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On the Methodology of Informal Rigour: Set Theory, Semantics, and Intuitionism

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On the Methodology of Informal Rigour: Set Theory, Semantics, and Intuitionism On the methodology of informal rigour: Set theory, semantics, and intuitionism MWPMW 2December2020 Notre Dame Walter Dean University of Warwick Hidenori Kurokawa Kanazawa University What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Context and assumptions I Georg Kreisel: “On informal rigour and completeness proofs” (1967b) I Problems in the Philosophy of Mathematics (Lakatos 1967, ed.) I Speakers: Bernays, Easley, Kreisel, Kalmar, Korner¨ , Mostowski, A. Robinson, Szabo. I Attendees: Bar-Hillelú, Carnap, Dummett, Hacking, Harsanyi´ , Heytingú, Hintikka, Kyburg, Luce, Jeffrey, Kleeneú, Kuhn, Myhill†, Popper, Quine, Suppes, Salmon, Tarski, Williams. I Assumptions for this talk: I Most of your are familiar with “Informal rigour” (1967b) wrt: 1) Kreisel’s ‘squeezing’ argument about first-order validity. 2) Kreisel’s argument that CH has a definite truth value. I Fewer are familiar with Kreisel’s Creating Subject argument about intuitionistic analysis and the status Markov’s principle. I There is no general received understanding of what Kreisel meant by ‘informal rigour’ or how it applies to these cases. 2/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Outline and goals I Goals for this talk: 1) What is ‘informal rigour’? – a precise answer. 2) Two of Kreisel’s core examples wrt to 20th c. phil of maths. i) the Creating Subject argument ii) the “La Pr´edicativit´e” argument 3) Convince you that the method is still relevant. i) Additional applications – e.g. squeezing non-classical validity? ii) An informally rigorous argument that CH is true or false? iii) Conceputal realism? (Kreisel vs Carnap vs G¨odel vs ...) I Other teasers: I Kreisel was the original ‘K’intheBHK interpretation. I Tait’s (1981) analysis of finitism –i.e.PRA – is a response to Kreisel’s (1960b, 1965) earlier analysis – i.e. PA. I In his work on predicative definability (1960a/61/62a) and lawless sequences (1958b/68) Kreisel anticipated the method of forcing. 3/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi What is informal rigour: Short answer Two slogans: I Kreisel: Informal rigour = ‘philosophical proof’. I Our gloss: Informal rigour = conceptual analysis mediated by mathematical theorems. 1967a: “Mathematical logic: what has it done for the philosophy of mathematics?” Successes of mathematical logic Time and again it has turned out that traditional notions in philosophy have an essentially unambiguous formulation when one thinks about them ... [A]lso, when so formulated by essential use of mathematical logic, they have non trivial consequences for the analysis of mathematical experience ... Among them are the well known cases (i) the notion of mechanical process, its stability in the sense that apparently different formulations lead to the same results ... (ii) the notion of aggregate which is analysed by means of the hierarchy (theory) of types, and, of course (iii) the notions of logical validity and logical inference which are analysed ... by means of (first order) predicate logic ... [T]he results are important as object lessons: once one has seen the simple considerations [wrt (iii)] concerning Godel’s¨ completeness theorem one cannot doubt the possibility of philosophical proof or, as one might put it, of informal rigour. (1967a, p. 202) 4/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi What is informal rigour: Beginning of longer answer Claim:All 7 instances of (mid-)Kreisel’s application of informal rigour Ø can be subsumed under a single scheme and sub-scheme: IRS: Reflection on common (or intuitive)andnovel concepts is combined with a mathematical theory to resolve an open question. SS: A common concept analyzed by showing that it is ‘squeezed between’ two precise concepts. The ‘old fashioned’ idea is that one obtains rules and definitions by analyzing intuitive notions and putting down their properties ... Informal rigour wants (i) to make this analysis as precise as possible ... in particular to eliminate doubtful properties of the intuitive notions when drawing conclusions ... and (ii) to extend this analysis [by] not to leav[ing] undecided questions which can be decided by full use of evident properties of these intuitive notions. Below the principal emphasis is on intuitive notions which do not occur in ordinary mathematical practice (so-called new primitive notions) ... (1967b, p. 138) 5/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Formal versus informal rigour Kreisel 1987a,b, 1989 Late- Kreisel 1972-… Foundational standpoints Predicativism Finitism Intuitionism Mid- Kreisel 1965, 1967 a,b,c (1958c/60a/60b/62a) (1951/60b/65/70a) (1958a/62b/68/70b) Kreisel 1963 - 1971 Creating Predicative Choice Standard FO Continuum Informal Subject vs definability sequence rigour Validity Hypothesis Finitist nonstandard Extensional Markov’s function models definiteness Intuitionistic Principle validity Finitist Explication Predicative proof Early- Positivism proof Kreisel <latexit sha1_base64="ABnCY77D28RVYW+3vbLNnsywIAs=">AAACDnicbVC7TsMwFHXKq5RXgJHFoqrEVCUIBGMFC2OR6ENqospxblqrjhPZDlIV9QtY+BUWBhBiZWbjb3DbDKXlyJaOzrn32vcEKWdKO86PVVpb39jcKm9Xdnb39g/sw6O2SjJJoUUTnshuQBRwJqClmebQTSWQOODQCUa3U7/zCFKxRDzocQp+TAaCRYwSbaS+XfMCGDCRUzNDTbDnLRwQYaH37apTd2bAq8QtSBUVaPbtby9MaBaD0JQTpXquk2o/J1IzymFS8TIFKaEjMoCeoYLEoPx8ts4E14wS4iiR5gqNZ+piR05ipcZxYCpjoodq2ZuK/3m9TEfXfs5EmmkQdP5QlHGsEzzNBodMAtV8bAihkpm/YjokklBtEqyYENzllVdJ+7zuXtad+4tq46aIo4xO0Ck6Qy66Qg10h5qohSh6Qi/oDb1bz9ar9WF9zktLVtFzjP7A+voFTmubow==</latexit> CARNAP 1950 - 1962 <latexit sha1_base64="+t67ic7Q0oF0WFkFYpCbZykeQbU=">AAACD3icbVDJSgNBEO2JW4xb1KOXwaB4McyIosdoPHiMYBbIxNDTqUma9Cx014hhmD/w4q948aCIV6/e/Bs7i6CJDwoe71VRVc+NBFdoWV9GZm5+YXEpu5xbWV1b38hvbtVUGEsGVRaKUDZcqkDwAKrIUUAjkkB9V0Dd7ZeHfv0OpOJhcIODCFo+7Qbc44yiltr5fcen2HO9xLkEgfTWbtupg3CPyWH6Y5XP03a+YBWtEcxZYk9IgUxQaec/nU7IYh8CZIIq1bStCFsJlciZgDTnxAoiyvq0C01NA+qDaiWjf1JzTysd0wulrgDNkfp7IqG+UgPf1Z3DE9W0NxT/85oxemethAdRjBCw8SIvFiaG5jAcs8MlMBQDTSiTXN9qsh6VlKGOMKdDsKdfniW1o6J9UrSujwuli0kcWbJDdskBsckpKZErUiFVwsgDeSIv5NV4NJ6NN+N93JoxJjPb5A+Mj29r/Zzm</latexit> 1 FIM HYP ∆1-CA KLEENE KLEENE 1 1 COHEN Pre- FOL SOL HPC PA ZF HEYTING Kreisel Formalism ROBINSON <latexit sha1_base64="ABnCY77D28RVYW+3vbLNnsywIAs=">AAACDnicbVC7TsMwFHXKq5RXgJHFoqrEVCUIBGMFC2OR6ENqospxblqrjhPZDlIV9QtY+BUWBhBiZWbjb3DbDKXlyJaOzrn32vcEKWdKO86PVVpb39jcKm9Xdnb39g/sw6O2SjJJoUUTnshuQBRwJqClmebQTSWQOODQCUa3U7/zCFKxRDzocQp+TAaCRYwSbaS+XfMCGDCRUzNDTbDnLRwQYaH37apTd2bAq8QtSBUVaPbtby9MaBaD0JQTpXquk2o/J1IzymFS8TIFKaEjMoCeoYLEoPx8ts4E14wS4iiR5gqNZ+piR05ipcZxYCpjoodq2ZuK/3m9TEfXfs5EmmkQdP5QlHGsEzzNBodMAtV8bAihkpm/YjokklBtEqyYENzllVdJ+7zuXtad+4tq46aIo4xO0Ck6Qy66Qg10h5qohSh6Qi/oDb1bz9ar9WF9zktLVtFzjP7A+voFTmubow==</latexit> BOURBAKI … -1949 BROUWER POINCARÉ Formal HILBERT & PRA Realism rigour BERNAYS 2 GÖDEL ZF PA2 Key Formal Informal Rigour system ZERMELO Concepts DEDEKIND RUSSELL Structures TURING Statements Natural Cumulative THEORIES Mechanical FREGE number type Positions procedure FIGURES structure structure Methods What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi The Informal Rigour Scheme I Three types of concepts: I Common (or ‘intuitive’): C1, C2,... (e.g mechanical, valid) I Precise: fi1,fi2,... (e.g. recursive, true in all models) I Novel (or ‘new’): N1, N2,... (e.g. truth, the CS has evidence for) I Let Ï be an open questions about common or precise concepts. 1) Reflection on concepts and practice yields: i) Γ1 expresses relations between common concepts. ii) Γ2 bridging principles btw common and precise concepts. iii) Γ3 bridging principles btw novel, common, precise concepts. 2) TP a mathematical theory in the precise language – e.g. ZF. 3) TK =TP +Γ1 +Γ2 +Γ3. 4) Philosophical Theorem: T Ï or T Ï. K „ K „¬ I Example: Kreisel’s CH argument. I Def (Ï) is the common property of mathematical definiteness. 2 s 2 s I Kreisel argues Ï(Def (Ï) Z = Ï or Z = Ï). ’ … | 2 | 2 ¬ I Zermelo’s Categoricity Thm allows us to conclude Def (CH). 7/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Brouwer on the Creating Subject Brouwer suggested that the notion of a Creating Subject [CS] – or idealized mathematician –canbeusedwithin intuitionisitic mathematics. Let [A]beamathematicalassertionthatcannot be tested,i.e.forwhichno method is known to prove either its absurdity or the absurdity of its absurdity. Then the creating subject can, in connection with the assertion A,createan infinitely proceeding sequence of rational numbers a1a2a3,... (1948, p. 478) I A cannot be tested iffno method is known for proving A or A. ¬ ¬¬ I There are untested statements – e.g. Goldbach’s conjecture. (Kreisel called this an empirical fact.) I Brouwer made use of untested statements together with the CS in his weak counterexamples – i.e. implausibility arguments which illustrate why certain principles should not be constructively provable. E.g. (LEM) A A ‚¬ (Apart) x y(x = y x#y) ’ ’ ” æ (MP) x(A(x) A(x)) ( xA(x) xA(x)) ’ ‚¬ æ ¬¬÷ æ÷ (GMP) –( x–(x)=0 x–(x)=0) ’ ¬¬÷ æ÷ 8/25 What is ‘informal rigour’? Creating Subject La Pr´edicativit´e Envoi Brouwer on non-identity and apartness Within intuitionistic mathematics ... n I Reals are given by sequences of rationals rn s.t. rn rn+1 < 2≠ . È Í | ≠ | I A distinction is drawn between 1) Non-identity: x = y –i.e.equalityofx, y is constructively absurd. ” 2) Apartness: x#y – i.e. there is a positive difference between x and y. I Classically x = y and x#y are equivalent. ” I Can we show that the following is not constructively provable?
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