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Leibniz and Kant on Mathematical and Philosophical Knowledge

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The Natural Philosophy of Leibniz

Part of the book series: The University of Western Ontario Series in Philosophy of Science ((WONS,volume 29))

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Abstract

Kant’s comments on Leibniz are often marginal in form, but always essential in substance. It is in these comments that Kant distances himself from the philosophical tradition and establishes a new orientation in philosophy in an important way. This is also true of the reference to Leibniz in the (pre-critical) Prize Essay (An Inquiry into the Distinctness of the Fundamental Principles of Natural Theology and Morals, 1764), Kant’s answer to the problem of the application of mathematical proof to the field of metaphysics posed by the Berlin Royal Academy of Sciences. The reference is of epistemological significance with respect to the system of the sciences. Here, Kant makes in a pragmatic form the distinction between mathematical and philosophical knowledge, which when it is later presented in the Critique of Pure Reason, in a systematically more elaborated form, forms an essential part of the ‘transcendental doctrine of method’. The opposing party is, as Kant makes clear, Leibniz with his Identification of both kinds of knowledge. From a systematic point of view, different ideals of knowledge and their realization in different disciplines -- paradigmatically given in conceptions of Leibniz and Kant --are at stake.

I would like to thank Robert E. Butts (The University of Western Ontario. London. Canada) for reading and making stimulating comments on the original version of this paper while staying at the University of Konstanz as Visiting Professor In 1983. He also helped me with the English text. In section 3 I have used some material from my “Substance and Its Concept In Leibniz” (see footnote 65). and In section 2 some material from my “The Philosopher’s Conception of Mathesis Universalis from Descartes to Leibniz” (Annals of Science 36. 1979. pp. 593–610) first read at a conference in Benmlller’s Inn on Lake Huron In 1978 -- organized by Robert E. Butts, patron of Leibnlzlan and Kantian scholarship.

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Notes

  1. Untersuchung über die Deutlichkeit der Grundsätze der natürlichen Theologie und der Moral [1764], commonly referred to as Prize Essay, I. Kant, Gesammelte Schriften, published by the Königlich Preussische Akademie der Wissenschaften, Berlin 1902ff. (cited hereafter as Acad.-Ed.), II, p. 281. I use here the translation of Lewis White Beck (Immanuel Kant, Critique of Practical Reason And Other Writings in Moral Philosophy, Chicago 1949).

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  2. Prize Essay, Acad.-Ed. II, p. 283.

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  7. Prize Essay, Acad.-Ed. II, p. 277.

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  14. Prize Essay, Acad.-Ed. II, p. 295.

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  26. Regula IV, Oeuvres X, p. 373.

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  27. Regula IV, Oeuvres X, p. 373.

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  29. La géométrie, Oeuvres VI, p. 369; compare Regula TV, Oeuvres X, p. 393. Descartes’s disparaging remark in the Discours that the ‘analysis of the old’ and the ‘algebra of the modern’ are useless (“the first always being bound to the view of figures in such a way that one cannot exercise the mind without exhausting the imagination, the second submitting us to the coercion of certain rules and signs in such a way that it has created a confused and dark art which constricts the mind and not a science which improves it”, Discours II, Oeuvres VI, p. 17f.), should not be taken seriously in this connection. It is part of his autobiographical style, which is used to convey the impression that he could learn nothing anywhere (see W. Kamiah, ‘Der Anfang der Vernunft bei Descartes autobiographisch and historisch’, Archiv für Geschichte der Philosophie 43 [1961], pp. 70–84).

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  30. Secundae Responsiones, Oeuvres VII, p. 156.

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  31. Secundae Responsiones, Oeuvres VII, p. 155.

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  32. Secundae Responsiones, Oeuvres VII, pp. 155f.

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  33. Secundae Responsiones, Oeuvres VII, p. 156.

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  36. Secundae Responsiones, Oeuvres VII, p. 157.

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  37. Secundae Responsiones, Oeuvres VII, p. 157.

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  38. Compare Secundae Responsiones, Oeuvres VII, p. 155.

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  39. Compare Meditationes de prima philosophia III, Oeuvres VII, p. 35.

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  40. To elucidate the analytic structure of the Meditationes, the reconstruction of the argumentative structure of the ‘cogito ergo sum’ which I have given elsewhere, might be helpful (Neuzeit und Aufklärung. Studien zur Entstehung der neuzeitlichen Wissenschaft und Philosophie, Berlin and New York 1970, pp. 382ff.), compare in addition, with regard to the analysis — synthesis distinction, my article on ‘mathesis universalis’ mentioned in note* (pp. 601ff.).

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  42. C., p. 155.

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  44. Compare A. Arnauld and P. Nicole, La logique, ou l’art de penser…, Paris 1662, repr. under the title L’art de penser, ed. B. v. Freytag Löringhoff and H.E. Brekle, Stuttgart-Bad Cannstatt 1965, pp. 303–308.

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  45. As to the often changing terminology in Leibniz and to the classification of the analytic and the synthetic method within the structure of a mathesis universalis compare R. Kauppi, Über die Leibnizsche Logik. Mit besonderer Berücksichtigung des Problems der Intension und der Extension, Helsinki 1960 (Acta Philosophica Fennica, Fasc. XII), pp. 14ff.

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  46. C., p. 557.

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  47. “Duas partes invenio Artis inveniendi, Combinatoriam et Analyticam; Combinatoria consistit in arte inveniendi quaestiones; Analytica in arte inveniendi quaestionum solutiones. Saepe tarnen fit ut quaestionum quarundam solutiones, plus habeant Combinatoriae quam analytice” (C, p. 167 [about 1669])

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  48. See L. Couturat, La logique de Leibniz, Paris 1901, pp. 177ff.

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  49. “Analytica seu ars judicandi, mihi quidem videtur duabus fere regulis tota absolvi: (1) Ut nulla vox admittitur, nisi explicata, (2) ut nulla propositio, nisi probata” (Nova methodus discendae docendaeque jurisprudentiae [1667], Sämtliche Schriften und Briefe, ed. Preussische Akademie der Wissenschaften, Darmstadt and Leipzig [later on Berlin and Leipzig] 1923ff. [cited hereafter as Acad.-Ed.), 6.1, p. 279).

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  50. Discours touchant la méthode de la certitude et de l’art d’inventer pour finir les disputes et pour faire en peu de temps des grands progrès, Philos. Schr. VII, pp. 180, 183.

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  51. See Die Leibniz-Handschriften der Königlichen öffentlichen Bibliothek zu Hannover, ed. E. Bodeman, Hannover 1889, p. 97. For the conceptual vagueness with respect to the relation between the mathesis universalis and the projected encyclopaedia see my Neuzeit und Aufklärung, pp. 435ff.

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  52. ‘Nova methodus pro maximis et minimis’, Acta Eruditorum 3 (1684), pp. 467–473 (Mathematische Schriften, I-VII, ed. C.I. Gerhardt, Berlin and Halle 1894–1863 [cited hereafter as Math. Schr.], V, pp. 220–226). Integrals were introduced two years later: ‘De geometria recondita et analysi indivisibilium atque infinitorum’, Acta Eruditorum 5 (1686), pp. 292–300 (Math. Schr. V, pp. 226–233).

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  53. For a synopsis of the arithmetical calculus and different stages of an algebraic calculus see my Neuzeit und Aufklärung, pp. 440ff., also see K. Dürr, ‘Die mathematische Logik von Leibniz’, Studia Philosophica 7 (1947), pp. 87–102;

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  54. N. Rescher, ‘Leibniz’s Interpretation of His Logical Calculi’, The Journal of Symbolic Logic 19 (1954), pp. 1–13.

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  55. The pertinent texts are included in the excellent edition by G.H.R. Parkinson: Leibniz: Logical Papers. A Selection. Translated and Edited with an Introduction, Oxford 1966.

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  56. Philos. Schr. VII, p. 206.

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  57. De organo sive arte magna cogitandi, C, p. 430, compare C, pp. 220, 435, also Philos. Schr. VII, pp. 185, 199.

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  58. Die Leibniz-Handschriften in der Königlichen öffentlichen Bibliothek zu Hannover, pp. 80f.

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  59. See Ch. Thiel, ‘Leibnizsche Charakteristik’, in: Enzyklopädie Philosophie und Wissenschaftstheorie II, ed. J. Mittelstrass, Mannheim and Wien and Zürich 1984, pp. 580f.

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  61. See Confessio naturae contra atheistas [1669], Acad.-Ed. 6.1, p. 490.

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  62. Système nouveau de la nature et de la communication des substances, Philos. Schr. IV, p. 482.

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  63. See ‘De linea isochrona, in qua grave sine accelcratione descendent, et de controversia cum Dn. Abbate de Conti’, Acta Eruditorum 8 (1689), p. 198 (Math. Schr. V, p. 237).

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  64. Système nouveau…, Philos. Schr. IV, pp. 483.

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  65. The expression ‘monad’, as Leibniz uses it, in all probability comes from the Kabbala denudata (I-II, Sulzbach 1677/1684), edited by C. Knorr von Rosenroth and known to Leibniz. In any case it is clear that he had first learned the term from F.M. van Helmont or Anne Conway (see Nouveaux essais sur l’entendement humain II, Acad.-Ed. 6.6, p. 72; letter of August 24, 1697 to T. Burnett, Philos. Schr. III, p. 217); compare the entries by van Helmont (Ad fundamenta Cabbalae…Dialogus, I/2, p. 309f., and H. More (Fundamenta philosophiae 12, I/2, p. 294). Leibniz, in spring and summer 1696, had several discussions in Hanover with van Helmont, whom he had known since 1671. In these discussions he learned about the philosophical and cabbalistical studies of Anne Conway. In 1690 van Helmont published Anne Conway’s Principia philosophiae (Opuscula philosophica, quibus continentur Principia philosophiae antiquissimae & recentissimae. Ac Philosophia vulgaris refutata, Amsterdam 1690); this work appeared two years later, translated back into English, under the title: The Principles of the Most Ancient and Modern Philosophy Concerning God, Christ, and the Creatures, London 1692. See C. Merchant, ‘The Vitalism of Anne Conway: Its Impact on Leibniz’s Concept of the Monad’, Journal of the History of Philosophy 17 (1979), pp. 255–269, and ‘The Vitalism of Francis Mercury van Helmont: Its Influence on Leibniz’, Ambix 26 (1979), pp. 170–183;

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  71. See Philos. Schr. VII, p. 300.

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  72. Discours de métaphysique §8, Philos. Schr. IV, pp. 432f.

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  73. Cat. 5.2a11–13.

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  74. According to Leibniz’s principle of the identity of indiscernibles (principium identitatis indiscernibilium) there can be no two individuals who correspond in all their properties. See K. Lorenz, ‘Die Begründung des principium identitatis indiscernibilium’, in: Akten des Internationalen Leibniz-Kongresses Hannover, 14–19 November 1966, III (Erkenntnistheorie - Logik - Sprachphilosophie - Editionsberichte), Wiesbaden 1969 (Studia Leibnitiana Supplementa III), pp. 149-159.

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  81. See Introductio ad Encyclopaediam arcanam, C, p. 513; Primae veritates, C, p. 519.

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  83. See Discours de métaphysique §33, Philos. Schr. IV, pp. 458f.; Monadology §78, Philos. Schr. VI, p. 620.

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  108. This has been pointed out in detail by F. Kambartel (Erfahrung und Struktur. Bausteine zu einer Kritik des Empirisumus und Formalismus, Frankfurt 1968, 1976 pp. 115ff.)

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  109. F. Kaulbach (‘Schema, Bild und Modell nach den Voraussetzungen des Kantischen Denkens’, Studium Generale 18 [1965], pp. 464–479;

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  110. Philosophie der Beschreibung, Köln and Graz 1968, pp. 228ff., 291ff.).

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  114. For the justification of this view see my“Uber’transzendental”, in: E. Schaper and W. Vossenkuhl (eds.), Bedingungen der Möglichkeit und transzendentales Denken, Stuttgart 1984, pp. 158–182.

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  115. Critique of Pure Reason B 152ff. Kant emphasizes here the constructivist character of mathematical intuition: “We cannot think a line without drawing it in thought, or a circle without describing it. We cannot represent the three dimensions of space save by setting three lines at right angles to one another from the same point. Even time itself we cannot represent, save in so far as we attend, in the drawing of a straight line (which has to serve as the outer figurative representation of time), merely to the act of the synthesis of the manifold whereby we successively determine inner sense, and in so doing attend to the succession of this determination in inner sense. Motion, as an act of the subject (not as a determination of an object), and therefore the synthesis of the manifold in space, first produces the concept of succession -- if we abstract from this manifold and attend solely to the act through which we determine the inner sense according to its form” (Critique of Pure Reason B 154f.).

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  116. “spatium…ordo coexistentium phaenomenorum, ut tempus successivorum” (letter of June 16, 1712 to B. Des Bosses, Philos. Schr. II, p. 450). Thus, with regard to the isotropy and homogeneity of space in which no direction and no place can be discriminated by mathematical or physical criteria, Leibniz speaks of abstract space as “the order of situations, when they are conceived as being possible” (fifth letter to S. Clarke, Philos. Schr. VII, p. 415). For the concept of relational space in Leibniz the fact is crucial that space, as a system of relations, possesses the same ideal status already claimed for the concept of physical bodies in the analysis of the continuum, and continued by introduction of the concept of formal atoms.

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  117. In Newton’s mechanics this concept serves for the definition of absolute movement and also for the definition of inertial movement. The concept had also been originally advocated by Kant (Von dem ersten Grunde des Unterschiedes der Gegenden im Raume [1768], Acad.-Ed. II, pp. 375–384.

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  118. Critique of Pure Reason B 754.

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  125. Prolegomena zu einer jeden künftigen Metaphysik, die als Wissenschaft wird auftreten können [1783], Pref., Acad.-Ed. IV, B 263, §§4f., Acad.-Ed. IV, pp. 274ff. For a clear methodological distinction between analysis and synthesis in Kant see §111 of his Logik: “Die analytische Methode ist der synthetischen entgegengesetzt. Jene fängt von dem Bedingten und Begründeten an und geht zu den Principien fort (a principiatis ad principia), diese hingegen geht von den Principien zu den Folgen oder vom Einfachen zum Zusammengesetzten. Die erstere könnte man auch die regressive, so wie die letztere die progressive nennen” (Acad.-Ed. IX, p. 149).

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  126. See my “’Über’transzendental” (footnote 107) and my ‘Rationale Rekonstruktion der Wissenschaftsgeschichte’, in: P. Janich (ed.), Wissen Schaftstheorie und Wissenschaftsforschung, München 1981, pp. 89-111, 137–148.

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  1. Department of Philosophy, University of Konstanz, Germany

    Jürgen Mittelstrass

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  2. Department of Philosophy, University of Toronto, Canada

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Mittelstrass, J. (1985). Leibniz and Kant on Mathematical and Philosophical Knowledge. In: Okruhlik, K., Brown, J.R. (eds) The Natural Philosophy of Leibniz. The University of Western Ontario Series in Philosophy of Science, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5490-8_9

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  • DOI: https://doi.org/10.1007/978-94-009-5490-8_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8923-4

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