Abstract
In an envelope of material relating to his work on the translation and revision of the Dialectica paper in 1968, Gödel kept a note that is in shorthand but in which one immediately notices the longhand name ‘Leibniz’. When transcribed and put into context, the note allows one to show that Leibniz was a source of inspiration for Gödel’s revision of the Dialectica Interpretation.
First accepted for publication as van Atten Forthcoming. Copyright © 2014 Mark van Atten.
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Notes
- 1.
Note MvA. Gödel writes ‘ungewissen’. If one understands ‘axiom’ in its foundational meaning of a statement of an immediate evidence (about the primitive terms), this qualification will be oxymoronic. But I take it that Gödel is here referring to its laxer use in ordinary mathematics (even when directed at truth).
- 2.
Gödel Papers 9b/148, 040495. Transcription Eva-Maria Engelen, Robin Rollinger, and MvA; translation MvA. The bars and underlinings are Gödel’s. We have worked from the microfilm edition of the Gödel papers. Consultation of the original document might allow full transcription of the middle part.
- 3.
Note MvA. Eva-Maria Engelen remarks that, although it is not excluded that instead of the shorthand for ‘betrachten’, Gödel wrote that for ‘bestimmen’, the former makes better sense.
- 4.
Als ich das Manuskript vornahm, um es abtippen zu lassen, fand ich in der philosophischen Einleitung (d.h. den ersten \(3\frac{1} {2}\) Seiten in den Dialectica) vieles schon im Urtext so unbefriedigend u. lückenhaft dargestellt, dass ich zahlreiche Ergänzungen u. Änderungen für unbedingt nötig hielt. Schliesslich habe ich dann diesen Teil gänzlich umgearbeitet u. den Umfang auf mehr als das doppelte vermehrt. Anfangs April wurde ich dann krank … (Gödel 2003, 260)
- 5.
This idea goes back to Aristotle, but in Leibniz’ logic and metaphysics takes on a systematic and pivotal role. See Grosholz and Yakira (1998, Sect. II.2.1) for differences between Aristotle’s and Leibniz’ views on predication.
- 6.
A notorious case of such a demonstration by unfolding definitions is Leibniz’ alleged proof that an infinite whole cannot exist because the whole is greater than the part. In van Atten (2011) I argue that not only is that argument incorrect, as Russell has shown, but Leibniz had all the means to see this. In his Russell paper of 1944, Gödel wrote:
Nor is it self-contradictory that a proper part should be identical (not merely equal) to the whole, as is seen in the case of structures in the abstract sense. The structure of the series of integers, e.g., contains itself as a proper part. (Gödel 1944, 139)
Among other things, Gödel says here that it is consistent that an equality relation holds between a proper part and the whole. That entails a rejection of Leibniz’s argument.
- 7.
Of course, Gödel was able to give precise reasons why primary truths need to be recognised whose truth is not a matter of their syntax.
- 8.
Propositiones autem identicas necessarias esse constat, sine omni terminorum intellectu sive resolutione, nam scio A esse A, quicquid demum intelligatur per A. Omnes autem propositiones quarum veritatem ex terminorum demum resolutione et intellectu patere necesse est, demonstrabiles sunt per eorum resolutionem, id est per definitionem. Hinc patet, Demonstrationem esse catenam definitionum. Nam in demonstratione alicujus propositionis non adhibentur nisi definitiones, axiomata (ad quae hoc loco postulata reduco), theoremata jam demonstrata et experimenta. Cumque theoremata rursus demonstrata esse debeant, et axiomata omnia exceptis identicis demonstrari etiam possint, patet denique omnes veritates resolvi in definitiones, propositiones identicas et experimenta (quanquam veritates pure intelligibiles experimentis non indigeant) et perfecta resolutione facta apparere, quod catena demonstrandi ab identicis propositionibus vel experimentis incipiat, in conclusionem desinat, definitionum autem interventu principia conclusioni connectantur, atque hoc sensu dixeram Demonstrationem esse catenam definitionum. (Leibniz 1875–1890, 1:194)
- 9.
For a connection between Leibniz’ notion of infinite analysis and Gödel’s 1944 paper on Russell, see Charles Parsons’ introduction to the latter (Gödel 1990, 115–116).
- 10.
To be distinguished from ‘reductive’ as in ‘reductive proof theory’, which studies particular relations between formal systems. Closely related, on the other hand, is the use of ‘reduction’ to indicate the reduction of terms, in a formal system, to normal form. In his introduction to the Dialectica paper in the Collected Works, Troelstra writes that ‘In view of Gödel’s choice of terminology (“reductive proof”) in note n1 [h1], it is tempting to think that he had something like a term model, defined via reductions, in mind. But there is no conclusive evidence for this.’ (Gödel 1990, 234) Indeed, for the reason explained in the main text I believe that the primary reason for Gödel’s choice of that term is the use Leibniz made of it.
- 11.
In what is probably the first draft of note h of Gödel (1972) (‘k’ in Gödel’s original marking), Gödel had written ‘reductive or analytical provability’, and then crossed out ‘or analytical’ (Gödel Papers 9b/142, 040452, 2). Similarly, there is a draft for that footnote in which it is said of ‘reductively provable’ that it is ‘a concept which closely approaches Kant’s meaning of “analytic” ’ (Gödel Papers 9b/145, 040458, 2 for the reference to the insertion and item 040462, k(2) ∘ for its text.). Gödel also claimed this in conversation with Kreisel, as recalled in Kreisel’s letter to Gödel of February 19, 1972 (Gödel Papers 2a/94, 011289), and in Kreisel (1987, 118) – in both cases followed by Kreisel’s objection that in proofs of propositions ∀xA(x) may occur functions of unbounded type that are not contained in the definition of A.
- 12.
Duplex est analysis, una communis per saltum qua utuntur in Algebra, altera peculiaris quam voco reductricem, quae longe elegantior est, sed parum cognita. (Leibniz 1875–1890, 7:297)
- 13.
idque in analysi per saltum, cum ipsa problema solvere ordimur nullis aliis praesuppositionis. Eodem modo et synthesis est per saltum cum a primis oriendo omnia necessaria percurrimus ad nostrum usque problema. Sed per gradum Analysis est, cum problema propositum revocamus ad facilius et hoc rursus ad facilius, et ita porro, donec veniamus ad id quod est in potestate. (Leibniz 1903, 351)
- 14.
Note MvA. Indeed, ‘reduco’ is the standard Latin translation of Aristotle’s ‘ἀνάγω’ (here in its meaning of to ‘lead back’, ‘to refer back’) when he speaks of transforming an imperfect syllogism into one in which all information needed to see its validity has been made explicit (Prior Analytics 29b1), or of the transformation of an argument into syllogistic form (ibid., 46b40).
- 15.
Analysis pura quae nihil syntheseos habet, est Anagogica, in qua semper procedimus per incognita retro, nempe reducendo problema propositum ad aliud facilius, et hoc iterum ad aliud. (Leibniz 1903, 558)
- 16.
- 17.
Note MvA. See Nouveaux Essais, book II, Chap. 32, Sect. 1 (Leibniz 1875–1890, 5:250).
- 18.
Note that Leibniz does not characterise the identical propositions purely formally. To a rare text in which Leibniz does say that the unprovability of an axiom is seen by the senses (Leibniz 1903, 186), Couturat adds the footnote ‘Cet appel à l’évidence sensible n’est guère conforme au rationalisme leibnitien’.
- 19.
Nouveaux Essais, book IV, Chap. 2, Sect. 1: ‘Deux est un et un, Trois est deux et un, Quatre est trois et un, et ainsi de suite. Il est vray qu’il y a là-dedans une enonciation cachée que j’ay déja remarquée, savoir que ces idées sont possibles: et cela se connoist icy intuitivement, de sorte qu’on peut dire, qu’une connoissance intuitive est comprise dans les definitions lorsque leur possibilité paroist d’abord. Et de cette maniere toutes les definitions adequates contiennent des verités primitives de raison et par consequent des connoissances intuitives. Enfin on peut dire en general que toutes les verités primitives de raison sont immediates d’une immediation d’idées’ (Leibniz 1875–1890, 5:347).
- 20.
Gödel Papers, 6b/70, 030096, 25–26. Transcription Robin Rollinger, Eva-Maria Engelen, in collaboration with other members of Gabriella Crocco’s group; based on earlier work by Cheryl Dawson.
Bem[erkung] (Gr[ammatik]): Es gibt 3 Arten Def[initionen] aufzufassen:
-
1.
Als Aussagen der Form: a ist sinn- und bedeutungsgleich mit b (wobei durch Variablen in a und b unendlich viele Fälle zusammengefaßt werden können), d.h. als bloß typogr[aphische] Abkürzungen.
-
2.
Als Aussagen der Form: a ist bedeutungsgleich mit b und ist ein Name des mit b Beschriebenen. |
-
3.
Als Aussagen der Form ϕ(a), d.h. Beschreibungen (dann muß aber Existenz und Eindeutigkeit bewiesen werden). In diesem Sinn könnten die Ax[iome] der Geometrie Def[initionen] der Grundbegriffe sein.
-
1.
- 21.
Gödel Papers, 6b/70, 030096 (Max X), 70–73 and 79–85.
- 22.
(For the transcribers, see Footnote 20.) 71: ‘2. Every proposition expresses a containment, analytic ones the containment of the predicate in the subject, synthetic ones the containment of “Being” in the combination subject-predicate.’ (‘2. Jeder Satz drückt ein Enthaltensein aus, bei analytischen das Enthaltensein des Präd[ikats] im Subjekt, bei synth[etischen] das Enthaltensein des “Seins” in der Kombination Subj[ekt]-Präd[ikat]’); 73: ‘Truth (according to Leibniz) = Relation between subject and predicate, more precisely an “inesse”.’ (‘Wahrheit (nach Leibniz) = Verhältnis des Subj[ekts] und Präd[ikats], genauer ein “inesse”.’)
- 23.
Generales Inquisitiones, Sect. 56: ‘Verum in genere sic definio, Verum est A, si pro A ponendo valorem, et quodlibet quod ingreditur valorem ipsius A rursis ita tractando ut A, si quidem id fieri potest, numquam occurat B et non-B seu contradictionem. Hinc sequitur ut certi simus veritatis vel continuandam esse resolutionem usque ad primo vera aut saltem jam tali processu tractata, aut quae constat esse vera, vel demonstrandum esse ex ipsa progressione resolutionis, seu ex relatione quadam generali inter resolutiones praecedentes et sequentem, nunquam tale quid occursurum, utcunque resolutio continuetur’ (Leibniz 1903, 370–371).
- 24.
The soundness proof is given in full detail in Troelstra (1973, Sect. 3.5.4).
- 25.
- 26.
‘Sed in Propositione affirmativa particulari non est necesse ut praedicatum in subjecto per se et absolute spectato insit seu ut notio subjecti per se praedicati notionem contineat, sed sufficit praedicatum in aliqua specie subjecti contineri seu notionem alicujus [exempli seu] speciei subjecti continere notionem praedicati; licet qualisnam ea species sit, non exprimatur.’ (Leibniz 1903, 55)
- 27.
‘das Probl[em] von Sein und Haben für Ex[istenz]sätze wird gelöst.’ For further details on that note, see van Atten (2014).
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Acknowledgements
The quotations from Gödel’s notebooks 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. I am greatly indebted to Eva-Maria Engelen and Robin Rollinger for their transcriptions of the shorthand. 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.
I am grateful to Göran Sundholm for discussion of earlier versions of this note.
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van Atten, M. (2015). Gödel’s Dialectica Interpretation and Leibniz. 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_4
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