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Working Foundations - ’ 91

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Bridging the Gap: Philosophy, Mathematics, and Physics

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 140))

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Abstract

There is currently a general malaise about the logical approach to the foundations of mathematics. One main reason is that foundational thought in this century has been dominated by a few global views about the nature of mathematics — logicism, formalism, platonism, and constructivism — each of which has proved to be defective in substantial ways, while nothing else has come to take their place. In recent years there has been an increasingly steady barrage of criticism directed against these all too familiar positions. Some of that criticism has been very sophisticated, coming from within the field of logic itself. Other criticism has come from mathematicians who would jettison the whole approach via logic and formal systems. Still more specifically on the mathematical side have been movements to replace logical foundations by purely mathematical foundations (e.g., using category theory)1.

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References

  • Aberth, O.: 1980, Computable Analysis, McGraw-Hill.

    Google Scholar 

  • Aczel, P.: 1977, ‘An introduction to inductive definitions’, in Barwise (1977), pp. 739–782.

    Google Scholar 

  • Aczel, P.: 1987, Lectures on Nonwellfounded Sets, CSLI Lecture Notes, No. 9, Center for Study of Language and Infonnation, Stanford.

    Google Scholar 

  • Ax, J. and Kochen, S.: 1965, ‘Diophantine problems over local fields I, II’, Amer. J. Math. 87,605–30.

    Google Scholar 

  • Barendregt, H.: 1977, ‘The type free lambda calculus’, in Barwise (1977), pp. 1091–1132.

    Google Scholar 

  • Barendregt, H.P.: 1984, The Lambda Caleulus, its Syntax and Semantics (2nd ed.), North-Holland.

    Google Scholar 

  • Barwise, J. (ed.): 1977, Handbook of Mathematical Logic, North-Holland.

    Google Scholar 

  • Barwise, J. and Etchemendy, J.: 1987, The Liar: An Essay in Truth and Circularity, Oxford University Press.

    Google Scholar 

  • Beeson, M.: 1980, ‘Extensionality and choice in constructive mathematics’, Pacific J. Math. 88, 1–28.

    Google Scholar 

  • Beeson, M.: 1981, ‘Formalizing constructive mathematics: why and how?’, in Lecture Notes in Math 873, 146–190.

    Article  Google Scholar 

  • Beeson, M.: 1982, ‘Problematic principles in constructive mathematics’, in Logic Colloquium 1980, North-Holland, pp. 11–55.

    Google Scholar 

  • Beeson, M.: 1985, Foundations of Constructive Mathematies: Metamathematical Studies, Springer-Verlag.

    Google Scholar 

  • Bishop, E.: 1967, Foundations of Construetive Analysis, McGraw-Hill.

    Google Scholar 

  • Bishop, E. and Bridges, O.: 1985, Constructive Analysis, Springer-Verlag.

    Google Scholar 

  • Bishop, E. and Cheng, H.: 1972, ‘Constructive measure theory’, AM.S. Memoirs No. 116.

    Google Scholar 

  • Bridges, D.: 1979, Constructive Functional Analysis, Pitman.

    Google Scholar 

  • Brouwer, L.E.J.: 1975, Collected Works, I. North-Holland.

    Google Scholar 

  • Buchholz, W., Fefennan, S., Pohlers, W., and Sieg, W.: 1981, Iterated inductive definitions and subsystems of analysis: recent proof-theoretical studies, Lecture Notes in Math. 897.

    Google Scholar 

  • Buchholz, W. and Schütte, K.: 1988, Prool Theory of Impredicative Subsystems of Analysis, Bibliopolis.

    Google Scholar 

  • Cherlin, G.: 1976, Model theoretic algebra: selected topics, Lecture Notes in Math. 521.

    Google Scholar 

  • Cohen, P.J.: 1966, Set Theory and the Continuum Hypothesis, Benjamin.

    Google Scholar 

  • Cohen, P.J.: 1969, ‘Decision procedures for real and p-adic fields’, Comm. Symp. Pure Appl. Math. 22, 131–151.

    Article  Google Scholar 

  • Davis, P.J. and Hersh, R.: 1981, The Mathematical Experience, Birkhäuser.

    Google Scholar 

  • Dreben, B. and Denton, J.: 1966, ‘A supplement to Herbrand’,J. Symbolie Logic 31, 393–398.

    Article  Google Scholar 

  • Eldof, P.: 1973, ‘Lefschetz’s principle and local functors’, Proc. Amer. Math. Soc. 37, 333–339.

    Google Scholar 

  • Feferman, S.: 1968, ’systems of predicative analysis II. Representations of ordinals’ J. Symbolic Logic 33,193–220.

    Article  Google Scholar 

  • Feferman, S.: 1968a, ‘Autonomous transfinite progressions and the extent of predicative mathematics’, in Logic, Methodology and Philosophy of Science. III, North-Holland, pp. 121–135.

    Google Scholar 

  • Feferman, S.: 1969, ’Set-theoretical foundations of category theory’ (with an Appendix by G. Kreisel), in Lecture Notes in Math 106, 201–247.

    Article  Google Scholar 

  • Feferman, S.: 1970, ‘Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis’, in Intuitionism and Proof Theory, North-Holland, pp. 303–326.

    Google Scholar 

  • Feferman, S.: 1972, ‘Infinitary properties, local functors, and systems of ordinal functions’, in Lecture Notes in Math 255, 63–97.

    Article  Google Scholar 

  • Feferman, S.: 1975, ‘A language and axioms for explicit mathematics’, in Lecture Notes in Math 450,87–139.

    Article  Google Scholar 

  • Feferman, S.: 1977, ‘Theories of finite type related to mathematical practice’, in Barwise (1977), pp. 913–971.

    Google Scholar 

  • Feferman, S.: 1977a, ‘Inductive schemata and recursively continuous functionals’, in Logic Colioquium, 1976, North-Holland, pp. 373–392.

    Google Scholar 

  • Feferman, S.: 1977b, ‘Categorical foundations and foundations of category theory’, in Logic, Foundations of Mathematics and Computability Theory, Reidel, pp. 149–169.

    Google Scholar 

  • Feferman, S.: 1978,‘A more perspicuous formal system for predicativity’, in Konstruktionen versus Positionen I, Walter de Gruyter, pp. 87–139.

    Google Scholar 

  • Feferman, S.: 1979, ‘Constructive theories of functions and classes’, in Logic Colioquium, 1978, North-Holland, pp. 159–224.

    Google Scholar 

  • Feferman, S.: 1979a, ‘What does logic have 10 tell us about mathematical proofs?’, The Mathematical Intelligencer 2, 20–24.

    Article  Google Scholar 

  • Feferman, S.: 1981, ‘The logic of mathematical discovery vs. the logical structure of mathematics’, in PSA 1978 2, 309–327. (Phil. Sci. Assoc.).

    Google Scholar 

  • Feferman, S.: 1982, ‘Inductively presented systems and the formalization of metamathematics’, in Logic Colioquium, 1980, North-Holland, pp. 95–128.

    Google Scholar 

  • Feferman, S.: 1982a, ‘Monotone inductive definitions’, in The L.EJ. Brouwer Centenary Symposium, North-Holland, pp. 77–89.

    Google Scholar 

  • Feferman, S.: 1984, ‘Toward useful type-free theories, I’,J. Symbolic Logic 49, 75–111.

    Article  Google Scholar 

  • Feferman, S.: 1984a, ‘Foundational ways’, in Perspectives in Mathematics, Birkhäuser, pp. 147–158.

    Google Scholar 

  • Feferman, S.: 1987, ‘Infinity in mathematics: is Cantor necessary?’, in L‘infinito nelta scienza, Inst della Encyclopedia Italiana.

    Google Scholar 

  • Feferman, S.: 1988, ‘Hilbert’s prograrn relativized: proof-theoretical and foundational reductions’, J. Symbolic Logic 53, 364–384.

    Article  Google Scholar 

  • Feferman, S.: 1988a, ‘Weyl vindicated: Das Kontinuum 70 years later’, in Temi e prospeuive della logica e della filosofia delta scienza contemporanee, CLUEB, pp. 59–93.

    Google Scholar 

  • Feferman, S.: 1989, ‘Finitary inductively presented logics’, in Logic Colioquium 1988, Elsevier,pp. 191–220.

    Google Scholar 

  • Feferman, S.: 1990, ‘Polymorphic typed lambda-calculi in a type-free axiomatic framework’, Logic and Computation, Contemporary Mathematics 106, pp. 101–136.

    Google Scholar 

  • Fefennan, S.: 1991, ‘Reftecting on incompleteness’, J. Symbolic Logic 56,1–49.

    Article  Google Scholar 

  • Fenstad, J.E.: 1980, General Recursion Theory. An Axiomatic Approach, Springer-Verlag.

    Google Scholar 

  • Fitting, M.C.: 1981, Fundamentals ofGeneralized Recursion Theory, North-Holland.

    Google Scholar 

  • Friedman, H.: 1971, ‘Algorithmic procedures, generalized Turing algorithrns and elementary recursion theories’, in Logic Colloquium, 1969, North-Holland, pp. 361–390.

    Google Scholar 

  • Friedman, H.: 1975, ’some systems of second order arithmetic and their use’, Proc. Int. Cong. Math., Vancouver, 1974 1, 235–24235–242

    Google Scholar 

  • Friedman, H.: 1976, ’systems of second order arithmetic with restricted induction’ (abstracts), J. Symbolic Logic 41, 557–559.

    Article  Google Scholar 

  • Friedman, H.: 1977, ’set-theoretic foundations for constructive analysis’, Ann. Math. 105, 1–28.

    Article  Google Scholar 

  • Friedman, H.: 1980, ‘A strong conservative extension of Peano arithmetic’, The Kleene Symposium, North-Holland, pp. 113–122.

    Google Scholar 

  • Friedman, H., McAloon, K., and Simpson, S.G.: 1982, ‘A finite combinatorial principle which is equivalent to the 1-consistency of predicative analysis’, in Patras Logic Symposium, North Holland, pp. 197–230.

    Google Scholar 

  • Friedman, H., Simpson, S.G., and Smith, R.: 1983, ‘Countable algebra and set existence axioms’, Ann. Pure and Applied Logic 25, 141–181.

    Article  Google Scholar 

  • Gentzen, G.: 1969, The Collected Papers of Gerhard Gentzen, North-Holland.

    Google Scholar 

  • Girard, J.-Y.: 1981, ‘Π1 2-logic, Part I: Dilators’, Ann. Math. Logic 21, 75–219.

    Article  Google Scholar 

  • Gödel, K.: 1959, ‘Consistency-proof for the generalized continuum-hypothesis’, Proc. Nat. Acad. Sci. U.S.A. 25, 220–224.

    Article  Google Scholar 

  • Gödel, K.: 1940, ‘The consistency of the continuum hypothesis’, Ann. Math. Studies No. 3, The Princeton Univ. Press; rev., notes added 1951, 1966.

    Google Scholar 

  • Gödel, K.: 1958, ‘Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes’, Dialectica 12, 280–287.

    Article  Google Scholar 

  • Gödel, K.: 1986, Collected Works, Vol. I. Publications 1929-1936, ed. by S. Fefennan, J.W. Dawson, Jr., S.C. Kleene, G.H. Moore, R.M. Solovay, and J. van Heijenoort, Oxford University Press.

    Google Scholar 

  • Gödel, K.: 1990, Collected Works, Vol. II. Publications 1938-1974, ed. by S. Fefennan et al., Oxford University Press.

    Google Scholar 

  • Goodman, N.D.: 1979, ‘Mathematics as an objective science’, Amer. Math. Monthly 86, 540–551.

    Article  Google Scholar 

  • Hersh, R.: 1979, ’some proposals for reviving the philosophy of mathematics’, Adv. Math. 31, 31–50.

    Article  Google Scholar 

  • Hilben, D. and Bernays, P.: 1959, ‘Grundlagen der Mathematik II’, Springer; rev. edn. 1970.

    Google Scholar 

  • Jäger, G. and Pohlers, W.: 1982, ‘Eine beweistheoretische Untersuchung von (Δ1 2-C A) + (B I) und verwandter Systeme’, Sitzungsber. Bayer. Akad. Wiss., Mat.-Nat. Klasse, 1–28.

    Google Scholar 

  • Jensen, R.: 1969, ‘On the consistency of a slight(?) modification of Quine’s New Foundations’,in Words and Objections: Essays on the Work of W.V.O. Quine, Reidel, pp. 278–291.

    Google Scholar 

  • Kanamori, A. and Magidor, M.: 1978, ‘The evolution of large cardinal axioms in set theory’, in Lecture Notes in Math 669, 99–275.

    Article  Google Scholar 

  • Kleene, S.C. and R.E. Vesley: 1965, The Foundations of 1ntuitionistic Mathematics, Espeeially in Relation to Recursive Functions, North-Holland.

    Google Scholar 

  • Kreisel, G.: 1958, ‘Mathematical significance of consistency proofs’, J. Symbolic Logic 23, 155–182.

    Google Scholar 

  • Kreisel, G.: 1967, ‘Mathematicallogic: what has it done for the philosophy of mathematics’, in Bertrand Russell, Philosopher of the Century, Allen and Unwin, pp. 201–272.

    Google Scholar 

  • Kreisel, G.: 1970, ‘Principles of proof and ordinals implicit in given concepts’, in Intuitionism and Proof Theory, North-Holland, pp. 489–516.

    Google Scholar 

  • Kreisel, G.: 1976, ‘What have we learned from Hilbert’s second problem?’, Proc. Sympos. Pure Math. XXVIII, 93–130 (A.M.S.).

    Google Scholar 

  • Kreisel, G.: 1977, ‘Review of Brouwer 1975’, Bull. A.M.S. 83, 86–93.

    Article  Google Scholar 

  • Kreisel, G. and Troelstra, A.S.: 1970, ‘Formal systems for some branches of intuitionistic analysis’, Annals. Math. Logic 1, 229–387.

    Google Scholar 

  • Lakatos, I.: 1976, Proofs and Refutations: the Logic of Mathematical Discovery, Cambridge University Press.

    Book  Google Scholar 

  • Lakatos, I.: 1978, Mathematics, Science and Epistemology: Philosophical Papers Vol. 2, Cambridge University Press.

    Google Scholar 

  • MacLane, S.: 1961, ‘Locally small categories and the foundations of mathematics’, in Infinitistic Methods, Pergamon, pp. 25–43.

    Google Scholar 

  • MacLane, S.: 1971, Categories for the Working Mathematician, Springer-Verlag.

    Google Scholar 

  • MacLane, S.: 1981, ‘Mathematical models: a sketch for the philosophy of mathematics’, Amer.Math. Monthly 88, 462–72.

    Google Scholar 

  • Martin, R.L. (ed.): 1984, Recent Essays on Truth and the Liar Paradox, Oxford University Press.

    Google Scholar 

  • Martin-Löf, P.: 1982, ‘Constructive mathematics and computer programming’, Logic, Methodology and Philosophy of Science VI, North-Holland.

    Google Scholar 

  • Martin-Löf, P.: 1984, Intuitionistic Type Theory, Bibliopolis.

    Google Scholar 

  • Mines, R., Richman, F., and Ruitenberg, W.: 1988, A Course in Constructive Algebra, Springer Verlag.

    Google Scholar 

  • Moore, G.: 1982, Zermelo’s Axiom of Choice, Springer.

    Google Scholar 

  • Moschovakis, Y.N.: 1977, ‘On the basic notions in the theory of induction’, in Logic,Foundations of Mathematics, and Computability Theory, Reidel, pp. 207–236.

    Google Scholar 

  • Moschovakis, Y.: 1991, ‘The formallanguage of recursion’, J. Symbolic Logic 54, 1216–1252.

    Article  Google Scholar 

  • Mosses, P.O.: 1990, ‘Denotational semantics’, in Handbook of theoretical Computer Science, Vol. B. Formal Models and Semantics, Elsevier, pp. 595–631.

    Google Scholar 

  • Myhill, J.: 1975, ‘Constructive set theory’, J. Symbolic Logic 40, 347–383.

    Article  Google Scholar 

  • Paris, J. and Harrington, L.: 1977, ‘A mathematical incompleteness in Peano Arithmetic’, in Barwise (1977), pp. 1133–1142.

    Google Scholar 

  • Pohlers, W.: 1989, Proof Theory: an introduction, Lecture Notes in Math 1407

    Google Scholar 

  • Pohlers, W.: 1991, ‘Proof theory and ordinal analysis’, Archive for Mathematical Logic 30,311–376.

    Article  Google Scholar 

  • Prawitz, O.: 1971, ‘Ideas and results in proof theory’, in Proc. Second Scandinavian LogicSymposium, North-Holland, pp. 235–307.

    Google Scholar 

  • Rathjen, J.: 1991, ‘Proof-theoretic analysis of KPM’, Archive for Mathematical Logic 30,377–403.

    Article  Google Scholar 

  • Robinson, A.: 1966, Non-Standard Analysis, North-Holland; rev. ed. 1974.

    Google Scholar 

  • Schütte, K.: 1978, Proof Theory, Springer.

    Google Scholar 

  • Scott, D.: 1972, ‘Continuous lattices’, in Lecture Notes in Math 274, 97–136.

    Article  Google Scholar 

  • Scott, D.: 1976, ‘Oata Types as lattices’, SIAM J. Comput. 5,522–587.

    Article  Google Scholar 

  • Sieg, W.: 1991, ‘Herbrand analyses’, Archivefor Mathematical Logic 30, 409–441.

    Article  Google Scholar 

  • Simpson, S.G.: 1984, ‘Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?’, J. Symbolic Logic 49, 783–802.

    Google Scholar 

  • Simpson, S.G.: 1987, ’subsystems of Z2 and reverse mathematics’, Appendix to Takeuti (1987), pp. 432–446.

    Google Scholar 

  • Simpson, S.G.: 1987a, ‘Unprovable theorems and fast growing functions’, in Logic and Combinatorics, Contemporary Mathematics 65, 359–394.

    Google Scholar 

  • Simpson, S.G.: 1988, ‘Partial realizations ofHilbert’s program’, J. Symbolic Logic 53,349–363.

    Google Scholar 

  • Stoy, J.E.: 1979, Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory, M.I.T. Press.

    Google Scholar 

  • Stroyan, K.D.: 1977, ‘Infinitesimal analysis of curves and surfaces’, in Barwise (1977), pp.197–231.

    Google Scholar 

  • Takeuti, G.: 1975, Proof Theory, North-Holland.

    Google Scholar 

  • Takeuti, G.: 1978, ‘A conservative extension ofPeano Arithmetic’, Part II of Two Applications of Logic to Mathematics,Princeton Univ. Press.

    Google Scholar 

  • Takeuti, G: 1987, Proof Theory (2nd. ed. with Appendixes by G. Kreisel, W. Pohlers, S. Simpson and S. Feferman), North-Holland.

    Google Scholar 

  • Troelstra, A.S.: 1977, ‘Aspects of constructive mathematics’, in Barwise (1977), pp. 973–1052.

    Google Scholar 

  • Troelstra, A.S. and van Dalen, D.: 1988, Constructivism in Mathematics (in two volumes), North Holland.

    Google Scholar 

  • Tymoczko, T. (ed.): 1986, New Directions in the Philosophy of Mathematics, Birkhäuser, Boston.

    Google Scholar 

  • Visser, A.: 1989, ’semantics and the liar paradox’, in Handbook of Philosophical Logic, Vol. IV,Reidel, pp. 617–706.

    Article  Google Scholar 

  • Wang, H.: 1974, From Mathematics to Philosophy, Routledge & Kegan Paul.

    Google Scholar 

  • Weyl, H.: 1918, Das Kontinuum, Veit

    Google Scholar 

  • Weyl, H.: 1987, The Continuum: A Critical Examination of the Foundation of Analysis (translation of Weyl (1918) by S. Pollard and T. Bole), Thomas Jefferson University Press.

    Google Scholar 

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  1. Departments of Mathematics and Philosophy, Stanford University, USA

    Solomon Feferman

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  1. Solomon Feferman
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  1. Department of Philosophy, University of Florence, Italy

    Giovanna Corsi  & Maria Luisa Dalla Chiara  & 

  2. Department of Theoretical Physics, University of Trieste, Italy

    Gian Carlo Ghirardi

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Feferman, S. (1993). Working Foundations - ’ 91. In: Corsi, G., Chiara, M.L.D., Ghirardi, G.C. (eds) Bridging the Gap: Philosophy, Mathematics, and Physics. Boston Studies in the Philosophy of Science, vol 140. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2496-6_5

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