Kant on the construction and composition of motion in the Phoronomy

Abstract

This paper examines the role of Kant's theory of mathematical cognition in his phoronomy, his pure doctrine of motion. I argue that Kant's account of how we can construct the composition of motion rests on the construction of extended intervals of space and time, and the representation of the identity of the part–whole relations the construction of these intervals allow. Furthermore, the construction of instantaneous velocities and their composition also rests on the representation of extended intervals of space and time, reflecting the general approach to instantaneous velocity in the eighteenth century.

Keywords::

• philosophy
• Kant
• philosophy of mathematics
• philosophy of science
• history of mathematics
• history of science

Acknowledgements

I thank the participants at a Workshop on Kant's Philosophy of mathematics and science at the Max Planck Institute for the History of Science in Dahlem, Germany, on 28 July 2014. I thank Vincenzo De Risi for organizing the workshop, and De Risi, Katherine Dunlop and Tal Glezer for especially helpful comments. I also thank Friedman for extended discussions on the role of mathematics in Kant's philosophy and for comments on the penultimate draft of this paper.

Notes

^1. This is made abundantly clear by a great deal of work on Kant's philosophy of science in the last two decades. Friedman (2013) has been particularly helpful to me in understanding the depth and sophistication of Kant's engagement with Newtonian science; this article is throughout indebted to it and would not have been possible without it.

^2. All references to Kant will be to the Akademie edition pagination by volume and page number, except references to Kant's Kritik der reinen Vernunft, which will follow the standard A/B form of reference. I follow the Michael Friedman's translation of the MFNS, and Paul Guyer and Allen Wood's translation of the CPR unless otherwise indicated.

^3. Natural philosophers did not always clearly express the distinction between speed as a scalar magnitude and velocity as a vector comprising both scalar magnitude and direction. “Velocity” is sometimes used for speed, or for the scalar magnitude of a velocity, though sometimes direction is also meant. My paper will focus almost exclusively on speed, though I too will sometimes refer to velocity.

^4. See Friedman (2013) for a clear, full treatment of the Phoronomy that takes into account its role in the MFNS as a whole.

^5. Thus, Kant would have articulated a sufficient but not a necessary condition of motion, and we cannot infer a lack of motion from a point being in the same place at two different moments. Because Kant is only giving an explication of motion and not a definition, this issue does not undermine his account, though it does underscore its limitations. The same could not be said about speed, however, which Kant classifies as a definition.

^6. Kant takes the two properties to be co-extensive, which indicates that he has in mind typical vibrations, oscillations and orbits that follow the same paths, and not closed paths in general, for which it is easy to imagine the properties coming apart. This reflects the fact that he is here concerned only with those motions for which speed is often used to describe frequency or period.

^7. Kant's example focuses on the instantaneous rest of the body at the turn-around point, and for that reason the ever-diminishing intervals take that spatial point as an endpoint. But a similar construction could be made for the speed of a body taking a particular instant of time as an endpoint. For simplicity I will sometimes refer below to the instantaneous speed at an instant, but I have both cases in mind.

^8. I say “apparent” since Abraham Robinson (1966) rehabilitated in finitesimals in his development of non-standard analysis, and showed how one can think of them in a non-contradictory way.

^9. The generalization also follows from the relativity of motion: if we consider the same body tossed in the air relative to a frame of reference that is moving downward 5 ft/sec relative to the original frame of reference, then the ratios of ever-diminishing intervals will approach an upward speed of 5 ft/sec relative to the new frame of reference. See Friedman (2013, 47). Of course, this must be the result if Kant's account is going be adequate; however, this way of establishing the relativity of instantaneous motion presupposes that motions can be composed, and Kant only demonstrates the composability of motions later in the Phoronomy, as we shall see. Thus, a generalization by considering instantaneous speed at other than the turn-around point within one reference frame is preferable. See note 15 below for further discussion.

^10. See Friedman (2013, 49 fn. 22 and fn 23, 51 fn 25) for discussion of this matter. If Newton's theory of fluxions gives priority to instantaneous velocities and velocities over intervals are secondary, Kant's approach on the interpretation I am offering would rule it out. Kant refers to infinitely small [unendlich klein] speed. (See the next footnote.) I do not think that Kant is committing himself to a speed that is simultaneously actually infinitely small and yet thought of as defined over an interval; he may simply mean indefinitely small. But I cannot delve further into this issue here. I would like to thank Friedman for helpful comments on this point.

^11. Note that Kant refers to a motion with “infinitely small [unendlich klein]” speed. See the previous note.

^12. I would like to thank Michael Friedman for pressing me on this issue and for continued discussion. I will return to it again in Section 5.4.

^13. I would like to thank Michael Friedman for prompting me to make this clear.

^14. There is a very direct sense in which we have considered the motion of one and the same point moving 2 ft/sec relative to the moving frame and 4 ft/sec relative to the rest frame. This seems to be sufficient to demonstrate the composability of motion. Nevertheless, we do not represent the motion of the moving frame of reference as a motion of one and the same point quite as directly. Is that a problem? Every point of space relative to the moving reference frame is moving at 2 ft/sec, so the path traversed by the moving point is moving at 2 ft/sec. We represent any motion by representing it as traversing a path, and this seems sufficient.

^15. As I mentioned in footnote 9, one might argue that Kant's characterization of instantaneous motion generalizes to any instantaneous motion simply by appealing to the relativity of motion to a frame of reference; the instantaneous motion at a point in one frame of reference will be the same as instantaneous rest of the point in a frame of reference that moves uniformly with the velocity of that instantaneous motion. But that way of generalizing the construction of instantaneous rest assumes the composability of motions, which Kant has not yet demonstrated. Accounting for the composition of instantaneous motions in the way described here avoids this problem.

^16. Recall from Section 2.2 that Kant sometimes uses “body” rather than “point” in the Phoronomy, but only to anticipate later portions of the Metaphysical Foundations and to make the exposition “less abstract and more comprehensible” (Kant 2004, 4: 480). In my view, the same sort of thing is going on here: a concrete example that appeals to doubling a speed is much more vivid and comprehensible, but the point Kant is making does not depend on the relation of equality thereby presupposed.

Additional information

Funding

This work was supported by a University of Illinois at Chicago Institute for the Humanities Faculty Fellowship for 2013–2014, for which I am most grateful.