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Kurt Gödel Philosopher-Scientist

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Gabriella Crocco
, 
Eva-Maria Engelen

New Readings in Gödel’s Philosophy

Gödel’s Dialectica Interpretation and Leibniz

Mark van Atten

Texte intégral

  • 1 © 2014 Mark van Atten. The present paper originated in research done for a longer paper on Gödel an (...)

1In 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.1 The writing is hard to read; by a coincidence, the parts that could be transcribed are the two parts whose importance Gödel indicated with vertical bars:

  • 2 [NB Gödel writes “ungewissen”. If one understands “axiom” in its foundational meaning of a statemen (...)
  • 3 9b/148, item 040495. Transcription Eva-Maria Engelen, Robin Rollinger, and MvA; translation MvA. Th (...)

Note2
Note
3

2Note first the dates Gödel wrote above this note, January to April 1968. In his letter to Bernays of December 20, 1967, Gödel had announced he would have the translation of the 1958 version of his Dialectica paper typed early January; but in his letter of May 16, 1968, Gödel explained why this did not happen:

  • 4 “[Eva-Maria Englelen remarks that, althoug it is not excluded that instead of the shorthand of ‘bet (...)
  • 5 “Als ich das Manuskript vornahm, um es abtippen zu lassen, fand ich in der philosophischen Einleitu (...)

As I took up the manuscript in order to have it typed, I found in the philosophical introduction (i.e., in the first 3 1/2 pages in Dialectica much even in the original text (that was) presented in such an unsatisfying and fragmentary way that I considered numerous supplementary remarks and changes as absolutely necessary. In the end, then, I completely rewrote that part and doubled its size. Then at the beginning of4 April I fell ill [...] (Gödel, 2003, p. 261)5

3Hence the period in which the note on Leibniz was written coincides with the period during which Gödel tried to rewrite the philosophical introduction to the Dialectica paper intuitionistically, a revision that, after this letter to Bernays, was only to be abandoned again; Gödel then decided to add further, explicatory footnotes (letter to Bernays, December 17, 1968, Gödel (2003, p. 264)); this led to a new version in 1970, which Bernays received from Dana Scott (Gödel, 2003, p. 280/281). (Bernays would remark on a wrong preposition in note k, but in the remainder of their correspondence, he never commented on the notion of reductive proof itself. It is also clear that he did not actually get to see the later version of 1972 anymore, in which note k was developed further; see also Feferman’s remarks in Gödel (2003, p. 73).) It seems, then, that the idea to justify the Dialectica Interpretation in specifically intuitionistic terms–documented in Van Atten (2014)–and the idea to justify it in terms of reductive proof both occurred to Gödel in early 1968, and that he decided to work on the former first. When that attempt did not lead to a satisfactory result, he returned to the development of the notion of reductive proof.

4Leibniz’ (development of an) account of truth is rich and intricate, and there is a wealth of literature; for extensive (recent) discussion and further references, I refer to Ishiguro (1990), Grosholz and Yakira (1998), Rauzy (2001), and Lenzen (2004). The following is only meant to recapitulate the minimum needed to exhibit the connection between Leibniz’ account of truth and Gödel’s notion of reductive proof. The two fundamental tenets of Leibniz’ account can be stated as follows:

  • 6 This idea goes back to Aristotle, but in Leibniz’ logic and metaphysics takes on a systematic and p (...)

1. “Praedicatum inest subjecto”: Propositions of the form “A is B”, if true, are true on the ground that the notion (or concept) of the predicate B is in (“inest”) the notion of the subject A; the latter contains (“continet”) the former.6
2. To propositions that are not of that first form correspond propositions that are, in such a way that truth of the former is accounted for in terms of truth of the latter.

  • 7 A notorious case of such a demonstration by unfolding definitions is Leibniz’ alleged proof that an (...)
  • 8 Of course, Gödel was able to give precise reasons why primary truths need to be recognized whose tr (...)

5That the notion of A is contained in that of B is shown by unfolding their definitions, until a so-called identical proposition is reached, and this simultaneously demonstrates the corresponding proposition “A is B”.7 Hence the analogy that Gödel mentions between primitive concept and axiom, defined concept and theorem, and between definition and proof. As identical propositions Leibniz recognized all instances of “A is A”, “A is not non-A”, and the like; more generally, “propositiones per se notas” (Leibniz, 1903, Generales Inquisitiones, section 61, p. 372).8 For the present purpose, a particularly important passage can be found in Leibniz’ letter to Conring of March 19, 1678:

  • 9 “Propositiones autem identicas necessarias esse constat, sine omni terminorum intellectu sive resol (...)

But we know that identical propositions are necessary propositions without any understanding or analysis of their terms, for I know that A is A, whatever may be understood by A. All propositions, however, whose truth must be shown by further analysing and understanding their terms are demonstrable by such analysis, that is, by definitions. So it is clear that demonstration is a chain of definitions [catenam definitionem]. For in the demonstration of any proposition, nothing is used but definitions, axioms (with which I here include postulates), theorems which have been demonstrated previously, and observations. Since the theorems again must themselves be demonstrated, and axioms, except for identities, can also all be demonstrated, it follows that all truths can be resolved into definitions, identical propositions, and observations–though purely intelligible truths do not need observations. After the analysis has been completed, it will become manifest that the chain of demonstration begins with identical propositions or observations and ends in a conclusion but that the beginning is connected with the conclusion through intervening definitions. In this sense I said that a demonstration is a chain of definitions. (Leibniz, 1956, p. 187)9

6Leibniz does not use the term “analogy” here, but he does in related passages, e.g., Leibniz (1903, p. 377), and a conspicuous use of it (which Gödel had seen: 10a/30, item 050125 “gelesen”) is in a footnote by Couturat to a passage at p. 352 of Leibniz (1903): “Remarquer l’analogie établie ici entre les concepts et les propositions, ou entre les propositions et les inférences”.

  • 10 For a connection between Leibniz’ notion of infinite analysis and Gödel’s 1944 paper on Russell, se (...)

7According to Leibniz, in the case of truths of reason, these chains of definition are finite (Leibniz, 1903, pp. 1, 19, 408), while contingent truths only admit of infinite analysis.10

8The notion of “reductive proof” that Gödel introduced in the revisions of Dialectica is clearly modelled after Leibniz’ notion of demonstration just explained:

A narrower concept of proof (than Heyting’s), which may be called “reductive proof” and which, roughly speaking, is defined by the fact that, up to certain trivial supplementations, the chain of definitions of the concepts occurring in the theorem together with certain axioms about the primitive terms forms by itself a proof, i.e., an unbroken chain of immediate evidences. (Gödel, 1972, p. 275, n.h1)

  • 11 To be distinguished from “reductive” as in “reductive proof theory”, which studies particular relat (...)
  • 12 In what is probably the first draft of note h of Gödel (1972) (“k” in Gödel’s original marking), Gö (...)

9Gödel characterizes proof in this sense as a chain of definitions, which is Leibniz’ “catenam definitionem” in the letter to Conring. Moreover, also Gödel’s terminological choice to call these proofs “reductive” seems to go back to Leibniz:11,12

  • 13 “Duplex est analysis, una communis per saltum qua utuntur in Algebra, altera peculiaris quam voco r (...)

Analysis is of two kinds. The common type advances by leaps and is used in algebra. The other is special and far more elegant but less well known; I call it “reductive” analysis. (Leibniz, 1956, p. 233)13

10Leibniz does not explain that term further there, but elsewhere he writes:

  • 14 idque in analysi per saltum, cum ipsa problema solvere ordimur nullis aliis praesuppositionis. Eode (...)

Analysis is through a leap, when we begin to solve the problem itself, with no other assumptions. In the same way also synthesis is through a leap, when at the very beginning we pass from all necessary truth to our problem. But analysis is by degrees when we reduce [revocamus] the proposed problem to an easier one, and this to an easier still, etc., until we arrive at one which is within our power.’ (Leibniz, 1956, p. 234, supplemented)14

11and

  • 15 [Indeed, “reduco” is the standard Latin translation of Aristotle’s “ἀνάγω” (here in its meaning of (...)
  • 16 “Analysis pura quae nihil syntheseos habet, est Anagogica, in qua semper procedimus per incognita r (...)

Pure analysis, which contains no syntheses, is anagogical,15 in which we always proceed backward through the unknown, reducing [reducendo] the problem that was proposed to another, simpler one, and that again to another. (Leibniz, 1903, p. 558)16

12Naturally, both Leibniz and Gödel take care of the condition that, to be able to conceive of chains of definitions as proofs of theorems, one has to ensure that the objects defined exist.

  • 17 See, besides the quotation following in the main text, Generales Inquisitiones, sections 144 and 14 (...)

13On Leibniz’ understanding of essential propositions, this means that it has to be shown that the object in question is possible (exists in some possible world).17 Thus, in the New Essays he writes:

  • 18 [Nouveaux Essais, book II, ch. 32, §1 (Leibniz, 1875-1890, vol. 5, p. 250).]
  • 19 [Note that Leibniz does not characterize the identical propositions purely formally. To a rare text (...)
  • 20 Nouveaux Essais, book IV, ch. 2, §1 (Leibniz, 1875-1890, vol. 5, p. 347). Translation MvA. “Deux es (...)

Two is one and one, three is two and one, four is three and one, and so on. It is true that there is a statement hidden in there which I have already remarked on,18 namely that these ideas are possible: and that is known here intuitively, so that one can say that intuitive knowledge is contained in definitions if their possibility appears first.19 And in this way, all adequate definitions contain primitive truths of reason and hence intuitive knowledge. Finally, one can say in general that all primitive truths of reason are immediate through an immediation of ideas.20

14Gödel puts it as follows:

Note that in this context a definition is to be considered as a theorem stating the existence and unicity of an object satisfying certain conditions. (Gödel, 1972, p. 275n. h1)

15He had already remarked on this use of definition in 1943 or early 1944, in notebook Max X:

16Remark (Grammar): There are 3 ways to conceive of definitions:

  1. As propositions of the form: a has the same sense and reference as b (where by using variables in a and b infinitely many cases can be subsumed), i.e., as merely typographical abbreviations.

  2. As propositions of the form: a has the same reference as b and is a name of the object described by b. |.

    • 21 6b/70, item 030096 (Notebook Max X, 12. III.1943–27.I.1944), pp. 25-26. Transcription Robin Rolling (...)

    As propositions of the form φ (a), i.e., descriptions (but then existence and uniqueness have to be proved). In this sense the axioms of geometry could be definitions of the basic concepts.21

  • 22 Max X, pp. 70-73 and 79-85.
  • 23 (For the transcribers, see footnote 21.) P. 71: “2. Every proposition expresses a containment, anal (...)

17This was written only two years after the Yale lecture of 1941, and further on in the same notebook there are a number of remarks on Leibniz,22 occasioned, as Gödel annotates, by his reading of Couturat’s edition Leibniz (1903), and including glosses on Leibniz’ notions of “inesse” and of truth.23 But the context is not that of the functional interpretation, and it seems Gödel did not connect proofs in that interpretation to ideas about definitions before the note of 1968 quoted at the beginning of this note.

18Another point that both Leibniz and Gödel make is that, for the notion of reductive proof to be of epistemic use, we do not have to insist on actually producing reductive proofs, but may instead reason about them in ways that assure that such a proof can be produced. In Leibniz, this is stated as follows:

  • 24 Translation MvA. “Verum in genere sic definio, Verum est A, si pro A ponendo valorem, et quodlibet (...)

Truth in general I define as follows: A is true, if when putting a value for A, and treating whatever is contained in the value of A itself in turn like A, if this can be done, B and non-B, that is, contradiction, never occurs. From this follows that to be certain of a truth, either the resolution must be continued until first truths (or at least to those that have already been dealt with in such a process, or of which it has been established that they are true), or it must be demonstrated from the progression itself of the resolution, that is from a certain general relation between the preceding resolutions and the next, that such a thing will never occur, in whatever way the resolution is continued. (Leibniz, 1903, Generales Inquisitiones, section 56, pp. 370-371)24

19Gödel writes:

Note that it is not claimed that the proofs of T’ are reductive. This is true only in certain cases, in particular for the proofs of the axioms of T and of the individual cases of the rules of T... What is claimed is only that no other concept of proof than that of reductive proof occurs in the propositions and proofs of T’, except, of course, insofar as any theorem P in intuitionism means: A proof of P has been given. (Gödel, 1972, p. 276, n.h3)

  • 25 The soundness proof is given in full detail in Troelstra (1973, section 3.5.4).

20An essential step in obtaining a reductive proof, if there is one, will be to apply the steps of the soundness proof of HA under the Dialectica Interpretation (sketched by Gödel on p. 280 and its note n) to the non-reductive proof at hand, thereby obtaining the witnessing term and reducing the use of logic to the minimum that is reductive.25 In such a case, Gödel’s soundness proof precisely plays the role of what Leibniz called a “demonstration from the progression itself of the resolution” that this resolution would, if continued in full detail, terminate successfully.

  • 26 Gödel’s reading notes to Russell’s book can be found in folder 10a/27. (NB The entry in the Finding (...)

21In The Philosophy of Leibniz, Russell objected to what I above identified as the second tenet in Leibniz’ account of truth; Russell argued that Leibniz’ insistence on the subject-predicate form as the fundamental one makes it impossible to account for the truth of relational propositions, and indeed for mathematical truth, to the extent that numerical propositions involve a plurality of subjects (Russell, 1900, p. 12ff). Russell’s objections are unfounded, as for example Ishiguro has shown (Ishiguro, 1990, ch. 12, in particular pp. 102-103), and there is no indication that Gödel was concerned about them.26

22However, Leibniz did recognize that an affirmative particular proposition (e.g., “Someone is learned”) cannot be correlated to a subject-predicate proposition without further ado:

  • 27 “Sed in Propositione affirmativa particulari non est necesse ut praedicatum in subjecto per se et a (...)

But in an affirmative particular proposition it is not necessary that the predicate is in the subject of itself and considered absolutely, or that the notion of the subject of itself contains the notion of the predicate, but it suffices that the predicate be contained in some species of the subject or that the notion of some example or species of the subject contains the notion of the predicate; although what sort of species that is, is not expressed. (Leibniz, 1903, p. 55, trl. MvA)27

  • 28 “das Probl[em] von Sein und Haben für Ex[istenz] sätze wird gelöst”. For further details on that no (...)

23In the case of Heyting Arithmetic, Gödel’s Dialectica Interpretation precisely supplies a witness for propositions of the form ∃xP (x), predicate logic’s rendition of the affirmative particular proposition with an indefinite term in Leibniz’ term logic. In a note of as late as 1974, without mentioning Leibniz’ name, Gödel lists as one of the advantages of his Dialectica Interpretation that “the problem of being and having for existential propositions is solved”.28 In light of the above, this should be read as a reference to Leibniz’ theory of truth; taking “being” and “having” here to be references to the notions of inesse and continere.

24Acknowledgements. 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 anwers 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 realized 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, and for the suggestion to separate it from van Atten 2014, of which it originally was an appendix.

Bibliographie

References

ATTEN van, Mark (2011) “A note on Leibniz’ argument against infinite wholes”, British Journal for the History of Philosophy, 19 (1) pp. 121-129.

ATTEN van, Mark (2014) “Gödel and intuitionism”, in Constructivity and Computability in Historical and Philosophical Perspective, ed. by Jacques Dubucs and Michel Bourdeau, Dordrecht: Springer, pp. 169-214.

GÖDEL, Kurt (1944) “Russell’s mathematical logic”, The Philosophy of Bertrand Russell., ed. by Paul Schilpp, Evanston: Northwestern University, 1944, pp. 123-153. Also in Gödel (1990), pp. 119-141.

GÖDEL, Kurt (1958) “Über eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes”, Dialectica, 12, pp. 280-287. Also, with English translation, in Gödel (1990), pp. 240-251.

GÖDEL, Kurt (1972) “On an extension of finitary mathematics which has not yet been used”, Revised and expanded translation of Gödel (1958), first published in Gödel (1990), pp. 271-280.

GÖDEL, Kurt (1990) Publications 1938-1974, vol. 2 of Collected Works, ed. by S. Feferman et al., Oxford: Oxford University Press.

GÖDEL, Kurt (2003) Correspondence A-G, vol. 4 of Collected Works, ed. by S. Feferman et al., Oxford: Oxford University Press.

GROSHOLZ, Emily and ELKANAN, Yakira, eds., (1998) Leibniz’s Science of the Rational, Stuttgart: Franz Steiner. (Studia Leibnitiana Sonderheft 26).

ISHIGURO, Hidé (1990) Leibniz’s Philosophy of Logic and Language (2nd ed), Cambridge: Cambridge University Press.

KREISEL, Georg (1987) “Gödel’s excursions into intuitionistic logic”, in Weingartner and Schmetterer (1987), pp. 65-179.

LEIBNIZ, Gottfried W., (1875-1890) Die philosophischen Schriften von Gottfried Wilhelm Leibniz, ed. by C.I. Gerhardt, Berlin: Weidmann. Vols. 1-7.

LEIBNIZ, Gottfried W., (1903) Opuscules et fragments inédits, ed. by L. Couturat, Paris: PUF.

LEIBNIZ, Gottfried W., (1956) Philosophical Papers and Letters, ed. by L. Loemker, Dordrecht: Reidel.

LENZEN, Wolfgang (2004) Calculus Universalis. Studien zur Logik von G.W. Leibniz, Paderborn: Mentis Verlag.

RAUZY, Jean-Baptiste (2001) La doctrine leibnizienne de la vérité, Paris: Vrin.

RUSSELL, Bertrand (1900) A Critical Exposition of the Philosophy of Leibniz, Cambridge: Cambridge University Press.

TROELSTRA, Anne S., (1973) Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Berlin: Springer.

WEINGARTNER, Paul and SCHMETTERER, Leopold, eds., (1987) Gödel Remembered. Salzburg 10-12 July 1983, Napoli: Bibliopolis.

Notes

1 © 2014 Mark van Atten. The present paper originated in research done for a longer paper on Gödel and intuitionism, van Atten 2014.

2 [NB 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).]

3 9b/148, item 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.

Image 10000000000003CC00000134367C4FAAE57AD4EA.jpg

4 “[Eva-Maria Englelen remarks that, althoug it is not excluded that instead of the shorthand of ‘betrachten’, Gödel wrote that for ‘bestimmen’, the former makes better sense.]”

5 “Als ich das Manuskript vornahm, um es abtippen zu lassen, fand ich in der philosophischen Einleitung (d.h. den ersten 3 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, p. 260)

6 This idea goes back to Aristotle, but in Leibniz’ logic and metaphysics takes on a systematic and pivotal role. See Grosholz and Yakira (1998, section II.2.1) for differences between Aristotle’s and Leibniz’ views on predication.

7 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, p. 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.

8 Of course, Gödel was able to give precise reasons why primary truths need to be recognized whose truth is not a matter of their syntax.

9 “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, vol. I, p. 194)

10 For a connection between Leibniz’ notion of infinite analysis and Gödel’s 1944 paper on Russell, see Charles Parsons’ introduction to the latter in Gödel 1990 pp. 115-116.

11 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 Dialectia 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, p. 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.

12 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” (9b/142, item 040452, p. 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’” (9b/145, item 040458, p. 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 (2a/94, item 011289), and in Kreisel (1987, p. 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.

13 “Duplex est analysis, una communis per saltum qua utuntur in Algebra, altera peculiaris quam voco reductricem, quae longe elegantior est, sed parum cognita”. (Leibniz, 1875-890, vol. VII, p. 297)

14 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, p. 351)

15 [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).]

16 “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”.

17 See, besides the quotation following in the main text, Generales Inquisitiones, sections 144 and 146 (Leibniz, 1903, p. 391, 392), and, for comments, Ishiguro (1990, pp. 183-187).

18 [Nouveaux Essais, book II, ch. 32, §1 (Leibniz, 1875-1890, vol. 5, p. 250).]

19 [Note that Leibniz does not characterize 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, p. 186), Couturat adds the footnote “Cet appel à l’évidence sensible n’est guère conforme au rationalisme leibnitien”.]

20 Nouveaux Essais, book IV, ch. 2, §1 (Leibniz, 1875-1890, vol. 5, p. 347). Translation MvA. “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”.

21 6b/70, item 030096 (Notebook Max X, 12. III.1943–27.I.1944), pp. 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.

22 Max X, pp. 70-73 and 79-85.

23 (For the transcribers, see footnote 21.) P. 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]”); p. 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’”.)

24 Translation MvA. “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”.

25 The soundness proof is given in full detail in Troelstra (1973, section 3.5.4).

26 Gödel’s reading notes to Russell’s book can be found in folder 10a/27. (NB The entry in the Finding aid for 10q/38 (Gödel 2003, p. 544) mistakenly states “Bertrand Russell” where the Leibniz translator Charles William Russell is meant.)

27 “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”.

28 “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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Auteur

Is Senior Researcher at CNRS (National Center for Scientific Research), affiliated with the Institute Sciences, Normes, Décision (CNRS/Paris 4). He is the author of Brouwer Meets Husserl: On the Phenomenology of Choice Sequences (Springer, 2007) and of Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer (Springer, 2015).

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