Abstract
After a brief survey of Gödel’s personal contacts with Brouwer and Heyting, examples are discussed where intuitionistic ideas had a direct influence on Gödel’s technical work. Then it is argued that the closest rapprochement of Gödel to intuitionism is seen in the development of the Dialectica Interpretation, during which he came to accept the notion of computable functional of finite type as primitive. It is shown that Gödel already thought of that possibility in the Princeton lectures on intuitionism of Spring 1941, and evidence is presented that he adopted it in the same year or the next, long before the publication of 1958. Draft material for the revision of the Dialectica paper is discussed in which Gödel describes the Dialectica Interpretation as being based on a new intuitionistic insight obtained by applying phenomenology, and also notes that relate the new notion of reductive proof to phenomenology. In an appendix, attention is drawn to notes from the archive according to which Gödel anticipated autonomous transfinite progressions when writing his incompleteness paper.
Originally published as van Atten 2014. Copyright © 2014 Springer Science+Business Media.
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Notes
- 1.
December 2, 1966.
- 2.
More on that visit below.
- 3.
Hence, after his Princeton lecture course on intuitionism of Spring 1941; according to Gödel’s letter to Bernays of February 6, 1957, these were held at the Institute for Advanced Study (i.e., not in the Mathematics Department) (Gödel 2003, 144). IAS Bulletin no. 9, of April 1940, gives as dates for the Spring Term of the academic year 1940–1941 February 1 to May 1. In the letter to Bernays, Gödel mentions that there exists no transcript of the course. However, his own lecture notes still exist, and are kept in the archive in Gödel Papers, 8c/121, 040407 and 8c/122, 040408. There are related notes in 8c/123, 040409. Also the notes in 6a/54, 030077, ‘Beweis d[er] Gültigkeit d[er] int[uitionistischen] Ax[iomen]’ belong with these.
- 4.
- 5.
Gödel Sammlung, LQH0236598. Translation MvA. ‘Jetzt habe ich noch eine grosse Bitte an Dich: Könntest Du die folgenden beiden Bücher von L.E.J. Brouwer beim Antiquarium K.F. Koehler (Leipzig, Täubchenweg 21) für mich bestellen? 1. Over de Grondslagen der Wiskunde Katalog 115 No 487 2. Wiskunde, Waarheid, Werkelijkheid Groningen 1919. Es sind kleine Bücher, die bloss ein paar Mark kosten werden. Man sagt mir dass Deutsche Buchhandlungen ohne weiteres Bücher an ausländische Adressen (wahrscheinlich auf Gefahr des Empfängers) versenden, wenn Sie von einem Inländer bestellt u[nd] bezahlt werden. Andrerseits kann man von hier aus durch Buchhandlungen nichts aus Deutschland bestellen. Natürlich mache ich die Bestellung bloss für den Fall, dass die Bücher vorrätig sind. Sie suchen zu lassen käme zu teuer.’
- 6.
In a letter of August 3, 1947 (Gödel Sammlung, item LQH0237199), he does ask Rudolf to find out in a bookstore whether anything had been published since 1941 by or about Leibniz.
- 7.
- 8.
The texts of Heyting’s lectures are held in the Heyting papers at the Rijksarchief Noord-Holland in Haarlem, items V57 (December 11), V57A (December 10), and V57B (December 9).
- 9.
As Sundholm (1983, 159) points out, in logical-mathematical contexts, ‘interpretation’ has come to refer to the interpretation of one formal theory in another. In contrast, the so-called Proof Interpretation (also known as BHK-Interpretation) is not an interpretation in this mathematical sense, but a meaning explanation. Gödel’s Dialectica Interpretation, on the other hand, indeed is one. Note that this immediately shows that the Proof Explanation and the Dialectica Interpretation differ in kind. Of course, a mathematical interpretation may devised because one has a particular meaning explanation in mind for the formulas it yields; this was Gödel’s foundational aim with the Dialectica Interpretation.
- 10.
William Howard (email to MvA, February 1, 2013) recalls:
Re the issue of impredicativity in BHK: Gödel and I did not discuss this issue explicitly, but it was implicit in some of our discussions of my little theory of constructions (the formulæ-as-types paper [Howard (1980)], which then existed in the form of handwritten document, which I had sent to Gödel as part of my application for my sabbatical at the IAS, 1972–1973). Yes, he had obviously read the little paper.
- 11.
[[The relevant form here is \(\forall \alpha _{0,1}\neg \neg \exists xA(\alpha,x) \rightarrow \forall \alpha _{0,1}\exists xA(\alpha,x)\) where A is primitive recursive, and α 0, 1 ranges over choice sequences (not: lawless sequences) chosen from {0, 1}.]]
- 12.
- 13.
[[Now Toledo 2011, 206.]]
- 14.
- 15.
In line 14 on p. 498 of van Atten and Kennedy 2009, read ‘23’ for ‘12’.
- 16.
5 3/4–8 1/2 Uhr Gödel. Über Unerschöpflichkeit der Mathematik (siehe besonderes Blatt). Er ist durch Brouwers Wiener Vortrag zu diesen Gedanken angeregt worden. Die Mathematik ist nicht restlos formalisierbar. Er scheint recht zu haben. (Köhler 2002a, 92)
- 17.
‘Wir lassen als legitime Mathematik gewisse Überlegungen über die Grammatik einer Sprache, die vom Empirischen spricht, zu. Wenn man eine solche Math[ematik] zu formulieren versucht, so gibt es bei jeder Formalisierung Probleme, die man einsichtig machen und in gewöhnlicher Wortsprache ausdrücken, aber nicht in der betroffenen formalisierten Sprache ausdrücken kann. Daraus folgt (Brouwer), dass die Math[ematik] unerschöpflich ist: man muss immer wieder von neuem aus dem “Born der Anschauung” schöpfen. Es gibt daher keine Characteristica universalis für die gesamte Math[ematik], und kein Entscheidungsverfahren für die gesamte Math. In irgend einer abgeschlossenen Sprache gibt es nur abzählbar viele Ausdrücke. Das Kontinuum tritt nur in der “gesamten Math.” auf. …Wenn wir nur eine Sprache haben, und über sie nur “Erläuterungen” machen können, so sind diese Erläuterungen unausschöpfbar, sie bedürfen immer wieder neuer Anschauung.’ (Carnap and Gödel 2002, 110) Note that Köhler, unlike Wang, does not explicitly identify this as the ‘separate sheet’ mentioned in the diary note; but both give the same date for it.
- 18.
- 19.
- 20.
On the otherwise empty back, Gödel wrote ‘Brouwer bar theorem’; that English term was introduced only in Brouwer (1954A). But it is not excluded that Gödel made these notes before or in 1942 and then added that jotting on the back later.
- 21.
For a detailed analysis of the analogy between forcing and intuitionistic logic, see Fitting (1969). In fact, Cohen’s development of forcing after his initial discovery was influenced by this analogy, when Dana Scott pointed out to him how it could be used to simplify his treatment of negation; see Scott’s foreword to Bell (1985). Scott there also mentions the anticipation of forcing in Kreisel (1961).
- 22.
In a draft note for the revision of the Dialectica paper (Gödel Papers, 9b/148.5, 040498.59), Gödel wrote: ‘Finally I wish to note that the definition of a proof as an unbroken chain of immediate evidences should be useful also for Heyting’s interpretation of logic. In particular A ⊃ B can then be defined simpler, namely by requiring that a proof of A ⊃ B is a finite sequence P i of propositions ending with B and such that each P i ≠ A is immediately evident, either by itself, or on the basis of some of the preceding propositions.’ A proof of the Bar Theorem based on that explanation of ⊃ has not yet been found; compare Gödel’s Footnote d (Gödel 1990, 272).
- 23.
Gödel Papers, 5c/19, 030025, 12–15 in the backward direction (Arbeitsheft 7). This note is labeled ‘Gentzen’. In the index to the Arbeitshefte (Gödel Papers, 5c/12, 030016) the reference to this note is the first entry under the heading ‘Interpr[etation] d[er] int[uitionistischen] Logik’, and has ‘(heur[istisch])’ written after it. It contains, for example, a version of the proof for the validity of modus ponens interpreted by functionals. This must be the note that Wang describes, without a specific reference, in note 13 on p. 47 of Wang (1987). The date ‘1./I.1941’ is on top of p. 12 (backward direction) of Arbeitsheft 7. On the same page, before the note ‘Gentzen’, there is one on ‘Rosser Wid[erspruchs]fr [eiheits] Bew[eis]’, with a horizontal line in between; that the date also holds for the second item on the page is very likely because it is also the date of another note, headed ‘Jede F[u]nct[ion] d[es] eigentl[ich] intuit[ionistischen] Systems ist berechenbar’. That note is in the notebook Resultate Grundlagen III (Gödel Papers, 6c/85, 030118, 188–191) and states the date ‘1./I.1941’. It begins with a reference to p. 34 (backward direction) of Arbeitsheft 7, which is where the formal system Σ is defined.
- 24.
A.M. Turing hat bekanntlich mit Hilfe des Begriffs einer Rechenmaschine eine Definition des Begriffs einer berechenbaren Funktion erster Stufe gegeben. Aber wenn dieser Begriff nicht schon vorher verständlich gewesen wäre, hätte die Frage, ob die Turingsche Definition adäquat ist, keinen Sinn. (Gödel 1958, 283n2)
- 25.
Man kann darüber im Zweifel sein, ob wir eine genügend deutliche Vorstellung vom Inhalt dieses Begriffs haben, aber nicht darüber, ob die weiter unten angegebenen Axiome für ihn gelten. Derselbe scheinbar paradoxe Sachverhalt besteht auch für den der intuitionistischen Logik zugrunde liegenden Begriff des inhaltlich richtigen Beweises. (Gödel 1958, 283n1)
- 26.
Compare Gödel’s claim, in the Lecture at Zilsel’s from 1938, that the axioms of the subsystem of Heyting’s logic presented there are, when interpreted intuitionistically, ‘actually plausible’ (tatsächlich plausibel; (Gödel *1938a, 100/101)).
- 27.
Notes he wrote to it when reading it in Gerhardt’s edition can be found in Gödel Papers, 10a/35. As Sundholm reminded me, the same distinctions are explained by Leibniz also in an earlier text of the same period (published in the same volume of Gerhardt’s edition), the ‘Meditationes de cognitione, veritate, et ideis’ of 1684 (Leibniz 1875–1890, 4:422–426). That text actually served as a basis for the section in the Discours. Leibniz takes up the theme again in the Nouveaux Essais of 1704 (but published posthumously, in 1765), Book II, Chap. XXXI, Sect. 1 (Leibniz 1875–1890, 5:247–248).
- 28.
There is a direct and documented relation between ideas of Leibniz and the revisions of the Dialectica Interpretation; see van Atten (Forthcoming).
- 29.
This is most explicit in the 1972 version, for the more specific interpretation in terms of reductive proof, which Gödel says is ‘constructive and evident in a higher degree than Heyting’s’ (Gödel 1972, 276n(h)).
- 30.
- 31.
Note MvA. On November 15, 1940, Gödel had lectured at Brown University on the consistency of the Continuum Hypothesis.
- 32.
Ich habe jetzt wi[eder] eine Menge zu tun, da ich eine Vorlesung halte u[nd] ausserdem wieder zu einem Vortrag eingeladen bin wobei in beiden Fällen das Thema meine allerletzte Arbeiten sind, die ich noch nicht einmal für mich selbst genau zu Papier gebracht habe.
- 33.
- 34.
See also Gödel (1995, 188).
- 35.
Note MvA. Perhaps Gödel uses the plural here because he is thinking of intuitionistic logic as it figures in different theories.
- 36.
See also Gödel (1995, 189).
- 37.
See also Gödel (1995, 189).
- 38.
Two or three words that are difficult to read; perhaps ‘which we try’?
- 39.
I have not attempted to reconstruct, from the Arbeitshefte, how far Gödel got. But he evidently did not succeed: in conversation with Kreisel in 1955, he mentioned the assignment of ordinals as an open problem (Kreisel 1987, 106), and, although it was solved for a special (but in a sense sufficient) case in Howard (1970), he did so again in a telephone conversation with Tait in 1974. But, as Tait remarks, to exploit such an assignment in a proof of normalisation, PRA together with induction up to ε0 are required, so it could not serve Gödel’s foundational aim (Tait 2001, 116 and its n39). (See Kanckos 2010 for a version of Howard’s proof in the setting of Natural Deduction.)
- 40.
‘Vielleicht kommt man in der Math[ematik] deswegen nicht weiter (und gibt es so viele ungelöste Probl[eme]), weil man sich auf Ext[ensionen] beschränkt – daher auch das Gefühl der Enttäuschung bei manchen Theorien, z.B. dem Aussagenkalkül und der Formalisierung überhaupt.’ (Gödel Papers, 6b/67, 030090, 198) Transcription Cheryl Dawson and Robin Rollinger.
- 41.
- 42.
Compare also Gödel to Bernays, September 30, 1958: ‘Kreisel told me that in your lectures in England you discussed the combinatorial concept of set in detail. I very much regret that nothing about that has appeared in print. Conceptual investigations of that sort are extremely rare today.’ (Gödel 2003, 157) (‘Kreisel erzählte mir, dass Sie in Ihren Vorträgen in England den kombinatorischen Mengenbegriff näher besprochen haben. Ich habe sehr bedauert, dass darüber nichts in Druck erscheinen wird. Begriffliche Untersuchungen dieser Art sind ja heute äusserst selten’, Gödel 2003, 156.)
- 43.
In a letter to Gödel of June 17, 1960, written after a visit to him, Sigekatu Kuroda wrote: ‘It was my great pleasure also that I heard from you that you are studying Husserl and you admired his philosophy, which was the unique philosophy that I devoted rather long period and effort in my youth. I hope I have a chance some day to speak with you about Husserl. As you are doing now, I would like to recollect Husserl’s philosophy after returning to my country.’ (Gödel Papers, 01/99, 011378) Note that by that time Kuroda had published philosophical and technical work on intuitionistic logic, notably Kuroda (1951), in which he moreover says (p. 36) that he shares Brouwer’s view that mathematics is an activity of thought that is independent of logic and based on immediate evidence that is intuitively clear. Without further sources it is of course impossible to tell whether Gödel and Kuroda discussed phenomenology and intuitionism in relation to one another, but Kuroda’s letter gives the impression that they had not.
- 44.
Personal communication from Georg Kreisel, letter to MvA, January 10, 2005.
- 45.
For more on ontological descriptivism, Brouwer’s exploitation of it, and its contrast to meaning-theoretical approaches to mathematics such as Dummett’s or Martin-Löf’s, see Sect. 5 of Sundholm and van Atten (2008).
- 46.
Note MvA. See also Gödel (1972, 272n(d)): ‘Unfortunately, however, no satisfactory constructivistic proof is known for either one of the two principles [i.e., Brouwer’s bar induction and Spector’s generalisation to finite types]’.
- 47.
Howard, story 20. In an email to me of January 26, 2013, William Howard adds that this was the only occasion during his conversations with Gödel (which took place during Howard’s year at the IAS, 1972–1973) that the topic of the Bar Theorem and of bar induction came up.
- 48.
I thank Dirk van Dalen for letting me photocopy the purple-ink duplicate he received from Scott in Oxford.
- 49.
The material is rich, and should also be studied with other questions in mind, and from other perspectives. To my mind, in particular D68 would have deserved to be included in the Collected Works as well.
- 50.
This was also Heyting’s view: ‘I must protest against the assertion that intuitionism starts from definite, more or less arbitrary assumptions. Its subject, constructive mathematical thought, determines uniquely its premises and places it beside, not interior to classical mathematics, which studies another subject, whatever subject that may be’ (Heyting 1956, 4).
- 51.
A.M. Turing hat bekanntlich mit Hilfe des Begriffs einer Rechenmaschine eine Definition des Begriffs einer berechenbaren Funktion erster Stufe gegeben. Aber wenn dieser Begriff nicht schon vorher verständlich gewesen wäre, hätte die Frage, ob die Turingsche Definition adäquat ist, keinen Sinn. (Gödel 1958, 283n2)
- 52.
Moreover, Gödel will have known the observation by Skolem, Heyting, and Péter that in constructivism, ‘computable function’ cannot be taken to mean ‘recursive function’. See Skolem (1955, 584), a paper to which my attention was drawn by Coquand (2014); Heyting (1958, 340–341), which appeared in the same special issue of Dialectica as Gödel’s paper; Péter (1959). Heyting is the one who emphasises the alternative of taking that notion as primitive. Tait (2006, 212–213) holds that the fact that a definition would be circular shows that there is a problem with the idea of constructive evidence for the computability of a function. To my mind, that is not correct, but I will not develop this point here. See also Kreisel’s review Kreisel (1969a) of Tait (1967).
- 53.
- 54.
It is also possible to avoid CS, formally, by using the Brouwer-Kripke Schema BKS instead, usually formulated as \(\exists \alpha (\exists n\alpha (n) = 1 \leftrightarrow A)\) (but the parenthetical qualification in Footnote 55 below also holds here: BKS should really be formulated as two rules with parameters P and α = α P ). However, from the intuitionistic point of view, the known justification of BKS also justifies CS. Versions using BKS were given by Gielen (as quoted in de Swart 1976b, 35) and Dragálin (1988, 134–135); Gielen’s construction is closest to Van Dalen’s. The (weaker) point that BKS and Church’s Thesis are incompatible was first made in print by Myhill (1966, 296–297), and taken up in the influential (Troelstra 1969, 100).
- 55.
E.g., Troelstra and van Dalen (1988, 1:236), in particular: \(A \leftrightarrow \exists n(\boldsymbol{\square }_{n}A)\). Intuitionistically, this is not difficult to justify; see the discussions of the topic in Dummett (2000b, Sect. 6.3), and van Atten (2004a, Chap. 5). (By the considerations in Sundholm and van Atten (2008), and also in Sundholm (2014), the principle cited should in fact be presented as a pair of (proof, not inference) rules, rather than as a bi-implication as understood in Natural Deduction. Note that the explanation usually given of the principle as cited is in effect that of the rules.)
- 56.
The equivalence would be best understood as an extensional one, so as to forestall paradoxes that might appear if one would straightforwardly render the sentential operator \(\boldsymbol{\square }_{n}\) by a provability predicate. Alternatively, one could use BKS instead of CS to construct the function, as mentioned in Footnote 54. I thank Albert Visser for raising this issue and for his Répondez!.
- 57.
es scheint mir jetzt, nach reiflicher Überlegung, dass die Wahlfolgen etwas Anschauliches u[nd] daher im Hilbertschen Sinn Finites sind, wenn auch Hilbert selbst vielleicht anderer Meinung war. (Gödel 2003, 268)
- 58.
This corresponds to Gödel Papers, 9b/148, 040498.
- 59.
Hilbert wollte Wahlfolgen wohl nicht zulassen? Mir scheinen sie durchaus anschaulich zu sein, aber den Finitismus nicht wesentlich zu erweitern. (Gödel 2003, 270)
- 60.
These phenomenological projects were overlooked in the research for van Atten and Kennedy (2003), to which this part of the present paper should be considered an addendum.
- 61.
Note MvA. In his Footnote 13, Gödel refers to Heyting (1934, 14).
- 62.
Note MvA. ‘Primitive recursive functions of finite type over the natural numbers’ (Gödel Papers, 9b/141, 040450, 24 (1968))
- 63.
Note MvA. References left open by Gödel.
- 64.
Note MvA. Reference left open by Gödel.
- 65.
Note MvA. Above ‘concepts’, Gödel wrote: ‘entities’.
- 66.
Note MvA. Also: ‘speaking (as intuitionists …do) of thoughts as occurrences in spacetime reality (instead of their content) the objectivation (in the statements of the theory) of abstract entities and existential assertions about them are avoided and, moreover, the content of the thoughts to be admitted, although itself something abstract, always refers to something concrete, namely other thoughts or symbols or actions’ Gödel Papers, 9b/148.5, between 040498.39 and 040498.43.
- 67.
Unreadable word; ‘comb[inations]’?
- 68.
‘Der reelle Bestand der Wahrnehmung und ihr transzendentes Objekt’ and ‘Reelle und intentionale Erlebniskomponenten. Das Noema’. Translations taken from Husserl (1983); the second one is modified.
- 69.
Note MvA. Page reference left open by Gödel.
- 70.
Gödel may have read this before 1962, the year the original German manuscript was reprinted in the Husserliana edition (Husserl 1962); there is a library slip (Gödel Papers, 9c/22, 050103) requesting the relevant volume (17: ‘P to Planting of Trees’) of the 14th edition of the Britannica of 1929. There are also some reading notes in the same folder. For a different connection between Gödel and the Britannica article, see van Atten and Kennedy 2003, Sect. 6.1 (Sect. 6.6.1 in this volume).
- 71.
Der Weg in die phänomenologische Transzendentalphilosophie von der Psychologie aus.
- 72.
1968 is the copyright year. That is not necessarily the year the book became available.
- 73.
The project of a non-empirical (e.g., ‘a priori’, ‘rational’ or ‘transcendental’) psychology has a long tradition (e.g., Wolff, Kant); for Gödel, Husserl’s version will have been attractive because it is closely related to transcendental phenomenology, to which Husserl considered it to be propaedeutic.
- 74.
‘Strictly speaking the construction of intuitive mathematics in itself is an action and not a science; it only becomes a science …in a mathematics of the second order, which consists of the mathematical consideration of mathematics or of the language of mathematics’ (Brouwer 1975, 61n1). (‘Eigenlijk is het gebouw der intuïtieve wiskunde zonder meer een daad, en geen wetenschap; een wetenschap …wordt zij eerst in de wiskunde der tweede orde, die het wiskundig bekijken van de wiskunde of van de taal der wiskunde is.’ (Brouwer 1907, 98n))
- 75.
Note MvA. In English in the original.
- 76.
Note MvA. I translate ‘Vorstellung’ as ‘image’ here, because that is the term Gödel uses in these manuscripts when writing in English. Spiegelberg (1965), a work that Gödel owned (2nd ed.) and knew well, translates it as ‘representation’, and a popular alternative is ‘presentation’. (NB Cairns’ recommendation, published in 1973, for the broadest Husserlian sense is ‘(mental) objectivation’ (Cairns 1973, 131).) I take it that Gödel’s choice of ‘image’ is motivated by a wish to avoid special terminology as much as possible, so as to avoid making his philosophical remarks seem more dependent on a particular philosophy than they are. To Wang he said, ‘I am cautious and only make public the less controversial parts of my philosophy’ (Wang 1996, 235). Similary, Wang remarks that ‘Gödel’s desire to shun conflict also affected his published work. He would make great efforts to present his ideas in such a form that people with different perspectives could all appreciate them (in different ways)’ (Wang 1996, 235). (I thank Nuno Jerónimo for locating these comments.)
- 77.
Note MvA. Here the list stops, at the bottom of the left half of the page, and the right half of the page begins with a new remark.
- 78.
(Gödel Papers, 9b/148, 040492). Transcription Eva-Maria Engelen; translation MvA. ‘Gr[undlagen]: Es ist unglaublich, wie sämtliche wichtigen ph[ilosophischen] und psych[ologischen] Probleme bei genauer Behandlung meines Systems T aktualisiert [werden] und wie viele wichtige Distinct[ionen] klar werden: zum Beispiel: Evokation der Vorstellung eines Verfahrens und Anwendung des Verfahrens; Vorstellung einer Regel und Regel (man sieht wie ‘flimsy’ die erstere und wie ‘ehern’ die letztere ist); Resultate der Zwischenschritte und Operationen der Zwischenschritte; Operation im Sinn einer geistigen Handlung und eines mat[hematischen] Objekts (kurz: Regel, Vorstellung der Regel, Anwendung der Regel, Vorstellung der Anwendung der Regel); Def[initions-]Verfahren, um die Funktionen von T zu erhalten, Verfahren um die einzelnen Funktionen von T zu berechnen;’
- 79.
As Feferman points out in his introduction to the Gödel-Bernays correspondence, it is noteworthy that in this philosophically rich exchange, Husserl is never discussed (Gödel 2003, 66n(ax)). (Gödel mentions phenomenology once, on August 11, 1961 (Gödel 2003, 193).) Bernays will have known of Gödel’s enthusiasm for Husserl early on: Bernays was in Princeton from November 1959 to April 1960, and came back for shorter visits around Easter 1961, in May 1961, and in the Spring of 1965.Footnote 80 A short text presented in 1963 – in the middle of this period of visits – and published the next year, ‘Begriffe des Phänomenologischen und das Programm der phänomenologischen Philosophie’ (Bernays 1964), shows Bernays rather critical of Husserlian phenomenology, in particular of epoché, the possibility to see essences, and its foundational character. In his letters to Gödel, Bernays never mentions this text or the objections formulated in it; one possible explanation is that they had dealt with the topic in their conversations.
- 80.
See letters 26–31; 34–36; 38–39; and 52–53, respectively.
- 81.
Die Sinnklärung besteht hier darin, daß man die betreffenden Begriffe schärfer ins Auge faßt, indem man die Aufmerksamkeit in einer bestimmten Weise dirigiert, nämlich auf unsere eigenen Akte bei der Verwendung dieser Begriffe, auf unsere Mächte bei der Vollführung unserer Akte, etc. Man muß sich dabei klar darüber sein, saß diese Phänomenologie nicht eine Wissenschaft im selben Sinn ist wie die andere Wissenschaften. Sie ist vielmehr (oder sollte jedenfalls sein) ein Verfahren oder Technik, welches in uns einen neuen Bewußtseinszustand hervorbringen soll, in dem wir die von uns verwendeten Grundbegriffe unseres Denkens detaillieren oder andere bisher uns unbekannte Grundbegriffe erfassen.’ (Gödel *1961/?, 382)
- 82.
That seems to have been the kind of reason why, in the published version of the supplement to the second edition of his Cantor paper, Gödel left out a hopeful reference to phenomenology that is present in the draft (Gödel Papers, 8c/101, 040311), while at the same time recommending Husserl to logicians in conversation. See van Atten and Kennedy (2003, 466); Sect. 6.6 in this volume.
- 83.
Elsewhere, I have argued that Gödel’s program to employ transcendental phenomenology to found classical mathematics is misguided (van Atten 2010); Sect. 12.3 in this volume. But, by the positive argument in the same paper, his attempt to use it to enrich intuitionism, whether eventually successful or not, makes perfectly good sense.
- 84.
konstruktiv erkennbare. (Gödel 1958, 282)
- 85.
Hier könnte wohl der Leser stutzen, da doch Ihr Verfahren bezweckt, den Begriff des intuitionistischen Beweises zu vermeiden. Es scheint mir jedoch, dass Sie de facto hier diesen Begriff auch gar nicht brauchen un dass es nur einer geeigneten Umformulierung bedarf, um dieses zum Ausdruck zu bringen. (Gödel 2003, 201)
- 86.
Note MvA. ‘CFI’ is Gödel’s abbreviation in D68 for ‘computable functional of finite type’ (Gödel Papers, 9b/141, 040450, 23–24 (1968)).
- 87.
‘Falsch’, in shorthand.
- 88.
In note k of D72 (Gödel 1990, 275n(h)), however, he quoted that part of the definition as ‘constructively evident or demonstrable’; I assume this was left in inadvertently.
- 89.
Gödel studied his Russell paper when working on the Dialectica paper. This is clear from the remarks on the loose sheet inserted with one of the four offprints that Gödel owned, offprint D in the ‘Textual notes’ in the Collected Works, Gödel 1990, 315–322; the remarks in question are on p. 320 and p. 321. NB Correction: The note on the title page of D does not say ‘gelesen bis p. 135 oben’ (Gödel 1990, 320), but ‘gelesen bis p. 138 oben’.
- 90.
Note MvA. The typescript erroneously has ‘XI’, as had the original publication of Gödel’s Russell paper (Gödel 1944, 134).
- 91.
Note MvA. ‘[die] unvermeidlichen “self reflexivities” ’.
- 92.
Note MvA. Presented, in the later version of D72, as note h in Gödel (1972).
- 93.
Ich danke Ihnen auch bestens für Ihren Brief über die Frage, ob der allgemeine intuition[istischen] Beweisbegriff für die intuition[istischen] Interpretation meines Systems T nötig ist (was meine Interpretationder logischen Operatoren erkenntnistheoretisch wertlos machen würde). Ich glaube, dass das nicht der Fall ist, sondern dass ein viel engerer (u[nd] im Prinzip) entscheidbarer Beweisbegriff genügt, den ik in Note k der Übersetzung meiner Dialectica [arbeit] eingeführt u[nd] ‘reduktive Beweisbarkeit’ genannt habe. Aber das im einzelnen befriedigend durchzuführen, ist nicht ganz leicht, hauptsächlich wegen der nicht eliminierbaren Imprädikativität auch dieses engeren Beweisbegriffes, welch mit der von Ihnen erwähnten Imprädikativität des Funktionsbegriffes nahe zusammenhängt. Est ist zweifelhaft, ob die Durchführung die Mühe lohnen würde. Ich habe mich daher bis jetzt nicht dazu entschliessen können, obwohl die weitere Verfolgung dieser Fragen vielleicht wesentlich zur Aufklärung der Grundlagen des Intuitionismus beitragen könnte. (Gödel 2003, 300)
- 94.
I write ‘at a certain level of generality’, because this compatibility may or may not be preserved when making one’s conception of phenomenology more specific. In van Atten (2010) I argue that, in particular, if one’s conception of phenomenology is that of the transcendental Husserl (of, roughly, the 1920s and 1930s), then intuitionistic mathematics is compatible with, and moreover part of, phenomenology, whereas classical mathematics is neither.
- 95.
In the archive, these are not kept with the drafts and galleys for the revised Dialectica paper, but in a folder named ‘Dialectica interpretation’ under the heading ‘Other loose manuscript notes’.
- 96.
Note MvA. There are notes in which Gödel writes antecedents as ‘Red(p)’, for ‘p is reductively provable’ (which for given p is decidable, or should be once the notion of reductive proof has been sufficiently clarified).
- 97.
Note MvA. The antecedent is almost unreadable, but it seems safe to say that Gödel here gives an example of an implication in T.
- 98.
Ground objects and (x) is therefore an operation binding ground variables that leads to objects.
- 99.
Note MvA. P is the system that was going to be named T′ in D72 (see the manuscript for D70, Gödel Papers, 9b/142, 040452, insertion to note k3. In the circulated typescript, this is the footnote on p. 13).
- 100.
Gödel Papers, 11b/6, 060039. Transcription Eva-Maria Engelen, Robin Rollinger, and MvA. Translation MvA.
- 101.
Grundobjekte und (x) ist also eine Grund variablen bindende Operation, welche zu Objekte[n] führt.
- 102.
Note MvA. The part from ‘where’ to the end is a later insertion: Gödel Papers, 9b/145, 040462, k(2) +.
- 103.
- 104.
I.e., if the arguments are computable.
- 105.
Wenn die Begriffe ‘berechenbare Funktion vom Typus t 0’, ‘berechenbare Funktion vom Typus t 1’, …, ‘berechenbare Funktion vom Typus t k ’ (wobei k ≥ 1) bereits definiert sind, so wird eine berechenbare Funktion vom Typus \((t_{0},t_{1},\ldots,t_{k})\) definiert als eine immer ausführbare (und als solche konstruktiv erkennbare) Operation, welche jedem k-tupel berechenbarer Funktionen der Typen \(t_{0},t_{1},\ldots,t_{k}\) eine berechenbare Funktion vom Typus t 0 zuordnet. Dieser Begriff ist als unmittelbar verständlich zu betrachten, vorausgesetzt dass man die Begriffe ‘berechenbare Funktion vom Typus t i ’ \((i = 0, 1,\ldots,k)\) bereits verstanden hat.
- 106.
- 107.
Gödel Papers, 10a/40, 050136. Transcription Eva-Maria Engelen; translation MvA. The bars and underlining are Gödel’s.
11. II. 74
[A] Meine Dial[ectica] Arbeit mit dem Begriff des reduktiven Beweis[es] gibt keine die Parad[oxien] ausschließende Interpretation (daher die Fundierung nicht wesentlich besser als Heyting und zwar deswegen, weil zum Beispiel der allgemeine Begriff der berechenbaren zahlentheoretischen Funktion vorkommt und dieser von irgendeiner Def[initions]-Kette spricht (also die Def[inition] x ∈ a ≡ ∼ x ∈ x kann vorkommen). Der Unterschied ist nur, dass der Begriff Evidenz nur auf Richtigkeit einer Def[inition] nicht auf Richtigkeit eines Beweises angewendet wird. Das heißt also, sie schließen nicht die ‘vastness’ des betracht[eten] Bereichs aus wie das Begriffe ‘zahlentheoretische Evidenz’, ‘typentheoretische Evidenz’, ‘Evidenz hinsichtlich Funktion endlichen Typs’ etc. tun. Diese Begriffe als Grundbegriffe sind ‘vage’. Aber vielleicht kann man präzise die erlaubten Sätze definieren (und diese wären dann eine konstruierte Menge wie die natürlich[en] Zahl[en], Gentzen), aber der Begriff des zahlentheoretisch sinnvollen Satzes würde \(\vert \) den Begriff des zahlentheoretisch sinnvollen Beweis[es] voraussetzen, da er B enthalten kann, also zirkulär [ist]. Was also geleistet wird, ist dreierlei:
[B]
-
1.)
‘richtiger Beweis’ ersetzt durch ‘richtige Def[inition]’,
-
2.)
der Beweis ist mat[hematisch] direkter (es werden viele ‘Verschlingungen’ vermieden,
-
3.)
das Probl[em] von Sein und Haben für Ex[istenz]sätze wird gelöst.
[C] Es könnte da irgendein Normalform-Th[eorem] für Beweise folgen (aus 3.)), aus welchem der Bar Ind[uktion] folgen könnte ?? Die Unmöglichkeit eine absolute Unbeweisbarkeit zu beweisen, könnte folgen aus einer Idealisierung der Beweise mit gewissen Grundbegriffen und dann könnte man Beweise definieren als mat[hematische] Beweise mit diesen Mitteln und das würde genügen für den Widerspruchsfreiheitsbeweis. Aber all das, damit es Sinn hat, setzt voraus, dass man der Parad[oxie] ∼ x ∈ x aufgelöst hat.
-
1.)
- 108.
Given this influence, it would be interesting also to look at Gödel’s notion of reductive proof in relation to his remarks on analyticity of mathematics and Leibniz at the end of his Russell paper of 1944 (quoted in Sect. 11.3.5.2), and to the brief exchange on this in the Gödel-Bernays correspondence (Gödel 2003, 194, 200, and also p. 57 of the introduction); but I will not do this here.
- 109.
As is clear from the subject headings in Gödel’s mathematical Arbeitshefte, conveniently listed in Dawson and Dawson (2005), Gödel closely studied Brouwer’s interpretation of analysis. At the time of writing this, Jan von Plato has announced a talk (at the conference in Aix-en-Provence in July 2013) on these notes in relation to Gödel’s thoughts about Gentzen’s work, equally documented in these notebooks, in particular with an eye on the question to what extent Gödel may have anticipated Spector’s result. I will therefore not attempt to say more about the matter here.
- 110.
- 111.
As Sundholm urged me to do.
- 112.
It is remarkable that Kreisel in his long paper on Gödel and intuitionism (Kreisel 1987) refers to neither his own, nor Goodman’s, nor Scott’s work on the theory of constructions.
- 113.
See Footnote 10 above. In the same email referred to there, William Howard also recalls the following conversation with Gödel, probably in their first meeting during that sabbatical:
Gödel: ‘You should extend your theory of constructions to transfinite types in such a way as to get a functional interpretation of set theory (ZFC).’
Me: ‘I made such an attempt a couple of years ago and concluded that, to carry this out, I would have to learn more set theory.’
Gödel: ‘So, do it.’
Me: ‘Learning a sufficient amount of set theory appears to be a daunting task. There are a lot of papers.’
Gödel: ‘Very little of a substantial nature has been done. In fact, if you just read my two papers, that may be sufficient.’
At that point, he got up, walked across the room to a filing cabinet, pulled out reprints of the two papers (Proc. Nat. Acad. Sciences 1938, 1939) [Gödel 1938, 1939a] and handed them to me, saying, ‘Here is what you should read. You may keep these.’ Howard, story 5, p. 83
William Howard comments (in the same email):
Presumably what he had in mind in his first remark was that if my little theory of constructions is extended to transfinite types, in a natural way, as far into the transfinite as possible, the resulting theory would provide an interpretation of a part of ZFC (or a constructive version of a part of ZFC) which would be significantly weaker than ZFC itself. Hence one would have shown an essential limitation on what could be achieved by Brouwer’s ideas. In other words, do to Brouwer’s program what Gödel had done to Hilbert’s program. At least, that was my impression at the time. I seem to recall that he actually said something to that effect, but I don’t have any quotation, in my notes for Amy [as part of the preparation for the article Shell-Gellasch 2003], of him saying that to me.
Probably in the Spring of 1973, Gödel encouraged Howard to read Girard’s thesis (1972a) to get some ideas towards such an extension to transfinite types. (Details in Howard, story 16, p. 110.) Aczel’s interpretation of CZF in Martin-Löf’s Constructive Type Theory (Aczel 1978) may be seen as an execution of this project.
- 114.
‘Der Zeitpunkt des Erscheinens scheint mir weniger wichtig zu sein als die Textverbesserungen.’ (Gödel 2003, 290)
- 115.
It had appeared in February or March, and by March 25 at the latest (Gödel 1995, 518).
- 116.
Note MvA. Compare Gödel’s formulation in his letter to Constance Reid of March 22, 1966: ‘Moreover, the question remains open whether, or to what extent, it is possible, on the basis of a formalistic approach, to prove “constructively” the consistency of classical mathematics, i.e., to replace its axioms about abstract entities of an objective Platonic realm by insights about the given operations of our mind’ (Gödel 2003, 187). The quotation marks around the word ‘constructively’ are there, it seems, to distinguish its sense from that in which a proper part of classical mathematics is constructive; see also Sect. 11.3.5.3.
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Acknowledgements
This is the revised and much extended text of the talk with the same title given at the conference ‘Calculability and constructivity: historical and philosophical aspects’ of the International Union of the History and Philosophy of Science (Joint Session of the Division of Logic, Methodology and Philosophy of Science and of the Division of the History of Science and Technology), Paris, November 18, 2006. Much of that talk was derived from a manuscript that has in the meantime appeared as part of the present author’s contribution to van Atten and Kennedy 2009 (written in 2005). Other versions of that talk were presented at the plenary discussion ‘Gödel’s Legacy’ at the ASL European Summer Meeting in Nijmegen, August 2, 2006 and at seminars in Nancy (2005), Tokyo (2006), Utrecht (2006), and Aix-en-Provence (2007). I am grateful to the respective organisers for the invitations, and to the audiences for their questions, criticisms, and comments.
The quotations from Gödel’s notebooks and lecture notes appear courtesy of the Kurt Gödel Papers, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA, on deposit at Princeton University. I am grateful to Marcia Tucker, Christine Di Bella, and Erica Mosner of the Historical Studies-Social Science Library at the IAS for their assistance in finding answers to various questions around this material. In the study of Gödel’s notes in Gabelsberger shorthand, I have been able to consult Cheryl Dawson’s transcriptions, which she generously made available to me; these were also useful to Robin Rollinger and Eva-Maria Engelen, to whom I am greatly indebted for additional, speedy help with the shorthand, also concerning previously untranscribed passages. Access to the microfilm edition of the Kurt Gödel Papers was kindly provided to Rollinger, Engelen and me by Gabriella Crocco. The present paper is realised as part of her project ‘Kurt Gödel philosophe: de la logique à la cosmologie’, funded by the Agence Nationale de Recherche (project number BLAN-NT09-436673), whose support is gratefully acknowledged.
Gödel’s letters to his brother quoted here are part of a collection of letters that was found in 2006. I am grateful to Matthias Baaz and Karl Sigmund for bringing this correspondence to my attention, and for providing me with photocopies. These letters have been deposited at the Wienbibliothek im Rathaus, Vienna. The quotations appear courtesy of the Kurt Gödel Papers, The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ, USA.
I am grateful to Dirk van Dalen, Georg Kreisel, Albert Visser, and, in particular, William Howard and Göran Sundholm, for comments, references, criticisms and discussion. An anonymous referee wrote a helpful report on an earlier version. William Howard kindly granted permission to quote from the reminiscences he generously shared with me; additional material comes from the collection Stories (Howard) that he prepared for Amy Shell-Gellasch, who used a selection for her article Shell-Gellasch 2003. Those notes are now held at the Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin, as part of the William Howard Oral History Collection, 1973, 1990–2003. The Archives of American Mathematics hold the copyright; quotations are by permission. I thank Carol Mead of the Archives for her help and advice concerning this material and its use.
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Appendix: Finitary Mathematics and Autonomous Transfinite Progressions
Appendix: Finitary Mathematics and Autonomous Transfinite Progressions
Naturally, the draft notes for the revision of the Dialectica paper also contain remarks that are not concerned with intuitionism as such, but with finitary mathematics.
In support of the admission of abstract objects, note also that it is altogether illusory to try to eliminate abstractions completely, whatever the science in question may be. Even finitism in its strictest form does contain them, since every general concept is an abstract entity (although not necessarily an abstract concept, which term is reserved for concepts referring to something abstract). The difference between finitism and the envisaged extension of it only is that in the former abstractions occurring are only used, but are not made objects of the theory. So the question is not whether abstractions should be admitted, but only which ones and in what sense. It seems reasonable, at any rate, to admit as object of the investigation anything which is admitted for use. This leads to something like the hierarchy described in footn[ote] 7. (Gödel Papers, 9b/148.5, 040498.31)
The example referred to at the end is that of autonomous transfinite progressions, which Gödel describes in Footnote 2 on p. 281 of the 1958 version and Footnotes 4 and f of the 1972 version. On both occasions he refers to the formal work that appeared in print in Kreisel (1960, 1965); but in D68, he moreover writes that he had arrived at this idea when writing his incompleteness paper Gödel (1931), and had considered it finitary:
That in Mon[ats]H[efte für] Math[ematik und] Phys[ik] 38 (1931), p. 197 I said that finitary mathematics conceivably may not be contained even in formalized set theory is due to the fact that, contrary to Hilbert’s conception, I considered systems obtained by reflection on finitary systems to be themselves finitary. (Gödel Papers, 9b/141, 040450, 4F (1968))
and, in a different version with the title ‘Kreisel’s hierarchy’,
How far in the series of ordinals this sequence of systems reaches is unknown. Evidently it is impossible to give a constructive definition and proof for its precise limit, since this ordinal would then itself be an admissible sequence of steps. When in Mon[ats]H[efte für] Math[ematik und] Phys[ik] 38 (1931) p. 197 I was speaking of ‘conceivably’ very powerful finitary reasoning, I was really thinking of this hierarchy, overlooking the fact that from a certain point on (and, in fact, already for rather small ordinals) abstract concepts are indispensable for showing that the axioms of the system are valid, even though they need not be introduced in the systems themselves. (Gödel Papers, 9b/146, 040477)
An evaluation of the reliability and importance of these remarks will have to take into account that Gödel is not writing shorthand notes for himself here, but is drafting passages in longhand towards a paper meant for publication. Also, the fact that Gödel did not mention the idea of this hierarchy when he addressed the topic of possible finitary proofs that are not formalizable in Principia in his letter to Herbrand of July 25, 1931 (Gödel 2003a, 22–23), not long after the publication of the incompleteness paper,Footnote 115 could well be explained by a quick discovery of his own oversight.
In the 1960s Gödel was inclined to think that the limit of finitary mathematics is ε0. He saw support for this in arguments proposed by Kreisel, Tait, and Bernays; for a discussion of this matter, I refer to Sects. 2.4 and 3.4 of Feferman’s introduction to the Gödel-Bernays correspondence in Gödel 2003 and to Tait 2006. Here I add the following element. In D72, Gödel says that Kreisel’s ‘arguments would have to be elaborated further in order to be fully convincing’, and mentions that ‘Kreisel’s hierarchy can be extended far beyond ε0 by considering as one step any sequence of steps that has been shown to be admissible’ (Gödel 1990, 274n(f)). In one of the draft notes he actually endorses that idea:
Kreisel himself says on p. 177 [of Kreisel 1965] under 3.621: ‘the only support for taking ε0 …as a bound is empirical’. I was formerly myself leaning towards Kreisel’s conjecture. But today it seems much more probable to me that the limit of idealized Finitism is quite large. (Gödel Papers, 9b/145, insertion for p. 12)
Feferman has raised the possibility that ‘Gödel wanted it seen as one of the values of his work in 1958 and 1972 that the step to the notions and principles of the system T would be just what is needed to go beyond finitary reasoning in order to capture arithmetic’ (Gödel 2003, 74). That suggestion finds corroboration in the following passage:
I do not wish to say that every math[ematical] concept which is non-finitary must nec[essarily] be called abstract, let alone that it must be abstract in the special sense explained below. But I don’t think that there is any other ext[ension] of finitism which preserves Hilbert’s idea of justifying the infinite of the Platonistic elem[ents] of math[ematics] in terms of what is finite, concretely given & precisely knowable. Note that in contradist[inction] to Plat[onistic] entities, precise thoughts about things that are or can in principle be concretely given & precisely known are themselves something concretely given & precisely knowable.Footnote 116 If this ext[ension] of finitism is combined with a training in this kind of int[uition], something in character very close to finitary evidence but much more powerful may result. (Gödel Papers, 9b/147, 040486)
This same passage may also serve to address Tait’s suggestion that Gödel, by extending Hilbert’s finitary position with thought contents or structures, ‘simply doesn’t see the “finite” in “finitary” ’ (Tait 2010, 93). Gödel emphasises that the same criterion that leads Hilbert, who considers only space-time intuition, to a restriction to configurations of a finite number of objects, allows for further, different objects when applied to thoughts, given a correspondingly wider notion of intuition. To hold that everything which is concretely given and precisely knowable is thereby, in a numerical sense or otherwise, finite, is to follow an old tradition.
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van Atten, M. (2015). Gödel and Intuitionism. In: Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer. Logic, Epistemology, and the Unity of Science, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-10031-9_11
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