Context

Kant’s account of rotation is easier to grasp if we look at it against a historical backdrop. That foil is Newton’s critique of Descartes. In Principles of Philosophy, Descartes gave a doctrine of true motion, which he defined as a change in a body’s metric relations to other bodies that immediately surround it. Newton in Principia then showed that Descartes’ definition was inadequate, because (inter alia) it failed to make sense of rotation. In a Scholium to the Definitions, Newton proved Descartes wrong in an empirical setting, with a mass of water set in circular motion. Newton took it for granted that we have physical criteria for telling whether the water really rotates or not. He then explained that, based on Descartes’ definition, water counts as truly rotating when, in fact, it is physically at rest; and it counts as truly at rest when, in fact, it shows the physical signs of true rotation.Footnote ¹⁴ Thus, Descartes’ definition of true motion anti-correlates with the physical behavior of material objects; hence, a science based on that definition will eo ipso be inadequate. In contrast, Newton’s own definition – absolutism, or the view that true motion consists in change of place in Absolute Space – always correlates, infallibly, with the physical behavior of bodies that happen to be truly rotating. In sum, absolutism gets it right on physical grounds, whereas Descartes’ version of relationism gets it wrong. And so, any other species of relationist doctrine must ensure that it correlates adequately with the physical behavior of bodies in true spin motion. My critical discussion below (of Kant’s analysis of rotation) takes Newton’s lesson to heart and spells out its quantitative details.

Finally, a note of caution. In Newton’s Scholium, circular motion – spin – shows up twice, in two different contexts, and for different purposes. One is the bucket experiment just described. The other is a thought experiment known as the ‘rotating globes scenario.’ Newton used the former to make a point in metaphysics, namely, that Descartes’ definition of true motion is inadequate, whereas his own definition is adequate. Additionally, he used the latter to make a point in epistemology, namely, that we can have knowledge of true rotation (its quantity and direction) even without evidence from any actual change between bodies.Footnote ¹⁵ Kant explicitly acknowledged just Newton’s second case (the rotating-globes scenario), and never discussed the first. Below, I do that on his behalf.

True Spin

Consider two descriptions of apparent motions happening at the same time. (1) Emily sits on the earth and looks at the stars, from dusk to dawn. Over hours, the stars as a whole appear to spin around her location. (2) Jack sits on Alpha Centauri and looks toward Emily. Over hours, she appears to spin around the earth’s axis.Footnote ¹⁶ Separately, Jack and Emily write down their individual observations in quantitative terms, namely, as changes of angular distance over time (Figure 1).

Figure 1aLong description

The illustration depicts the Earth’s night side with a dotted circle on the right labeled Earth through the night. Two lines extend from the center: one pointing to a dotted star labeled star at dusk and the other to a star labeled star at dawn. A curved downward arrow connects the two stars, indicating the transition from dusk to dawn. There is a mark on the Earth’s dotted line at the line to the star at dawn.

Figure 1bLong description

An illustration of Earth with a dotted circle on the right, indicating its rotation with a curved upward arrow. Two lines extend from the center, and the first one points to a star. Above this first line, on the Earth’s dotted line, a point is labeled Earth at dawn, and another point below the second line is labeled Earth at dusk.

Figure 1 The same apparent rotation (viz. finite angular displacement) as observed from two Kantian ‘relative spaces.’

As descriptions of apparent motion, the two are equivalent: neither contains any kinematic quantity that the other does not. However, from Copernicus to Kant everyone thought there is a difference between (1) and (2). Namely, one of the two systems truly and really spins, whereas the other just seems to spin, but does not really do so; in reality, it does not rotate at all.Footnote ¹⁷ And, Huygens and Newton had taught how to detect true spin – by means of dynamical criteria, due to force-like effects present when true spin occurs, but absent in cases of merely apparent spin.Footnote ¹⁸ In sum, after 1673 there was universal consensus that true spin and apparent spin are genuinely distinct states, even though they are kinematically equivalent; and we can tell the difference by means of objective facts.

Still, agreement that true spin is a real state put pressure on natural philosophers to answer a metaphysical question: what is the nature of true spin? What does it consist in? Just as with true motion, two schools of thought emerged as follows. Absolutism: true spin consists in circular motion with respect to an immaterial container, the Absolute Space of Newton and Euler. Relationism: true spin consists in motion relative to a privileged matter frame, or ‘relative space.’

Now for a crucial aspect. Choosing a position had consequences and came with obligations. Specifically, Newton and Huygens had created theories that gave determinate quantitative answers to the following question: How fast and in what direction does a true-rotating body move? Answer: the speed of true spin equals w²r, where w is the angular velocity around the axis of rotation, and r is the distance to that axis. The direction of true spin is tangential to the trajectory at that point.Footnote ¹⁹ Their answer was sound and final – no one thought that it could be ignored or rejected.

Newton and Euler chose absolutism, which gave them an easy way to make sense of true spin. The formula is valid only when the spin is described from an inertial frame, and Absolute Space is one by design. In contrast, relationists faced enormous difficulties. There are many – very many – relative spaces such that, if the spin is referred to them, the formula becomes trivially false.Footnote ²⁰ So, a responsible relationist must choose very carefully how she states her position, or else she runs afoul of true rotation. Specifically, she must solve two tasks: first, explain relative to which material frame spin counts as true motion; second, show that her relationist analysis agrees with the direction and quantity of true spin as given earlier. This is the background to Kant’s account of rotation, easily the most difficult part of his Phenomenology. I turn to it now.

Kant on True Spin

In line with his general approach in Phenomenology, he accepts that true rotation is a genuine state of motion, and he assigns it a modal category, namely, actuality:

Saying ‘is an actual motion’ means we have conclusive evidence that some bodies are in true spin; and by ‘illusion’ he means that, based on the evidence, the judgment ‘this relative space really rotates’ is false.

Behind his thought are situations like the two observers mentioned earlier. The same kinematic content (viz., angular motion) can be described as the body spinning relative to the matter-frame where Jack sits; or as a motion of his frame around the body on which Emily sits. In classical physics, evidence from centrifugal effects shows that Emily spins truly and Jack does not. Kant’s theorem codifies just that kind of univocal ascription of true spin motion.

Now he faces up to a hard problem. Often, we can establish true spin from facts that make no reference to an external frame, and involve no kinematic relation to it. In fact, we could prove the actual existence of true spin even if no external frame exists at all; and Kant knows it.Footnote ²¹ That puts tremendous pressure on his relationism about rotation; here is why:

1. 1. Some bodies are in states of true spin. [the early modern consensus]

2. 2. True spin consists in actual kinematic change relative to an external frame. [by Kant’s theorem II of Phenomenology]

3. 3. We can establish that a thing spins truly even though no actual change obtains relative to an external frame. [cf. Kant’s falling-stone scenario]

4. 4. Conclusion: Theorem II is false.

Then what does true rotation consist in? What is the nature of objective circular motion? He realizes the gravity of his situation: “So, a motion – which is a change of external relations in space – can be given empirically, even though this space is not itself given, and is no object of experience. This is a paradox deserving to be solved” (558).

Now for Kant’s solution. To explain it well, I must introduce two concepts; they are implicit in his treatment of true spin. One concept is ‘opposite parts’ (see Figure 2).

Figure 2 In a system that spins, any two pieces of matter lying in the same plane, equidistant from the axis of rotation, count as ‘opposite parts’ in Kant’s sense. For instance, A and B; or G and H. In his analysis, true rotation consists in opposite parts (latently) moving relative to each other.

Figure 2Long description

The technical drawing comprises two circular illustrations. The first shows four shaded horizontal planes along a slightly right-tilted axis. On the second plane, two pieces of matter labeled A and B are positioned at either end. On the fourth plane, two pieces of matter labeled G and H are placed on either side of the axis. The second illustration depicts a vertical line labeled C at the top and D at the bottom, passing through the center point O. A horizontal line segment labeled A B intersects C D at point E. Points A and B lie on the circle. Dashed lines connect points A and B to O. Near point D, a horizontal line segment intersects C D between points G and H. Points G and H do not touch the circle.

In a body or system K in true spin, any two matter bits count as opposite parts if they lie in the same plane, diametrically across from each other, normal to the axis of spin. The other concept is ‘latent motion,’ which I explicate thus:

1. ▪ If two matter bits are actually at mutual rest, but they would move relative to each other, provided the forces between them were counterfactually turned off, then the bits are in latent motion.

2. ▪ But, if they would keep the same relative distance (even with the forces turned off), the bits are at latent rest.

The concept of latent motion is not Kant’s explicit phrase; it is my term. Still, it makes good sense of his answer to the ‘paradox’ of rotation, which he gives in a General Remark to Phenomenology. In essence, I construe his solution as follows. True spin consists in latent motion, which is a relation of matter to matter. Namely, if a system rotates truly, its ‘opposite parts’ are in latent motion relative to each other, pairwise. That is my reconstruction of Kant’s difficult train of thought behind his official resolution of the ‘paradox’ of true spin.

But if a system appears to spin and yet its parts are not in latent motion, it displays the ‘illusion’ of rotation, though it really is at rotational rest.

Assessment

Before we evaluate for soundness Kant’s treatment of rotation, recall his basic commitment, as a relationist: All true motion is a privileged relation to matter. There are four assumptions behind this thesis. Actuality: All true motion consists in an actual change of kinematic relations. Frame-dependence: True motion is always a relation to a relative space, or matter frame. Externality: All true motion is relative to a material space that is external to the moving body. Cognizability: True motion is knowable only as a relation of the body to a material space. For motion to be given “even as appearance, we need an empirical representation of the space relative to which the mobile must change its relation,” or else it is not a cognizing of a motion (559).

Now these commitments extend to rotation: it too amounts to a “change of external relations in space.” But his grappling with the ‘paradox’ of true spin strains them to the breaking point. In particular, his official solution requires him to quietly violate all of these commitments, as follows. Not every true motion is an actual change of relations: True spin is not actual, just latent. Not all true motion is a relation to a space: Some are relations of bodies to bodies, for example, true spin (and action by contact too). Not every motion is an external relation of the body, to another matter: True spin is internal to the body, qua relation between its parts. Not all true motion requires a sensible-material space for us to relate it to: True spin is cognizable even when no such space is given in experience (Figure 3).

Figure 3 (left) The latent motion of two parts K, G of a spinning body as Kant analyzes it. The parts’ mutual distance would grow if the physical bond that holds them were dissolved. The increase would be along the straight line between them, and nonuniform. (It is a quadratic function of time.) Note that the latent-moving parts do not form an angle with each other, so it is meaningless to speak of their motion having an angular speed. (right) The motion of the two parts K and G as seen from a true inertial frame Nwtn or by an observer at rest in Newton’s Absolute Space.

Figure 3Long description

The left part shows three vertical double-headed arrows labeled G on the top and K at the bottom. The arrows are of increasing lengths from left to right. The right part shows a shaded circle. Along the circumference of the circle, on the left labeled K, arrows labeled k are shown downwards; and on the right labeled G, arrows labeled g are shown upwards. These forces act in opposite directions. Below is a three-axis graph with axes labeled w, t, and n meeting at the origin N.

Even evaluated more narrowly, his treatment of rotation falters not a little. Recall that a condition of adequacy for any analysis of rotation was that it vindicate the quantity and direction of true spin. But his analysis secures neither. For one, the quantity includes an angular speed, w. However, in the latent motion that Kant privileged in his analysis, the concept of a change of angle is not well defined: Opposite parts do not make an angle with each other. For another, true spin is tangent to the trajectory, whereas the direction of latent motion is perpendicular to the orbit (see Figure 3) – an unwelcome result.

In fairness, his mixed success is typical of early modern struggles to analyze rotation without resort to Absolute Space; since that challenge was so difficult, I gave it separate exposition here. Kant certainly did no worse than his fellow relationists on this count.Footnote ²² The broader question is, was it worth discarding Newton’s space if the alternative is so costly?

Occasion

His first robust outline of a matter theory was Monads. To understand his aims and premises there, it helps to know his context. First, the Berlin Academy in the 1740s had been home to debates between defenders and opponents of ‘monads,’ a family of views claiming that genuine substance is a partless simple.Footnote ³¹ Second, there was a proof that space (qua entity for geometric theorizing) was infinitely divisible. The version that he knew was by John Keill, but there were others. Third, some prominent local figures had turned impenetrability into a basic, irreducible attribute of material substance. Some had even placed it below extension itself, in the order of grounding.Footnote ³² Fourth and last, Wolff and his school had split Leibniz’s idealist legacy into a dual ontology that included ‘elements,’ a dis-mentalized species of substance. They were metaphysical simples with no representational powers at all, not even the petites perceptions of Leibniz’s late doctrine. Having stripped them of all mindlike attributes, Wolff then called them ‘physical monads.’Footnote ³³ Admittedly, not all German Rationalists turned dualist; some stayed true to monadic idealism. Still, both Wolffians and their Leibnizian dissenters agreed that simplicity is a nonnegotiable constraint on substance metaphysics. That is, no doctrine is acceptable unless it makes substance into a partless monas, or ‘simple being.’Footnote ³⁴

These ambient facts entail a task, or problem, which Kant takes up officially in Monads. The problem is that geometry and metaphysics cannot be ‘combined,’ that is, used jointly to ground natural philosophy. Namely, each discipline entails an individual thesis that negates the other. Consider these inferential chains. [geometry] Space-divisibility entails that any finite bit of extension has finite proper parts. In turn, each part has its own extended parts; and so on, to no end. So, no extended simple or partless unity obtains in space. [metaphysics] Genuine substance is partless. There exist located bodies, and they are composites of material ‘elements.’ So, there are matter simples, or partless unities in space.

Evidently, these two conclusions are incompatible, which puts their entailing disciplines in conflict. Prima facie, then, combining metaphysics and geometry yields a conceptual nutcracker than crushes natural philosophy by denying it a coherent notion of body.

Substance

Kant’s way out of this predicament is a type of material substance invulnerable to that nutcracker. To signal that it is simple but not mindlike, he calls it ‘physical monad.’ Precisely put, his monad is an object with three causal powers, which he calls ‘forces.’ They are, respectively, the force of inertia, of repulsion, and attraction. The latter two are actions at a distance. He posits that they obey power laws and speculates that repulsion is an inverse-cube force, and attraction is inverse-square.Footnote ³⁵

A physical monad has two ‘spaces’ associated with it. Namely, it exerts different causal powers over different-sized regions. The spaces are:

1. ▪ place – a point-sized region where inertia resides and the two forces originate. He calls it the “place” taken up by the “mere positing” of a substance; that is, by just asserting it to exist, without regard to its causal activities (1: 483).

2. ▪ sphere of activity – a finite-sized, spherical volume. Its bounding surface S is the set of points where the monad’s repulsive and attractive forces balance each other.Footnote ³⁶ Anywhere below this surface there is net repulsion: Any other monad must expend momentum to reach below S, and when it comes to rest there, it will be scattered back. At the center of S, the repulsive force becomes infinite. Not by fiat, but because of the law of repulsive force.

I offer this distinction to elucidate Kant’s thesis that “any monad not only is in space, but it also fills space,” namely, a tiny volume.Footnote ³⁷ So, two spaces are associated with a physical monad: the one that the monad ‘is in’ and the one that it ‘fills.’ In sum, a monad is in a space the size of a point, and it fills a space the size of a small sphere.

This conceptual innovation lets Kant solve the problem he uncovered. To help see it, I introduce a pair of operations on things with space properties:Footnote ³⁸

1. ▪ s-divisibility; slicing – to conceive a mental plane intersecting a material volume. Mutatis mutandis for material surfaces or lines.

2. ▪ b-divisibility; breaking – to conceive that adjoining parts (of a matter volume) have become separated by a net distance. Alternatively, to destroy some proper part of that volume, whether by physical causes or by divine action.

Their logical relations are as follows. ‘Breaking’ entails ‘slicing’: Any possible b-division conceptually requires an act of s-division. However, ‘slicing’ does not entail ‘breaking.’ Not every possible s-division can result in a b-division. This is shown by counterexamples; the best known and strongest was Absolute Space.Footnote ³⁹ The physical monad is another. In effect, Kant exploits the conceptual asymmetry between ‘slicing’ and ‘breaking’ to show that we can have a substance that is not b-divisible.

To understand his solution, it is crucial to keep in mind the two associated spaces; in particular, that monadic ‘place’ is just the size of a point, and ‘sphere of activity’ is a finite volume. Both are indivisible in just the right respects. (1) The place is indivisible in every relevant sense: It cannot be ‘sliced,’ ‘broken,’ or ruptured. Recall that ‘slicing’ requires mentally producing a lesser-dimensional geometric object inside the thing to be ‘sliced’ (i.e., a plane slices a volume, a line slices a plane, etc.). But a point is a zero-dimensional object. So, it is conceptually impossible to ‘slice’ it. Nor is it possible to ‘break’ it, as a point has no proper parts to separate or annihilate. Therefore, we cannot ‘break’ the resident monad at that place either. (2) The monad’s sphere of activity is divisible, but in a safe sense: it can be ‘sliced’ but not ‘broken.’ That sphere is really just a spherical field of acceleration; thus neither it nor its parts are movable independently of their generating source, that is, the point-sized monad qua source of the repulsion field.Footnote ⁴⁰ So, the monad’s filled space is b-indivisible. These ideas seem the natural reading of his claim that

Thereby Kant’s physical monad avoids the threat of space-divisibility-cum-substantial-simplicity that he set out to avoid.

Body

As a corollary bonus, his solution yields an answer to a problem besetting monadologies (such as Wolff’s) that had refused Leibniz’s idealist way out of it. That problem was the origin of body.

Leibniz, Wolff, and their successors believed that matter is continuous. And Leibniz knew that the mathematical continuum (and any piece of finite extension) cannot be assembled from points. So, if monads are extensionless and pointlike, whereas a body is extended and continuous, then a body cannot result from aggregating monads. Leibniz solved this problem elegantly, by switching to thoroughgoing idealism about substance.Footnote ⁴¹ Wolff, whose ‘physical monads’ were not mindlike, had no coherent solution to this problem.

Kant found another solution, as unprecedented as it is ingenious. The matter theory in Monads entails that a body arises by physical composition, as follows. Any two or more physical monads can form an equilibrium configuration. Intuitively, that obtains when a monad sits on the bounding surface of another monad’s sphere of activity (Figure 4, left). Namely, when A rests at a distance from B where B’s attractive and repulsive forces balance each other. Let that distance be d. Whenever two or more physical monads come to be at d units away from each other, they will remain at relative rest. That is their equilibrium configuration. The configuration is fairly stable: When outside forces push or pull on any monad F in the configuration, F entrains the other monads in it, and they vibrate (relative to each other) around their positions of equilibrium.

Figure 4 Bodies composed of monads. Left: B sits at the distance where A’s repulsion and attraction add up to zero. Center and right: Monads M, N, P are endowed with respectively unequal forces (of attraction or repulsion), whereas V, T, and W’s respective forces are of the same strength.

In this picture, a body is any set of physical monads in an equilibrium configuration. In modern terms, it is a lattice of mass-points. That way of composing a body counts as physical because Kant-monads do that – they reach mutual equilibrium – by means of physical forces (of attraction and repulsion) that they carry ab initio.

Finally, note an important fact. The young Kant discarded the Leibniz-Wolff tenet that bodies are continuous matter. Bodies assembled from Kant-monads are discrete: Between any two neighboring monads there is a small but finite distance of empty space – or at least it is empty of mass, though is it ‘filled’ with two acceleration fields of attraction and repulsion, to be sure. Still, at this stage in his thought, bodies just appear continuous. At the length-scales of human perception, a body’s mass seems distributed continuously over the volume it takes up. But at microscopic scales and below, bodies are structured as lattices of mass-points.Footnote ⁴² This fact is worth signaling, because it separates the young Kant not just from his German predecessors, but also from his later views on matter, to which I now turn.

The Nature of Matter

Kant’s transcendental program notwithstanding, a key part of his late picture comes from conceptual analysis. He seeks a list of Teilbegriffe, or parts that jointly make up the concept of matter. In a nutshell, he takes them to be three, as follows:

${\text{‹matter› =}}^{\text{def}}\text{mobile, impenetrable, and inert extension}$

Remarkably stable across his Critical years, this list is just the minimal account, as it were. Starting from it, some complex reasoning leads Kant to further determinations (of matter) co-extensive with the list; I summarize them at the end of the subsection. Now I discuss his concept-parts, introduced by single ‘explications,’ at the outset of his chapters.

Mobility

In the nature of matter, the first ingredient is mobility: “matter is the movable in space” (480). Two aspects about this notion deserve some discussion. First, mobility counts as a basic attribute for him. Namely, it is explanatorily fundamental relative to the other essential properties of matter: “[I]f anything should be an object of outer sense, its basic determination must be motion, for only through motion can that sense be affected.”Footnote ⁴³ Kant means his reductivism weakly, as just explanatory reduction: All other properties co-natural to matter are connected explanatorily to mobility, be it as causes of motion (e.g., the original forces of matter) or as its generic effects (e.g., being movent by transferring momentum).Footnote ⁴⁴

Second, in Foundations mobility is an equivocal notion: Kant in effect has two concepts of motion, and they are distinct. One is fit for single points of matter, the other works with extended bodies. The former concept is implicit and presupposed by his definition of speed as C=S/T.Footnote ⁴⁵ This quantity implies that motion is change (over time) in the Cartesian coordinates of a free particle. His other concept is “change of outer relations to a given space” (482). But he gives no indication what those relations may be, if they are quantifiable, and how. So, we do not know if his second motion-concept can support a mathematized theory of matter in motion. This is a pity, because his picture of matter in Dynamics requires much stronger quantities of motion than the rudimentary kinematics he allowed himself in Phoronomy.Footnote ⁴⁶

In sum, the nature of matter is to be mobile, but in two senses. One is implicit and quantitative, the other is explicit but just verbal-qualitative.

Impenetrability

Kant uses ‘impenetrability’ and ‘space filling’ synonymously, though he prefers the latter. What he really means by space-filling is that any body takes up a finite volume to the exclusion of any other body, at any given instant. Among the early moderns he is alone in trying to explain impenetrability.Footnote ⁴⁷ That is, he rejects the view (then common) that no two bodies can collocate, because of a primitive, analytic-conceptual property constitutive of body, namely, their impenetrability. To the contrary, he argues, impenetrability is universal but derivative: Failure of collocation is an effect induced by a material property explanatorily upstream from it. The property is a force of repulsion: “Matter fills its space through repulsive force” (499). The force is causally responsible for matter ‘filling space,’ namely, taking up a volume and resisting its penetration; it does so by way of repelling any other body seeking to enter that filled volume.

Since failure of collocation obtains universally, and because repulsion is responsible for its obtaining, it follows that matter is ‘originally’ endowed with a force to repel other matter bits universally: “matter is impenetrable, namely, through its original expansive force.” He counts this result as a theorem (503).

Further, if exerting repulsive force is universal to matter, so is having a force of attraction, he claims: “Repulsive force belongs to the essence of matter just as much as attractive force; neither one can be left out from the concept of matter” (511). He thinks attraction is co-essential to matter because he infers it from the fact of repulsion being essential – via his Balance Argument, an inference he had devised in his youth.

Inertia

Another property Kant ascribes to matter is being inert. In the century after Huygens and Newton, natural philosophers had turned inertia into a generous term denoting three distinct patterns of behavior, as follows. No single body accelerates itself: All changes to its velocity are due to outside agencies. Conservation of linear motion: absent exogenous actions, a body in translation will stay in that state. Resistance to force: Bodies respond to force differentially – a given force will accelerate two different bodies by different amounts.Footnote ⁴⁸ Kant’s notion of inertia keeps the first two strands, though not the third (see below).

Take the fact of null self-acceleration first; it is his second law of mechanics: “All change in matter has an external cause” (543). He considers it a metaphysical principle, and purports to derive it by deductive argument from generic facts about the ontology of body. His key fact is that “matter has no purely internal determinations nor grounds of determination.” So, he concludes, no piece of matter can cause itself to change its state of motion. From this result he infers bluntly, as if it were an easy corollary, the second fact: A body “persists in its state of rest or of motion in the same direction at the same speed” if left to itself.Footnote ⁴⁹

More alarmingly, Foundations kept no trace of his earlier thought – a real insight – that inertia is primarily resistance to exogenous action. That creates a problem for his metaphysics of matter.

Body

His ideas above yield a concept of body, understood as “a matter within determinate boundaries, hence one that has a shape” (525). Put modernly, a Kant-body is a finite volume K of matter bounded by a determinate surface S. K has all the properties that Foundations assigns to matter in general: It is mobile, impenetrable, inert, it carries momentum, and has objective states of motion. He says little about what makes shape S determinate. Presumably, he would think it is the surface such that, at every point on it, the body’s overall force of repulsion balances its net attraction.Footnote ⁵⁰

For more light on his view, I offer a contrastive account between his early and Critical notions of body, seen from two angles. One is mass distribution. The young Kant thought that bodies were discrete, but later he changed his mind, and argued that bodies really are continuous.Footnote ⁵¹ Another one is part–whole relations. Over three decades, he came to rethink his view of them entirely. In Monads, the parts are prior to the whole, explanatorily and ontologically. They explain why the body that they make up has the parameters that it has: because its quantities are just arithmetic sums of the determinate parameters that its component monads have. And they subsist – individually, down to single monads – before and after the contingent body that they make up has come in and gone out of existence. Speaking traditionally, physical monads are actual parts of bodies. In Foundations, however, bodies have just potential parts; and every part depends on its body in two ways. Explanatorily, parameter-values associated with any part obtain as ‘limitations’ of whole-body parameters.Footnote ⁵² Ontologically, any part depends (for its becoming actual) on acts of division, physical or just mental, done on the body.

Additivity

Q has a quantity structure if it is an additive property: For any aggregate of matter-parts, its Q is the arithmetic sum of the Q values associated with each of its parts.Footnote ⁶¹

At this juncture, his philosophy of mathematics becomes relevant. To vindicate condition (1), he must show that addition for Q is constructable a priori, that is, we can represent in pure intuition the operation of adding Qs into a Q-sum. That is because in Foundations he gave himself the task of showing that another physical property (namely, velocity) counts as a quantity because it is ‘composable’ in intuition, or additive: “To construct concepts, it is required … that the condition of their constructing must not be a concept that cannot be at all given a priori in intuition. This much we must establish wholly a priori and intuitively on behalf of applied mathematics. … The question [in Phoronomy] is how the concept of velocity is constructed as a magnitude.” But, Q is likewise a property of matter that needs mathematizing. So, it too must be shown to be composable in pure intuition.Footnote ⁶²

Asserting Q of matter-bits depends in part on the architecture of matter. Namely, ascribing determinate Q-values to parts of matter – then making claims about their sum – depends on one’s matter theory: How matter at fundamental levels is distributed in space constrains the mathematical form of Q-statements about matter. In more intuitive terms, a summation claim about mass points differs (in regard to mathematical structure) from a summation claim about matter qua physical continuum. Consider how we express mathematically that a body’s Q is the sum of the Q-magnitudes associated with its parts. Let K and G be two bodies, and K be made of j discrete point-particles, while G is physically continuous; and let q_i be the quantity of matter associated with each particle in K, whereas q denotes the density of quantity-of-matter for each element of volume in the continuous body G. Here is how we say that K’s quantity Q is the finite sum of its discrete parts:

$Q_K={\sum }_j{q}_j$(1).

And for G, a continuous body that takes up a volume V in space:

$Q_G={\int }_{V}q\text{d}V$(2).

Put in our terms, one is a Riemann sum over a finite set, the other is a volume integral, or infinite sum of infinitely small elements. Kant in 1786 thinks that matter is continuous – so, representing a body’s quantity of matter involves a Type-2 expression.

Now I must raise a worry: What is q? We know the answer – it is the mass density (at a point), namely, the ratio of inertial mass over the volume it occupies. But what would he say it denotes? It turns out that Kant gave the wrong answer, in effect denying himself the chance of giving the correct one. He says:

For transparency, I put his point in modern terms. Kant above claims that, in an extended body, its quantity of substance is the volume-integral of the velocity, because he thinks that two points in a continuous body differ at most in their degrees of speed, and nothing else. This makes clear how he went astray: In reality, points in a physical continuum can differ from each other in respect of their velocities and also in the values of inertial-mass density at those points. That is the insight behind formula (2). It is that, in an extended body, its total mass is the sum of the inertial-mass values at every point in the body. These values can vary from point to point, even for a body in pure translation (i.e., whose points all have the same velocity); and they differ in kind from the velocity-values that obtain at every point in a moved body. Thus, a look at how he explains quantifying matter confirms my verdict above that Foundations lacks a proper concept of mass.

The right conception is that mass is an exact synonym of inertia: Mass is the numeric aspect of a body’s being inert by resisting linear acceleration. Mass is the quantity of linear inertia at a point.Footnote ⁶³ But, this synonymy seems absent from Foundations. Kant there uses ‘mass’ for the output of summing over corporeal parts (541). Let B be an extended body, and grant Kant his notion of an ‘independently movable’ part: Any part of B that (by means of applied forces) we could cause to move separately from B; for instance, by detaching it from B. Tacitly, he means the least such part.Footnote ⁶⁴ Then by ‘mass’ Kant in Foundations means the sum, or total amount, of the independently movable parts in a body. But that is not what classical physics means by mass. Hence, Kant mass and inertial mass are distinct concepts. His term seems quite close to the meaning of mass in the phrase ‘mass noun.’ In his usage, mass is a feature of the whole, not a quantitative property (the degree of resistance) of every part in it. And, thereby he forgot to explain what gets added, when we compute the mass of an aggregate: What is the intensive magnitude (at every point) that we integrate over, so as infer to the mass of the whole body?

Lastly, note that Kant would not incur this problem, had he kept his old theory from Monads. In his youth he had a ready answer to our query about the meaning of q_j above. He would have said that q_j measures the ‘force of inertia’ associated with a single physical monad j.

Measure

Condition (2) requires some backdrop.Footnote ⁶⁵ Just like ‘force,’ ‘temperature,’ and ‘charge,’ ‘mass’ is a theoretical term: It denotes a property not given in perception. With such properties, we must infer to their presence and size from other properties that are given to the senses. In particular, making size claims about theoretical properties requires us to rely – indispensably, as evidence for such claims – on facts about the behavior of perceptible counterparts for them: Some sensible property that tracks changes in the theoretical one at issue. The name for this generic approach is ‘measuring by proxy.’ Through it, we use measurements of the sensible property qua epistemic proxies for size claims about the perceptually unavailable theoretical property.

For this approach to be sound, the theoretical term at issue must have robust proxies. Specifically, since mass is a non-perceptual feature of body, we must secure some empirical property that meets two criteria. (i) The property is universally accessible, namely, it allows us to measure its value for any body, no matter how big or how far. (ii) The empirical property must vary linearly with the theoretical property for which it stands as a proxy.

In this regard, mass is a happy case, for it has a proxy both robust and eminently perceptible by Kantian criteria. That proxy is a species of space-quantity. More exactly, it is acceleration, namely, change of space-distance with respect to time, twice. Acceleration is special, because its aptness qua measurement-proxy for mass is guaranteed nomically – by Newton’s laws of motion. In particular, the Second Law in conjunction with the law of universal gravitation entails that for any body K, its mass correlates with the acceleration it causes on any other body G; and also with its own acceleration, whenever K interacts with any part of the world. That is,

$(m_K\approx a_G)$(3),

$m_K\approx a_K$(4).

Here I must clarify an aspect. Recall my point that, to be a suitable proxy measure for mass, an empirical property must be universal.Footnote ⁶⁶ Acceleration meets that criterion, yet not by itself and unconditionally, but because of a force: There is one (and just one) causal power that induces accelerations fit to stand in as proxy measures of masses. That power is gravity. It is a universal force: It reaches wherever matter may exist. And, it is direct: It needs no intermediary to cause the proxy-accelerations that correlate with Q. These features of gravity secure condition (2) for Q to count as a legitimate quantification of matter, with empirical import – because it is the Law of Universal Gravity that underwrites the nomic correlations (3) and (4) above. In a nutshell, Kant takes his Q notion, connects it with static weight (namely, the corporeal property that on Earth we measure by weighing an object), and then extends the concept of terrestrial weight

I explain these matters in Appendix A; see also Friedman Reference Friedman2013.

Assessment

Friedman’s construal (of Kant showing Q to have an apt empirical measure) is compelling, inspired, and without precedent. But if we inspect it closely, we discover that it rests on three key premises, two of them left tacit. These premises are: Newton’s Second and Third Law; a statement nowadays called the Equivalence Principle; and the Law of Universal Gravity. I explain them below, and their challenge for Kant.

The Third Law is needed to guarantee that we can measure a body’s ‘quantity of matter’ Q from its own acceleration.Footnote ⁶⁷ But that law needs Newton’s Second Law to give it meaning; without it, the Third Law is an empty statement – it lacks empirical content in terms of motion. Friedman’s account needs Kant to rely on the Second Law twice. First, the law ensures that Q, the quantity of matter in a body K, is proportional to b, the acceleration that K induces (by its ‘dynamical’ force of attraction) in some other body, G. Second, to then infer that b is proportional to a, the acceleration that K in turn suffers from G acting on it reciprocally, as the Third Law dictates.

But this raises some worries. The Second Law is entirely absent from Kant’s metaphysical foundations. Not only does he not state it, but it is doubtful that he could have incorporated it. Consider its content:

$f_{impressed}=m·a$

The law really says that a force’s effect is proportional to m, the inertial mass of the body on which it acts. But again, Kant does not have a notion of inertial mass in his Critical metaphysics of nature. A fortiori, even less room does he have for a principle that depends on inertial mass for its very meaning, namely, the Second Law.Footnote ⁶⁸

Friedman’s reading must rely on another key assumption. It is the Equivalence Principle: A body’s inertial mass is equal to its ‘heaviness,’ or weight. That principle is needed to justify why we measure mass reliably by relying on gravity. Specifically, using gravity (e.g., by weighing a body) discloses how much ‘heaviness’ a body has: its weight, or “endeavor to move in the direction of greater gravity” around it (518). But weight (qua response to gravity) and inertial mass are distinct properties – conceptually and physically. And so, we need a principled justification that weight tracks mass, namely, that every act of accurately measuring a body’s weight informs reliably us what the body’s inertial mass is. That justification is called the Weak Equivalence Principle.Footnote ⁶⁹ It says that, for any single body, its (Kantian) weight and its inertial mass are equal: Their ratio is 1:1. However, the Equivalence Principle is an empirical fact, not an a priori thesis. Newton had confirmed it by experiment, and in the 1700s no one had any purely philosophical warrant for it. Therefore, as we assess Kant’s philosophical explanation of how we quantify matter, we must not lose sight of the empirical assumptions on which he relies.

The third key assumption is that gravity is universal, and its quantitative strength is as given by Newton in Principia.Footnote ⁷⁰ Kant believes it beyond doubt, as Friedman has shown amply. Still, we may ask about the grounds for his confidence. There are two avenues. Kant could have accepted the Law of Universal Gravity based on Newton’s evidence for it. Now the Briton’s warrant was a long inductive argument, relying on orbital parameters inferred from decades of empirical data. That sort of evidence makes Newton’s law of gravity count as a posteriori knowledge; then we might wonder why Kant thinks that it belongs in the metaphysics of body. And it makes his case (that matter is quantifiable) ultimately contingent, not a priori certain, because it depends crucially on a contingent fact, namely, the kinematic effects of gravity in our solar system (they were Newton’s evidence for his law).

Alternatively, suppose Kant relied not on Newton’s own evidence but on stronger facts. Namely, on identifying gravity with his ‘original attraction,’ a metaphysical force for which he claims a priori warrant.Footnote ⁷¹ He suggests as much himself: “The effect of universal attraction – which matter exerts on any other matter, directly and at any distance – we call gravity. The endeavor to move in the direction of greater gravity is weight” (518). Clearly, identifying the two forces would obviate the worries I raised above. But, what justifies his thinking that they are identical – that gravity and ‘original attraction’ are the same force-species? What is the evidence for that? Not only is it a mystery, but, to further confound the reader, Kant in the next breath censures the thought that Newtonian gravity may rest on a priori warrant, which original attraction does: “We must not venture any law of force, attractive or repulsive, based on a priori conjectures. In fact, we must infer everything from data of experience. Even the universal attraction (as the cause of gravity), as well as its governing law” (534; my emphasis). What to make of these conflicting statements is for future exegesis.

The upshot of my discussion is that Kant’s philosophical account of quantification – the application of mathematics to material bodies – requires some concepts that he lacks, for example, inertial mass; and some premises that are empirical, hence extraneous to metaphysics. More work is needed for us to discern exactly just how much quantification can be vindicated from his available, official resources.

In sum, a constant of Kant’s long grappling with the constitution of matter was his rejection of extension-first doctrines. By that I mean the family of early modern views (by Descartes, Malebranche, d’Alembert, Euler, and Lambert) for whom extension was explanatorily basic, and force – if they admitted it – was an episodic action that occurs when impenetrable extendeds make contact. Kant was a force-first theorist. Against the neo-Cartesians, he proposed a picture in which corporeal extension is derivative: It obtains from force, namely, from the mutual balancing of two forces (attraction and repulsion) that matter has ‘originally’ and exerts at all times. That matter has these two forces indispensably is the Leitfaden running through his natural philosophy.

Look underneath it, however, and you will see very significant changes. Based on my glosses, three stand out. First, in the 1750s Kant had not integrated explanatorily his matter theory (from Monads) and his dynamics of interaction (from Motion). By 1786, he had found a way to bring them together, though obscurely.Footnote ⁷² Second, his views on the microstructure of matter go through a seismic shift. On the most coherent and charitable construal, Monads is a theory of discrete mass-points; they are the actual parts of bodies.Footnote ⁷³ In contrast, Foundations argues that matter is a physical continuum, and so all corporeal parts are potential, down to infinitesimal volume elements. Third, and most innovatively, the mature Kant became sensitive to the general problem of quantification in natural philosophy. In the Critical decade, we see him absorbed in explaining how physical theory associates quantity-concepts with pictures of matter. That project is a daunting task for any philosopher, including him. To the modern exegete, his work on quantifying matter reveals deep tension between his philosophy of mathematics and his ontology of body.Footnote ⁷⁴ Commendably, Kant did not shirk from these hard problems. In the end, then, his metaphysics of matter made visible progress, though at great cost.

Purity

I begin with an elucidation. Kant has a phrase, ‘pure natural science,’ that is as important to him as it is opaque. The term ‘pure’ is notoriously equivocal in his doctrine – he applies it both to sub-propositional representations and also to knowledge items. Thus, Kantian purity could be semantic or epistemological. So, claiming that there is a pure science of nature has two construals:

1. ▪ s-pure science – knowledge that incorporates non-empirical concepts. Their semantic content is not gained by acquaintance, abstraction from perceptions, idealization from material setups, or other a posteriori avenues for concept acquisition.

2. ▪ e-pure science – knowledge confirmed by non-empirical methods or a priori sources of evidence.

It is uncertain whether pure natural science is expounded in Foundations or lies outside of Kant’s book, downstream from it. Suppose that he did expound it there. What kind of pure science is it, then?

Older exegetes sought to argue that it is s-pure knowledge.Footnote ⁷⁶ Specifically, that it is knowledge inferred from ‹matter›, a concept whose content is a priori, allegedly, because it is given by explicit definition from space and time, and they are a priori representations. These scholars think that perception contributes to pure science just trivially, by securing that ‹matter› refers. Still, that contribution leaves the concept s-pure in the sense above; hence so must be the science that follows from it.

This reading cannot be true. Strong evidence, available early in Foundations, speaks against it. Kant, in Phoronomy, explains that he means ‹motion› in the sense of temporal change in kinematic relations with respect to some relative space. But the latter notion is a posteriori:

In effect, ‘relative space’ is a concept obtained from sensibility, that is, from perceived volumes of matter in fairly stable configurations. Then it counts as s-impure; and so do ‘force,’ ‘mass,’ and ‘action,’ the other basic concepts in Foundations. Then a theory built on them cannot be s-pure.

Patterns of Evidential Reasoning

If Foundations does not deal in s-pure science, a more promising route opens if we consider the other option: Some natural science might count as epistemologically pure. I go down this path next, to see how far it goes. That is, I present here the patterns of a priori inference available in his doctrine, so as to decide how much of his reasoning they can account for.