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EUROCAST 2017: Computer Aided Systems Theory – EUROCAST 2017 pp 49–65Cite as

Kurt Gödel: A Godfather of Computer Science

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10671))

Abstract

We argue that Kurt Gödel exercised a major influence on computer science. Although not immediately involved in building computers, he was a pioneer in defining central concepts of computer theory. Gödel was the first to show how the precision of the formal language systems of Frege, Peano and Russell could be put to work to prove important facts about those language systems themselves, with important consequences for mathematics. As Hilbert’s collaborator, Paul Bernays put it, in his famous Incompleteness Proof Gödel did the “homework” that the people in Göttingen working on Hilbert’s Program to prove the consistency of mathematics missed. (Hilbert’s Metamathematics assumed that all mathematical proofs could be treated as coding problems, and enciphering is applied arithmetic.) The core of Gödel’s Proof also gave exact definitions of the central concept of arithmetic, namely of the recursion involved in mathematical induction (with help from the great French logician Jacques Herbrand). This immediately led to whirlwind developments in Göttingen, Cambridge and Princeton, the working headquarters of major researchers: Paul Bernays and John von Neumann; Alan Turing; and Alonzo Church, respectively. Turing’s and von Neumann’s ideas on computer architecture can be traced to Gödel’s Proof. Especially interesting is the fact that Church and his lambda calculus was the main influence on John McCarthy’s LISP, which became the major language of Artificial Intelligence.

The title of this article is a play on Gödel’s name. In the Austrian and Bavarian dialect of German, “Göd” means godfather, and “Gödel” means literally “little godfather”, although it is actually used to refer only to godmothers in the regions of its use. In any case, Kurt Gödel did take a godfatherly interest in children, in particular to Oskar Morgenstern’s son in Princeton.

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Notes

  1. 1.

    Frege [1]; English translation by Bauer-Mengelberg in [2]. Whitehead and Russell [3].

  2. 2.

    Gödel was a “convinced Platonist” in 1925 already. Cf. Köhler [4]; and Wang [5], (esp. pp. 17f.).

  3. 3.

    NB: Gödel also believed in such completeness! Despite his Incompleteness Proof, Gödel always thought that human reason was (in principle) capable of solving all problems; this couldn’t be done in any single system, but intuition goes beyond all formal systems anyway—just as Brouwer claimed.

  4. 4.

    Hilbert’s most famous call for completeness, including an explicit opposition to (DuBois-Reymond’s) “Ignoramibus”, came at the end of the introduction to his list of “Mathematische Probleme” (published in [6]).

  5. 5.

    Hilbert [7]; reprinted with notes by Bernays in [8]; translated into English by Ewald in [9] and in Mancosu [10]. An authoritative evaluation of Hilbert’s Program is by Kreisel [11]; revised and reprinted in Benacerraf and Putnam [12].

  6. 6.

    Leibniz’s encoding method was first published by Louis Couturat (1903), which Gödel may not have known in Vienna; but Gödel almost certainly knew Lewis [13] where Couturat’s discovery of Leibniz’s encoding method is sketched on pp. 11f.

  7. 7.

    An accessible general survey of Gödel’s Proof is provided by Casti and DePauli [14]. The proof itself was in Gödel [15]; translated into English by van Heijenoort in [2]); reprinted with detailed commentary by Kleene in Feferman et al. [16]. Useful information about Gödel’s life and work appeared in Köhler et al. [17] and Buldt et al. [18].

  8. 8.

    Takeuti [19] (p. 26).

  9. 9.

    Introduced by Peano [20]. Frege [34] and then Dedekind [35] had the axioms beforehand; for commentary cf. footnote 20.

  10. 10.

    Finsler [21].

  11. 11.

    This is easy to explain: within Hilbert’s Program, provability was a strictly finitary notion, hence arithmetizable within Peano Arithmetic. In contrast, truth (e.g. the truth of Gödel’s sentence G) always occupies a standpoint from outside the system and is thus not representable within it. Of course there are stronger notions of provability which approach truth at an upper limit. Cf. the references below to Gentzen (footnote 14). Turing’s Ordinal Logics clarifies the situation more pointedly; cf. footnote 32.

  12. 12.

    Much later Jeff Paris and Leo Harrington found a “natural” sentence of Peano Arithmetic (in Ramsey Theory) also unprovable: “A Mathematical Incompleteness in Peano Arithmetic”, in Barwise [22].

  13. 13.

    Thus John von Neumann, cited by Goldstine [23], p. 174; cf. footnote 35.

  14. 14.

    Gentzen [24, 25].

  15. 15.

    Kleene [26], Rogers [27], Davis [28]. The last has been called “one of the few real classics in computer science” in an advertising blurb.

  16. 16.

    Davis [29] (pp. 120f.).

  17. 17.

    Oswald Veblen had early established an Afternoon Tea at exactly three o’clock every weekday, “we explain to each other what we do not understand” (J. Robert Oppenheimer); cf. Dyson [30] p. 90.

  18. 18.

    See [31, 32]; translated as Recursive Functions, Academic Press, New York 1967. (NB: “Zs” is pronounced in Hungarian like the French “j”.) Péter refers “recursive function” to Dedekind [35]; important here Sect. 9, “126. Satz der Definition durch Induktion”.

  19. 19.

    Soare [33].

  20. 20.

    The recursion concept of Dedekind [35] is the same as Vererbung [inheritance, a property of certain relations, in particular that of inheritance or ancestry] of Frege [34]. Dedekind acknowledged Frege’s priority in capturing all the axioms of Arithmetic and in precisely characterizing mathematical induction. The axioms are now named after Peano [20]—who had admittedly taken them from Dedekind! (The difference was “merely” that Peano used a symbolic language, much of which was taken over by Whitehead and Russell [3]). So Frege was twice cheated out of the credit he deserves for being the first to formally specify the axioms of arithmetic.

  21. 21.

    Von Neumann (János/“Jancsi” in Hungary, Johann in Germany, John/“Johnny” in the US) had believed very strongly in Hilbert’s Program and was otherwise strongly involved with Hilbert’s multifarious interests (e.g. axiomatizing parts of Physics: von Neumann wrote the classic Mathematische Grundlagen der Quantenmechanik [36]). In [37], Gödel used von Neumann’s axioms in a version prepared by Paul Bernays, called NBG in the literature (for Neumann-Bernays-Gödel). NBG extends Set Theory to include “classes” (which were predicate-extensions, what Cantor called, in effect, “sometimes inconsistent sets” because they can not automatically be elements of others sets (or better of any class at all) by the rules of their construction. Today we call these classes proper-classes.). Gödel’s Incompleteness Proof [15] had devastated von Neumann rather much, explaining von Neumann’s chivalrous, boundless and everlasting awe of Gödel. In 1953, von Neumann told those IAS professors opposing Gödel’s promotion to Permanent Member: “If Gödel doesn’t deserve a professorship at the Institute for Advanced Study, none of us do”.

  22. 22.

    At the time of Gödel’s lectures, the IAS was located in Fine Hall on Princeton’s main campus together with the mathematicians; only later did the IAS move to Fuld Hall on Princeton’s west side, where Gödel later had an office with a picture window and a stunning view of well-tended trees. In Fine Hall, famous mathematicians such as Oswald Veblen, Hermann Weyl, John von Neumann, Marston Morse, Deane Montgomery, later John Nash, had offices.

  23. 23.

    Kurt Gödel: “On Undecidable Propositions of Formal Mathematical Systems”, notes from lectures given in English at the Institute for Advanced Study, Princeton, hectographed and distributed by Kleene and Rosser, first published in Davis [38]. Gödel discusses Herbrand’s “private communication” in his footnote 34 (written before the 1965 publication by Davis), still indicating doubt about the “adequacy” of general recursiveness as the correct explication; but in Gödel’s ultimate Postscriptum in Davis (1965), Gödel claims Turing’s definition is “unquestionably adequate‘”.

  24. 24.

    Herbrand [39]; English translation by van Heijenoort [2]. Herbrand had written a letter to Gödel on 7 April 1931 concerning recursion, translated by Wilfried Sieg, with detailed notes, in Feferman et al. [40].

  25. 25.

    Church [41, 42].

  26. 26.

    Turing [43, 44]. The equivalence proofs by Kleene, Rosser, and Turing are all included in the collection by Davis [38].

  27. 27.

    This sounds dangerous, and sure enough, Kleene found contradictions which he and Church worked hard to avoid in later versions of the \(\lambda \)-calculus. Kleene himself resorted to different formalisms in his own work on recursion.

  28. 28.

    Curry and Feys [45], Curry et al. [46]. For a survey, cf. Bimbó [47].

  29. 29.

    McCarthy [48, 49]. In McCarthy’s history of LISP at http://www-formal.stanford.edu/jmc/history/lisp/node3.html, drafted in 1996, he writes “Another way to show that LISP was neater than Turing machines was to write a universal LISP function and show it is briefer and more comprehensible than the description of a universal Turing machine. This was the LISP function EVAL[e, a], which computes the value of a LISP expression e—the second argument a being a list of assignments of values to variables. (a is needed to make recursion work.)...”. “Russell noticed that EVAL could serve as an interpreter for LISP, promptly hand coded it, and we now had a programming language with an interpreter...”.

  30. 30.

    Wang [5], Sect. 6.1.

  31. 31.

    Kreisel [50, 51].

  32. 32.

    Turing [52]; reprinted in Davis [38], and in Copeland [53]. In his introduction to Turing, Copeland compares Turing’s Ordinal Logics very nicely with Gödel’s and Church’s work. For example, he quotes Turing’s claim that, with Ordinal Logics, “one can approximate ‘truth’ by ‘provability’ as well as you please.”.

  33. 33.

    Laplace [54].

  34. 34.

    Kolmogorov [55], Arnol’d [56], Broer [57].

  35. 35.

    This claim is well supported by Goldstine [23] (Goldstine was the principal engineer for von Neumann’s 1951 IAS machine in Princeton; then he worked for IBM). A later authority is Aspray [58].

  36. 36.

    Originally established by Oswald Veblen during World War I to allow US mathematicians to contribute to the American war effort, giving work to the young Norbert Wiener, among others.

  37. 37.

    Copeland ([53], pp. 21ff.) provides several citations from von Neumann himself describing how Turing’s work influenced his thinking on computer design, in particular storing programs in memory.

  38. 38.

    In 1947 it was converted to stored-program control after the example of EDVAC; and it was upgraded to 100 words of (IBM) magnetic-core memory in 1953 before the plug was pulled in 1955.

  39. 39.

    “1. von Neumann [59]. Entirely reprinted in Stern [60]; also excerpted in Brian Randell [61]. 2. Burks et al. [62]”; reprinted in von Neumann [63]; also excerpted in Randell [61]. (This was typed up in Fuld Hall 219, the office next to Gödel’s.). Davis [29] (pp. 191ff.) justly makes Turing out to be the “real father” of stored-program architecture; cf. von Neumann’s own credits to Turing mentioned in footnote 37. But between Leibniz and Turing there came Gödel [15]: Gödel numerals encoded programs (i.e. proof procedures) as data already in 1931.

  40. 40.

    A cathode-ray tube (CRT) patented 1946 in the UK by Freddie Williams and Tom Kilburn. (Thanks to my nephew, Gordon Cichon of the Computer Science Department at the University of Munich, for pointing out this considerable difference between Turing Machines and von Neumann Machines.).

  41. 41.

    RCA employed Vladimir Zworykin, the leader of RCA’s television-development team for David Sarnoff, hence RCA was a leader in CRT technology.

  42. 42.

    Distressing Eckert, IBM negotiated an advisorship contract with von Neumann 1 May 1945 already.

  43. 43.

    Aspray [58] (p. 93). Since the 701 was contracted for by the Defense Department, it was called the “Defense Calculator”. Its business sibling were the IBM 702 and the famous IBM 650, the first mass-produced computer, all influenced by the MANIAC.

  44. 44.

    Mauchly was senior, but Eckert was more entrepreneurial.

  45. 45.

    It’s interesting that the “violently anti-communist” von Neumann (self-description) put such faith in the public sector, which libertarians (e.g. Friedrich von Hayek) nowadays hardly consider, wishing everything to be privatized. An explanation may lie in the fact that von Neumann was born in the Austrian Monarchy, where the public sector since Empress Maria Theresia and her son, the Emperor Josef II, traditionally played a strong role, as it does in France; whereas Great Britain and its colonies in North America early on became laissez-faire in the Whig tradition. (However, the libertarian Hayek also was born in that same monarchy, and his ancestor was also ennobled by Emperor Franz Josef I.). Despite the negative ruling by the patent lawyers on the EDVAC, Eckert and Mauchly could have tried suing, but that would have been expensive, and they would be facing their ex-employer.

  46. 46.

    Various parts of the story of the lawsuit are told by 1. Burks [64]. 2. Stern [60]. 3. Mollenhoff [65]. 4. Aspray [58].

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  1. University of Vienna, Vienna, Austria

    Eckehart Köhler & Werner Schimanovich

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    Alexis Quesada-Arencibia

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Köhler, E., Schimanovich, W. (2018). Kurt Gödel: A Godfather of Computer Science. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2017. EUROCAST 2017. Lecture Notes in Computer Science(), vol 10671. Springer, Cham. https://doi.org/10.1007/978-3-319-74718-7_7

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