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On causality as the fundamental concept of Gödel’s philosophy

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Abstract

This paper proposes a possible reconstruction and philosophical-logical clarification of Gödel’s idea of causality as the philosophical fundamental concept. The results are based on Gödel’s published and non-published texts (including Max Phil notebooks), and are established on the ground of interconnections of Gödel’s dispersed remarks on causality, as well as on the ground of his general philosophical views. The paper is logically informal but is connected with already achieved results in the formalization of a causal account of Gödel’s onto-theological theory. Gödel’s main causal concepts are analysed (will, force, enjoyment, God, time and space, life, form, matter). Special attention is paid to a possible causal account of some of Gödel’s logical concepts (assertion, privation, affirmation, negation, whole, part, general, particular, subject, predicate, necessary, possible, implication), as well as of logical antinomies. The problem of mechanical and non-mechanical procedures in the work with and on concepts is addressed in terms of Gödel’s causal view.

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Notes

  1. A formalization of a causal account of Gödel’s onto-theology is proposed in Kovač (2012, 2015), in connection with the discussion (philosophical and technical) on the so-called modal collapse.

  2. Many reflections from Wang (1987, 1996) and the reflections from Toledo (2011) stem from the 1970s; the manuscript Max X was written from March 1943 to January 1944, Phil XIV from July 1946 to May 1955, and Max XV from May 1955 and thereafter. For the datation of Max Phil manuscripts, see Crocco and Engelen (2016b). In light of further investigation and publication of Gödel’s manuscripts, we can expect new clarifications of Gödel’s related views.

  3. According to Gödel, “true philosophy” (Wang 1996, 9.3.16 p. 307, 9.3.21 p. 308) cannot be communicated to others as a “propositional knowledge”—what can be communicated is only the method used, “attitude of mind” (Wang 1996, 5.3.30–31 pp. 169–170, 5.3.35 and 5.3.37 p. 171). Moreover, the publication of “true philosophy” would be “contrary to the world” and quite unsafe for the author (Wang 1996, 9.3.16 p. 307, cf. 5.3.8 p. 166). Although aiming at “true philosophy”, Gödel does not claim ever to have arrived at some absolute philosophical truths (Wang 1996, 5.3.30 p. 170). Yet, he held the opinion that philosophers such as Kant and Husserl arrived at “true philosophy” (“absolute knowledge”, “superscience”) but did not publish it. See footnote 126 below, Wang (1996, pp. 166, 167, 169–171, 307, 308), and Kovač (2008, pp. 149–150). For Leibniz, see Wang (1996, p. 307) and Wang (1987, pp. 103–104, 224, 311).

  4. See Wang (1996, 9.2.19 p. 300, 9.2.22 p. 301).

  5. We will sometimes call primitive concepts “basic” concepts, to possibly avoid the strong connotation of their definite irreducibility. In Gödel’s usage, “primitive concepts”, “primitive notions” (Grundbegriffe; cf. “main categories”, Wang 1996, 5.3.7 p. 166), are, first, undefined concepts of some theory with axioms that “follow from the meaning of the primitive terms under consideration” (Gödel 1995f, p. 321). Secondly, “primitive concepts” are also concepts philosophically considered to be fundamental, independently of given axiomatic systems (“fundamental concepts”; Gödel 1995a, p. 337, 1990d, p. 121). Cf. Russell’s distinction between “undefined” and “indefinable ideas” (Gödel 1986–2003, vol. 1, p. 44, introductory note by B. Dreben and J. van Heijenoort). In general, Gödel allows for a possibility that primitive concepts could once become defined in terms of “more fundamental” or “undefinable” ones (“irreducible to anything more fundamental”; Gödel 1990d, p. 139). Regarding the “concept” itself, as will be shown later, its intended meaning is “just generality” (Sect. 3.2.4) but seems to be connected with some essential difficulties regarding its possible axiomatic description (see Sect. 3.3).

  6. However, with respect to the theory of concepts (logic proper), Wang reports Gödel as saying, “[W]e know what the primitives of the theory are which cannot be reduced to anything more primitive” (Wang 1996, 8.5.13 p. 272). Gödel uses the term ‘logic’ sometimes in a broader sense, comprising theory of concepts (intensional logic) as well as mathematical logic (extensional, including set theory), and sometimes narrowly, in the sense of the theory of concepts. We will focus on logic in the latter (occasionally considered as strict) sense. See Wang (1996, 8.4.18 p. 268, 8.6.1 p. 274). For the inclusive sense of ‘logic’, see also in Gödel (1995a, p. 334, ftn. 2).

  7. When citing Gödel’s original pagination, ‘p’ for ‘page’ will be omitted.

  8. See footnote 90 below.

  9. Cf. Wang (1996, p. 108) and Wang (1987, p. 217).

  10. Cf. Wang (1987, p. 217). See Kovač (2015, pp. 163–164).

  11. See Tieszen (2016, pp. 455–456). For Gödel’s critique of Husserl’s subjectivism, see van Atten (2015, pp. 122–124) or van Atten and Kennedy (2003); cf. Wang (1987, p. 219).

  12. “My theory ...  is like the monadology by Leibniz in its general structure” (Wang 1996, 0.2.1 p. 8, emphasis added).

  13. Cf. also the distinction between the category of causality and a subjective concept of causality, limited to the “effect in time” Gödel (2016, XIV 45). On Gödel’s philosophy of time in relation to Einsteins relativity theory, see extensively in Yourgrau (1999, 2005).

  14. For Kant, all categories have formal-logical origin, since they originate from basic logical forms of judgements (“table of judgements”; see Kant 1968, B105, translated in Kant 1998). As applied to what is given in an intuition, categories receive “transcendental-logical” (i.e., ontological) status.

  15. E.g., “space” is missing, while “enjoyment” could be assumed as implicit in “life” in the sense of remark (3).

  16. We follow van Atten and Kennedy (2003) and van Atten (2015) in having ‘will’ instead of ‘well’. For the causal meaning of these concepts, see Sects. 2.2.1 (“will”, “good”), 3.2.4, 3.2.5 and 3.4 (“object”); “relation” seems to be included in the meaning of the causation itself (cf. Ursachenrelation in Gödel 2016, XIV 18, but also “causality” as a sort of “relation” in Kant 1968, B106).

  17. On one place, Wang terms the most general concepts “categories”: “The basic categories of G’s ontology (i.e., the ‘things’ or all that which exists) are objects and concepts” (Wang 1987, p. 193). See Remark in Sect. 3.2.4 below.

  18. See Wang (1996, 9.2.20–22 pp. 300–301) with Gödel’s examples from Plato and Kant on geometric method. For Husserl’s “Ideation” (Wesensschauung), see Husserl (2002, pp. 10, 138–139); cf. Wang (1987, 5.3.7 p. 166, 7.1.15 p. 218). In van Atten 2015, pp. 128–129 (see van Atten and Kennedy 2003, pp. 459–460), Leibniz’s role is especially pointed out.

  19. “The fundamental principles are concerned with what the primitive concepts are and also their relationship” (Wang 1996, 9.1.30 p. 297, cf. pp. 289, 244). “In a theory concepts and axioms must be combined” (Wang 1996, 9.3.10 p. 306).

  20. Cf. “This whole consideration incidentally shows that the philosophical implications of the mathematical facts explained do not lie entirely on the side of rationalistic or idealistic philosophy, but that in one respect they favor the empiricist viewpoint. ...[I]t suggests that the situation in mathematics is not so very different from that in the natural sciences” (Gödel 1995f, p. 313 and ftn. 20). Cf. also Gödel’s remark on “empirical content” as pertaining to the existence of “something”, possibly also in mathematics (Gödel 2016, XV, p. 64).

  21. Van Atten expressed the distinction between Gödel’s intended “phenomenological psychology” (“a priori psychology”) and empirical psychology (on “a priori psychology”, see also Toledo 2011, p. 206) in the following way: although the first one “describes mental phenomena”, it “is not concerned with individual concrete facts but with invariant forms they instantiate and which delineate the range of possible concrete facts” (van Atten 2015, p. 213).

  22. What Gödel expected from his study of Husserl’s phenomenology was a sound foundation (“thorough and systematic beginning”, Wang 1996, p. 171, 9.2.6 p. 298) for a possible systematic discovery of primitive concepts (van Atten 2015, pp. 127–128/van Atten and Kennedy 2003). This in distinction to Kant’s, as Gödel thought, “sloppy architectonic”, which he criticized on several occasions, e.g., in (Gödel 1986–2003, vol. 4, letter to Bernays from 11 May 1961, pp. 186–188). Cf. Wang (1987, p. 226). However, it should be mentioned that the paradigmatic work on the “completeness” of Kant’s table of judgements by Klaus Reich (Reich 1948; Husserl being one of his teachers) seems to offer a clue to an insight into deeper foundations of Kant’s architectonic.

  23. Gödel often uses the term ‘structure’ in a general causal sense of “holding together” (zusammenhalten) and “combining” (verbinden) parts in a whole. For example, “Der Unterschied zwischen einem Haufen und einer Struktur ist, dass bei der Struktur die Teile durch die Tatsache ‘verbunden’ sind und diese Verbindungen selbst zur Struktur gehören” (Crocco et al. 2017, 20); “The difference between a heap and a structure is that, in the structure, the parts are ‘combined’ by the fact and these combinations themselves belong to the structure” [our translation]). Gödel also speaks of “the structure of a concept” (Crocco et al. 2017, 80), as well as of logical concepts as the structure and the “putty” that holds together parts of a given concept (footnote 87 below). In addition, there are examples like “living structure” (Example 1 in Sect. 2.3 below) and “structure of the world” in a causal or some other general sense (Wang 1996, 8.6.10 p. 275, 5.3.8 p. 166; Gödel 1995d, p. 404; Crocco et al. 2017, 90). Correspondingly, we can understand the five causal components of remark (3) as constituents of a causal whole (“structure”), in which they are “combined” and “held together”.

       It should be noted that Husserl investigates “structures” of consciousness, noetic and noematic structures, structure of “life-experience” (Erlebnis), where ‘structure’ has a general meaning of ‘building’ (Bau, Aufbau), with a causal sense methodologically excluded (Husserl 1995, 2002). Of course, Gödel used the concept of structure much prior to his study of Husserl. In addition, he sometimes expresses his interest for something more than pure structures (“But ‘the study of structure’ is a confession that we don’t know what the things are”, Wang 1996, 9.1.10 p. 292). According to the analyses of this paper, this “more” could have a causal sense.

  24. As will be shown, Gödel’s concept of causation essentially involves a teleological aspect of “directedness” towards an end (“good”).

  25. “[T]he basic form of consciousness distinguishes between an intentional object and our being pointed (gerichtet) toward it in some way (feeling, willing, cognizing)” (Wang 1996, 5.3.28 p. 169). Cf. also footnote 61 below. In Max XV, Gödel makes a distinction between “will” and “willing” (Wollen, related to “ought”, Sollen; Gödel 2016, XV, p. 66).

  26. Since this is a significant note giving a broader context of Gödel’s causal philosophy, it is given here, for convenience, more extensively: “Gott schuf die Dinge so, dass sie wieder ihrerseits etwas ‘erschaffen’ können (darauf beruht letzten Endes alle Wirkung). Vielleicht aber besteht darin der Sündenfall, dass sie nicht in diesen Zustand verharren, in dem sie Gott ursprünglich erschaffen hatte (d.h. dass sie diesen Zustand nicht gewollt\({<}\hbox {haben}{>}\), sondern ‘etwas Besseres’...)” (Crocco et al. 2017, 12–13, emphasis added; cf. also Gödel 2016, XIV 64). “God created things so that they at their turn again ‘create’ something (in the end, all effect is based on this). However, the Fall may consist in this that they do not persist in the state in which God originally created them (i.e., that they did not want this state, but ‘something better’)” (cf. Mertens 2016, pp. 196–197, modified translation).

  27. Cf. “Will is the opposite of reason” (Wang 1996, 9.3.15 p. 307). Also, will could be ineffective (Crocco 2016, pp. 140–141). However, the final perspective is ultimately positive: “Dieses Bessere erreichen sie auch schließlich, wenn alle Zeit zu Ende ist” (Crocco et al. 2017, 13; “They eventually reach this Better when all time is finished” [our translation]).

  28. In Phil XIV, whence remark (3) is taken, “better” is an explanation of “value”. The concepts of value—of “positive” and “perfection”—are crucial in Gödel’s ontological argument: “The ontological proof must be grounded on the concept of value (p better than \({\sim }p\)) and on the axioms” (Gödel 1995d, pp. 432–433, also 434–435). On the “perfecting” (Vervollkommnung) and “perfection” (Vollkommenheit), which are implicitly connected with will as its intentional “object”, see Crocco et al (2017, 7–8).

  29. Cf. independency of the “accidental structure of the world” in Gödel (1995d, p. 404, from 1970).

  30. “[D]ie Einheit ist das Wesentliche des Dinges (bedeutet das aber vielleicht nur, dass man alles als ‘eines’ betrachten kann)” (Gödel 2016, XIV 81; “[T]he unity is what is essential of a thing (this may probably only mean that everything can be considered as ‘one’)” [our translation]).

  31. Gödel warns of inconsistency in Kant’s moral philosophy: what has no objective meaning should be subjectively (for the purpose of morals) assumed (Wang 1996, 5.3.38–39 pp. 171–172).

  32. See Plato (1964–1978, VI, Republic 508E–509B, 517C), where “good” is taken to be the cause of cognition, truth, and being (and seems to be understood as a god). Also, knowledge and truth are said to be “good-like”,

    figure d

    (Republic 509A).

  33. See Wang (1996, 9.1.9 p. 292, 9.1.22 p. 295, 5.3.28 p. 169).

  34. In Mertens (2016) it can be seen that consciousness in general has a high position in Gödel’s ontological hierarchy (Max XI, from 1944): logical concepts, space and time, matter, life, consciousness, angels, ..., God. However, as in Leibniz, this does not exclude unconscious states in a monad; on “unconscious pain” and “unconscious life-experiences”, see Gödel (2016, XIV 65). Cf. Leibniz (1978a, §§14, 20–21 pp. 608–609, 610) and Leibniz (1978b, §4 pp. 599–600).

  35. Cf. Gödel’s reflections from the 1970s, where wish, as a kind of force, is characterized by the directedness to “being something” (Wang 1996, 9.4.7 p. 312) and “to realize something” (Wang 1996, 9.4.3 p. 311).

  36. Presumably, non-interrelated parts could act each in its own way, dissolving the whole to which they belong.

  37. See also Bernard (2016, pp. 100–104).

  38. See Bernard (2016, p. 98). According to Leibniz, some laws of motion need a teleological explanation; cf. Leibniz (1978b, §11 p. 603).

  39. “Der Sündenfall wäre also ein Fall ‘in die zeitliche Existenz’ wegen der Unersättlichkeit. Aber das, was die Dinge selbst erschaffen, damit sind sie wieder nicht zufrieden etc.” (Crocco et al. 2017, 13; “The Fall would be a fall ‘into temporal existence’ because of insatiability. However, things are again not satisfied with what they themselves create etc.” [our translation]). Cf. also “satiatedness” in the remark from Max X quoted in footnote 56.

  40. Cf. the following passage from the context of (3): “[T]hat as many [things] as possible will come to be—...this is the ultimate ground of variety (variatio delectat)” (Gödel 1995d, pp. 432–433, translation modified; Gödel 2016, XIV 104).

  41. See Gödel (2016, XIV 76), where Gödel explains that, according to the “good principle”, what was separated in an opposition becomes united, and he calls this unification (Vereinigung) harmony, in distinction to separation, which is disharmony. Cf. also in Gödel (2016, XIV 103) and (1995d, pp. 432–433) that “harmony” implies “more being than disharmony”. For the harmony in nature, see, for instance, Gödel’s example of the correspondence of a “whole ‘organism’ ” of an “electron event” to each small disturbance (Crocco et al. 2017, 10), the example of an accordance in a “dipole”, where “the closer both parts are, the stronger the pairs would act” (Crocco et al. 2017, 5; Bernard 2016), and the following example of accordance: “at the same angle, there is the same color” (Gödel 1995d, pp. 432–433; 1995d, XIV 104). Noticeably, Gödel complains of a lack of sufficient understanding of the concept of harmony (Gödel 2016, XV, pp. 66–67).

  42. Gödel implicitly relates enjoyment, as the opposite of sadness, with the beautiful: “[I]f there were a completely hopeless sadness, there would be nothing beautiful in it” (Wang 1996, on 27.2.1950 p. 43). Also, for example, “Slezak simply leaves out all the nonpretty, since it is not enjoyable to write about them” (Wang 1996, on 17.3.1962 p. 44).

  43. See Gödel’s letter to Bernays from 1961 (Gödel 1986–2003, IV, pp. 186–188).

  44. Cf. “[E]lementary particles are a lower form of mind” (Wang 1996, 9.4.12 p. 314; cf. 6.2.4 p. 191).

  45. “Remark (Theology): The reflection: according to the Principle of Sufficient Reason the world must have a cause. This must be necessary in itself [an sich notwendig] (otherwise it would require a further cause).” (Gödel 1995d, pp. 430–431, Max XI; see Wang 1996, p. 120). Because of the irrationality of will, we could understand that the complex: will–force–enjoyment, determined by the intended goals, does not, in general, possess some such “necessity in itself” that would guarantee the realization of positive values.

  46. Cf. also Gödel’s remark that God is not definable without the concepts of value (“Gott [ist] nicht ohne Wertbegriffe definierbar”, Gödel 2016, XIV 118). At the same time, “God” sometimes appears on Gödel’s lists of primitive concepts (see Sect. 2.1 above). This indicates that the meaning of the concept “God”, for Gödel, may not be exhausted by the above explicit definition (although this definition may be understood as a kind of description in causal terms by means of the concept of positivity). Accordingly, Gödel notes that we do not see as far through His essence (“weil wir das Wesen Gottes nicht so weit durchschauen”; Crocco et al. 2017, 62) (where what is the essence of God, according to Gödel’s onto-theology, is precisely “to be God”). One further hint is that, although God is defined by having all positive properties, He is simple, in some not fully understood sense: “Das ‘Einfache’ an Gott is vielleicht, dass nur eine Art von Gut in höchster \(\textit{Intens}{<}{} \textit{it}\ddot{a}{} \textit{t}\, \textit{ist}{>}\)?” (Crocco et al. 2017, 70; “The ‘simple’ on God is perhaps that only one sort of good is in the highest intensity?” [our translation]).

  47. Cf. “weil wir durch unseren Willensfreiheit am Schöpfen Gottes teilnehmen (Gott schafft den Menschen sich zum Bilde)” (Gödel 2016, XIV 77; “because we participate through our freedom of will on God’s creation (God creates the man as an image of Himself)” [our translation]). Cf. the remark quoted in footnote 26 above. The idea of “created creators” in Gödel’s theological view was drawn to my attention by Paul Weingartner. For his interpretation (in connection with Thomas Aquinas), see Weingartner (2016, p. 466).

  48. See footnote 1 above.

  49. “Das Wahrnehmen des ‘Fließens’ ist die direkte Wahrnehmnug der Erzeugung oder Ursachenrelation” (Gödel 2016, XIV 18; “The perception of ‘flow’ is the direct perception of the generating or causal relation” [our translation]).

  50. See Gödel (1990b) and Yourgrau (1999, 2005). Also: “Causation is unchanging in time and does not imply change. ...  Change is subjective in the Einstein universe” (Wang 1996, 9.5.9 p. 320); “As we present time to ourselves, it simply does not agree with fact” (Wang 1996, 5.3.23 p. 168). In distinction, for Leibniz, the linear order of time is based on the structure of spatio-causal events (see Leibniz 1863, p. 18, and Futch 2008, pp. 118–125).

  51. “Being near = possibility of influence” (“Möglichkeit der Einwirkung”) (Gödel 1995d, pp. 434–435; 2016, XIV 104). “Räumliche Nähe bedeutet offenbar: gegenseitige Beeinflussbarkeit” (Gödel 2016, XIV 19; “Spatial nearness means obviously: mutual influenceability” [our translation]). For the role of nearness in an “electron event”, see Crocco et al (2017, 13–14).

  52. Cf. also Gödel’s remark: “We understand space only through the drive of the objects in space; otherwise we have no idea what space is” (Wang 1996, 9.1.10 p. 292).

  53. For Leibniz’s understanding of time and space, see Futch (2008, pp. 118–125, 152–160).

  54. As reproduced by Wang: “Will and enjoyment lead to life and affirmation and negation” (Wang 1996, 9.4.16 p. 315).

  55. “Das Leben is offenbar eine unvollkommene Struktur, welche daher Materia von außen anzieht [nämlich Sauerstoff, Kohlen\(\textit{hydr<ate>, Amino}\)säuren] und diese in Struktur aufnimmt. Die neue Struktur übt offenbar wieder eine ‘Zerfallungskraft’ auf sich selbst aus, sodass Harnstoff and Kohlensäure abgegeben werden” (Crocco et al. 2017, 7). (“Life is obviously an imperfect structure which because of that attracts matter from the outside [namely oxygen, carbohydrates, and amino acids] and receives them into structure. The new structure again obviously exerts a ‘dissociation force’ on itself, so that urea and carbonic acids are emitted.” Cf. translations in Mertens 2016 and Bernard 2016, here slightly modified).

  56. See also in an already mentioned remark from Max X: “Aber vielleicht gibt es die Speise, die für alle Zeit satt macht? Und den Atemzug, der alles weitere Atmen überflüssig macht” (Crocco et al. 2017, 7–8; “However, maybe there is food that satiates for all time? And the breath that makes all further breathing superfluous”; see translations in Bernard 2016, p. 101, and Mertens 2016, p. 196, here slightly modified).

  57. “Life force is a primitive element of the universe and it obeys certain laws of action” (Wang 1996, 6.2.12 p. 193).

  58. Gödel does not argue for animism, since for him, for example, the physical world “is inanimate” (Wang 1996, 9.1.22 p. 295). But as soon as we have some whole built of matter, an additional, possibly animate, component is needed. “For something to be a whole, it has to have an additional object, say, a soul or a mind” (Wang 1996, 9.4.12 p. 314; cf. “In the case of matter, for something to be whole, it has to have an additional object”, Wang 1996, 6.2.9 p. 192).

  59. If “drive” is understood as involving “force”, Gödel’s reflection about the understanding of space by means of the drive of objects in space (see footnote 52 above) could also be seen as implying form–matter relationship. A sort of acceptance of Aristotelian primitive concepts does not contradict Gödel’s Platonism. Besides “form” and “matter”, concepts “substance” and “accident”, too, can normally be found in Gödel’s lists of primitive concepts. In an analysis of a possible Gödelian sense of these concepts it should be abstracted from Aristotelian realism, where concepts are only “parts or aspects of things”, since Gödel, Platonically, accepts the separate objective “world” of concepts, which are “things” but not objects. See “Gibbs lecture” in Gödel (1995f, p. 321) and Gödel’s distinction between objects and concepts (here, Sects. 2.1 and 3.2.4, Remark).

  60. See footnote 25 above.

  61. Cf. Crocco et al (2017, 59), with reference to Leibniz: “[M]an richtet die Aufmerksamkeit nur auf das, was dem, was man will, günstig ist” (emphasis added; “[O]ne directs the attention only on what is favourable to what one wants” [our translation]).

  62. E.g., Gödel, opposing Herbart, refers to “Leibnizian force” in the sense of the inner “principle of change” in monads (Gödel 2016, XIV 83). Cf. Wang’s interpretation: “Objects are, in the first place, the monads, whose appetition is force” (Wang 1996, pp. 312, cf. pp. 311, 310; appetition as “the tendency to go from state to state”). However, in Leibniz’s Monadology, drives are conceived as “imitations” of God’s will, which is, in turn, an infinite and perfect drive (Leibniz 1978a, §48 p. 615). As mentioned above, attraction and repulsion are non-Leibnizian aspects of Gödel’s monadology (Bernard 2016). The similar holds of “interactions of monads”, if what is meant is not just an “ideal influence”, since, according to Leibniz, no outer cause can influence the interior of a monad (Leibniz 1978a, §11 p. 608, §51 p. 615). On monads “without windows”, see Gödel’s remark in Max X (Crocco et al. 2017, 34).

  63. Also, according to Leibniz, there is under God’s government no good action without reward, and no bad action without a punishment (Leibniz 1978a, §90 p. 622).

  64. “Die räumlichen Beziehungen (auch die makroskopischen) bedeuten objectiv nichts, sondern gelten nur mit Bezug auf uns, und zwar a) sie bedeuten wenigstens stat.\({<}\hbox {isch}{>}\) etwas Objektives\(^{*}\) (\(^{*}\)z.B. gegenseitig Empfinden des Naheseins)(aber Ausnahme z.B. wenn zwei Wesen voneinander träumen)” (Gödel 2016, XIV 69). (“The space relations (the macroscopic ones, too) mean nothing objective, but hold merely with respect to us, and that, a) they mean at least statistically something objective\(^{*}\) (\(^{*}\)e.g., mutual sensation of nearness) (but the exception, e.g., when two beings dream about one another)” [our translation]).

  65. Taking into account Gödel’s remarks before the 1970s, “fact” should not be just “synthetic fact concerning sensations” (“empirical fact”), but also a fact “consisting in relations between concepts” (Gödel 1995a, p. 351, ftn. 41, pp. 355, 356) and a mathematical fact (Gödel 1995a, p. 337).

  66. See, e.g., Gödel (1995f, p. 312, with ftn. 18) and Crocco et al (2017, 29).

  67. Apparently, Gödel is in his ontology of force and fact primarily focused on the world of empirical facts (Wang 1996, 9.4.9 p. 312). However, it can be seen that, according to Gödel, concepts, too, i.e., “generalities” are a “fundamental aspect of the world” (Wang 1996, 9.1.24 p. 295). We will return to this in the following section, where we will outline a possible Gödelian causal account of the “conceptual world” (Wang 1996, 9.2.10 p. 299, 9.4.18 p. 316).

  68. “A fulfilled wish is a union of wish and fact” (Wang 1996, 9.4.3 p. 311).

  69. See Wang (1996, 9.4.9–10 pp. 312–313).

  70. Cf. “that the conclusion is implied by the premises is itself an objective fact concerning the primitive terms of logic” (Gödel 1995a, p. 350, ftn. 40, emphasis added). As expressed in 1975, concepts “are related each in the other and form the ‘conceptual space’ ” (Wang 1996, 4.4.7, p. 149).

  71. For example, in his dialogue Phaedo, as indicated by a reviewer. See Plato (1964–1978, I, Phaedo 100B–101C) . Platonic “idea” is, for Gödel, the Platonic counterpart of “concept” (Wang 1996, 8.6.18 p. 277, cf. 5.3.15 p. 167).

  72. Cf. Plato (1964–1978, VII, Sophist 239D-241B, 263B-C).

  73. Cf. expressions like “a cautiously platonistic point of view” (Gödel 1986–2003, vol. 4, letter to Bernays, pp. 308–309), “a kind of Platonism, which cannot satisfy any critical mind” (Gödel 1995e, p. 50).

  74. “[W]e don’t perceive the concepts of ‘concept’ and of ‘class’ with sufficient distinctness, as is shown by the paradoxes” (Gödel 1990d, pp. 139-140). According to Gödel, Russell brought to light “the amazing fact that our logical intuitions (i.e., intuitions concerning such notions as: truth, concept, being, class, etc.) are self-contradictory” (Gödel 1990d, p. 124; cf. Wang 1996, 8.5.1 p. 269).

  75. “When we formulate the paradoxes in terms of concepts clearly defined for everything, we don’t see what is wrong” (Wang 1996, 8.6.23 p. 278).

  76. In a remark from 1975 according to Wang, Gödel says: “The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have. Even ‘all’, as studied in predicate logic, is less clear, since we don’t have an overview of the whole world” (Wang 1996, 4.4.12 p. 150).

  77. Cf. Max X: “Die hellsten Sterne [deutlichsten Begriffe] scheinen nicht die grössten [objektiv grundlegendsten] zu sein, sondern diese werden durch Konstruktion aus den deutlichsten gewonnen [ein Weg, z. B. der Nominalismus]” (Crocco et al. 2017, 3; “The brightest stars [the most distinct concepts] do not seem to be the biggest ones [objectively most fundamental], but they are obtained by construction from the most distinct ones [one way, e.g., nominalism]” [our translation]).

  78. See Gödel (1995c, pp. 382–385), Gödel (1990a, pp. 272–273), and Gödel (1990c).

  79. A broad discussion about the logical role of will, decision, and approval/disapproval developed in the second half of the 19th and the first half of the twentieth century. A good insight into the problem could be obtained from Gödel’s philosophy professor in Vienna, Heinrich Gomperz (1897, pp. 64–70).

  80. This clearly shows a weakening of ordinary Platonistic causality of “ideas”, and on the other hand, a causal strengthening of Kant’s and phenomenological transcendentalism—a characteristic middle position Gödel aims at, e.g., in Gödel (1995c).

  81. Thus, Gödel not only endorsed Kant’s idea of the subjectivity of time, which he extended (on the ground of the relativity theory) to the clear non-objectivity of time (see Gödel 1990b). He also stood for a version of temporal “schematism” applied to primitive logical concepts (for Kant, see especially Kant 1968, B 183–184). Cf. below on subject-predicate relationship and implication. However, Gödel emphasizes that this is only the form of the subjective aspect of logic (“thinking”), in distinction to the “form of objectively logical relationships” (Crocco et al. 2017, 2).

  82. See Crocco et al (2017, 44, 67, 69) and Gödel (2016, XIV 99). On the distinction between concepts and Kantian (inexhaustible, direction-giving) ideas, see Wang (1996, pp. 268–269, 6.1.13 p. 188).

  83. In the cosmological sense of the affirmation, the being as such should be affirmed in order to start establishing the structure of the world. Cf.: “The affirmation of being is the cause of the world [Das Bejahen des Seins ist die Ursache der Welt]. The first creature: to being is added the affirmation of being. ...as many [things] as possible will come to be ...” (Gödel 1995d, pp. 432–433, slightly modified translation, and Gödel 2016, XIV 104).

  84. E.g., Gödel speaks of the “causation in mathematics, in the sense of, say, a fundamental theorem causing its consequences” (Wang 1996, 9.5.6 p. 320). Similarly, Wang reports that Gödel “...once said to me that there is a sense of cause according to which axioms cause theorems. It seems likely that Gödel has in mind something like Aristotle’s conception of cause or aitia which includes both causes and reasons” (Wang 1996, p. 120). Of course, in accordance with a reviewer’s remark, one should trace the idea of reasons as causes further back to Plato. Cf., in particular, the stress on the unstable, changing and moving character of arguments (“reasons”), for instance, in the dialogue Euthyphron (1964–1978, I, 11C-E); Gödel refers to this dialogue in the 1970s (Toledo 2011).

    It is a natural consequence of Platonistic objectivism to conceive reasons as causes, since the implication relation between reasons and consequents should also be understood as something objective, and hence, most naturally, as causation. Concepts, reasons, and causes, as well as logic, epistemology, and ontology, get intrinsically and systematically interconnected (the possibility of an ontological proof is one of the consequences).

  85. See Gödel (1995h), Artemov and Yavorskaya (2011) and Fitting (2014). For a causal interpretation, see (Kovač 2015).

  86. “Whole and part ...are most fundamental in our conceptual system” (Wang 1996, 9.1.24 p. 295).

  87. “Insbesondere bilden bei der Analyse eines Begriffes in seine Bestandteile die logischen Begriffe ‘den Kitt’, welcher sie zusammenhält. In diesem Sinn sind sie das ‘Formale’, nämlich die Struktur der Bestandteile” (Crocco et al. 2017, 1, ftn.). (“In particular, in the analysis of a concept into its components, the logical concepts build the ‘putty’ which holds them together. In this sense, they are the ‘formal’, that is, the structure of the components” [our translation]). Since, obviously, whole is a sort of numerical (not distributive) identity, Gödel counts it to “extensional” unity (characterized by “contact” or “partition”) (Gödel 2016, XV, pp. 57, 62).

  88. Harmony implies “more being” than disharmony, which, in turn, means “cancellation” of being—if parts are in opposition (“disharmony”), their being is cancelled (“der Gegensatz der Teile hebt ihr Sein auf”, Gödel 1995d, p. 432, Phil IX 104). For the relatedness of harmony to enjoyment, see Sect. 2.2.3 above.

  89. It should be noted that the causal meaning of the whole-part relationship is evident in Kant’s corresponding category of “reciprocal influence” (Wechselwirkung), “deduced” from the logical form of disjunctive judgement as a relation of (exclusive) disjunctive whole and its disjuncts.

  90. “Generalities are just a fundamental aspect of the world. It is a fundamental fact of reality that there are two kinds of reality: universals and particulars (or individuals)” (Wang 1996, 9.1.24 p. 295).

  91. Cf. Max IX on concepts as what “holds the world together” (“hält ...zusammen”) in Crocco (2016, p. 140, 142).

  92. Gödel says that “a concept represents repetition of objects”, but also that mere “multiplicity (or repetition) is mathematics, which does not take primary place in this scheme” (Wang 1996, 9.4.5–6 p. 312).

  93. “Property = cause of the difference of things” (Gödel 1995d, pp. 432–435; 2016, XIV 105). This certainly also reflects the Platonistic view of things as depending on (participating in) properties as concepts (Plato’s “ideas”).

  94. “[A] concept A applies to something B (which may also be a concept)” (Wang 1996, 8.6.18 p. 277). On self-applicability of concepts, see Wang (1996, 8.6.3 p. 274, 8.6.23 p. 278).

  95. See Wang (1996, 9.1.25 p. 295, 9.1.27–28 p. 296, 8.6.25 p. 279).

  96. According to the previous analyses, unity is in general a goal intended by will, and its realization should be immanent to an applied “holding” force. Let us add that there is a connection of Gödel’s temporal schematism with Kant’s. In Kant’s schematism of categories, “substance” (corresponding to “subject”) is schematized by the persistence (unchangeability) in time, and “accident” (corresponding to predicate) by changeability (Kant 1968, B 183). Of course, past is unchangeable, and future seems to be changeable.

  97. See Crocco et al. (2017, 71).

  98. The concept “possible, not necessary” is also encountered, in the sense of “compatibility” (cf. verträglich) of \(\phi (A)\) as well as \(\lnot \phi (A)\) with the essence of the same “being” (Gödel 2016, XIV 5). Cf. Sect. 3.2.1 above.

  99. See, e.g., Crocco et al (2017, 41).

  100. See footnote 51.

  101. “When a truth is necessary, its reason can be found by analysis, resolving it into simpler ideas and simpler truths until we reach the primitives. ...And there are, finally, simple ideas, whose definition cannot be given. There are also axioms and postulates, in brief, primitive principles, which cannot be proved and which need no proof. And these are identical propositions, whose opposite contains an explicit contradiction” (cf. Leibniz 1978a, Monadologie, §§33, 35 p. 612; translation from Leibniz 1989, p. 217).

  102. “[T]he short period of misery may even be necessary for the whole” (Wang 1996, 9.4.20 p. 317).

  103. On connections between the so-called “modal collapse” and Gödel’s version of a “slingshot” argument from 1944, see Kovač and Świętorzecka (2015).

  104. See Sect. 3.5 on the non-reducibility of mind to a machine.

  105. See Gödel (2016, XIV 53, 117; cf. XV, p. 65).

  106. Similarly, for Kant, the antecedent and the consequent in a “hypothetical judgement” have only the value of logical possibility (“problematic” modality) (Kant 1968, B 100). Also, the temporal schema of the category of causality (corresponding to the hypothetical judgement) is a regular succession in time (Kant 1968, B 183).

  107. See footnote 14.

  108. Cf. footnote 61.

  109. “By a function ...I understand the unity of the action of ordering different representations under a common one” (Kant 1968, B 93). Kant speaks, correspondingly, of the “function of unity” in a judgement containing a predicated concept—for example Kant (1968, B 94).

  110. Cf. Gödel’s expression like “forming of unities” (Wang 1996, 8.3.1 p. 260) or: “we form our ideas” of objects (Gödel 1990e, p. 268, emphasis added). Cf. pairs force–space and form–matter in Sects. 2.2.5 and 2.2.7.

  111. Gödel understands “I” (Ich) as an active (“efficient”) principle (das wirkende Prinzip). See Engelen (2016, p. 179, ftn. 33, Max VI 381). For the active intellect, see footnote 113.

  112. “Kantian intuition is too weak a concept of idealization of our real intuition. I prefer a strong concept of idealization of it. ...Understanding a primitive concept is by abstract intuition” (Wang 1996, 7.1.13 p. 217). Cf. also in Gomperz (1897, pp. 86–103).

  113. “I believe there is a causal connection in the perception of concepts. ... Already noûs in Aristotle is a causal affair”; “the active intellect works on the passive intellect which somehow shadows what the former is doing and helps us as a medium” (Wang 1996, 7.3.14 p. 235, 6.1.22 p. 189, emphasis added). Eva-Maria Engelen has shown that Gödel’s reception of Aristotle’s doctrine of the “intellect” is mediated through Thomas Aquinas’ commentary on Aristotle’s De Anima; see Engelen (2013). Cf. also: “Concepts are not the moving force of the world but may act on the mind in some way” (Wang 1996, 4.4.7 p. 149, emphasis added). There is Husserl’s, to some extent analogous, distinction between “active” and “passive genesis” of a transcendental subject (“I”); i.e, new objects are being “constituted” by active genesis on the ground of objects already given in the passive genesis. Although Husserl concedes, in the broadest sense of the word, “causal” character of the genesis of the transcendental “I”, he prefers, in parallel with psychology, to speak about “motivation”, this also “in contrast” to causality as pertaining to reality, which “transcends” the phenomenological sphere (Husserl 1995, §38 pp. 79–82, §37 p. 77; 2002, p. 89 with ftn. 1).

  114. “Out of objectivity we define objects in different ways. Faced with objectivity, how to single out objects is your own child. ...[I]n physics objects are almost uniquely determined by objectivity, if you want to do it in the ‘natural’ way” (Wang 1996, 9.2.40 p. 303). However, “there is a large gap between objectivity and objects: given the fact of objectivity, there may be other possibilities of selecting objects which we don’t know yet” (Wang 1996, 9.2.43 p. 304). This might be a late shape of an earlier view (Max IX) on concepts as “choosing” their objects (Crocco 2016).

  115. “Some pluralities can be thought together as unities, some cannot. Hence, there must be something objective in the forming of unities” (Wang 1996, 8.3.1 p. 260). Correspondingly, Gödel mentions “absolutely infinite or inconsistent multitudes” leading to a contradiction, in distinction to a “consistent multitude” (Wang 1996, 8.3.6 p. 261).

  116. For the importance of Turing’s informal but rigorous clarification of the concept of computability by means of a Turing machine, see in Kennedy (2014): the precise general notion of a formal system was established only on the ground of Turing’s “sharpening of intuition” in his “informal analysis” of the concept of mechanical procedure. See, e.g., Gödel (1990c) and Gödel (1986, p. 195, note from 1963).

  117. See Wang (1996, 6.1.8 pp. 186–187) and Gödel (1995f, p. 310). Cf. “mind is not mechanical”; i.e., “evident” mathematical axioms “cannot be embodied in a finite rule” (Wang 1996, 6.1.10 p. 187). Here is an essential difference from Leibniz: Leibniz’s “universal characteristic”, according to Gödel, cannot exist, because “any systematic procedure for solving problems of all kinds must be nonmechanical” (Wang 1996, 6.3.16 p. 202). However, see Gödel (1990d, pp. 140–141).

  118. “We perceive objects and understand concepts. Understanding is a different kind of perception: it is a step in the direction of reduction to the last cause” (Wang 1996, 7.3.12 p. 235). Cf. also: “Sets are objects but concepts are not objects” (Wang 1996, 7.3.12 p. 235).

  119. “Effective thinking” consists in “introspection and correct thinking”, and it mainly consists in paying attention to “what you have to disregard”: “you have to know what to leave out; this is the essence of effective thinking” (Wang 1996, 9.2.4 p. 298). See also Wang (1996, 9.2.19 p. 300, 9.2.22 p. 301).

  120. “[A]sking the right questions on the basis of mechanical procedure” (Wang 1996, 6.3.15 p. 200).

  121. It is not a concept itself that is the effect of understanding, but the axiomatic explication of the concept. Cf. “[I]dealization ...is not the cause of the concepts” (Wang 1996, 9.2.19 p. 300).

  122. See Wang (1996, 7.3.5 p. 233, 7.3.13–15 p. 235). This may be connected with Gödel’s idea that our brain could be a “computing machine connected with a spirit” (Wang 1996, 6.1.19 p. 189; 1996, 6.1.14 p. 193).

  123. Cf. “The reason ...  as an advisory being inherent to us”, which would be in a sense, “the ‘object’ or at least the cause of our conceptual cognition”. Further, the “advisory reason” (active intellect,

    figure f

    ) appears, in a way, as an “an emissary of God” (Crocco et al. 2017, 86, our translation and emphasis).

  124. According to Leibniz, perception, however indistinct, is “inexplicable” by a machine, because what is found in the interior of a machine are just “pieces that push on one another” (Leibniz 1978a, §17 p. 609, emphasis added).

  125. Cf.

    figure g

    as “highest liveliness [höchste Lebendigkeit] (namely person)” (Crocco et al. 2017, 87).

  126. “The understanding of the system of primitive terms and their relationships cannot be transferred from one person to another. The purpose of reading Husserl should be to use his experience to get to this understanding more quickly” (Toledo 2011, p. 200). See also Wang (1996, 5.3.28–31 pp. 169–170). For the phenomenological approach, independently of the causality concept, see, e.g., section 6 of Tieszen (2016).

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Acknowledgements

This work was supported by the Croatian Science Foundation under the project IP-2014-09-9378. Works of Kurt Gödel used with permission of the Institute for Advanced Study. Unpublished Copyright Institute for Advanced Study. All rights reserved. The author is grateful to Professor Gabriella Crocco (Aix-Marseille University) for permission to quote Gödel’s Max Phil notebooks from the transcription of Max Phil X in Crocco et al. (2017), from the draft transcriptions of notebooks Phil XIV and Max XV in Gödel (2016), and from Crocco and Engelen (2016a), as well as to add or slightly modify the translations of the quotations. In particular, thanks are due to Gabriella Crocco and to Anaïs Mauriceau (Library Granger-Guillermit at the Aix-Marseille University) for the availability of the above-mentioned draft transcription (Gödel 2016). The author would like to thank the audiences of a seminar at the Department of Philosophy of the Cardinal S. Wyszyński University, Warsaw, May 2013, and of the conference Kurt Gödel Philosopher: From Logic to Cosmology, Aix-en-Provence, 11–13 July 2013, where initial drafts of the paper were presented, for valuable comments and discussions.

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Kovač, S. On causality as the fundamental concept of Gödel’s philosophy. Synthese 197, 1803–1838 (2020). https://doi.org/10.1007/s11229-018-1771-2

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