emeritus professor of philosophy at the University of Chicago, where he served as a faculty member from 1972 to 1996, and as department chair from 1981 to 1987....
(1986), "Truth and Proof: The Platonism of Mathematics", Synthese 69 (1986), 341-370. Reprinted, pp. 142–167 in W.D. Hart (ed., 1996). - Tarski, A. (1983), Logic, Semantics, Metamathematics: Papers from 1923 to 1938, J.H. Woodger (trans.), Oxford University Press, Oxford, UK, 1956. 2nd edition, John Corcoran (ed.), Hackett Publishing, Indianapolis, IN, 1983.
- Tymoczko, ThomasThomas TymoczkoA. Thomas Tymoczko was a philosopher specializing in logic and the philosophy of mathematics. He taught at Smith College in Northampton, Massachusetts from 1971 until his untimely death....
(1998), New Directions in the Philosophy of Mathematics, Catalog entry? - Ulam, S.M. (1990), Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators, A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA.
- van Heijenoort, JeanJean Van HeijenoortJean Louis Maxime van Heijenoort was a pioneer historian of mathematical logic. He was also a personal secretary to Leon Trotsky from 1932 to 1939, and from then until 1947, an American Trotskyist activist.-Life:Van Heijenoort was born in Creil, France...
(ed. 1967), From Frege To Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA. - Wigner, Eugene (1960), "The Unreasonable Effectiveness of Mathematics in the Natural SciencesThe Unreasonable Effectiveness of Mathematics in the Natural SciencesThe Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist Eugene Wigner...
", Communications on Pure and Applied MathematicsCommunications on Pure and Applied MathematicsCommunications on Pure and Applied Mathematics is a scientific journal which is associated with the Courant Institute of Mathematical Sciences. It publishes original research originating from or solicited by the institute, typically in the fields of applied mathematics, mathematical analysis, or...
13(1): 1-14. Eprint - Wilder, Raymond L.Raymond Louis WilderRaymond Louis Wilder was an American mathematician, who specialized in topology and gradually acquired philosophical and anthropological interests.-Life:...
Mathematics as a Cultural System, Pergamon, 1980.
External links
- The London Philosophy Study Guide offers many suggestions on what to read, depending on the student's familiarity with the subject:
- R.B. Jones' philosophy of mathematics page
- The Philosophy of Real Mathematics Blog
- Kaina Stoicheia by C. S. Peirce.