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Platonism

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A '''mathematical object''' is an [[abstract concept]] arising in [[mathematics]].<ref>''[[Oxford English Dictionary]]'', s.v. “[[doi:10.1093/OED/1545759557|Mathematical (''adj.''), sense 2]],” September 2024. "''Designating or relating to objects apprehended not by sense perception but by thought or abstraction''."</ref> Typically, a mathematical object can be a value that can be assigned to a [[Glossary of mathematical symbols|symbol]], and therefore can be involved in [[formulas]]. Commonly encountered mathematical objects include [[number]]s, [[Expression (mathematics)|expressions]], [[shape]]s, [[function (mathematics)|functions]], and [[set (mathematics)|sets]]. Mathematical objects can be very complex; for example, [[theorem]]s, [[proof (mathematics)|proofs]], and even [[theory (mathematical logic)|formal theories]] are considered as mathematical objects in [[proof theory]].

In [[Philosophy of mathematics]], the concept of "mathematical objects" touches on topics of [[existence]], [[Identity (philosophy)|identity]], and the [[Nature (philosophy)|nature]] of [[reality]].<ref>{{Citation |last1=Rettler |first1=Bradley |title=Object |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/object/ |access-date=2024-08-28 |edition=Summer 2024 |publisher=Metaphysics Research Lab, Stanford University |last2=Bailey |first2=Andrew M. |editor2-last=Nodelman |editor2-first=Uri}}</ref> In [[metaphysics]], objects are often considered [[Entity|entities]] that possess [[Property (philosophy)|properties]] and can stand in various [[Relation (philosophy)|relations]] to one another.<ref name=":0">{{Cite book |last1=Carroll |first1=John W. |title=An introduction to metaphysics |last2=Markosian |first2=Ned |date=2010 |publisher=Cambridge University Press |isbn=978-0-521-82629-7 |edition=1. publ |series=Cambridge introductions to philosophy |location=Cambridge}}</ref> Philosophers debate whether mathematical objects have an independent existence outside of [[Thought|human thought]] ([[Metaphysical realism|realism]]), or if their existence is dependent on mental constructs or language ([[idealism]] and [[nominalism#Mathematical nominalism|nominalism]]). Objects can range from the [[Concrete object|concrete]]: such as [[physical objects]] usually studied in [[applied mathematics]], to the [[Abstraction (mathematics)|abstract]], studied in [[pure mathematics]]. What constitutes an "object" is foundational to many areas of philosophy, from [[ontology]] (the study of being) to [[epistemology]] (the study of knowledge). In mathematics, objects are often seen as entities that exist independently of the [[physical world]], raising questions about their ontological status.<ref>[[John P. Burgess|Burgess, John]], and Rosen, Gideon, 1997. ''A Subject with No Object: Strategies for Nominalistic Reconstrual of Mathematics''. [[Oxford University Press]]. {{isbn|0198236158}}</ref><ref>{{Citation |last1=Falguera |first1=José L. |title=Abstract Objects |date=2022 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/abstract-objects/ |access-date=2024-08-28 |edition=Summer 2022 |publisher=Metaphysics Research Lab, Stanford University |last2=Martínez-Vidal |first2=Concha |last3=Rosen |first3=Gideon}}</ref> There are varying [[School of thought|schools of thought]] which offer different perspectives on the matter, and many famous mathematicians and philosophers each have differing opinions on which is more correct.<ref>{{Citation |last=Horsten |first=Leon |title=Philosophy of Mathematics |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/philosophy-mathematics/ |access-date=2024-08-29 |edition=Winter 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>

== In philosophy of mathematics ==

=== Quine-Putnam indispensability ===
[[Quine–Putnam indispensability argument|Quine-Putnam indispensability]] is an argument for the existence of mathematical objects based on their [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|unreasonable effectiveness]] in the [[natural science]]s. Every [[Branches of science|branch of science]] relies largely on large and often vastly different areas of mathematics. From [[Physics|physics']] use of [[Hilbert space]]s in [[quantum mechanics]] and [[differential geometry]] in [[general relativity]] to [[biology]]'s use of [[chaos theory]] and [[combinatorics]] (see [[Mathematical and theoretical biology|mathematical biology]]), not only does mathematics help with [[prediction]]s, it allows these areas to have an elegant [[Language of mathematics|language]] to express these ideas. Moreover, it is hard to imagine how areas like quantum mechanics and general relativity could have developed without their assistance from mathematics, and therefore, one could argue that mathematics is ''indispensable'' to these theories. It is because of this unreasonable effectiveness and indispensability of mathematics that philosophers [[Willard Quine]] and [[Hilary Putnam]] argue that we should believe the mathematical objects for which these theories depend actually exist, that is, we ought to have an [[ontological commitment]] to them. The argument is described by the following [[syllogism]]:<ref>{{Citation |last=Colyvan |first=Mark |title=Indispensability Arguments in the Philosophy of Mathematics |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/mathphil-indis/ |access-date=2024-08-28 |edition=Summer 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref><blockquote>([[Premise]] 1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.

(Premise 2) Mathematical entities are indispensable to our best scientific theories.

([[Logical consequence|Conclusion]]) We ought to have ontological commitment to mathematical entities</blockquote>This argument resonates with a philosophy in [[applied mathematics]] called [[Metaphysical naturalism|Naturalism]]<ref>{{Citation |last=Paseau |first=Alexander |title=Naturalism in the Philosophy of Mathematics |date=2016 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/naturalism-mathematics/ |access-date=2024-08-28 |edition=Winter 2016 |publisher=Metaphysics Research Lab, Stanford University}}</ref> (or sometimes Predicativism)<ref>{{Citation |last=Horsten |first=Leon |title=Philosophy of Mathematics |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/philosophy-mathematics/ |access-date=2024-08-28 |edition=Winter 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref> which states that the only [[Authority|authoritative]] standards on existence are those of [[science]].

=== Schools of thought ===

==== Platonism ====
[[File:Plato-raphael.jpg|thumb|Plato depicted in ''[[The School of Athens]]'' by [[Raphael Sanzio]]]]
[[Platonism]] asserts that mathematical objects are seen as real, [[Abstract and concrete|abstract entities]] that exist independently of human [[thought]], often in some [[Platonic realm]]. Just as [[physical object]]s like [[electron]]s and [[planet]]s exist, so do numbers and sets. And just as [[Statement (logic)|statements]] about electrons and planets are true or false as these objects contain perfectly [[Property (philosophy)|objective properties]], so are statements about numbers and sets. Mathematicians discover these objects rather than invent them.<ref>{{Citation |last=Linnebo |first=Øystein |title=Platonism in the Philosophy of Mathematics |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/platonism-mathematics/ |access-date=2024-08-27 |edition=Summer 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref><ref>{{Cite web |title=Platonism, Mathematical {{!}} Internet Encyclopedia of Philosophy |url=https://iep.utm.edu/mathplat/ |access-date=2024-08-28 |language=en-US}}</ref> (See also: [[Philosophy of mathematics#Contemporary schools of thought|Mathematical Platonism]])

Some notable platonists include:

* '''[[Plato]]''': The ancient [[Ancient Greek philosophy|Greek philosopher]] who, though not a mathematician, laid the groundwork for Platonism by positing the existence of an abstract realm of perfect [[Theory of forms|forms]] or ideas, which influenced later thinkers in mathematics.
* '''[[Kurt Gödel]]''': A 20th-century logician and mathematician, Gödel was a strong proponent of mathematical Platonism, and his work in [[model theory]] was a major influence on [[Platonism#Modern platonism|modern platonism]]
* '''[[Roger Penrose]]''': A contemporary [[Mathematical physics|mathematical physicist]], Penrose has argued for a Platonic view of mathematics, suggesting that mathematical truths exist in a realm of abstract reality that we discover.<ref>{{Cite web |last=Roibu |first=Tib |date=2023-07-11 |title=Sir Roger Penrose |url=https://geometrymatters.com/sir-roger-penrose/ |access-date=2024-08-27 |website=Geometry Matters |language=en-US}}</ref>

==== Nominalism ====
[[Nominalism]] denies the independent existence of mathematical objects. Instead, it suggests that they are merely [[convenient fiction]]s or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects do not have an existence beyond the symbols and concepts we use.<ref>{{Citation |last=Bueno |first=Otávio |title=Nominalism in the Philosophy of Mathematics |date=2020 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/nominalism-mathematics/ |access-date=2024-08-27 |edition=Fall 2020 |publisher=Metaphysics Research Lab, Stanford University |author-link=Otávio Bueno |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref><ref>{{Cite web |title=Mathematical Nominalism {{!}} Internet Encyclopedia of Philosophy |url=https://iep.utm.edu/mathematical-nominalism/ |access-date=2024-08-28 |language=en-US}}</ref>

Some notable nominalists include:

* '''[[Nelson Goodman]]''': A philosopher known for his work in the philosophy of science and nominalism. He argued against the existence of abstract objects, proposing instead that mathematical objects are merely a product of our linguistic and symbolic conventions.
* '''[[Hartry Field]]''': A [[Contemporary philosophy|contemporary philosopher]] who has developed the form of nominalism called "[[fictionalism]]," which argues that mathematical statements are useful fictions that do not correspond to any actual abstract objects.<ref>{{Cite book |last=Field |first=Hartry |url=http://dx.doi.org/10.1093/acprof:oso/9780198777915.001.0001 |title=Science without Numbers |date=2016-10-27 |publisher=Oxford University Press |doi=10.1093/acprof:oso/9780198777915.001.0001 |isbn=978-0-19-877791-5}}</ref>

==== Logicism ====
[[Logicism]] asserts that all mathematical truths can be reduced to [[logical truth]]s, and a''ll'' objects forming the subject matter of those branches of mathematics are logical objects. In other words, mathematics is fundamentally a branch of [[logic]], and all mathematical concepts, [[theorem]]s, and truths can be derived from purely logical principles and definitions. Logicism faced challenges, particularly with the Russillian axioms, the Multiplicative axiom (now called the [[Axiom of choice|Axiom of Choice]]) and his [[Axiom of infinity|Axiom of Infinity]], and later with the discovery of [[Gödel's incompleteness theorems]], which showed that any sufficiently powerful [[formal system]] (like those used to express [[arithmetic]]) cannot be both [[Completeness (logic)|complete]] and [[Consistency|consistent]]. This meant that not all mathematical truths could be derived purely from a logical system, undermining the logicist program.<ref>{{Citation |last=Tennant |first=Neil |title=Logicism and Neologicism |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/logicism/ |access-date=2024-08-27 |edition=Winter 2023 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>

Some notable logicists include:

* '''[[Gottlob Frege]]''': Frege is often regarded as the founder of logicism. In his work, ''[[The Foundations of Arithmetic|Grundgesetze der Arithmetik]]'' (Basic Laws of Arithmetic), Frege attempted to show that arithmetic could be derived from logical axioms. He developed a formal system that aimed to express all of arithmetic in terms of logic. Frege's work laid the groundwork for much of [[modern logic]] and was highly influential, though it encountered difficulties, most notably [[Russell's paradox]], which revealed inconsistencies in Frege's system.<ref>{{Cite web |title=Frege, Gottlob {{!}} Internet Encyclopedia of Philosophy |url=https://iep.utm.edu/frege/ |access-date=2024-08-29 |language=en-US}}</ref>
* '''[[Bertrand Russell]]''': Russell, along with [[Alfred North Whitehead]], further developed logicism in their monumental work ''[[Principia Mathematica]]''. They attempted to derive all of mathematics from a set of [[logical axioms]], using a [[type theory]] to avoid the paradoxes that Frege's system encountered. Although ''Principia Mathematica'' was enormously influential, the effort to reduce all of mathematics to logic was ultimately seen as incomplete. However, it did advance the development of [[Formal Logic|mathematical logic]] and [[analytic philosophy]].<ref name="Glock 2008 p. 1">{{cite book |last=Glock |first=H.J. |url=https://books.google.com/books?id=WnvhtAEACAAJ |title=What is Analytic Philosophy? |publisher=Cambridge University Press |year=2008 |isbn=978-0-521-87267-6 |page=1 |access-date=2023-08-28}}</ref>

==== Formalism ====
[[Formalism (philosophy of mathematics)|Mathematical formalism]] treats objects as symbols within a [[formal system]]. The focus is on the manipulation of these symbols according to specified rules, rather than on the objects themselves. One common understanding of formalism takes mathematics as not a body of propositions representing an abstract piece of reality but much more akin to a game, bringing with it no more ontological commitment of objects or properties than playing [[ludo]] or [[chess]]. In this view, mathematics is about the consistency of formal systems rather than the discovery of pre-existing objects. Some philosophers consider logicism to be a type of formalism.<ref>{{Citation |last=Weir |first=Alan |title=Formalism in the Philosophy of Mathematics |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/formalism-mathematics/ |access-date=2024-08-28 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>

Some notable formalists include:

* '''[[David Hilbert]]''': A leading mathematician of the early 20th century, Hilbert is one of the most prominent advocates of formalism as a foundation of mathematics (see [[Hilbert's program]]). He believed that mathematics is a system of formal rules and that its truth lies in the consistency of these rules rather than any connection to an abstract reality.<ref>{{Cite book |last=Simons |first=Peter |url=https://books.google.com/books?id=mbn35b2ghgkC&q=formalism |title=Philosophy of Mathematics |publisher=Elsevier |year=2009 |isbn=9780080930589 |pages=292 |language=en |chapter=Formalism}}</ref>
* '''[[Hermann Weyl]]''': German mathematician and philosopher who, while not strictly a formalist, contributed to formalist ideas, particularly in his work on the foundations of mathematics.<ref>{{Citation |last1=Bell |first1=John L. |title=Hermann Weyl |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/weyl/ |access-date=2024-08-28 |edition=Summer 2024 |publisher=Metaphysics Research Lab, Stanford University |last2=Korté |first2=Herbert |editor2-last=Nodelman |editor2-first=Uri}}</ref> [[Freeman Dyson]] wrote that Weyl alone bore comparison with the "last great universal mathematicians of the nineteenth century", [[Henri Poincaré]] and [[David Hilbert]].<ref name="Dyson">{{cite journal |author=[[Freeman Dyson]] |date=10 March 1956 |title=Prof. Hermann Weyl, For.Mem.R.S. |journal=Nature |volume=177 |issue=4506 |pages=457–458 |bibcode=1956Natur.177..457D |doi=10.1038/177457a0 |s2cid=216075495 |quote="He alone could stand comparison with the last great universal mathematicians of the nineteenth century, Hilbert and Poincaré. ... Now he is dead, the contact is broken, and our hopes of comprehending the physical universe by a direct use of creative mathematical imagination are for the time being ended." |doi-access=free}}</ref>

==== Constructivism ====
[[Constructivism (philosophy of mathematics)|Mathematical constructivism]] asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a [[contradiction]] from that assumption. Such a [[proof by contradiction]] might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the [[existential quantifier]], which is at odds with its classical interpretation.<ref>{{Citation |last1=Bridges |first1=Douglas |title=Constructive Mathematics |date=2022 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/mathematics-constructive/ |access-date=2024-08-28 |edition=Fall 2022 |publisher=Metaphysics Research Lab, Stanford University |last2=Palmgren |first2=Erik |last3=Ishihara |first3=Hajime |editor2-last=Nodelman |editor2-first=Uri}}</ref> There are many forms of constructivism.<ref>[[Anne Sjerp Troelstra|Troelstra, Anne Sjerp]] (1977a). "Aspects of Constructive Mathematics". ''Handbook of Mathematical Logic''. '''90''': 973–1052. [[Doi (identifier)|doi]]:10.1016/S0049-237X(08)71127-3</ref> These include [[L. E. J. Brouwer|Brouwer]]'s program of [[intuitionism]], the [[finitism]] of [[David Hilbert|Hilbert]] and [[Paul Bernays|Bernays]], the constructive recursive mathematics of mathematicians [[Nikolai Aleksandrovich Shanin|Shanin]] and [[Andrey Markov (Soviet mathematician)|Markov]], and [[Errett Bishop|Bishop]]'s program of [[constructive analysis]].<ref>[[Errett Bishop|Bishop, Errett]] (1967). ''Foundations of Constructive Analysis''. New York: Academic Press. {{ISBN|4-87187-714-0}}.</ref> Constructivism also includes the study of [[Constructive set theory|constructive set theories]] such as [[CZF|Constructive Zermelo–Fraenkel]] and the study of philosophy.

Some notable constructivists include:

* '''[[L. E. J. Brouwer]]''': Dutch mathematician and philosopher regarded as one of the greatest mathematicians of the 20th century, known for (among other things) pioneering the [[intuitionism|intuitionist]] movement to mathematical logic, and opposition of David Hilbert's formalism movement (see: [[Brouwer–Hilbert controversy]]).
* '''[[Errett Bishop]]''': American mathematician known for his work on analysis. He is best known for developing [[constructive analysis]] in his 1967 ''Foundations of Constructive Analysis'', where he proved most of the important theorems in [[real analysis]] using constructivist methods.

==== Structuralism ====
[[Structuralism (philosophy of mathematics)|Structuralism]] suggests that mathematical objects are defined by their place within a structure or system. The nature of a number, for example, is not tied to any particular thing, but to its role within the system of [[arithmetic]]. In a sense, the thesis is that mathematical objects (if there are such objects) simply have no intrinsic nature.<ref>{{Cite web |title=Structuralism, Mathematical {{!}} Internet Encyclopedia of Philosophy |url=https://iep.utm.edu/m-struct/ |access-date=2024-08-28 |language=en-US}}</ref><ref>{{Citation |last1=Reck |first1=Erich |title=Structuralism in the Philosophy of Mathematics |date=2023 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/structuralism-mathematics/ |access-date=2024-08-28 |edition=Spring 2023 |publisher=Metaphysics Research Lab, Stanford University |last2=Schiemer |first2=Georg |editor2-last=Nodelman |editor2-first=Uri}}</ref>

Some notable structuralists include:

* '''[[Paul Benacerraf]]''': A philosopher known for his work in the philosophy of mathematics, particularly his paper "What Numbers Could Not Be," which argues for a structuralist view of mathematical objects.
* '''[[Stewart Shapiro]]''': Another prominent philosopher who has developed and defended structuralism, especially in his book ''Philosophy of Mathematics: Structure and Ontology''.<ref>''Philosophy of Mathematics: Structure and Ontology''. Oxford University Press, 1997. {{ISBN|0-19-513930-5}}</ref>

=== Objects versus mappings ===
[[File:Function_color_example_3.svg|thumb|In mathematics, a [[Map (mathematics)|'''map''' or '''mapping''']], is a function in the general sense; here as in the association of any of the four colored shapes in X to its color in Y.<ref>{{Cite book |last=Halmos |first=Paul R. |author-link=Paul Halmos |url=https://books.google.com/books?id=x6cZBQ9qtgoC |title=Naive set theory |date=1974 |publisher=Springer-Verlag |isbn=978-0-387-90092-6 |series=Undergraduate texts in mathematics |location=New York |pages=30}}</ref>]]
[[Gottlob Frege|Frege]] famously distinguished between ''[[Function (mathematics)|functions]]'' and ''objects''.<ref>{{Cite journal |last=Marshall |first=William |date=1953 |title=Frege's Theory of Functions and Objects |url=https://www.jstor.org/stable/2182877#:~:text=Functions,%20however,%20cannot%20exist%20apart,are%20independent%20or%20self-sufficient. |journal=The Philosophical Review |volume=62 |issue=3 |pages=374–390 |doi=10.2307/2182877 |jstor=2182877 |issn=0031-8108|url-access=subscription }}</ref> According to his view, a function is a kind of ‘incomplete’ [[entity]] that [[Map (mathematics)|maps]] arguments to values, and is denoted by an incomplete expression, whereas an object is a ‘complete’ entity and can be denoted by a singular term. Frege reduced [[Property (philosophy)|properties]] and [[Relation (philosophy)|relations]] to functions and so these entities are not included among the objects. Some authors make use of Frege's notion of ‘object’ when discussing abstract objects.<ref>{{Citation |last=Hale |first=Bob |title=Abstract objects |encyclopedia=Routledge Encyclopedia of Philosophy |date=2016 |url=http://dx.doi.org/10.4324/9780415249126-n080-1 |access-date=2024-08-28 |place=London |publisher=Routledge|doi=10.4324/9780415249126-n080-1 |isbn=978-0-415-25069-6 |url-access=subscription }}</ref> But though Frege's sense of ‘object’ is important, it is not the only way to use the term. Other philosophers include properties and relations among the abstract objects. And when the background context for discussing objects is [[type theory]], properties and relations of higher type (e.g., properties of properties, and properties of relations) may be all be considered ‘objects’. This latter use of ‘object’ is interchangeable with ‘entity.’ It is this more broad interpretation that mathematicians mean when they use the term 'object'.<ref>{{Citation |last1=Falguera |first1=José L. |title=Abstract Objects |date=2022 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/abstract-objects/ |access-date=2024-08-28 |edition=Summer 2022 |publisher=Metaphysics Research Lab, Stanford University |last2=Martínez-Vidal |first2=Concha |last3=Rosen |first3=Gideon}}</ref>

==See also==
* [[Abstract object]]
* [[Exceptional object]]
* [[Impossible object]]
* [[List of mathematical objects]]
* [[List of mathematical shapes]]
* [[List of shapes]]
* [[List of surfaces]]
* [[List of two-dimensional geometric shapes]]
* [[Mathematical structure]]

==Notes==
{{notelist}}

==References==
'''Citations'''
{{reflist}}

'''Further reading'''
* Azzouni, J., 1994. ''Metaphysical Myths, Mathematical Practice''. Cambridge University Press.
* Burgess, John, and Rosen, Gideon, 1997. ''A Subject with No Object''. Oxford Univ. Press.
* [[Philip J. Davis|Davis, Philip]] and [[Reuben Hersh]], 1999 [1981]. ''The Mathematical Experience''. Mariner Books: 156–62.
* [[Bonnie Gold|Gold, Bonnie]], and Simons, Roger A., 2011. ''[https://books.google.com/books?id=wPhwJdjI-dIC&q=%22mathematical+object%22 Proof and Other Dilemmas: Mathematics and Philosophy]''. Mathematical Association of America.
* Hersh, Reuben, 1997. ''What is Mathematics, Really?'' Oxford University Press.
* [[Anna Sfard|Sfard, A.]], 2000, "Symbolizing mathematical reality into being, Or how mathematical discourse and mathematical objects create each other," in Cobb, P., ''et al.'', ''Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools and instructional design''. Lawrence Erlbaum.
* [[Stewart Shapiro]], 2000. ''Thinking about mathematics: The philosophy of mathematics''. Oxford University Press.

==External links==
*''[[Stanford Encyclopedia of Philosophy]]'': "[http://plato.stanford.edu/entries/abstract-objects Abstract Objects]"—by Gideon Rosen.
*Wells, Charles. "[https://abstractmath.org/MM//MMMathObj.htm Mathematical Objects]".
*[https://web.archive.org/web/20100717234001/http://theory.cs.uvic.ca/amof/ AMOF: The Amazing Mathematical Object Factory]
*[https://web.archive.org/web/20100611203942/http://www.math.cuhk.edu.hk/exhibit/ Mathematical Object Exhibit]

{{Mathematical logic}}
{{Authority control}}

[[Category:Mathematical objects| ]]
[[Category:Category theory]]
[[Category:Mathematical concepts]]
[[Category:Mathematical Platonism]]

Mathematical object - Revision history

imported>AQIL TIJANI: /* Platonism */