View source for Philosophy:Definitions of mathematics

{{Short description|Aspect of the philosophy of mathematics}}
[[Mathematics]] has no generally accepted definition. Different schools of thought, particularly in [[Philosophy:Philosophy|philosophy]], have put forth radically different definitions. All proposed definitions are controversial in their own ways.<ref name=Mura/><ref name=Runge/>

== Early definitions ==
[[Biography:Pythagoras|Pythagoras]] stated "All is number. Number rules the universe", paraphrased by [[Biography:Plato|Plato]], from which [[Philosophy:Platonism|Platonism]] historically was the main mathematics school of thought and is still large.  His student [[Biography:Aristotle|Aristotle]] defined mathematics as "the science of quantity", and this definition prevailed until the 18th century. In his classification of the sciences, he further distinguished between [[Arithmetic|arithmetic]], which studies discrete quantities, and [[Geometry|geometry]] that studies continuous quantities. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart.<ref name=Franklin>{{Cite book |last=Franklin |first=James |editor-last=Irvine |editor-first=Andrew D. |date=2009 |chapter=Aristotelian Realism |title=Philosophy of Mathematics |chapter-url=https://books.google.com/books?id=mbn35b2ghgkC&pg=PA104 |url-status=live |publisher=Elsevier B.V. |pages=104–106 |isbn=978-0-08-093058-9 |archive-url=https://web.archive.org/web/20150906134402/https://books.google.com/books?id=mbn35b2ghgkC&pg=PA104 |archive-date=September 6, 2015 |access-date=July 1, 2020}}</ref>  [[Philosophy:Peripatetic school|Peripatetic]]/Aristotelian [[Philosophy:Philosophy of mathematics#Mathematical realism|Realism]] influenced most modern Realism.

[[Biography:Auguste Comte|Auguste Comte]]'s definition tried to explain the role of mathematics in coordinating phenomena in all other fields:<ref>Arline Reilein Standley, ''Auguste Comte,'' p. 61. Twayne Publishers (1981).</ref>

<blockquote>The science of indirect measurement.<ref name=Cajori>{{cite book|author=Cajori, Florian|title=A History of Mathematics|publisher=American Mathematical Society (1991 reprint)|year=1893|isbn=978-0-8218-2102-2|pages=[https://books.google.com/books?id=mGJRjIC9fZgC&pg=PA285 285–86]|}}</ref> <small>[[Biography:Auguste Comte|Auguste Comte]] 1851</small></blockquote>

The "indirectness" in Comte's definition refers to determining quantities that cannot be measured directly, such as the distance to planets or the size of atoms, by means of their relations to quantities that can be measured directly.<ref>Auguste Comte, ''The Philosophy of Mathematics,'' tr. W.M. Gillespie, [https://archive.org/details/philosophymathe01comtgoog/page/n35 <!-- pg=25 --> pp. 17–25]. Harper & Brothers, New York (1851).</ref>

== Greater abstraction and competing philosophical schools ==
In the 19th century, as mathematics branched out into abstract topics such as [[Group theory|group theory]] and [[Projective geometry|projective geometry]], which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.<ref name=Cajori />

Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different [[Philosophy:Philosophy of mathematics|philosophy of mathematics]]. However, each has its own flaws, none have achieved mainstream consensus, and all three appear irreconcilable.<ref name=Snapper>{{Cite journal |doi=10.2307/2689412 |title=The Three Crises in Mathematics: Logicism, Intuitionism, and Formalism |journal=Mathematics Magazine |date=September 1979 |first=Ernst |last=Snapper |volume=52 |issue=4 |pages=207–16 |jstor=2689412}}</ref>

=== Logicism ===
With mathematicians pursuing greater rigor and more abstract [[Philosophy:Foundations of mathematics|foundations]], some proposed defining mathematics purely in terms of [[Deductive reasoning|deduction]] and [[Logic|logic]]: 

<blockquote>Mathematics is the science that draws necessary conclusions.<ref name=eves>[https://books.google.com/books?id=-UzKwHWzdesC Foundations and fundamental concepts of mathematics By Howard Eves] page 150</ref> <small>[[Biography:Benjamin Peirce|Benjamin Peirce]] 1870</small></blockquote>

<blockquote>All Mathematics is Symbolic Logic.<ref name=Russell>Bertrand Russell, ''The Principles of Mathematics,'' [https://archive.org/details/principlesmathe00russgoog/page/n36 p. 5]. University Press, Cambridge (1903)</ref> <small>[[Biography:Bertrand Russell|Bertrand Russell]] 1903</small></blockquote>

Peirce did not think that mathematics is the same as logic, since he thought mathematics makes only hypothetical assertions, not [[Philosophy:Categorical proposition|categorical]] ones.<ref>Carl Boyer, [[Biography:Uta Merzbach|Uta Merzbach]], ''A History of Mathematics,'' [https://books.google.com/books?id=BokVHiuIk9UC&pg=PT426 p. 426]. John Wiley & Sons (2011).</ref> Russell's definition, on the other hand, expresses the logicist view without reservation.<ref name=Snapper />

=== Intuitionism ===
Rather than characterize mathematics by deductive logic, intuitionism views mathematics as primarily about the construction of ideas in the mind:<ref name=Snapper />

<blockquote>The only possible foundation of mathematics must be sought in this construction under the obligation carefully to watch which constructions intuition allows and which not.<ref name="van Atten">{{Cite encyclopedia |last=van Atten |first=Mark |title=The Development of Intuitionistic Logic |date=November 8, 2017 |url=https://plato.stanford.edu/entries/intuitionistic-logic-development/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |edition=Winter 2017 |publisher=Metaphysics Research Lab, Stanford University |access-date=January 29, 2022}}</ref> <small>[[Biography:L. E. J. Brouwer|L. E. J. Brouwer]] 1907</small></blockquote>

<blockquote>... intuitionist mathematics is nothing more nor less than an investigation of the utmost limits which the intellect can attain in its self-unfolding.<ref name="van Atten" /> <small>Arend Heyting 1968</small></blockquote>

Intuitionism sprang from the philosophy of mathematician L. E. J. Brouwer and also led to the development of a modified [[Intuitionistic logic|intuitionistic logic]]. As a result, intuitionism has generated some genuinely different results that, while coherent and valid, differ from some theorems grounded in classical logic.<ref name=Snapper />

=== Formalism ===
One other perspective, formalism, de-emphasizes logical or intuitive meanings altogether, grounding mathematics instead in its [[Symbol|symbol]]s and [[Philosophy:Syntax|syntax]] rules for manipulating them:<ref name=Snapper />

<blockquote>Mathematics is the science of formal systems.<ref>{{cite book |title=Outlines of a Formalist Philosophy of Mathematics |author1=Haskell Brooks Curry |publisher=Elsevier |year=1951 |isbn=978-0-444-53368-5 |page=56 |url=https://books.google.com/books?id=tZHrBQgp1bkC |}}</ref> <small>[[Biography:Haskell Curry|Haskell Curry]] 1951</small></blockquote>

=== Other views ===
Aside from the definitions above, other definitions approach mathematics by emphasizing the element of pattern, order or structure. For example:

<blockquote>Mathematics is the classification and study of all possible patterns.<ref name="Sawyer">{{cite book | url=https://books.google.com/books?id=6wF4SklxcnkC&pg=PA12 | title=Prelude to Mathematics | publisher=Penguin Books | author=Sawyer, W.W. |  year=1955 | pages=12 | isbn=978-0486244013}}</ref> <small>[[Biography:Walter Warwick Sawyer|Walter Warwick Sawyer]], 1955</small></blockquote>

Yet another approach is to make [[Abstraction (mathematics)|abstraction]] the defining criterion:

<blockquote>Mathematics is a broad-ranging field of study in which the properties and interactions of idealized objects are examined.<ref>{{Cite web|url=http://mathworld.wolfram.com/Mathematics.html|title=Mathematics|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-10-18}}</ref></blockquote>

== Contemporary general reference works ==
Most contemporary reference works define mathematics by summarizing its main topics and methods:

<blockquote>The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra. <ref>{{Cite OED|mathematics}} [https://web.archive.org/web/20180210180226/https://en.oxforddictionaries.com/definition/mathematics mathematics]</ref><small>Oxford English Dictionary, 1933</small></blockquote>

<blockquote>The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols.<ref>{{Cite American Heritage Dictionary|mathematics}}</ref> <small>American Heritage Dictionary, 2000</small></blockquote>

<blockquote>The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects.<ref>{{Britannica|369194|Mathematics}}</ref> <small>[[Encyclopædia Britannica]], 2006</small></blockquote>

== Playful, metaphorical, and poetic definitions ==
Bertrand Russell wrote this famous tongue-in-cheek definition, describing the way all terms in mathematics are ultimately defined by reference to undefined terms:

<blockquote>The subject in which we never know what we are talking about, nor whether what we are saying is true.<ref>{{Citation |last=Russell |first=Bertrand |date=1901 |title=Recent Work on the Principles of Mathematics |journal=International Monthly |volume=4}}</ref> <small>[[Biography:Bertrand Russell|Bertrand Russell]] 1901</small></blockquote>

Many other attempts to characterize mathematics have led to humor or poetic prose:

<blockquote>A mathematician is a blind man in a dark room looking for [[Philosophy:Black cat analogy|a black cat which isn't there.]]<ref>"Pi in the Sky", John Barrow</ref> <small>[[Biography:Charles Darwin|Charles Darwin]]</small><ref>{{cite book |title=The G.O.D. Experiments: How Science Is Discovering God In Everything, Including Us |edition=illustrated |first1=Gary E. |last1=Schwartz |publisher=Simon and Schuster |year=2007 |isbn=978-0-7434-7741-3 |page=209 |url=https://books.google.com/books?id=fJ18OrGBHsQC}} [https://books.google.com/books?id=fJ18OrGBHsQC&pg=PA209 Extract of page 209]</ref></blockquote>

<blockquote>A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.<ref>{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/Quotations/Hardy.html|title=Quotations by Hardy|website=www-history.mcs.st-andrews.ac.uk|access-date=2019-10-18}}</ref> <small>[[Biography:G. H. Hardy|G. H. Hardy]], 1940</small></blockquote>

<blockquote>Mathematics is the art of giving the same name to different things.<ref name=eves/> <small>[[Biography:Henri Poincaré|Henri Poincaré]]</small></blockquote>

<blockquote>Mathematics is the science of skillful operations with concepts and rules invented just for this purpose. [this purpose being the skillful operation ....]<ref>Wigner, Eugene P. (1960). "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," ''Communications in Pure and Applied Sciences'', 13(1960):1–14. Reprinted in ''Mathematics: People, Problems, Results,'' vol. 3, ed. Douglas M. Campbell and John C. Higgins, [https://books.google.com/books?id=FcgB818WAQgC&pg=PA116 p. 116]</ref> <small>[[Biography:Eugene Wigner|Eugene Wigner]]</small></blockquote>

<blockquote>Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its contour defined: it is limitless as that space which it finds too narrow for its aspirations; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life, which seems to slumber in each monad, in every atom of matter, in each leaf and bud cell, and is forever ready to burst forth into new forms of vegetable and animal existence.<ref name=StewartFrom>"From Here to Infinity", Ian Stewart</ref> <small>[[Biography:James Joseph Sylvester|James Joseph Sylvester]]</small></blockquote>

<blockquote>What is mathematics? What is it for? What are mathematicians doing nowadays? Wasn't it all finished long ago? How many new numbers can you invent anyway? Is today's mathematics just a matter of huge calculations, with the mathematician as a kind of zookeeper, making sure the precious computers are fed and watered? If it's not, what is it other than the incomprehensible outpourings of superpowered brainboxes with their heads in the clouds and their feet dangling from the lofty balconies of their ivory towers? Mathematics is all of these, and none. Mostly, it's just different. It's not what you expect it to be, you turn your back for a moment and it's changed. It's certainly not just a fixed body of knowledge, its growth is not confined to inventing new numbers, and its hidden tendrils pervade every aspect of modern life.<ref name=StewartFrom/> <small>[[Biography:Ian Stewart (mathematician)|Ian Stewart]]</small></blockquote>

== See also ==
* [[Philosophy:Philosophy of mathematics|Philosophy of mathematics]]
* [[Philosophy:Foundations of mathematics|Foundations of mathematics]]

== References ==
{{Reflist|refs=
<ref name="Mura">{{citation |last=Mura |first=Robert |date=December 1993 |title=Images of Mathematics Held by University Teachers of Mathematical Sciences |journal=Educational Studies in Mathematics |volume=25 |issue=4 |pages=375–385 |doi=10.1007/BF01273907 |jstor=3482762 |s2cid=122351146 }}</ref>
<ref name="Runge">{{citation |last1=Tobies |first1=Renate|last2=Neunzert |first2=Helmut |date=2012 |title=Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry |publisher=Springer |page=9 |isbn=978-3-0348-0229-1 |quote=It is first necessary to ask what is meant by ''mathematics'' in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form. |url=https://books.google.com/books?id=EDm0eQqFUQ4C&pg=PA9}}</ref>
}}

== Further reading ==
*{{citation |last1=Courant |first1=Richard |last2=Robbins |first2=Herbert |date=1996 |title=What Is Mathematics? |edition=2nd |publisher=Oxford University Press |isbn=978-0-19-510519-3|title-link=What Is Mathematics? }}
*{{citation |editor1-last=Gowers |editor1-first=Timothy |editor1-link=Timothy Gowers |editor2-last=Barrow-Green |editor2-first=June |editor3-last=Leader |editor3-first=Imre |editor3-link=Imre Leader |date=2008 |title=The Princeton Companion to Mathematics |publisher=Princeton University Press |isbn=978-0-691-11880-2|title-link=The Princeton Companion to Mathematics }}
*{{citation |last=Hersh |first=Reuben |date=1999 |title=What is Mathematics, Really? |publisher=Oxford University Press |isbn=978-0-19-513087-4 |url-access=registration |url=https://archive.org/details/whatismathematic00reub }}
*{{citation |last=Paulos |first=John Allen |date=1991 |title=Beyond Numeracy |journal=Nature |volume=359 |issue=6394 |pages=463–464 |publisher=Viking |isbn=978-0-670-83654-3|bibcode=1992Natur.359..463B |doi=10.1038/359463b0 |s2cid=30811417 }}
*{{citation |last=Stewart |first=Ian |date=1996 |title=From Here to Infinity |publisher=Oxford University Press |isbn=978-0-19-283202-3|title-link=From Here to Infinity (book) }}

[[Category:Philosophy of mathematics]]

{{Sourceattribution|Definitions of mathematics}}