User:Farkle Griffen/Draft for "Mathematical object": Difference between revisions - Wikipedia
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==== Quine-Putnam indispensability ==== |
==== Quine-Putnam indispensability ==== |
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[[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|natural sciences]]. 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|Hilbert spaces]] in [[quantum mechanics]] and [[differential geometry]] in [[general relativity]] to [[Biology]]'s use of [[Chaos theory|chaos thoery]] and [[statistics]] (see [[Mathematical and theoretical biology|Mathematical biology]]), not only does mathematics help with [[Prediction|predictions]], 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 indispensibility 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 |work=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. |
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(Premise 2) Mathematical entities are indispensable to our best scientific theories. |
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([[Logical consequence|Conclusion]]) We ought to have ontological commitment to mathematical entities</blockquote> |
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=== Schools of thought === |
=== Schools of thought === |
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==== Platonism ==== |
==== Platonism ==== |
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[[Platonism]] asserts that mathematical objects are seen as real, [[Abstract and concrete|abstract entities]] that exist independently of human [[thought]]. Just as [[Physical object|physical objects]] like [[Electron|electrons]] and [[Planet|planets]] 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 |work=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> |
[[Platonism]] asserts that mathematical objects are seen as real, [[Abstract and concrete|abstract entities]] that exist independently of human [[thought]]. Just as [[Physical object|physical objects]] like [[Electron|electrons]] and [[Planet|planets]] 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 |work=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> |
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Some some notable platonists include: |
Some some notable platonists include: |
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==== Nominalism ==== |
==== Nominalism ==== |
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[[Nominalism]] denies the independent existence of mathematical objects. Instead, it suggests that they are merely [[Convenient fiction|convenient fictions]] or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects don't 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 |work=The Stanford Encyclopedia of Philosophy |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}}</ref> |
[[Nominalism]] denies the independent existence of mathematical objects. Instead, it suggests that they are merely [[Convenient fiction|convenient fictions]] or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects don't 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 |work=The Stanford Encyclopedia of Philosophy |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}}</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> |
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Some notable nominalists incluse: |
Some notable nominalists incluse: |
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==== Formalism ==== |
==== Formalism ==== |
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[[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 philosphers consider logicism to be a type of formalism.<ref>{{Citation |last=Weir |first=Alan |title=Formalism in the Philosophy of Mathematics |date=2024 |work=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> |
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(This section is intensionally left blank and will be filled in later. If you believe you can add something to this section, please don't hesitate to do so) |
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Some notable formalists include: |
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* '''[[David Hilbert]]''': A leading mathematician of the early 20th century, Hilbert is one of the most prominent advocates of formalism. 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. |
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* '''[[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 |last=Bell |first=John L. |title=Hermann Weyl |date=2024 |work=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> |
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==== Constructivism ==== |
==== Constructivism ==== |
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[[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 |last=Bridges |first=Douglas |title=Constructive Mathematics |date=2022 |work=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 the program of [[intuitionism]] founded by [[Luitzen Egbertus Jan Brouwer|Brouwer]], 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 (identifier)|ISBN]] [[Special:BookSources/4-87187-714-0|<bdi>4-87187-714-0</bdi>]].</ref> Constructivism also includes the study of [[Constructive set theory|constructive set theories]] such as [[CZF|Constructive Zermelo–Fraenkel]] and the study of philosophy. |
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(This section is intensionally left blank and will be filled in later. If you believe you can add something to this section, please don't hesitate to do so) |
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==== Structuralism ==== |
==== Structuralism ==== |
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[[Structuralism (philosophy of mathematics)|Mathematical 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 |last=Reck |first=Erich |title=Structuralism in the Philosophy of Mathematics |date=2023 |work=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> |
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(This section is intensionally left blank and will be filled in later. If you believe you can add something to this section, please don't hesitate to do so) |
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Some notable structuralists include: |
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* '''[[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. |
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* '''[[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 (identifier)|ISBN]] [[Special:BookSources/0-19-513930-5|0-19-513930-5]]</ref> |
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=== Objects versus mappings === |
=== Objects versus mappings === |
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Revision as of 01:22, 28 August 2024
In philosophy of mathematics
Nature of mathematical objects
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Quine-Putnam indispensability
Quine-Putnam indispensability is an argument for the existence of mathematical objects based on their unreasonable effectiveness in the natural sciences. Every branch of science relies largely on large and often vastly different areas of mathematics. From physics' use of Hilbert spaces in quantum mechanics and differential geometry in general relativity to Biology's use of chaos thoery and statistics (see Mathematical biology), not only does mathematics help with predictions, it allows these areas to have an elegant 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 indispensibility 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:[1]
(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.
(Conclusion) We ought to have ontological commitment to mathematical entities
Schools of thought
Platonism
Platonism asserts that mathematical objects are seen as real, abstract entities that exist independently of human thought. Just as physical objects like electrons and planets exist, so do numbers and sets. And just as statements about electrons and planets are true or false as these objects contain perfectly objective properties, so are statements about numbers and sets. Mathematicians discover these objects rather than invent them.[2][3]
Some some notable platonists include:
- Plato: The ancient Greek philosopher who, though not a mathematician, laid the groundwork for Platonism by positing the existence of an abstract realm of perfect 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, believing in the existence of a mathematical reality.
- Roger Penrose: A contemporary 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.[4]
Nominalism
Nominalism denies the independent existence of mathematical objects. Instead, it suggests that they are merely convenient fictions or shorthand for describing relationships and structures within our language and theories. Under this view, mathematical objects don't have an existence beyond the symbols and concepts we use.[5][6]
Some notable nominalists incluse:
- 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 philosopher who has developed the form of nominalism called "fictionalism," which argues that mathematical statements are useful fictions that don't correspond to any actual abstract objects.
Logicism
Logicism asserts that all mathematical truths can be reduced to logical truths, and all 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, theorems, and truths can be derived from purely logical principles and definitions. Logicism faced challenges, particularly with the Russillian axioms, Axiom of Choice and his 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 complete and consistent. This meant that not all mathematical truths could be derived purely from a logical system, undermining the logicist program.[7]
Some notable logicists include:
- Gottlob Frege: Frege is often regarded as the founder of logicism. In his work, 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.
- 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 and influenced subsequent work in mathematical logic and the philosophy of mathematics.
Formalism
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 philosphers consider logicism to be a type of formalism.[8]
Some notable formalists include:
- David Hilbert: A leading mathematician of the early 20th century, Hilbert is one of the most prominent advocates of formalism. 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.
- 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.[9]
Constructivism
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.[10] There are many forms of constructivism.[11] These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of mathematicians Shanin and Markov, and Bishop's program of constructive analysis.[12] Constructivism also includes the study of constructive set theories such as Constructive Zermelo–Fraenkel and the study of philosophy.
Structuralism
Mathematical 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.[13][14]
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.[15]
Objects versus mappings
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Gödel and the incompleteness theorems
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In category theory
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In foundations of mathematics
Set theory
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Mathematical logic
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Type theory
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- ^ Colyvan, Mark (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Indispensability Arguments in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
- ^ Linnebo, Øystein (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Platonism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-27
- ^ "Platonism, Mathematical | Internet Encyclopedia of Philosophy". Retrieved 2024-08-28.
- ^ Roibu, Tib (2023-07-11). "Sir Roger Penrose". Geometry Matters. Retrieved 2024-08-27.
- ^ Bueno, Otávio (2020), Zalta, Edward N. (ed.), "Nominalism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Fall 2020 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-27
- ^ "Mathematical Nominalism | Internet Encyclopedia of Philosophy". Retrieved 2024-08-28.
- ^ Tennant, Neil (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Logicism and Neologicism", The Stanford Encyclopedia of Philosophy (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-27
- ^ Weir, Alan (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Formalism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
- ^ Bell, John L.; Korté, Herbert (2024), Zalta, Edward N.; Nodelman, Uri (eds.), "Hermann Weyl", The Stanford Encyclopedia of Philosophy (Summer 2024 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
- ^ Bridges, Douglas; Palmgren, Erik; Ishihara, Hajime (2022), Zalta, Edward N.; Nodelman, Uri (eds.), "Constructive Mathematics", The Stanford Encyclopedia of Philosophy (Fall 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
- ^ Troelstra, Anne Sjerp (1977a). "Aspects of Constructive Mathematics". Handbook of Mathematical Logic. 90: 973–1052. doi:10.1016/S0049-237X(08)71127-3
- ^ Bishop, Errett (1967). Foundations of Constructive Analysis. New York: Academic Press. ISBN 4-87187-714-0.
- ^ "Structuralism, Mathematical | Internet Encyclopedia of Philosophy". Retrieved 2024-08-28.
- ^ Reck, Erich; Schiemer, Georg (2023), Zalta, Edward N.; Nodelman, Uri (eds.), "Structuralism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Spring 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-08-28
- ^ Philosophy of Mathematics: Structure and Ontology. Oxford University Press, 1997. ISBN 0-19-513930-5