In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess."[1] According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics). In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.
Along with realism and intuitionism, formalism is one of the main theories in the philosophy of mathematics that developed in the late nineteenth and early twentieth century. Among formalists, David Hilbert was the most prominent advocate.[2]
The early mathematical formalists attempted "to block, avoid, or sidestep (in some way) any ontological commitment to a problematic realm of abstract objects."[1] German mathematicians Eduard Heine and Carl Johannes Thomae are considered early advocates of mathematical formalism.[1] Heine and Thomae's formalism can be found in Gottlob Frege's criticisms in The Foundations of Arithmetic.
According to Alan Weir, the formalism of Heine and Thomae that Frege attacks can be "describe[d] as term formalism or game formalism."[1] Term formalism is the view that mathematical expressions refer to symbols, not numbers. Heine expressed this view as follows: "When it comes to definition, I take a purely formal position, in that I call certain tangible signs numbers, so that the existence of these numbers is not in question."[3]
Thomae is characterized as a game formalist who claimed that "[f]or the formalist, arithmetic is a game with signs which are called empty. That means that they have no other content (in the calculating game) than they are assigned by their behaviour with respect to certain rules of combination (rules of the game)."[4]
Frege provides three criticisms of Heine and Thomae's formalism: "that [formalism] cannot account for the application of mathematics; that it confuses formal theory with metatheory; [and] that it can give no coherent explanation of the concept of an infinite sequence."[5] Frege's criticism of Heine's formalism is that his formalism cannot account for infinite sequences. Dummett argues that more developed accounts of formalism than Heine's account could avoid Frege's objections by claiming they are concerned with abstract symbols rather than concrete objects.[6] Frege objects to the comparison of formalism with that of a game, such as chess.[7] Frege argues that Thomae's formalism fails to distinguish between game and theory.
A major figure of formalism was David Hilbert, whose program was intended to be a complete and consistent axiomatization of all of mathematics.[8] Hilbert aimed to show the consistency of mathematical systems from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent (i.e. no contradictions can be derived from the system).
The way that Hilbert tried to show that an axiomatic system was consistent was by formalizing it using a particular language.[9] In order to formalize an axiomatic system, you must first choose a language in which you can express and perform operations within that system. This language must include five components:
It must include variables such as x, which can stand for some number.
It must have quantifiers such as the symbol for the existence of an object.
It must include equality.
It must include connectives such as ↔ for "if and only if."
It must include certain undefined terms called parameters. For geometry, these undefined terms might be something like a point or a line, which we still choose symbols for.
By adopting this language, Hilbert thought that we could prove all theorems within any axiomatic system using nothing more than the axioms themselves and the chosen formal language.
Gödel's conclusion in his incompleteness theorems was that you cannot prove consistency within any consistent axiomatic system rich enough to include classical arithmetic. On the one hand, you must use only the formal language chosen to formalize this axiomatic system; on the other hand, it is impossible to prove the consistency of this language in itself.[9]Hilbert was originally frustrated by Gödel's work because it shattered his life's goal to completely formalize everything in number theory.[10] However, Gödel did not feel that he contradicted everything about Hilbert's formalist point of view.[11] After Gödel published his work, it became apparent that proof theory still had some use, the only difference is that it could not be used to prove the consistency of all of number theory as Hilbert had hoped.[10]
Hilbert was initially a deductivist,[citation needed] but 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.
Haskell Curry defines mathematics as "the science of formal systems."[13] Curry's formalism is unlike that of term formalists, game formalists, or Hilbert's formalism. For Curry, mathematical formalism is about the formal structure of mathematics and not about a formal system.[13]Stewart Shapiro describes Curry's formalism as starting from the "historical thesis that as a branch of mathematics develops, it becomes more and more rigorous in its methodology, the end-result being the codification of the branch in formal deductive systems."[14]
Criticisms of formalism
Kurt Gödel indicated one of the weak points of formalism by addressing the question of consistency in axiomatic systems.
Bertrand Russell has argued that formalism fails to explain what is meant by the linguistic application of numbers in statements such as "there are three men in the room".[15]
^ abcdWeir, Alan (2015), "Formalism in the Philosophy of Mathematics", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Spring 2015 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-05-25
^Simons, Peter (2009). "Formalism". Philosophy of Mathematics. Elsevier. p. 292. ISBN9780080930589.
^Simons, Peter (2009). Philosophy of Mathematics. Elsevier. p. 293. ISBN9780080930589.
^Frege, Gottlob (1903). The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number. Chicago: Northwestern University Press. p. 183.
^Dummett, Michael (1991). Frege: Philosophy of Mathematics. Cambridge: Harvard University Press. p. 252. ISBN9780674319356.
^Dummett, Michael (1991). Frege: Philosophy of Mathematics. Cambridge: Harvard University Press. p. 253. ISBN9780674319356.
^Frege, Gottlob; Ebert, Philip A.; Cook, Roy T. (1893). Basic Laws of Arithmetic: Derived using concept-script. Oxford: Oxford University Press (published 2013). pp. § 93. ISBN9780199281749.
^Zach, Richard (2019), "Hilbert's Program", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Summer 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-05-25
^ abSnapper, Ernst (September 1979). "The Three Crises in Mathematics: Logicism, Intuitionism and Formalism" (PDF). Mathematics Magazine. 52 (4): 207–216. doi:10.1080/0025570X.1979.11976784.
^ abReid, Constance; Weyl, Hermann (1970). Hilbert. Springer-Verlag. p. 198. ISBN9783662286159.
^Gödel, Kurt (1986). Feferman, Solomon (ed.). Kurt Gödel: Collected Works: Volume I: Publications 1929-1936. Vol. 1. Oxford: Oxford University Press. p. 195. ISBN9780195039641.
^Carnap, Rudolf (1937). Logical Syntax of Language. Routledge. pp. 325–328. ISBN9781317830597.
^ abCurry, Haskell B. (1951). Outlines of a Formalist Philosophy of Mathematics. Elsevier. p. 56. ISBN9780444533685.
^Shapiro, Stewart (2005). "Formalism". The Oxford Companion to Philosophy. Honderich, Ted (2nd ed.). Oxford: Oxford University Press. ISBN9780191532658. OCLC 62563098.
^Bertrand Russell My Philosophical Development, 1959, ch. X.
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March 04, 2023
formalism, philosophy, mathematics, this, article, multiple, issues, please, help, improve, discuss, these, issues, talk, page, learn, when, remove, these, template, messages, neutrality, this, article, disputed, relevant, discussion, found, talk, page, please. 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 The neutrality of this article is disputed Relevant discussion may be found on the talk page Please do not remove this message until conditions to do so are met March 2019 Learn how and when to remove this template message This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations April 2016 Learn how and when to remove this template message Learn how and when to remove this template message In the philosophy of mathematics formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings alphanumeric sequences of symbols usually as equations using established manipulation rules A central idea of formalism is that mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game bringing with it no more commitment to an ontology of objects or properties than ludo or chess 1 According to formalism the truths expressed in logic and mathematics are not about numbers sets or triangles or any other coextensive subject matter in fact they aren t about anything at all Rather mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation or semantics In contrast to mathematical realism logicism or intuitionism formalism s contours are less defined due to broad approaches that can be categorized as formalist Along with realism and intuitionism formalism is one of the main theories in the philosophy of mathematics that developed in the late nineteenth and early twentieth century Among formalists David Hilbert was the most prominent advocate 2 Contents 1 Early formalism 2 Hilbert s formalism 3 Further developments 4 Criticisms of formalism 5 See also 6 References 7 External linksEarly formalism EditThe early mathematical formalists attempted to block avoid or sidestep in some way any ontological commitment to a problematic realm of abstract objects 1 German mathematicians Eduard Heine and Carl Johannes Thomae are considered early advocates of mathematical formalism 1 Heine and Thomae s formalism can be found in Gottlob Frege s criticisms in The Foundations of Arithmetic According to Alan Weir the formalism of Heine and Thomae that Frege attacks can be describe d as term formalism or game formalism 1 Term formalism is the view that mathematical expressions refer to symbols not numbers Heine expressed this view as follows When it comes to definition I take a purely formal position in that I call certain tangible signs numbers so that the existence of these numbers is not in question 3 Thomae is characterized as a game formalist who claimed that f or the formalist arithmetic is a game with signs which are called empty That means that they have no other content in the calculating game than they are assigned by their behaviour with respect to certain rules of combination rules of the game 4 Frege provides three criticisms of Heine and Thomae s formalism that formalism cannot account for the application of mathematics that it confuses formal theory with metatheory and that it can give no coherent explanation of the concept of an infinite sequence 5 Frege s criticism of Heine s formalism is that his formalism cannot account for infinite sequences Dummett argues that more developed accounts of formalism than Heine s account could avoid Frege s objections by claiming they are concerned with abstract symbols rather than concrete objects 6 Frege objects to the comparison of formalism with that of a game such as chess 7 Frege argues that Thomae s formalism fails to distinguish between game and theory Hilbert s formalism Edit David Hilbert A major figure of formalism was David Hilbert whose program was intended to be a complete and consistent axiomatization of all of mathematics 8 Hilbert aimed to show the consistency of mathematical systems from the assumption that the finitary arithmetic a subsystem of the usual arithmetic of the positive integers chosen to be philosophically uncontroversial was consistent i e no contradictions can be derived from the system The way that Hilbert tried to show that an axiomatic system was consistent was by formalizing it using a particular language 9 In order to formalize an axiomatic system you must first choose a language in which you can express and perform operations within that system This language must include five components It must include variables such as x which can stand for some number It must have quantifiers such as the symbol for the existence of an object It must include equality It must include connectives such as for if and only if It must include certain undefined terms called parameters For geometry these undefined terms might be something like a point or a line which we still choose symbols for By adopting this language Hilbert thought that we could prove all theorems within any axiomatic system using nothing more than the axioms themselves and the chosen formal language Godel s conclusion in his incompleteness theorems was that you cannot prove consistency within any consistent axiomatic system rich enough to include classical arithmetic On the one hand you must use only the formal language chosen to formalize this axiomatic system on the other hand it is impossible to prove the consistency of this language in itself 9 Hilbert was originally frustrated by Godel s work because it shattered his life s goal to completely formalize everything in number theory 10 However Godel did not feel that he contradicted everything about Hilbert s formalist point of view 11 After Godel published his work it became apparent that proof theory still had some use the only difference is that it could not be used to prove the consistency of all of number theory as Hilbert had hoped 10 Hilbert was initially a deductivist citation needed but 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 Further developments EditOther formalists such as Rudolf Carnap considered mathematics to be the investigation of formal axiom systems 12 Haskell Curry defines mathematics as the science of formal systems 13 Curry s formalism is unlike that of term formalists game formalists or Hilbert s formalism For Curry mathematical formalism is about the formal structure of mathematics and not about a formal system 13 Stewart Shapiro describes Curry s formalism as starting from the historical thesis that as a branch of mathematics develops it becomes more and more rigorous in its methodology the end result being the codification of the branch in formal deductive systems 14 Criticisms of formalism EditKurt Godel indicated one of the weak points of formalism by addressing the question of consistency in axiomatic systems Bertrand Russell has argued that formalism fails to explain what is meant by the linguistic application of numbers in statements such as there are three men in the room 15 See also EditQED project Mathematical formalism Formalized mathematicsReferences Edit a b c d Weir Alan 2015 Formalism in the Philosophy of Mathematics in Zalta Edward N ed The Stanford Encyclopedia of Philosophy Spring 2015 ed Metaphysics Research Lab Stanford University retrieved 2019 05 25 Simons Peter 2009 Formalism Philosophy of Mathematics Elsevier p 292 ISBN 9780080930589 Simons Peter 2009 Philosophy of Mathematics Elsevier p 293 ISBN 9780080930589 Frege Gottlob 1903 The Foundations of Arithmetic A Logico Mathematical Enquiry Into the Concept of Number Chicago Northwestern University Press p 183 Dummett Michael 1991 Frege Philosophy of Mathematics Cambridge Harvard University Press p 252 ISBN 9780674319356 Dummett Michael 1991 Frege Philosophy of Mathematics Cambridge Harvard University Press p 253 ISBN 9780674319356 Frege Gottlob Ebert Philip A Cook Roy T 1893 Basic Laws of Arithmetic Derived using concept script Oxford Oxford University Press published 2013 pp 93 ISBN 9780199281749 Zach Richard 2019 Hilbert s Program in Zalta Edward N ed The Stanford Encyclopedia of Philosophy Summer 2019 ed Metaphysics Research Lab Stanford University retrieved 2019 05 25 a b Snapper Ernst September 1979 The Three Crises in Mathematics Logicism Intuitionism and Formalism PDF Mathematics Magazine 52 4 207 216 doi 10 1080 0025570X 1979 11976784 a b Reid Constance Weyl Hermann 1970 Hilbert Springer Verlag p 198 ISBN 9783662286159 Godel Kurt 1986 Feferman Solomon ed Kurt Godel Collected Works Volume I Publications 1929 1936 Vol 1 Oxford Oxford University Press p 195 ISBN 9780195039641 Carnap Rudolf 1937 Logical Syntax of Language Routledge pp 325 328 ISBN 9781317830597 a b Curry Haskell B 1951 Outlines of a Formalist Philosophy of Mathematics Elsevier p 56 ISBN 9780444533685 Shapiro Stewart 2005 Formalism The Oxford Companion to Philosophy Honderich Ted 2nd ed Oxford Oxford University Press ISBN 9780191532658 OCLC 62563098 Bertrand Russell My Philosophical Development 1959 ch X External links Edit Media related to Formalism deductive at Wikimedia Commons Retrieved from https en wikipedia org w index php title Formalism philosophy of mathematics amp oldid 1100232718, wikipedia, wiki, book, books, library,
