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Péter on Church’s Thesis, Constructivity and Computers

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Connecting with Computability (CiE 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12813))

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Abstract

The aim of this paper is to take a look at Péter’s talk Rekursivität und Konstruktivität delivered at the Constructivity in Mathematics Colloquium in 1957, where she challenged Church’s Thesis from a constructive point of view. The discussion of her argument and motivations is then connected to her earlier work on recursion theory as well as her later work on theoretical computer science.

I would like to thank Marianna Antonutti-Marfori and Alberto Naibo for inviting me to the HaPoC Special Session on Church’s Thesis in Constructive Mathematics. I am also indebted to Alberto for his insightful discussions on this topic and his useful comments on earlier versions of this paper. I would also like to thank Kendra Chilson, Katalin Gosztonyi, Wilfried Sieg and Kristóf Szabó for their help and the anonymous reviewers for their suggestions. The writing of this paper was supported by the UK Engineering and Physical Sciences Research Council under grant EP/R03169X/1.

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Notes

  1. 1.

    Already in his ([12], p. 60) Kleene labeled the assertion that “Every effectively calculable function is general recursive” as a ‘thesis’ that was stated by Church (and implicitly by Turing) but did not use the phrase ‘Church’s Thesis’ anywhere in the paper. The phrase became widely used after it was popularized in Kleene’s [13].

  2. 2.

    Péter also concludes her popular book, Playing with Infinity [22], with the same thought. The last chapter explains Gödel’s incompleteness and Church’s undecidability results and raises the question whether we have “come up against final obstacles?” (p. 264). This is then answered quite forcefully by the very last paragraph of the book: “Future development is sure to enlarge the framework, even if we cannot as yet see how. The eternal lesson is that Mathematics is not something static, closed, but living and developing. Try as we may to constrain it into a closed form, it finds an outlet somewhere and escapes alive” (p. 265).

  3. 3.

    For a detailed description and analysis of Kalmár’s [8] visit Szabó’s [33]. In addition, Gosztonyi’s [4] discusses Péter and Kalmár’s shared views on mathematics and its education within the Hungarian mathematical culture as their broader context, while Máté’s [15] examines their philosophical views on mathematics.

  4. 4.

    The correspondence of Péter and Kalmár is in Hungarian; quotes are translated by the present author.

  5. 5.

    For examples, see Sieg’s ([28], pp. 558–559) quoting Church, and Kleene’s [11] where he states that “The notion of a recursive function of natural numbers, which is familiar in the special cases associated with primitive recursions, Ackermann-Péter multiple recursions, and others, has received a general formulation from Herbrand and Gödel. The resulting notion is of especial interest, since the intuitive notion of a ‘constructive’ or ‘effectively calculable’ function of natural numbers can be identified with it very satisfactorily” (p. 544).

  6. 6.

    Kleene brings up a related issue in his [10]: “The definition of general recursive function offers no constructive process for determining when a recursive function is defined. This must be the case, if the definition is to be adequate, since otherwise still more general “recursive” functions could be obtained by the diagonal process” (p. 738).

  7. 7.

    See Coquand’s [2] for a discussion of their relevant writings.

  8. 8.

    Here Péter remarks that since no “real” general recursive function is known, i.e. one that is general recursive but does not belong to any of the special types of recursive functions, it is not clear what the difference between the two interpretations could actually amount to.

  9. 9.

    See Sundholm’s ([30], pp. 13–14) on Church’s view on the Thesis and its relation to constructivism.

  10. 10.

    Heyting raises a rather similar concern about the circularity involved in the definition of recursive functions from a constructivist point of view in his [5] without referring to Péter’s [21].

  11. 11.

    The term “special recursive functions” is used here loosely, to refer to the collection of those recursive functions that are defined via a specific type of recursion (see above) and are seen as obviously finitely calculable and constructive functions.

  12. 12.

    Translated by Tamás Lénárt.

  13. 13.

    Kalmár’s travel report [7] on the Constructivity in Mathematics Colloquium also stands witness to these interests of his. In the second page of the report he mentions that: “[After the Colloquium I had] the opportunity to visit the ‘Mathematisch Centrum’ and take a look at their operating ARMAC electronic calculator. In addition I had scientific discussion with [Jurjen Ferdinand] Koksma, a mathematics professor, and his colleagues; and with [Adriaan] van Wijngaarden, an engineering professor, about the practical applications of the calculator, as well as about the logical machine under construction in Szeged.” (Translated by the present author.) In the report, Kalmár mistakenly refers to “A. Koksma.” On Kalmár’s description of the ARMAC computer as an electronic calculator, see footnote 14. For short descriptions of the ARMAC computer and the logical machine in Szeged, visit [35] and [32], respectively.

  14. 14.

    At the time computers were referred to as (high speed) electronic or digital calculators in Hungary.

  15. 15.

    To learn more about Péter’s and Kalužnin’s work on diagrams in the historical context of automata theory, visit [18]; for a short biography of Kalužnin see [31].

  16. 16.

    Later on the page Péter adds the following remark: “The above idealization (which will be assumed throughout in what follows) always arises if a general mathematical theory is applied to practical problems. This is often expressed by saying ‘the infinite is a useful approximation to the large but finite”’ (p. 9).

  17. 17.

    This is in stark contrast with Gödel’s view that the human mind infinitely surpasses any finite machine for which the “inexhaustibility” of mathematics or the possible future discovery of humanly effective but non-mechanical processes would provide an argument. See Sieg’s [29] for a detailed analysis of Gödel’s writings on this issue.

  18. 18.

    The greeting is stapled over, thus ‘dear’ is merely an educated guess here.

  19. 19.

    Sadly this list was not kept in [9].

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Author information

Authors and Affiliations

  1. University of Oxford, Oxford, UK

    Máté Szabó

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  1. Máté Szabó
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Editor information

Editors and Affiliations

  1. University of Lille, Lille, France

    Liesbeth De Mol

  2. Vakgroep Wiskunde, Ghent University, Ghent, Belgium

    Andreas Weiermann

  3. University of Göttingen, Göttingen, Germany

    Florin Manea

  4. Ghent University, Ghent, Belgium

    David Fernández-Duque

Appendix: The Amsterdam Colloquium

Appendix: The Amsterdam Colloquium

In the summer of 1956, Heyting sent Péter the following invitation (found in [9]) to the Amsterdam Colloquium, which was then in the early planning phase.

DearFootnote 18 Madam Péter,

The International Union for Logic, Methodology and Philosophy of the Sciences has charged the Netherlands Society for Logic and Philosophy of Sciences to organize in 1957 at Amsterdam a colloquium on “The different notions of constructivity in mathematics”. I hope to organize this colloquium in the summer of 1957, presumably in August. Subventions have been asked from UNESCO and from the Dutch government, by which probably we shall be able to pay a considerable part of the expenses of the participants.

The object of the colloquium will be to study the different notions of constructivity which have been proposed and the relations between them. I shall be please very much if you will participate in it. I join to this letter a list of persons whom I intend to invite.Footnote 19 However, this list, as the plan in general is in a preliminary state. I shall be thankful for suggestions w[h]ich you can give me. If the funds are sufficient, I should like to invite a number of young and promising mathematicians as auditors. You will also oblige me by mentioning names which can be considered in this respect, but I beg you to remember that it is by no means sure that such invitations will be possible.

Yours sincerely

A. Heyting

The working title of the Colloquium appears to be more pluralistic at this stage than the final Constructivity in Mathematics. Nevertheless, in the Preface of the Proceedings, Heyting states in a similar vein that “Several different notions of constructivity were discussed in these lectures” (p. 9).

Based on Péter and Kalmár’s correspondence [9] and Kalmár’s travel report [7], we know that Péter was among the first 18 logicians invited to the Colloquium, and Kalmár was later invited on her strong recommendation. Altogether 24 logicians received invitations to the event, 21 attended and 19 gave talks, while about the same number of “promising young mathematicians” were among the audience.

Originally the organizers offered to cover 60% of the costs of the attendees. However, since even the remaining costs were essentially insurmountable for the Hungarian scholars, the costs of Péter, Kalmár, and their student, the set theorist and combinatorist András Hajnal (later frequent Paul Erdős co-author) were almost entirely covered by the organizers.

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Szabó, M. (2021). Péter on Church’s Thesis, Constructivity and Computers. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_43

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