Continuity of change in Kant’s dynamics


Since his Metaphysische Anfangsgründe der Naturwissenschaft was first published in 1786, controversy has surrounded Immanuel Kant’s conception of matter. In particular, the justification for both his dynamical theory of matter and the related dismissal of mechanical philosophy are obscure. In this paper, I address these longstanding issues and establish that Kant’s dynamism rests upon Leibnizian, metaphysical commitments held by Kant from his early pre-Critical texts on natural philosophy to his major critical works. I demonstrate that, throughout his corpus and inspired by Leibniz, Kant endorses the a priori law of continuity of alteration as a truth of metaphysics, according to which all alterations in experience must occur gradually through all intervening degrees. The principle thus legislates against mechanical philosophy’s absolutely impenetrable atoms, as they would would involve instantaneous changes of velocity in impact. This reveals the metaphysical incoherencies in mechanical philosophy and leaves Kant’s own dynamical theory of matter, grounded on material forces, as the only viable approach to physical explanation. Subsequently, I demonstrate that Kant nevertheless made conceptual space in his system for the theoretical consideration of mechanical explanations, which makes manifest one of the positive roles that the faculty of reason can play with respect to natural science.

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Fig. 1


  1. 1.

    Throughout the paper, I also refer to metaphysical-dynamical mode of explanation as “dynamism” and “dynamic philosophy,” whereas the mathematical-mechanical mode is additionally denoted by “mechanism” and “mechanical philosophy.” Kant also holds that the latter is equivalent to atomism and corpuscular philosophy (MAN, 4:532f.). Generally, in Kant’s writing, “mechanism” and its forms have a somewhat wide array of meanings, depending on context. For a discussion and agglomeration of this array, see van den Berg (2014, pp. 69–75).

  2. 2.

    In different senses, Gottfried Wilhelm Leibniz, Herman Boerhaave, and Roger Boscovich are dynamists. For a discussion of Kant’s dynamism in relation to Boerhaave’s and Boscovich’s respective formulations, see Carrier (1990, pp. 176–180). For more on Boerhaave’s influence on Kant’s early physical theory, see Massimi (2011).

  3. 3.

    Although the review was anonymously penned, it is now known that Kästner was the author (Förster 2000, p. 183n.).

  4. 4.

    Westphal, in particular, echoes some of Kästner’s earlier skepticism regarding Kant’s argument for dynamism.

  5. 5.

    That said, for Brittan (1978, p. 156), dynamical philosophy also suffers from defects insofar as fundamental forces cannot be mathematically constructed and it is hence, at best, a hypothesis.

  6. 6.

    The details of Kant’s views on the mathematical construction of natural concepts are extremely obscure and controversial. For varying interpretations of his conception of the application of mathematics to natural science, see, for example, Brittan (1978, p. 103), Nayak and Sotnak (1995), Friedman (2013, pp. 26–33), McNulty (2014), and van den Berg (2014, pp. 15–51).

  7. 7.

    For more on Leibniz’s role in the vis viva controversy, see Iltis (1971). For additional Leibnizian arguments against Cartesian mechanics, see L (458–463, 538–543) and AG (117–138).

  8. 8.

    Ariew and Garber provide a visual depiction of the discontinuities Leibniz has in mind.

  9. 9.

    For another discussion of this objection to Cartesian mechanics, see Jorgensen (2009, pp. 229–232).

  10. 10.

    Carrier (1990, p. 176) notes that Boscovich, another dynamist, also traces his anti-mechanical position to the Leibnizian principle of continuity.

  11. 11.

    Wilson (1982) diagnoses Leibniz’s progressive falling out with atomism, observing that his anti-atomic stance developed from a confluence of factors and not merely his commitment to first principles, such as the law of continuity.

  12. 12.

    Christian Wolff and Alexander Baumgarten, with whom Kant was well acquainted, comprise two other forerunners to his views on continuity. In his Deutsche Metaphysik, Wolff asserts that nature takes no leaps, explaining that “all alterations of bodies occur bit by bit, and therefore all events come about bit by bit through determinate degrees” (1747, p. 425). In his Metaphysica, Baumgarten defends a version of the principle of continuity as a corollary of the principle of sufficient reason. He defines an “absolute leap” as an event lacking a proximate (immediate) sufficient ground and writes “That which would exist without any proximate sufficient ground [...] would exist through pure chance [...] and hence an absolute leap is impossible [...] and must not be posited in this or in any world” (Baumgarten 2013, p. 172). A violation of continuity, or a leap, is impossible for Baumgarten insofar as it would have no sufficient reason. Baumgarten, too, rejects the doctrine of atoms, though not explicitly on the grounds of continuity. He claims that the indivisibility of atoms is impossible (p. 180f.). For more on Kant’s views on continuity, especially in relation to Baumgarten’s cosmology, see Watkins (2001, pp. 76–81).

  13. 13.

    This claim is also found in Refl 40 (14:148), Refl 4666 (17:630), and V-Meta/Mrongovius (29:921).

  14. 14.

    Warren (2001b, pp. 22–30) explains that intensive magnitudes are essentially those whose degrees are determined through their effects. Hence the intensive magnitude of the sun that causes my sensation of brightness is determined through the intensity of my sensation. See also Jankowiak (2013) for an illuminating reading of the argument of the Anticipations of Perception.

  15. 15.

    In virtue of motion’s being an intensive magnitude—one not consisting of parts—Kant believes that composite motions cannot be represented merely through the concatenation of vectors (MAN 4:493–495). This would be to treat the composite motion as an extensive magnitude, or one composed of parts.

  16. 16.

    Warren (2001b, p. 69n.) suggests that one lacks justification for applying the concept of continuity, because one cannot experience all of the infinite states that make up a continuous change. Thus, that a particular change is continuous cannot be empirically verified. Warren’s objection, however, puts the cart before the horse. The law of continuity does not commit Kant to the idea that changes need be experienced as fully continuous (as Warren tacitly assumes). Rather, Kant’s claim that changes are fully continuous at most commits him to the idea that each individual state in a continuous change is in principle experienceable. An analogy with causation is useful here. Just as one need not experience the cause of an alteration to know that it has some temporally precedent cause, one need not experience each particular degree of alteration to know that the alteration is continuous. No empirical confirmation of continuity is required: we already possess a priori, metaphysical validation that all changes are continuous.

  17. 17.

    A similar proof of the law can also be found in \(\hbox {V-Meta}/\hbox {L}_{1}\) (28:201). Some commentators have expressed skepticism with regards to Kant’s law of continuity of alteration and its purportedly a priori basis (Bennett 1966, pp. 176–180; Guyer 1987, p. 204f.). I, like Paton (1936, pp. 2:284–289), maintain that Kant had ample ground for justifying the law of continuity, although an in-depth discussion of his critical proof would take us astray. In an unpublished manuscript, Jankowiak offers a detailed, illuminating examination of the warrant for the law of continuity of alteration and evaluates Kant’s various attempts to prove the principle. For my purposes, the crucial point is simply that Kant thinks the law of continuity to be an a priori, universal law of nature.

  18. 18.

    See Watkins (2013) for an argument for the thesis that the law of continuity of forms especially concerns species under a shared genus.

  19. 19.

    This ‘mechanical’ law of continuity is, I maintain, distinct from the mechanical law of continuity discussed in MAN (4:552). That mechanical law of MAN presupposes the dynamical theory of matter and furthermore concerns transfer of motion in impact.

  20. 20.

    Subsequently Kant cites Newton’s insight from the Opticks that light reflecting off of a mirror must slow down, rest, and accelerate away (similar references are also found in V-Meta/Herder (28:41f.) and \(\hbox {V-Meta}/\hbox {L}_{1}\) (28:204)). Although in this paper I especially emphasize the Leibnizian backdrop for Kant’s theory of nature, clearly Newton was also influential for Kant in this context.

  21. 21.

    Kant’s tentativeness appears to be a transitory. By the 70s and 80s, Kant’s strong endorsement of the law of continuity in its many forms is reinstated, as I have shown.

  22. 22.

    Pollok (2001, p. 237) denies that the continuity can bear argumentative weight in Kant’s argument for dynamism. According to Pollok, Kant’s appeal to the “striving [Bestrebung] to penetrate” in the first proposition of the Dynamics (MAN 4:497)—an infinitely small moment of acceleration—presupposes both the existence and continuity of the fundamental forces, which Kant has not yet proven at the outset of the Dynamics. This is entirely accurate; the argument against mechanism based on the moment of acceleration (and the mechanical law of continuity) presupposes Kant’s dynamical theory of matter (see note 19). However, Kant’s anti-mechanical stance may legitimately rest on the continuity of alteration (as I contend), for it is proven in the Second Analogy of KrV as a general feature of possible experience.

  23. 23.

    See also V-Meta/Mrongovius (29:921f.).

  24. 24.

    A proviso: elsewhere Kant suggests that the modes of explanation are a bit more various (V-Ph/Danziger, 29:104f.). First, he distinguishes physical explanations—which deal with causes and effects—from teleological explanations—which have to do with means and ends. Subsequently, he distinguishes three types of physical explanation: mechanical, dynamical, and organic, where according to the latter, “one explains something through the capacity for sensation of a being” (29:105). Because they involve sensory capacities, only a fraction of natural occurrences admits of organic explanation. So, when considering physical attributes exhibited by all matter, there are two viable modes of explanation: mechanical and dynamical.

  25. 25.

    Kant describes this line of thought most clearly in OP. There he explains that there are philosophical foundations of natural science—which explain motions via powers—and mathematical foundations—which explain powers via motions (see OP, 21:481f.). As Friedman (1992, pp. 222–242) has argued, this distinction is meant to be the same as that between dynamical and mechanical philosophy. Thus, this way of presenting the distinction is cleanest and makes clear the exhaustivity of the disjunction (there are only two possible directions of explanation between powers and motions). However, there is some controversy regarding the interpretation of these passages from OP and the identification of the two distinctions: Tuschling (1971, p. 90f.), Westphal (1995, pp. 409–415), and Edwards (2000, pp. 229–241) attack Friedman’s interpretation, while I support it (McNulty 2016, pp. 72–82).

  26. 26.

    Reflecting on similar passages, Warren (2010) argues that mechanical philosophy is, for Kant, explanatorily deficient, insofar it only explains motions by appeal to other motions. I discuss his interpretation below in Sect. 5.1.

  27. 27.

    Indeed, Leibniz himself calculates quantities of motion according to the Cartesian system and asks the reader to imagine a graph of these values in the “Specimen Dynamicum.”

  28. 28.

    Van den Berg (2014, p. 73f.) also conceives of Kant’s objection to mechanism as resting on explanatory considerations. He contends that were space-filling simply analytic of matter—that is, were matter to fill its space through its mere existence, as the mechanist would have it—we would lack a genuine explanation of it.

  29. 29.

    Elsewhere, Warren (2001b) has offered a different metaphysical understanding of Kant’s rejection of mechanical philosophy. He contends that, in Kant’s eyes, the mechanists attempt to explain material phenomena—paradigmatically, that of space-filling—via absolutely inner determinations. However, insofar as the mechanical account involves such inner determinations, it constitutes an attempt to describe things-in-themselves, and is thereby metaphysically deficient (pp. 70–73). This insightful account from Warren is well grounded in Kant’s texts, and reveals fascinating details of his views on the nature of things-in-themselves and the relation between metaphysics and physics. I, hence, mean not to dismiss Warren’s metaphysical characterization in order to replace it with my own. Rather, I maintain that both lines of thought are present in Kant’s anti-mechanical thought. Furthermore, I intend for my argument to correct Warren’s all-too-hasty dismissal of the law of continuity as the basis for Kant’s views (p. 70; see also 2001a, p. 112f.). As I have shown, considerations of continuity are central to Kant’s thought throughout his corpus and, especially, in the context of the metaphysical foundations for physics.

  30. 30.

    This is a comment on a passage from Baumgarten (2013, p. 180) in which he makes the same claim.

  31. 31.

    Considerations of explanatory success may, however, have influenced Kant’s claim that chemistry “must proceed dynamically instead of mechanically” in Refl 63 (14:481). Whereas Kant presents a priori proofs of the existence of the physical dynamical forces, he admits that that whether chemical forces—those “capable of effecting a complete dissolution of matter” (MAN, 4:530)—are actually found in nature is unknown. Thus, dynamism lacks the definitive metaphysical superiority vis-à-vis mechanism in the context of chemistry. I thus conjecture that the explanatory defects of mechanism highlighted by Warren may have influenced Kant’s recommendation for a chemistry based on forces. A bit more speculatively, Kant might have also thought that mechanical philosophy’s irregular laws (discussed above) entail that dynamical explanations are to be preferred in chemistry.

  32. 32.

    Schäfer (1966, pp. 83–85) also contends that the discrete (mechanical) and continuous (dynamic) representations of matter rest on ideas, for Kant. He supports his view by tracing these two approaches to the thesis and antithesis, respectively, of the Second Antinomy; Falkenburg (1995, p. 22f.) and Malzkorn (1998, pp. 392–409) echo this identification. I note only that in MAN, mechanism is treated as an explanatory thesis, distinct from either side of the Second Antinomy. Furthermore, Kant offers additional arguments against mechanism in MAN, revealing that he did not take the Second Antinomy to provide a conclusive disproof of this mode of explanation.

  33. 33.

    For accounts of these ideas and the contribution of reason to scientific cognition, see Buchdahl (1966, 1971), Wartenberg (1979), Rauscher (2011), and McNulty (2015).

  34. 34.

    See Friedman (1992, pp. 141–149; 2013, pp. 474–509). For more on Kant’s conception of absolute space, see Palter (1971).

  35. 35.

    This claim is also found in Refl 5341, from the late 70s or 80s (18:156). In the General Remark to the Phenomenology, Kant claims that absolute space is an idea of reason (MAN, 4:559), and absolute space, for Kant, is “[e]mpty space in the phoronomical sense” (4:563). However, the empty space relevant for our present concerns is empty in the dynamical sense—that is, space that does not, to any degree, resist the motion of incoming bodies.

  36. 36.

    Kant even goes as far as to claim that the regulative use of ideas of reason presupposes transcendental principles that are simultaneously necessary for the use of the understanding (KrV, A650f./B678f.). That said, there is a great deal of conflicting literature on the appropriate understanding of the transcendental status of reason’s principles, a small fragment of which includes Morrison (1989), Schiemann (1992), Grier (2001, pp. 263–301), Ostaric (2009), and Pickering (2011).

  37. 37.

    This sort of possibility is similar in a sense to Leibniz’s metaphysical picture (see Crockett 1999). Leibniz, too, disagrees with mechanism and holds that at the fundamental level reality consists of discrete simples (i.e. monads). Nevertheless, at the phenomenal level, bodies and their changes are continuous.

  38. 38.

    I also thank Jankowiak for raising the following difficulty. It appears problematic to claim that mechanical philosophy could be true in the noumenal realm, for it involves spatial content—its impenetrable and empty spaces—and space is a form of phenomena. While I agree that there is difficulty in sorting out where, exactly, ideas of absolutely impenetrable and empty spaces belong in Kant’s system, I assert that Kant countenanced such as regulative ideas, despite spatial content, is clear (as I demonstrated above). Moreover, in the General Remark to the Phenomenology, Kant claims unequivocally that absolute space is an idea of reason: “It cannot be an object of experience, for space without matter is no object of perception, and yet it is a necessary concept of reason, and thus nothing more than a mere idea” (MAN, 4:559). Kant thus allows for the formation and consideration of ideas that both refer to objects beyond the possibility of experience and contain spatial content. The nature of these ideas and their relations to noumena and phenomena are topics that demand further treatment but that also lie beyond the scope of the present paper.

  39. 39.

    This is not to say that Kästner, a notable critic of the law of continuity, would have been satisfied by this response.


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An earlier version of this paper was presented at the \(11{\mathrm{th}}\) International History of Philosophy of Science Conference. I thank the audience as well as my symposium co-presenters, Katherine Dunlop and Marius Stan, for their questions and comments. Sean Greenberg and Don Rutherford also provided me with helpful information on the dissemination of Leibniz’s dynamic theory. I am especially grateful to Tim Jankowiak for discussing Kant’s views on continuity with me, offering detailed feedback on the paper, and allowing me to read his draft manuscript examining Kant’s proofs of the law of continuity. Finally, the recommendations of two anonymous referees at Synthese greatly improved my argumentation.

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Correspondence to Michael Bennett McNulty.



In the paper, the following abbreviations are used for Kant’s works.

  • GSK = Gedanken von der wahren Schätzung der lebendigen Kräfte

  • KrV = Kritik der reinen Vernunft

  • MAN = Metaphysische Anfangsgründe der Naturwissenschaft

  • MonPh = Metaphysicae cum geometria iunctae usus in philosophia naturali, cuius specimen I. continet monadologiam physicam

  • MSI =De mundi sensibilis atque intelligibilis forma et principiis

  • NLBR = Neue Lehrbegriff der Bewegung und Ruhe

  • OP = Opus postumum

  • Refl = Reflexionen

  • V-Meta/Dohna = Metaphysik Dohna

  • V-Meta/Herder = Metaphysik Herder

  • V-Meta/L\(_{1}\) = \(Metaphysik \, L_{1}\)

  • V-Meta/Mrongovius = Metaphysik Mrongovius

  • V-Meta/Volckmann = Metaphysik Volckmann

  • V-Ph/Danziger = Danziger Physik

As is standard, references to KrV refer to the first and second edition page numbers (A/B). References to Kant’s other works are to the Akademie edition of his collected writings (Kant 1900) and include a volume and page number. For English translations, I have used those available in Cambridge University Press’ series of Kant’s writings, where available Kant (1997, 1998, 2003, 2004, 2012).

The following abbreviations are used for references to the English translations of Leibniz’s works.

  • AG = Philosophical Essays, ed. Ariew and Garber (Leibniz 1989)

  • L = Philosophical Papers and Letters, ed. Loemker (Leibniz 1956)

  • NE = New Essays, ed. Remnant and Bennett (Leibniz 1996).

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McNulty, M.B. Continuity of change in Kant’s dynamics. Synthese 196, 1595–1622 (2019).

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  • Immanuel Kant
  • Matter
  • Dynamics
  • Continuity
  • Mechanical philosophy
  • Gottfried Wilhelm Leibniz