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Informal and Absolute Proofs: Some Remarks from a Gödelian Perspective

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Abstract

After a brief discussion of Kreisel’s notion of informal rigour and Myhill’s notion of absolute proof, Gödel’s analysis of the subject is presented. It is shown how Gödel avoids the notion of informal proof because such a use would contradict one of the senses of “formal” that Gödel wants to preserve. This Gödelian notion of “formal” is directly tied to his notion of absolute proof and to the question of the general applicability of concepts, in a way that overcomes both Kreisel and Myhill’s conceptions. This paper aims to contribute to the present-day debate on informal and epistemic mathematics, focusing on what appears necessary for a better understanding of the issues at stake.

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Notes

  1. Gödel’s (1951) paper was known by Kreisel at the time, directly from Gödel’s Nachlass.

  2. See more on this subject in the next section.

  3. See the introduction of Charles Parsons in (Gödel 1995, p. 144).

  4. See next section for details.

  5. See (Goodman and Vesley 1987).

  6. On this subject, see Moore’s aphorism quoted by Myhill “the meaning of a sentence may be perfectly clear even if the analysis of that meaning is not” (Myhill 1960, p. 465).

  7. Note that the existence of such statements is excluded from the outset as impossible from an intuitionistic point of view, as we will see later.

  8. See Myhill’s assertion quoted above about his skepticism about the existence of a uniform informal method of proofs and for Kreisel see, in particular, Kreisel’s considerations on Gödel in (Kreisel 1987, p. 504).

  9. At least concerning Cantor’s notion of Absolute. This does not preclude a specific influence of physical concepts from Kreisel’s discussion on the notion of “informal”. Actually, Mark van Atten kindly showed me Kreisel’s letter addressed to him of the 10th of January 2005, and mentioned by him (van Atten 2015, pp. 204–205). From this letter it appears that the choice of the term ‘informal’ was suggested to Kreisel by the ideal of physical rigour : “Informal rigour was (for me, in the 1960’s) a straightforward transfer to logic [...] of the idea(l) of physical reasoning or—physical rigour— in rational mechanics (in which I was engaged in the 1940s during the world service in England)”.

  10. It is remarkable that Gödel quotes both his colleagues: (Kreisel 1960) is quoted in the Dialectica paper (Gödel 1995, p. 274, note 4), while Myhill’s work is mentioned by Gödel in his answer to a letter from Perlis in 1968 (Gödel 2003b, p. 156).

  11. See below on this point.

  12. For the translation of the term “inhaltlich”, Hodges and Watson propose “material”.

  13. Different versions of this paper were written between 1953 and 1959, but none of the six was published by Gödel during his lifetime. See Gödel (1995, pp. 332–362) for versions III and V and Rodriguez-Consuegra (1995) for versions II and IV.

  14. See for example (Weyl 1918), Chap. 1, Sect. 1.

  15. See (Wang 1996, p. 277) and in particular (Wang 1996, p. 295), fragment 9.1.26. See also (Gödel 2017, p. 1) for a different list of primitives.

  16. See for example (Gödel 2017, p. 66; 23–24) and (Crocco 2016) for related comments.

  17. See for example in (Wang 1996, pp. 271–272) the fragments 8.5.8–8.5.15.

  18. In 8.4.16, those who refuse to use such infinitary concepts, but still have an intensional analysis of the logical connectives, considering them as thought constructions, are called by Gödel empiricists. Gödel seems to object to the proponents of such empiricism that there is no reason to limit ourselves to the consideration of the logical concepts of classical logic, because we also have evidence for connectives such as many, most, some, necessarily, etc. (see Wang 1996, p. 266, fragments 8.4.12–8.4.13).

  19. See for example (Kant 1787, Architectonic of Pure Reason, p. 541, line 18).

  20. We do not agree, therefore, with the restricted sense that Parsons gives to “absolute” in his introduction to the 1946 paper (Gödel 1995, p. 145).

  21. The word “formal” here should be interpreted as “formal2”, that is, universally applicable to any set of the cumulative hierarchy.

  22. See point 3 in (Wang 1996, p. 316), or, in a corrected form, (Crocco 2016, p. 36).

  23. The analysis is given in (Turing 1939) and discussed in (Wang 1974, pp. 90–99).

  24. “There might exist methods for actualizing this development, which could form part of this procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing’s number of distinguishable states of mind) may converge toward infinity in the course of application of the procedure”. (Gödel 1995, p. 306).

  25. Wang presents the discussion on the role and nature of logic in Chap. 8 of his book, the discussion on the mind-machine problem in Chap. 6.

  26. Regarding the subscript, see Table 1.

  27. Regarding Gödel’s notion of ‘object’ in the seventies see in (Wang 1996, p. 296) fragment 9.1.27.

  28. The words inside square brackets are additions from Wang.

  29. See also Gödel’s note 14, Chap. 6 in (Wang 1974).

  30. See (Gödel 1961, pp. 381–382).

  31. (Gödel 1951, p. 305).

  32. Cartwright’s analysis is presented in the very eloquent introduction of her book, in these terms: “This book supposes that, as appearance suggests, we live in a dappled world, a world of different things, with different natures, behaving in different ways. The laws that describe this world are a patchwork, not a pyramid. They do not take after the simple, elegant and abstract structure of a system of axioms and theorems. Rather they look like –and steadfastly stick to looking like, –science as we know it: apportioned into disciplines, apparently arbitrarily grown up; governing different sets of properties at different levels of abstraction; pockets of great precision; large parcels of qualitative maxims resisting precise formulation; erratic overlaps; here and there, once in a while, corners that line up, but mostly ragged edges; and always the cover of laws just loosely attached to the jumbled world of material things. For all we know, most of what occurs in nature occurs by hap, subject to no law at all. What happens is most like an outcome of negotiation between domains than a logical consequence of a system of order. The dappled world is what for the most part, comes naturally: regimented behavior results from good engineering.” (Cartwright 1999, p. 1).

  33. In (Gödel 1961) “rightward” and “leftward” tendencies refer to schema of the possible philosophical Weltanshauungen considered by Gödel in order to discuss the modern development of the foundations of mathematics. To the leftward tendencies belong scepticism, materialism and positivism. To the rightward ones belong spiritualism, idealism and metaphysics.

  34. As we stressed in Sect. 2, semantics depends on language, whereas concepts as intensions do not depend on it, although they can be represented in it.

  35. On this subject, see (Girard 2006, 2016, pp. 31–33).

References

  • Antonutti-Marfori M (2010) Informal proofs and mathematical rigour. Stud Logica 96(2):261–272

    Article  Google Scholar 

  • Bernays P (1935) Sur le platonisme dans les mathématiques. L’enseignement mathématique, 34, 52–69. Translated into English as On Platonism in mathematics. In: Benacerraf P, Putnam H (1964) Philosophy of mathematics. Blackwell, Oxford

    Google Scholar 

  • Bueno O, Linnebo Ø. (eds) (2009) New waves in philosophy of mathematics. Palgrave Macmillan, London

    Google Scholar 

  • Cartwright N (1999) The Dappled world. A study of the boundaries of science. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Crocco G (2006) Gödel on concepts. Hist Philos Logic 27(2):171–191

    Article  Google Scholar 

  • Crocco G (2016) Sinn/Bedeutung and intension/extension in Gödel’s Max Phil IX. In: Crocco Engelen (2016), pp 127–153

  • Crocco G, Engelen E-M (eds) (2016) Kurt Gödel : philosopher-scientist. Presses Universitaires de Provence, Aix-en-Provence

    Google Scholar 

  • Dauben JW (1979) Georg Cantor: his mathematics and philosophy of the infinite. Harvard University Press, Harvard

    Google Scholar 

  • Davis M (ed) (1965) The undecidable: basic papers on undecidable propositions, unsolvable problems, computable functions. Dover Publications, Mineola

    Google Scholar 

  • Dosen K, Adzic M (2016) Gödel on deduction. On-line document available via Cornell University Library at: https://128.84.21.199/abs/1604.02757

  • Feferman S (1958) Ordinal logics re-examined, and on the strength of ordinal logics. J Symb Log 23:289–299

    Article  Google Scholar 

  • Gandy RO (1980) Church’s thesis and principles for mechanisms. In: Barwise J, Keisler J, Kunen K. (eds.) The Kleene Symposium. North-Holland. Pub. Co., Amsterdam, New York, pp 123–148

    Chapter  Google Scholar 

  • Gentzen G (1969) The collected Papers of Gerhard Gentzen. North-Holland Pub. Co., Amsterdam

    Google Scholar 

  • Giaquinto M (2007) Visual thinking in mathematics: an epistemological study. Oxford University Press, Oxford

    Book  Google Scholar 

  • Girard J-Y (2006) Le point Aveugle, Cours de Logique, vol 1. Hermann, Paris

    Google Scholar 

  • Girard J-Y (2016) Le Fantôme de la transparence. Allia, Paris

    Google Scholar 

  • Gödel K (1936) Über die Länge der Beweisen. Translated into English as On the Lengths of proofs. In (Gödel 1986), pp 396–399

  • Gödel K. (1944) Russell’s mathematical logic. In (Gödel 1990), pp 119–141

  • Gödel K (1946) Remarks before the Princeton bicentennial Conference on problems in mathematics. In (Gödel 1990), pp 150–153

  • Gödel K (1951) Some basic theorems in the foundations of mathematics and their implications. In (Gödel 1995), pp 304–323

  • Gödel K (1953; 1959) Is mathematics syntax of language? In (Gödel 1995), Version III, pp 334–356, Version V, pp 356–362

  • Gödel K (1958) On an hitherto unutilized extension of the finitary standpoint. In (Gödel 1990), pp 240–251

  • Gödel K (1961) The modern development of the foundations of mathematics in the light of philosophy. In (Gödel 1995), 374–387

  • Gödel K (1972a) Some remarks on the undecidability results. In (Gödel 1990), 305–306

  • Gödel K (1972b) On an extension of finitary mathematics which has not been used. In (Gödel 1990), pp 271–280

  • Gödel K (1986) Kurt Gödel collected works. Feferman S, Dawson I, Kleene S, Moore G, Solovay R, van Heijenoort J. (eds), vol. 1 Oxford University Press, Oxford

  • Gödel K (1990) Kurt Gödel collected works. Feferman S, Dawson J, Kleene S, Moore G, Solovay R, van Heijenoort J. (eds), vol. 2 Oxford University Press, Oxford

    Google Scholar 

  • Gödel K (1995) Kurt Gödel collected works. Feferman S, Dawson J, Kleene S, Moore G, Solovay R, van Heijenoort J. (eds), vol. 3 Oxford University Press, Oxford

    Google Scholar 

  • Gödel K (2003a) Kurt Gödel collected works. Feferman S, Dawson J, Kleene S, Moore G, Solovay R, van Heijenoort J. (eds), vol. 5, Oxford University Press, Oxford

    Google Scholar 

  • Gödel K (2003b) Kurt Gödel collected works. Feferman S, Dawson J, Kleene S, Moore G, Solovay R, van Heijenoort J. (eds), vol. 4 Oxford University Press, Oxford

    Google Scholar 

  • Gödel K (2017) Kurt Gödel Maxims and philosophical remarks. In: Crocco G, Van Atten M, Cantu P, Engelen E-M (eds), vol 10. Available via HAL at https://hal.archives-ouvertes.fr/hal-01459188

  • Goodman ND, Vesley RE (1987) Obituary: John R. Myhill (1923–1987). Hist Philos Log 8(2):243–244

    Article  Google Scholar 

  • Kant I (1787) Kritik der reinen Vernunft. Akademie Ausgabe III. Available at: https://korpora.zim.uni-duisburg-essen.de/kant/aa03

  • Kreisel G (1960) Ordinal logics and the characterization of informal concepts of proof. In Proceedings of the international congress of mathematicians 14–21 Aug. 1958, Cambridge University Press, Edinburgh, pp 289–299

  • Kreisel G (1967) Informal rigour and completeness proofs. In: Lakatos I (ed) Problems in the philosophy of mathematics. North-Holland Pub. Co., Amsterdam pp 138–157

    Chapter  Google Scholar 

  • Kreisel G (1987) Church’s thesis and the ideal of informal rigour. Notre Dame J. Formal Log 28:4:499–519

    Article  Google Scholar 

  • Lakatos I (1962) Infinite regress and the foundations of mathematics. Aristot Soc Suppl Vol 36:155–184

    Google Scholar 

  • Lakatos I (1976) Proofs and refutations. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Leitgeb H (2009) On formal and informal provability. In: Linnebo Ø., Bueno O (eds) New waves in philosophy of mathematics. Palgrave Macmillan, London

    Google Scholar 

  • Mancosu P (ed) (2008) The philosophy of mathematical practice. Oxford University Press, Oxford

    Google Scholar 

  • Myhill J (1960) Some remarks on the notion of proof. J Philos 57:14, 461–471

    Article  Google Scholar 

  • Quine WVO (1970) The philosophy of logic. Harvard University Press, Harvard

    Google Scholar 

  • Rav Y (1999) Why do we prove theorems ? Philos Math 7(1):5–41

    Article  Google Scholar 

  • Rodriguez-Consuegra F (1995) Kurt Gödel: unpublished philosophical essays. Birkhäuser, Basel

    Book  Google Scholar 

  • Turing A (1939) Systems of logic based on Ordinals. Proceedings of the London mathematical society, vol. 2-45, Issue 1, 61–228. Reprinted in (Davis 1965), 155–222

  • van Atten M (2015) Gödel and intuitionism. In: Essays on Gödel’s reception on Leibniz, Husserl, and Brouwer, Springer, pp 189–234

  • Wang H (1974) From mathematics to philosophy. MIT Press, London

    Google Scholar 

  • Wang H (1996) A logical journey from Gödel to philosophy. MIT Press, London

    Google Scholar 

  • Weyl H (1918) Das Kontinuum. Von Veit & Comp, Leipzig

    Google Scholar 

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  1. Aix Marseille University, CNRS, CEPERC UMR 7304, Aix-en-Provence, France

    Gabriella Crocco

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Crocco, G. Informal and Absolute Proofs: Some Remarks from a Gödelian Perspective. Topoi 38, 561–575 (2019). https://doi.org/10.1007/s11245-017-9515-3

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