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formalism philosophy of mathematics wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 early formalism 2 hilbert s formalism 3 further developments 4 criticism 5 see also 6 references 7 external links toggle the table of contents formalism philosophy of mathematics 21 languages العربية català чӑвашла deutsch español فارسی suomi gaeilge עברית հայերեն bahasa indonesia 日本語 la lojban 한국어 nederlands português русский српски srpski türkçe українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia view that mathematics does not necessarily represent reality but is more akin to a game formalism mathematics redirects here for other uses see formalism disambiguation for the field of study see mathematical logic formal logical systems in 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 mathematical statements are not about numbers sets triangles or any other mathematical objects in the way that physical statements are about material objects instead they are purely syntactic expressions formal strings of symbols manipulated according to explicit rules without inherent meaning these symbolic expressions only acquire interpretation or semantics when we choose to assign it similar to how chess pieces follow movement rules without representing real world entities this view stands in stark contrast to mathematical realism which holds that mathematical objects genuinely exist in some abstract realm formalism emerged as a response to foundational crises in mathematics during the late nineteenth and early twentieth centuries particularly concerns about paradoxes in set theory and questions about the consistency of mathematical systems it represents one of the three major philosophical approaches to mathematics developed during this period alongside logicism and intuitionism though formalism encompasses a broader spectrum of positions than these more narrowly defined views among formalists the german mathematician david hilbert was the most influential advocate developing what became known as hilbert s program to establish the consistency of mathematics through purely formal methods 2 early formalism edit the early mathematical formalists attempted to block avoid or sidestep in some way any ontological commitment to a problematic realm of abstract objects 3 german mathematicians eduard heine and carl johannes thomae are considered early advocates of mathematical formalism 3 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 3 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 4 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 5 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 6 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 7 frege objects to the comparison of formalism with that of a game such as chess 8 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 9 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 10 in order to formalize an axiomatic system a language must first be chosen in which operations can be expressed and performed 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 all theorems could be proven 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 one cannot prove consistency within any consistent axiomatic system rich enough to include classical arithmetic on the one hand only the formal language chosen to formalize this axiomatic system must be used on the other hand it is impossible to prove the consistency of this language in itself 10 hilbert was originally frustrated by gödel s work because it shattered his life s goal to completely formalize everything in number theory 11 however gödel did not feel that he contradicted everything about hilbert s formalist point of view 12 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 11 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 edit other formalists such as rudolf carnap considered mathematics to be the investigation of formal axiom systems 13 haskell curry defines mathematics as the science of formal systems 14 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 14 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 15 criticism edit 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 16 see also edit qed project formalized mathematics formal system references edit 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 1 2 3 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 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 cite book isbn date incompatibility help 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 1 2 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 1 2 reid constance weyl hermann 1970 hilbert springer verlag p 198 isbn 9783662286159 gödel kurt 1986 feferman solomon ed kurt gödel 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 cite book isbn date incompatibility help 1 2 curry haskell b 1951 outlines of a formalist philosophy of mathematics elsevier p 56 isbn 9780444533685 cite book isbn date incompatibility help 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 v t e philosophical logic critical thinking and informal logic analysis ambiguity argument belief bias credibility dialectic antithesis socratic method unity of opposites evidence explanation explanatory power fact fallacy list of fallacies hypothesis inquiry opinion parsimony occam s razor premise propaganda prudence razor reasoning relevance rhetoric rigor theory vagueness theories of deduction constructivism dialetheism fictionalism finitism formalism intuitionism logical atomism logicism nominalism platonic realism pragmatism realism category retrieved from https en wikipedia org w index php title formalism_ philosophy_of_mathematics oldid 1375397276 categories formalism deductive philosophy of mathematics hidden categories articles with short description short description is different from wikidata cs1 errors isbn date all articles with unsourced statements articles with unsourced statements from march 2019 commons category link is on wikidata this 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