Philosophy of mathematics  

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 +"[[2 + 2 = 5]]"--''[[Nineteen Eighty-Four]]'' (1949) by George Orwell
 +|}
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 +The '''philosophy of mathematics''' is the [[Branch (academia)|branch]] of [[philosophy]] that studies the assumptions, foundations, and implications of [[mathematics]], and to provide a viewpoint 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.
 +
 +The terms ''philosophy of mathematics'' and ''mathematical philosophy'' are frequently used interchangeably. The latter, however, may be used to refer to several other areas of study. One refers to a project of formalizing a philosophical subject matter, say, [[aesthetics]], [[ethics]], [[logic]], [[metaphysics]], or [[theology]], in a purportedly more exact and rigorous form, as for example the labors of [[scholasticism|scholastic]] [[theologian]]s, or the systematic aims of [[Gottfried Wilhelm Leibniz|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 to the foundations of mathematics taken by [[Bertrand Russell]] in his books ''[[The Principles of Mathematics]]'' and ''[[Introduction to Mathematical Philosophy]]''.
 +==Major themes==
 +===Mathematical realism===<!--Von Neumann universe links here-->
 +''Mathematical realism'', like [[philosophical realism|realism]] 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 the [[continuum hypothesis]] conjecture 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. Major forms of mathematical realism include [[Philosophy of mathematics#Platonism|Platonism]].
 +
 +===Mathematical anti-realism===
 +Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by [[correspondence theory of truth|corresponding]] to a special realm of immaterial or non-empirical entities. Major forms of mathematical anti-realism include [[Philosophy of mathematics#Formalism|Formalism]] and [[Philosophy of mathematics#Fictionalism|Fictionalism]].
 +
==See also== ==See also==
-*[[Meditations_on_First_Philosophy#Meditation_VI:_Concerning_the_Existence_of_Material_Things.2C_and_the_Real_Distinction_between_Mind_and_Body|Concerning existence of material things, and real distinction between mind and body (Descartes)]]+* [[Axiomatic set theory]]
-* [[Essence]]+* [[Axiomatic system]]
 +* [[Category theory]]
 +* [[Definitions of mathematics]]
 +* [[Formal language]]
 +* [[Formal system]]
 +* [[Foundations of mathematics]]
 +* [[Golden ratio]]
 +* [[History of mathematics]]
 +* [[Intuitionistic logic]]
* [[Logic]] * [[Logic]]
-* [[Metaethics]]+* [[Mathematical beauty]]
-* [[Nihilism]]+* [[Mathematical constructivism]]
-* [[Personal identity (philosophy)]]+* [[Mathematical logic]]
-* [[Philosophical logic]]+* [[Mathematical proof]]
-* [[Philosophical theology]]+* [[Metamathematics]]
-* [[Philosophy of Mathematics]]+* [[Model theory]]
-* [[Philosophy of physics]]+* [[Naive set theory]]
-* [[Pluralism (philosophy of mind)]]+* [[Non-standard analysis]]
-* [[Philosophical realism]]+* [[Philosophy of language]]
-* [[Substance theory]]+* [[Philosophy of science]]
-* [[Time travel]]+* [[Philosophy of probability]]
 +* [[Proof theory]]
 +* [[Rule of inference]]
 +* [[Science studies]]
 +* [[Scientific method]]
 +* [[Set theory]]
 +* ''[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]''
 +* [[Truth]]
 +*[[Ultimate ensemble]]
 + 
 +===Related works===
 +* ''[[The Analyst]]''
 +* [[Euclid's Elements|Euclid's ''Elements'']]
 +* [[Original proof of Gödel's completeness theorem|Gödel's completeness theorem]]
 +* ''[[Introduction to Mathematical Philosophy]]''
 +* ''[[New Foundations]]''
 +* ''[[Principia Mathematica]]''
 +* ''[[Charles Sanders Peirce bibliography#CP|The Simplest Mathematics]]''
 + 
 +{{col-end}}
 + 
 +===Historical topics===
 +* [[History and philosophy of science]]
 +* [[History of mathematics]]
 +* [[History of philosophy]]
 + 
{{GFDL}} {{GFDL}}

Revision as of 16:17, 3 May 2020

"2 + 2 = 5"--Nineteen Eighty-Four (1949) by George Orwell

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The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and to provide a viewpoint 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.

The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably. The latter, however, may be used to refer to several other areas of study. One refers to a project of formalizing a philosophical subject matter, say, aesthetics, ethics, logic, metaphysics, or theology, in a purportedly more exact and rigorous form, as for example the labors 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 to the foundations of mathematics taken by Bertrand Russell in his books The Principles of Mathematics and Introduction to Mathematical Philosophy.

Contents

Major themes

Mathematical realism

Mathematical realism, like realism 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; triangles, 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 the continuum hypothesis conjecture 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. Major forms of mathematical realism include Platonism.

Mathematical anti-realism

Mathematical anti-realism generally holds that mathematical statements have truth-values, but that they do not do so by corresponding to a special realm of immaterial or non-empirical entities. Major forms of mathematical anti-realism include Formalism and Fictionalism.

See also

Related works

Template:Col-end

Historical topics




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