Philosophy of mathematics
This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages)

Part of a series on  
Mathematics  



Mathematics Portal  
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities.
Major themes that are dealt with in philosophy of mathematics include:
 Reality: The question is whether mathematics is a pure product of human mind or whether it has some reality by itself.
 Logic and rigor
 Relationship with physical reality
 Relationship with science
 Relationship with applications
 Mathematical truth
 Nature as human activity (science, art, game, or all together)
Major themes
Reality
The connection between mathematics and material reality has led to philosophical debates since at least the time of Pythagoras. The ancient philosopher Plato argued that abstractions that reflect material reality have themselves a reality that exists outside space and time. As a result, the philosophical view that mathematical objects somehow exist on their own in abstraction is often referred to as Platonism. Independently of their possible philosophical opinions, modern mathematicians may be generally considered as Platonists, since they think of and talk of their objects of study as real objects (see Mathematical object).^{[1]}
Armand Borel summarized this view of mathematics reality as follows, and provided quotations of G. H. Hardy, Charles Hermite, Henri Poincaré and Albert Einstein that support his views.^{[2]}
Something becomes objective (as opposed to "subjective") as soon as we are convinced that it exists in the minds of others in the same form as it does in ours and that we can think about it and discuss it together.^{[3]} Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. In my opinion, that is sufficient to provide us with a feeling of an objective existence, of a reality of mathematics ...
Logic and rigor
Mathematical reasoning requires
The rules of rigorous reasoning have been established by the
For many centuries, logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians.^{}
Several methods have been proposed to solve the problem by changing of logical framework, such as
The problems of
It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simply a pleonasm. Where a special concept of rigor comes into play is in the socialized aspects of a proof. In particular, proofs are rarely written in full details, and some steps of a proof are generally considered as trivial, easy, or straightforward, and therefore left to the reader. As most proof errors occur in these skipped steps, a new proof requires to be verified by other specialists of the subject, and can be considered as reliable only after having been accepted by the community of the specialists, which may need several years.^{[6]}
Also, the concept of "rigor" may remain useful for teaching to beginners what is a mathematical proof.^{[7]}
Relationship with physical reality
Before the 19th century, the basic mathematical concepts, such as points, lines, natural numbers, real numbers (used for measurements), etc. were abstractions from the physical world, and it was commonly considered that it was sufficient for defining them.^{[c]}
As a consequence of this closeness to physical reality, mathematicians were very cautious when problems they want to solve led them to introduce new concepts that are not directly related the real world. These precautions are still reflected in modern terminology, where the numbers that are not quotient of natural numbers are called irrational numbers, originally meaning that reason cannot conceive them. Similarly, real numbers are the numbers that can be used for measurement, while imaginary numbers cannot.
During the 19th century, there were an active research for giving more precise definitions to the basic concepts resulting of abstraction from the real world; for example
After strong debates, axiomatic approach became eventually a de facto norm in mathematics. This mean that mathematical theories must be based on
This axiomatic approach has been applied to the whole mathematics, through
. The whole mathematics has been rebuilt inside this theory. Except if the contrary is explicitly stated, all modern mathematical texts use it as a foundation of mathematics.As a consequence, the relationship between mathematics and physical reality is no more a mathematical question, but the nature of this relationship remains a philosophical question that does not have any uncontroversial answer.
Relationship with sciences
Mathematics is used in most
There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is
Unreasonable effectiveness
The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner.^{[18]} It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced.^{[19]} Examples of unexpected applications of mathematical theories can be found in many areas of mathematics.
A notable example is the
In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of nonEuclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a nonEuclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four.^{[22]}^{[23]}
A striking aspect of the interaction between mathematics and physics is when mathematics drives research in physics. This is illustrated by the discoveries of the positron and the baryon In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.^{[2]}^{[24]}^{[25]}
History
The origin of mathematics is of arguments and disagreements. Whether the birth of mathematics was by chance or induced by necessity during the development of similar subjects, such as physics, remains an area of contention.^{[26]}^{[27]}
Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some^{[}
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 (unitlength) 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 reevaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were so traumatized by this discovery that they murdered Hippasus to stop him from spreading his heretical idea.^{[28]} 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 centuries.
Contemporary philosophy
A perennial issue in the philosophy of mathematics concerns the relationship between logic and mathematics at their joint foundations. While 20thcentury philosophers continued to ask the questions mentioned at the outset of this article, the philosophy of mathematics in the 20th century was characterized by a predominant interest in
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
Surprising and counterintuitive developments in formal logic and set theory early in the 20th century led to new questions concerning what was traditionally called the foundations of mathematics. As the century unfolded, the initial focus of concern expanded to an open exploration of the fundamental axioms of mathematics, the axiomatic approach having been taken for granted since the time of
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.^{[30]} As the 20th century progressed, however, philosophical opinions diverged as to just how wellfounded were the questions about foundations that were raised at the century's beginning. Hilary Putnam summed up one common view of the situation in the last third of the century by saying:
When philosophy discovers something wrong with science, sometimes science has to be changed—Russell's paradox comes to mind, as does Berkeley's attack on the actual infinitesimal—but more often it is philosophy that has to be changed. I do not think that the difficulties that philosophy finds with classical mathematics today are genuine difficulties; and I think that the philosophical interpretations of mathematics that we are being offered on every hand are wrong, and that "philosophical interpretation" is just what mathematics doesn't need.^{[31]}^{: 169–170 }
Philosophy of mathematics today proceeds along several different lines of inquiry, by philosophers of mathematics, logicians, and mathematicians, and there are many schools of thought on the subject. The schools are addressed separately in the next section, and their assumptions explained.
Contemporary schools of thought
Artistic
The view that claims that
Platonism
Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. The term Platonism is used because such a view is seen to parallel
A major question considered in mathematical Platonism is: Precisely where and how do the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, that is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One proposed answer is the
Fullblooded 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
Settheoretic realism (also settheoretic Platonism)^{}
Mathematicism
Max Tegmark's mathematical universe hypothesis (or mathematicism) goes further than Platonism in asserting that not only do all mathematical objects exist, but nothing else does. Tegmark's sole postulate is: All structures that exist mathematically also exist physically. That is, in the sense that "in those [worlds] complex enough to contain selfaware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world".^{[40]}^{[41]}
Logicism
Rudolf Carnap (1931) presents the logicist thesis in two parts:^{[42]}
 The concepts of mathematics can be derived from logical concepts through explicit definitions.
 The theorems of mathematics can be derived from logical axioms through purely logical deduction.
Gottlob Frege was the founder of logicism. In his seminal Die Grundgesetze der Arithmetik (Basic Laws of Arithmetic) he built up arithmetic from a system of logic with a general principle of comprehension, which he called "Basic Law V" (for concepts F and G, the extension of F equals the extension of G if and only if for all objects a, Fa equals Ga), a principle that he took to be acceptable as part of logic.
Frege's construction was flawed.
Modern logicists (like Bob Hale, Crispin Wright, and perhaps others) have returned to a program closer to Frege's. They have abandoned Basic Law V in favor of abstraction principles such as Hume's principle (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 onetoone correspondence). Frege required Basic Law V to be able to give an explicit definition of the numbers, but all the properties of numbers can be derived from Hume's principle. This would not have been enough for Frege because (to paraphrase him) it does not exclude the possibility that the number 3 is in fact Julius Caesar. In addition, many of the weakened principles that they have had to adopt to replace Basic Law V no longer seem so obviously analytic, and thus purely logical.
Formalism
Formalism holds that mathematical statements may be thought of as statements about the consequences of certain string manipulation rules. For example, in the "game" of Euclidean geometry (which is seen as consisting of some strings called "axioms", and some "rules of inference" to generate new strings from given ones), one can prove that the Pythagorean theorem holds (that is, one can generate the string corresponding to the Pythagorean theorem). According to formalism, mathematical truths are not about numbers and sets and triangles and the like—in fact, they are not "about" anything at all.
Another version of formalism is known as
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
Hilbert was initially a deductivist, but, as may be clear from above, he considered certain metamathematical methods to yield intrinsically meaningful results and was a realist with respect to the finitary arithmetic. Later, he held the opinion that there was no other meaningful mathematics whatsoever, regardless of interpretation.
Other formalists, such as Rudolf Carnap, Alfred Tarski, and Haskell Curry, considered mathematics to be the investigation of formal axiom systems. Mathematical logicians study formal systems but are just as often realists as they are formalists.
Formalists are relatively tolerant and inviting to new approaches to logic, nonstandard 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^{[}
Conventionalism
The French
Intuitionism
In mathematics, intuitionism is a program of methodological reform whose motto is that "there are no nonexperienced mathematical truths" (
A major force behind intuitionism was
is also rejected in most intuitionistic set theories, though in some versions it is accepted.In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science.
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 nonconstructive proofs, such as using proof by contradiction when showing the existence of an object or when trying to establish the truth of some proposition. Important work was done by Errett Bishop, who managed to prove versions of the most important theorems in real analysis as constructive analysis in his 1967 Foundations of Constructive Analysis.^{[46]}
Finitism
The most famous proponent of finitism was Leopold Kronecker,^{[47]} who said:
God created the natural numbers, all else is the work of man.
Ultrafinitism is an even more extreme version of finitism, which rejects not only infinities but finite quantities that cannot feasibly be constructed with available resources. Another variant of finitism is Euclidean arithmetic, a system developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets.^{[48]} Mayberry's system is Aristotelian in general inspiration and, despite his strong rejection of any role for operationalism or feasibility in the foundations of mathematics, comes to somewhat similar conclusions, such as, for instance, that superexponentiation is not a legitimate finitary function.
Structuralism
Structuralism is an
The ante rem structuralism ("before the thing") has a similar ontology to
The in re structuralism ("in the thing") is the equivalent of Aristotelian realism. Structures are held to exist inasmuch as some concrete system exemplifies them. This incurs the usual issues that some perfectly legitimate structures might accidentally happen not to exist, and that a finite physical world might not be "big" enough to accommodate some otherwise legitimate structures.
The post rem structuralism ("after the thing") is antirealist about structures in a way that parallels nominalism. Like nominalism, the post rem approach denies the existence of abstract mathematical objects with properties other than their place in a relational structure. According to this view mathematical systems exist, and have structural features in common. If something is true of a structure, it will be true of all systems exemplifying the structure. However, it is merely instrumental to talk of structures being "held in common" between systems: they in fact have no independent existence.
Embodied mind theories
The cognitive processes of patternfinding and distinguishing objects are also subject to neuroscience; if mathematics is considered to be relevant to a natural world (such as from realism or a degree of it, as opposed to pure solipsism).
Its actual relevance to reality, while accepted to be a trustworthy approximation (it is also suggested the evolution of perceptions, the body, and the senses may have been necessary for survival) is not necessarily accurate to a full realism (and is still subject to flaws such as illusion, assumptions (consequently; the foundations and axioms in which mathematics have been formed by humans), generalisations, deception, and hallucinations). As such, this may also raise questions for the modern scientific method for its compatibility with general mathematics; as while relatively reliable, it is still limited by what can be measured by empiricism which may not be as reliable as previously assumed (see also: 'counterintuitive' concepts in such as quantum nonlocality, and action at a distance).
Another issue is that one numeral system may not necessarily be applicable to problem solving. Subjects such as complex numbers or imaginary numbers require specific changes to more commonly used axioms of mathematics; otherwise they cannot be adequately understood.
Alternatively, computer programmers may use
As perceptions from the human brain are subject to illusions, assumptions, deceptions, (induced) hallucinations, cognitive errors or assumptions in a general context, it can be questioned whether they are accurate or strictly indicative of truth (see also: philosophy of being), and the nature of empiricism itself in relation to the universe and whether it is independent to the senses and the universe.
The human mind has no special claim on reality or approaches to it built out of math. If such constructs as Euler's identity are true then they are true as a map of the human mind and cognition.
Embodied mind theorists thus explain the effectiveness of mathematics—mathematics was constructed by the brain in order to be effective in this universe.
The most accessible, famous, and infamous treatment of this perspective is
Aristotelian realism
Aristotelian realism holds that mathematics studies properties such as symmetry, continuity and order that can be literally realized in the physical world (or in any other world there might be). It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized. For example, the number 4 is realized in the relation between a heap of parrots and the universal "being a parrot" that divides the heap into so many parrots.^{[50]}^{[51]} 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 that when an egg carton is opened, a set of three eggs is perceived (that is, a mathematical entity realized in the physical world).^{[52]} A problem for Aristotelian realism is what account to give of higher infinities, which may not be realizable in the physical world.
The Euclidean arithmetic developed by John Penn Mayberry in his book The Foundations of Mathematics in the Theory of Sets^{[48]} also falls into the Aristotelian realist tradition. Mayberry, following Euclid, considers numbers to be simply "definite multitudes of units" realized in nature—such as "the members of the London Symphony Orchestra" or "the trees in Birnam wood". Whether or not there are definite multitudes of units for which Euclid's Common Notion 5 (the whole is greater than the part) fails and which would consequently be reckoned as infinite is for Mayberry essentially a question about Nature and does not entail any transcendental suppositions.
Psychologism
Empiricism
Karl Popper was another philosopher to point out empirical aspects of mathematics, observing that "most mathematical theories are, like those of physics and biology, hypotheticodeductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."^{[54]} Popper also noted he would "admit a system as empirical or scientific only if it is capable of being tested by experience."^{[55]}
Contemporary mathematical empiricism, formulated by
Putnam strongly rejected the term "
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
For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy's Realism in Mathematics. Another example of a realist theory is the embodied mind theory.
For experimental evidence suggesting that human infants can do elementary arithmetic, see Brian Butterworth.
Fictionalism
Mathematical fictionalism was brought to fame in 1980 when
Having shown how to do science without using numbers, Field proceeded to rehabilitate mathematics as a kind of
Another fictionalist, Mary Leng, expresses the perspective succinctly by dismissing any seeming connection between mathematics and the physical world as "a happy coincidence". This rejection separates fictionalism from other forms of antirealism, which see mathematics itself as artificial but still bounded or fitted to reality in some way.^{[58]}
By this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about nonmathematical physics, and about
Social constructivism
Social constructivism sees mathematics primarily as a social construct, as a product of culture, subject to correction and change. Like the other sciences, mathematics is viewed as an empirical endeavor whose results are constantly evaluated and may be discarded. However, while on an empiricist view the evaluation is some sort of comparison with "reality", social constructivists emphasize that the direction of mathematical research is dictated by the fashions of the social group performing it or by the needs of the society financing it. However, although such external forces may change the direction of some mathematical research, there are strong internal constraints—the mathematical traditions, methods, problems, meanings and values into which mathematicians are enculturated—that work to conserve the historicallydefined 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 evolves, the status of previous mathematics is cast into doubt, and is corrected to the degree it is required or desired by the current mathematical community. This can be seen in the development of analysis from reexamination of the calculus of Leibniz and Newton. They argue further that finished mathematics is often accorded too much status, and folk mathematics not enough, due to an overemphasis on axiomatic proof and peer review as practices.
The social nature of mathematics is highlighted in its
Contributions to this school have been made by Imre Lakatos and Thomas Tymoczko, although it is not clear that either would endorse the title.^{[clarification needed]} More recently Paul Ernest has explicitly formulated a social constructivist philosophy of mathematics.^{[59]} 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,^{[60]} similar to but not quite the same as that associated with Alvin White;^{[61]} one of Hersh's coauthors, Philip J. Davis, has expressed sympathy for the social view as well.
Beyond the traditional schools
Unreasonable effectiveness
Rather than focus on narrow debates about the true nature of mathematical truth, 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.
Popper's two senses of number statements
Realist and constructivist theories are normally taken to be contraries. However, Karl Popper^{[62]} 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.^{[63]}
Philosophy of language
This section possibly contains original research. (February 2023) 
Innovations in the philosophy of language during the 20th century renewed interest in whether mathematics is, as is often said,^{[}
Mohan Ganesalingam has analysed mathematical language using tools from formal linguistics.^{}
Arguments
Indispensability argument for realism
This argument, associated with
 One must have ontologicalcommitments 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.^{}[66]
The justification for the first premise is the most controversial. Both Putnam and Quine invoke naturalism to justify the exclusion of all nonscientific 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 wellconfirmed theories. This puts the nominalist who wishes to exclude the existence of sets and nonEuclidean geometry, but to include the existence of quarks and other undetectable entities of physics, for example, in a difficult position.^{[66]}
Epistemic argument against realism
The
Field developed his views into
The argument hinges on the idea that a satisfactory
Another line of defense is to maintain that abstract objects are relevant to mathematical reasoning in a way that is noncausal, and not analogous to perception. This argument is developed by
A more radical defense is denial of physical reality, i.e. the mathematical universe hypothesis. In that case, a mathematician's knowledge of mathematics is one mathematical object making contact with another.
Aesthetics
This section needs additional citations for verification. (November 2015) 
Many practicing mathematicians have been drawn to their subject because of a sense of beauty 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
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 √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 bestknown 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.
Journals
See also
 Definitions of mathematics
 Formal language
 Foundations of mathematics
 Golden ratio
 Model theory
 Nonstandard analysis
 Philosophy of language
 Philosophy of logic
 Philosophy of science
 Philosophy of physics
 Philosophy of probability
 Rule of inference
 Science studies
 Scientific method
Related works
 The Analyst
 Euclid's Elements
 "On Formally Undecidable Propositions of Principia Mathematica and Related Systems"
 "On Computable Numbers, with an Application to the Entscheidungsproblem"
 Introduction to Mathematical Philosophy
 "New Foundations for Mathematical Logic"
 Principia Mathematica
 The Simplest Mathematics
Historical topics
Notes
 ^ This does not mean to make explicit all inference rules that are used. On the contrary, this is generally impossible, without computers and proof assistants. Even with this modern technology, it may take years of human work for writing down a completely detailed proof.
 ^ This does not mean that empirical evidence and intuition are not needed for choosing the theorems to be proved and to prove them.
 ^ Even if some people could consider that more accurates definitions were needed, they were unable to provide them.
References
 ^ Balaguer, Mark (2016). "Platonism in Metaphysics". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Spring 2016 ed.). Metaphysics Research Lab, Stanford University. Archived from the original on January 30, 2022. Retrieved April 2, 2022.
 ^ ISSN 1027488X.
 . 189303; also in Newman, J. R. (1956). The World of Mathematics. Vol. 4. New York: Simon and Schuster. pp. 2348–2364.
 (PDF) from the original on December 5, 2022. Retrieved November 21, 2022.
 (PDF) from the original on February 2, 2023. Retrieved November 11, 2022.
 ^ Perminov, V. Ya. (1988). "On the Reliability of Mathematical Proofs". Philosophy of Mathematics. 42 (167 (4)). Revue Internationale de Philosophie: 500–508.
 S2CID 149753721.
 . Retrieved November 16, 2022.
 . Retrieved November 17, 2022.
 ^ Frigg, Roman; Hartmann, Stephan (February 4, 2020). "Models in Science". Stanford Encyclopedia of Philosophy. Archived from the original on November 17, 2022. Retrieved November 17, 2022.
 ISBN 9783319724782. Retrieved November 17, 2022.
 ^ "The science checklist applied: Mathematics". Understanding Science. University of California, Berkeley. Archived from the original on October 27, 2019. Retrieved October 27, 2019.
 ISBN 9780750301060. Retrieved March 19, 2023.
 . Retrieved April 5, 2020.
 .
 .
 ^ Pigliucci, Massimo (2014). "Are There 'Other' Ways of Knowing?". Philosophy Now. Archived from the original on May 13, 2020. Retrieved April 6, 2020.
 from the original on February 28, 2011.
 .
 ^ Wagstaff, Samuel S. Jr. (2021). "History of Integer Factoring" (PDF). In Bos, Joppe W.; Stam, Martijn (eds.). Computational Cryptography, Algorithmic Aspects of Cryptography, A Tribute to AKL. London Mathematical Society Lecture Notes Series 469. Cambridge University Press. pp. 41–77. Archived (PDF) from the original on November 20, 2022. Retrieved November 20, 2022.
 ^ "Curves: Ellipse". MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on October 14, 2022. Retrieved November 20, 2022.
 ^ Mukunth, Vasudevan (September 10, 2015). "Beyond the Surface of Einstein's Relativity Lay a Chimerical Geometry". The Wire. Archived from the original on November 20, 2022. Retrieved November 20, 2022.
 JSTOR 20022840.
 .
 .
 ^ "Is mathematics discovered or invented?". University of Exeter. Archived from the original on 27 July 2018. Retrieved 28 March 2018.
 ^ "Math: Discovered, Invented, or Both?". pbs.org. 13 April 2015. Archived from the original on 28 March 2018. Retrieved 28 March 2018.
 ^ Morris Kline (1990), Mathematical Thought from Ancient to Modern Times, page 32. Oxford University Press.
 ^ Kleene, Stephen (1971). Introduction to Metamathematics. Amsterdam, Netherlands: NorthHolland Publishing Company. p. 5.
 ^ Mac Lane, Saunders (1998), Categories for the Working Mathematician, 2nd edition, SpringerVerlag, New York, NY.
 ^ *Putnam, Hilary (1967), "Mathematics Without Foundations", Journal of Philosophy 64/1, 522. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996).
 ^ "A Mathematician's Apology Quotes by G.H. Hardy". Archived from the original on 20210508. Retrieved 20200720.
 S2CID 4212863.
 ^ "Platonism in Metaphysics". Platonism in Metaphysics (Stanford Encyclopedia of Philosophy). Metaphysics Research Lab, Stanford University. 2016. Archived from the original on 20190428. Retrieved 20180824.
 ^ "Platonism in the Philosophy of Mathematics". "Platonism in the Philosophy of Mathematics", (Stanford Encyclopedia of Philosophy). Metaphysics Research Lab, Stanford University. 2018. Archived from the original on 20181125. Retrieved 20180817.
 ^ Ivor GrattanGuinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, Routledge, 2002, p. 681.
 ^ "Naturalism in the Philosophy of Mathematics". Naturalism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy). Metaphysics Research Lab, Stanford University. 2016. Archived from the original on 20180611. Retrieved 20180818.
 .
 .
 .
 ^ Tegmark (1998), p. 1.
 ^ ^{a} ^{b} Carnap, Rudolf (1931), "Die logizistische Grundlegung der Mathematik", Erkenntnis 2, 91121. 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).
 ^ Alexander Paseau; Fabian Pregel. Deductivism in the Philosophy of Mathematics. Metaphysics Research Lab, Stanford University.
 ^ Zach, Richard (2019), "Hilbert's Program", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2019 ed.), Metaphysics Research Lab, Stanford University, archived from the original on 20220208, retrieved 20190525
 ^ Audi, Robert (1999), The Cambridge Dictionary of Philosophy, Cambridge University Press, Cambridge, UK, 1995. 2nd edition. Page 542.
 ISBN 9784871877145
 H. M. Weber's memorial article, as quoted and translated in Gonzalez Cabillon, Julio (20000203). "FOM: What were Kronecker's f.o.m.?". Archivedfrom the original on 20071009. Retrieved 20080719. Gonzalez gives as the sources for the memorial article, the following: Weber, H: "Leopold Kronecker", Jahresberichte der Deutschen Mathematiker Vereinigung, vol ii (1893), pp. 531. Cf. page 19. See also Mathematische Annalen vol. xliii (1893), pp. 125.
 ^ ^{a} ^{b} Mayberry, J.P. (2001). The Foundations of Mathematics in the Theory of Sets. Cambridge University Press.
 ISBN 9780415960472.
 .
 . Retrieved 30 June 2021.
 ^ Maddy, Penelope (1990), Realism in Mathematics, Oxford University Press, Oxford, UK.
 ISBN 9780486200101.
 .
 .
 .
 ^ Field, Hartry, Science Without Numbers, Blackwell, 1980.
 ISBN 9780199280797.
 ^ Ernest, Paul. "Is Mathematics Discovered or Invented?". University of Exeter. Archived from the original on 20080405. Retrieved 20081226.
 ^ Hersh, Reuben (February 10, 1997). "What Kind of a Thing is a Number?" (Interview). Interviewed by John Brockman. Edge Foundation. Archived from the original on May 16, 2008. Retrieved 20081226.
 ^ "Humanism and Mathematics Education". Math Forum. Humanistic Mathematics Network Journal. Archived from the original on 20080724. Retrieved 20081226.
 ^ Popper, Karl Raimund (1946) Aristotelian Society Supplementary Volume XX.
 ^ Gregory, Frank Hutson (1996) "Arithmetic and Reality: A Development of Popper's Ideas". City University of Hong Kong. Republished in Philosophy of Mathematics Education Journal No. 26 (December 2011)
 ^ S2CID 14260721.
 ^ Yablo, S. (November 8, 1998). "A Paradox of Existence". Archived from the original on January 7, 2020. Retrieved August 26, 2019.
 ^ ^{a} ^{b} Putnam, H. Mathematics, Matter and Method. Philosophical Papers, vol. 1. Cambridge: Cambridge University Press, 1975. 2nd. ed., 1985.
 ^ Field, Hartry, 1989, Realism, Mathematics, and Modality, Oxford: Blackwell, p. 68
 ^ "Since abstract objects are outside the nexus of causes and effects, and thus perceptually inaccessible, they cannot be known through their effects on us" — Katz, J. Realistic Rationalism, 2000, p. 15
 ^ "Philosophy Now: "Mathematical Knowledge: A dilemma"". Archived from the original on February 7, 2011.
 ^ "Platonism in the Philosophy of Mathematics". Standard Encyclopaedia of Philosophy. Metaphysics Research Lab, Stanford University. 2018. Archived from the original on 20101204. Retrieved 20110213.
 ^ Review Archived 20110514 at the Wayback Machine of The Emperor's New Mind.
Further reading
 ISBN 9781107268135.
 Hart, W. D. (1996). Wilbur Dyre Hart (ed.). The Philosophy of Mathematics. Oxford University Press. .
 Irvine, A., ed. (2009). The Philosophy of Mathematics. Handbook of the Philosophy of Science. NorthHolland Elsevier. ISBN 9780080930589.
 .
 .
 .
External links
 Philosophy of mathematics at PhilPapers
 Philosophy of mathematics at the Indiana Philosophy Ontology Project
 Horsten, Leon. "Philosophy of Mathematics". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
 "Philosophy of mathematics". Internet Encyclopedia of Philosophy.
 The London Philosophy Study Guide Archived 20090923 at the Wayback Machine offers many suggestions on what to read, depending on the student's familiarity with the subject:
 Philosophy of Mathematics Archived 20090620 at the Wayback Machine
 Mathematical Logic Archived 20090125 at the Wayback Machine
 Set Theory & Further Logic Archived 20090227 at the Wayback Machine
 R.B. Jones' philosophy of mathematics page
 Philosophy of mathematics at Curlie
 Corfield, David. "The Philosophy of Real Mathematics – Blog".
 ISBN 9780253007810.