1. Introduction

In the preface to MAN,Footnote ¹ Kant claims that chemistry is a science but not a proper science, like physics. For Kant, a proper natural science is a systematic body of cognitions connected as grounds and consequents that satisfies three equivalent conditions.Footnote ² The science must have a priori laws, its cognitions must be apodictically certain and it must allow for the application of mathematics.Footnote ³ In regard to the latter condition, Kant asserts that ‘in any special doctrine of nature there can only be as much proper science as there is mathematics therein’ (Kant Reference Kant2004: 6; MAN, 4: 470). And since chemistry fails to satisfy this condition, it is an improper science.

For Kant, a science is proper when mathematics is applicable to it and mathematics is not applicable to chemistry. That much is clear; yet the manner that mathematics is supposed to appear in a natural science is anything but.

There are two central questions regarding Kant’s views on the application of mathematics in chemistry. First, what constitutes the application of mathematics to a proper natural science for Kant? As I will argue, answers to this question present in the literature fall short in two ways. Those of the first sort overlook the role of mathematical construction.Footnote ⁴ Those of the second sort take seriously the role of mathematical construction, but either fail to detail the relations amongst metaphysics, mathematics and natural science or make the application of mathematics condition too strong to satisfy.Footnote ⁵ Second, in what sense does chemistry fail to adequately allow for the application of mathematics? No adequate answer to this question is available in the literature. Few commentators give more than a cursory account of chemistry’s status in MAN; those that delve deeper, I will argue, unfortunately apply incorrect standards of mathematical application.Footnote ⁶

This paper explains why Kant believes that chemistry is an improper science in MAN. In section 2, I describe how mathematics is supposed to appear in a proper natural science. I contend that the concepts of a proper natural science must be coordinated with constructible, mathematical concepts to adequately allow for the application of mathematics. In section 3, I explain how mathematical coordinations are justified in the proper science of physics. I argue that Kant’s phoronomy a priori coordinates motions with mathematical constructions and that other physical attributes can be mathematically treated only insofar as they are reducible to the coordinated doctrine of motion. Finally, in section 4, I claim that chemistry is an improper science because it has no a priori principles of coordination and cannot be reduced to the doctrine of motion.

2. How to Apply Mathematics in Proper Science

In MAN, Kant demonstrates the possibility of a priori cognition in special natural science. Special natural sciences (or special doctrines of nature), for Kant, are systematic bodies of cognitions of objects determined by some empirical concept.Footnote ⁷ For example, physics is the science of the empirical concept of matter. It is possible to attain a priori knowledge of objects as such in the general doctrine of nature (or general metaphysics). However, Kant believes that mathematics makes possible a priori knowledge of objects determined by some empirical concept: ‘although a pure philosophy of nature in general, that is, that which investigates only what constitutes the concept of a nature in general, may indeed be possible even without mathematics, a pure doctrine of nature concerning determinate natural things (doctrine of body or doctrine of soul) is only possible by means of mathematics’ (Kant Reference Kant2004: 6; MAN, 4: 470).

There is a natural, weak reading of this mathematization requirement. This interpretation holds that what makes possible the application of mathematics to a science is the quantitative measurability of that science’s subject matter.Footnote ⁸ However, there is evidence against this interpretation. For instance, it leaves chemistry’s impropriety unexplained. In the textbook Kant used in association with the Danziger Physik (DP), Karsten presents a variety of quantitative, chemical measurements and devices for these measurements.Footnote ⁹ In addition, it is clear from DP that Kant was exposed to developments in statical or quantitative chemistry.Footnote ¹⁰ Since Kant knew that contemporaneous chemistry involved quantitative measurement and nevertheless believed it to inadequately allow for the application of mathematics, quantitative measurement cannot be the standard of mathematical application. Furthermore, as I noted above, Kant presents three criteria for the propriety of a science that he holds to be equivalent. Namely, a science is proper if and only if its laws are a priori, its cognitions are apodictically certain and it makes possible the application of mathematics. If the application of mathematics amounts to quantitative measurability of the science’s subject matter, the equivalence breaks down. Measurability clearly does not imply the apriority of the science’s laws or the certainty of its cognitions.Footnote ¹¹

This proposal fails because it makes the application of mathematics requirement too weak. In contrast, I claim that the application of mathematics to a science must involve mathematical construction: ‘in order to make possible the application of mathematics to the doctrine of body, which only through this can become natural science, principles for the construction of the concepts that belong to the possibility of matter in general must be introduced first’ (Kant Reference Kant2004: 8; MAN, 4: 472).Footnote ¹² To make sense of Kant’s conception of proper science, we must thus examine mathematical construction.

Kant’s doctrine of mathematical construction appears in The Discipline of Pure Reason in Dogmatic Use of KrV (Kant Reference Kant1998: 628–43; KrV, A712–38/B740–66). There Kant explains that mathematical cognitions depend upon construction of concepts, where ‘to construct a concept means to exhibit a priori the intuition corresponding to it’ (Kant Reference Kant1998: 630; KrV, A713/B741). Mathematical demonstrations are mediated by particular instances of mathematical concepts, either imagined or drawn. Nevertheless, so-derived knowledge can be a priori, for we can abstract from the particular empirical peculiarities of any individual imagined or drawn figure.

For Kant, mathematical concepts are grasped with their construction procedures; for example, to know the concept of triangle is to know how to produce particular triangles. Possessing a concept’s construction procedure then allows one to distinguish those features of the individual figure that are idiosyncratic to the mode of presentation from those that hold universally of all objects falling under the concept, that is, those that hold of an object in virtue of its being produced by that very construction procedure.

Take, for example, Euclid’s proposition 32 from book 1 of the Elements, discussed by Kant in the Discipline (Kant Reference Kant1998; 631–2; KrV, A716–17/B744–5). Proposition 32 is the theorem that the interior angles of a triangle sum to two right angles. According to Kant, the mathematician proves this theorem by producing a particular triangle. That is, she begins with three line segments, any two of which are longer than the third, and constructs a particular triangle △ABC (Figure 1). Afterwards, she completes the requisite additional constructions: extending AC to D and drawing CE parallel to AB. Then, using the angle equality theorems, she infers that $\angle$ BAC = $\angle$ ECD and that $\angle$ ABC = $\angle$ BCE. Finally, as $\angle$ ACB, $\angle$ BCE, and $\angle$ ECD sum to two right angles, so too do $\angle$ ACB, $\angle$ ABC and $\angle$ BAC. At this point, the mathematician has shown that the interior angles of the individual triangle △ABC sum to two right angles. Her possession of the construction procedure for triangles allows her to infer that this feature holds universally of triangles. She recognizes that no matter the lengths of the three line segments that make up the triangle (as long as any two of them are longer than the third) and no matter their configuration in the triangle, the succeeding constructions (extending AC to D and drawing CE parallel to AB) will show the triangle’s interior angles to sum to two right angles. Thus, the construction procedure allows the mathematician to secure the universality (and hence apriority) of the theorem.

Figure 1

Euclid’s 1.32

Kant analogously believes that spontaneously produced intuitions support inferences in proper natural science. For Kant, physics is the science of matter, conceived of as that which is movable in space (Kant Reference Kant2004: 15; MAN, 4: 480). So spontaneously produced intuitions must exhibit the content of the concepts of the science of matter, concepts such as motion, its composition and its communication. Indeed, throughout MAN, Kant presents principles of mathematical construction for the concepts of physics. For instance, in the Phoronomy, Kant explains how to mathematically represent a motion. We first recognize that motions have a magnitude and direction (Kant Reference Kant2004: 15, 22; MAN, 4: 480, 487). This recognition justifies us in representing motions with line segments, which then allows for the derivation of further principles regarding motion.

We derive further a priori facts of motion via the production of particular intuitions exhibiting the concepts of matter and motion. For instance, in the single phoronomic proposition, Kant explains how to compose motions (represented as line segments), deriving the parallelogram law (Kant Reference Kant2004: 26–9; MAN, 4: 490–3). If one is given two distinct line segments, AB and CD, that each represent a motion, one represents the combined motion as shown in Figure 2. One motion, say AB, is conceived of as occurring in a particular background space held to be stationary. Then we conceive of another space that is at motion with respect to the first. This second space’s motion is opposite to CD, such that a point that is stationary with respect to the first space will have motion CD with respect to the second. Then the motion AB with respect to the second space represents the composed motion (EF). With construction procedures in hand, we can thus derive a priori mathematical laws for natural science.

Figure 2

The Composition of Motions

However, there is a critical qualification to Kant’s claim that the concepts of a proper natural science must be constructed. Given Kant’s account of mathematical construction from the Discipline of KrV, the concepts characteristic of a special natural science cannot literally be constructible (in the strict sense).Footnote ¹³ As we noted previously, what distinguishes a special natural science from general metaphysics is that the former seeks a priori cognitions regarding a particular empirical concept; so, for instance, physics concerns the empirical concept of matter. However, empirical concepts are not amenable to mathematical treatment.

There are two conditions for mathematical construction that are not satisfied by empirical concepts.Footnote ¹⁴ First, possession of a mathematical concept is sufficient for the representation of an object falling under it (Kant Reference Kant1998: 638–9; KrV, A729–30/B757–8). This permits mathematical demonstrations to be a priori; merely possessing the concept of triangle allows the mathematician to spontaneously produce a triangle for utilization in a proof. Possessing an empirical concept, on the other hand, is clearly insufficient for the representation of an object falling under it. For instance, grasping the marks of the concept of tree does not allow one to produce a tree. One can only come across given trees in experience. Consequently, one cannot take the first step towards constructing an empirical concept: to wit, the spontaneous production of an intuition corresponding to the concept.

But even if one could spontaneously produce such instances, one could not use these instances to support inferences to a priori knowledge. Mathematical concepts are made, while empirical concepts are given (Kant Reference Kant1998: 637–9; KrV, A727–30/B755–8). To clarify the content of an empirically given concept, one can provide expositions, but never definitions.Footnote ¹⁵ That is, for an empirically given concept, we can only catalogue the marks of the concept that we have come upon; we can never be certain that we possess a complete list of marks. Such a complete, exhaustive list of marks of a concept would be that concept’s definition.

Mathematical concepts, on the other hand, are made: they contain an arbitrary synthesis of marks and are grasped with their definitions. In mathematics, ‘I can always define my concept: for I must know what I wanted to think, since I deliberately made it up, and it was not given to me either through the nature of the understanding or through experience’ (Kant Reference Kant1998: 638; KrV, A729/B757). So to grasp a mathematical concept is to grasp all its marks.Footnote ¹⁶ This feature is critical for the success of mathematical construction. In the course of a demonstration, the mathematician must be able to distinguish between those features of the construction that belong to the relevant mathematical concepts and those that are mere idiosyncrasies of the mode of presentation. In order to so distinguish the sorts of features of the diagram, she must possess exhaustive definitions of the relevant mathematical concepts. For this reason, one cannot use mathematical construction to gain a priori knowledge of empirical concepts. Since an empirical concept is given and indefinable, we cannot distinguish with certainty those features of a particular token that belong to the concept from those that are unique to that token.

For these reasons, we must think of the application of mathematics to a proper natural science as requiring a coordination Footnote ¹⁷ of the mathematical and natural rather than construction procedures for the empirical, natural concepts themselves.Footnote ¹⁸ That application depends upon coordination allows us to avoid the pair of difficulties that faced the construction of empirical concepts. For instance, in the aforementioned phoronomic case, that motions are coordinated with line segments allows one to spontaneously produce those singular intuitions that will support inferences to pieces of general, physical knowledge. In addition, since the concept of a line segment is mathematical, it is exhaustively definable, meaning that one can distinguish between those features of a phoronomic construction that are had merely by the mode of presentation.

3. Why Physics is a Proper Science

But how are we justified in adopting particular coordinations in proper science? This question is pressing, for if coordinations are supposed to be necessary, as they must be in the pure part of proper science, then they require a priori grounding. In this section, I explain how Kant a priori grounds the coordination of the mathematical and physical in MAN. This explanation will clarify the stringent conditions on coordination in proper science. I claim that there are two sorts of coordination principles in physics. Of the first sort is the coordination of composite motions in the phoronomic proposition. This coordination is a priori, I argue, in virtue of being necessary for the possibility of experiencing objects of outer sense. Of the second sort is the coordination of the filling of space in Kant’s dynamics. Kant shows that this phenomenon can be reduced to a sort of effect on motions, meaning that, in virtue of the coordination of composite motions from the phoronomy, the filling of space too can be coordinated with a mathematical construction. Hence, the coordinated doctrine of motion lays the groundwork for Kant’s mathematical physics.

To recognize this groundwork for mathematical physics, we first need to understand (another aspect of) Kant’s project in MAN. As Watkins (Reference Watkins1998: 577–8) observes, an objective of MAN is to show how experience of objects of outer sense is possible. To achieve this goal, Kant first claims that ‘[t]he basic determination of something that is to be an object of the outer senses had to be motion, because only thereby can these senses be affected’ (Kant Reference Kant2004: 12; MAN, 4: 476). Therefore physics, originally defined as the science of the objects of outer sense, is equally the science of matter, or the movable in space. Kant then explains the sense in which the concept of the movable in space is determinable by the categories. That is, in each chapter of MAN, Kant introduces a new determination of matter and argues that it corresponds with a categorial determination. This process secures the possibility of experiencing matter (and hence objects of outer sense) and also exhaustively produces all that may be a priori known about matter (Kant: Reference Kant2004: 10–12; MAN, 4: 474–6). KrV shows that an object of possible experience must be determinable according to the categories; MAN demonstrates the way in which the concept of matter is so-determinable, thus making objects of outer sense possible objects of experience.Footnote ¹⁹

For instance, the Phoronomy corresponds to the category of quantity; so it is supposed to present a determination of the concept of matter that is a magnitude. Kant argues that motion is the quantity of physics.

Here, Kant is explaining that motions can be coordinated with spaces: the motion of a point (which is what is at issue in phoronomy) is coordinated with a line segment. The line segment represents the space traversed (the direction and distance). Yet one also takes heed of the duration of the represented displacement. The selfsame segment AB represents a different motion if it is traversed in one second versus one minute.

As Friedman (Reference Friedman2012: 485–6) notes, to show that motions are magnitudes is just to show how two motions may be combined into another motion.Footnote ²¹ Hence, to demonstrate that motions are magnitudes (and thus to show how the concept of matter is determinable by the category of quantity), Kant coordinates the concept of a composite motion with a constructive procedure.

Therefore, the parallelogram law, discussed above, shows that the concept of matter is determinable by the category of quantity and also coordinates the concept of a composite motion with a particular mathematical construction.

These principles of coordination – between motions on one hand and mathematical constructions on the other – are necessary.Footnote ²² For matter to be a possible object of experience, it must be determinable by the categories, and so it must have a quantity. But for matter to have a quantity, composite motions must be coordinated with mathematical constructions as depicted in the parallelogram law. The phoronomic construction is consequently necessary for the possibility of experiencing matter. But, in physics the concept of matter stands in for the concept of object of the outer sense. The phoronomic proposition’s coordination of the physical and mathematical is therefore necessary for the possibility of experience of objects of outer sense. It is this sense in which the phoronomic coordination is non-arbitrary, a priori and necessary.Footnote ²³

This is the pure core of the doctrine of motion, which then allows for the coordination of other aspects of physics with mathematical constructions.

Thus, in the subsequent chapters of MAN, Kant reduces the quality, relation, and modality of objects of the outer sense to the pure core of the doctrine of motion. The quality and relation of matter are respectively the filling of space and the communication of motion, which are defined entirely in terms of its effects on motion. The modalities of matter are motions that accord with particular constraints. The central point is that the quality, relation and modality of matter can be represented in composite motions, whose mathematical coordinations are made possible by the phoronomic proposition.

A detailed consideration of mathematical coordination in the Dynamics of MAN is necessary to fill out this interpretation and to prepare the ground for our consideration of chemistry in section 4. In Kant’s dynamics, he claims that the quality of matter is the filling of space, that is, the resistance of a matter to other matters penetrating its space. The filling of space is conceived of entirely in terms of its effects on motion.

The reduction of the filling of space to an effect on motions opens the door to coordination of the filling of space with a mathematical construction. A body, α, resisting the penetration of another, β, via α’s filling of its space can be mathematically represented by the composition of β’s motion with a contrary motion that represents the strength of α’s resistance. That the composition of motions is constructible via the phoronomic proposition means that such a scenario too can be coordinated with a mathematical construction.

Now this coordination is a priori because it is necessary for matter to have a quality (and thus necessary for the possibility of experiencing objects of the outer sense). Qualities, for Kant, are intensive magnitudes, that is, continuous magnitudes capable of degrees (Kant Reference Kant1998: 290; KrV, A165/B208). As Daniel Warren (Reference Warren2001b: 22–31) has helpfully shown, intensive magnitudes are those that result from causal powers.Footnote ²⁴ That is, the crucial feature of that which falls under the category of quality is that it is a property whose quantity is measured not according to its parts, but according to the effects that it produces.Footnote ²⁵ Warren writes, ‘[t]hus, a reality (a sensible quality) is represented as a determinate magnitude through its causal relations to effects which are unproblematically quantifiable’ (2001b: 26). The phoronomic proposition justifies the claim that the filling of space is a quality by making composite motions ‘unproblematically quantifiable’ (for the effect of the filling of space is a sort of composite motion). Then, in proposition 2 of the Dynamics, Kant shows that the filling of space has a degree (Kant Reference Kant2004: 36; MAN, 4: 499). The filling of space is thus the quality of physics, and the associated coordination of this quality with a continuously variable quantity is consequently necessary.

The a priori grounding of mathematical coordination for proper science can only go so far. That the filling of a space can be represented in composite motions is necessary for matter to be a possible object of experience. The case is different for the fundamental repulsive and attractive forces. In propositions 2 and 5 of the Dynamics, Kant respectively argues that these two forces are essential to matter for it to fill space (Kant Reference Kant2004: 36, 46; MAN, 4: 499, 508–9). The repulsive force explains the diminution of motion into the space of that matter, while the attractive force offsets the repulsive force, which guarantees that matter does not disperse infinitely (as it would if only the repulsive force were essential to matter). This introduction of the attractive force does not depend upon the mere empirical fact that matter is not infinitely dispersed; rather, the infinite dispersal of matter would mean that the filling of space is not the quality of physics.Footnote ²⁶ Yet Kant coordinates neither fundamental force with a mathematical construction. Furthermore, he suggests that the coordination of the fundamental forces of attraction and repulsion is impossible in the General Remark to the Dynamics (Kant Reference Kant2004: 62–4; MAN, 4: 524–5). A number of commentators have been perplexed by this apparent lacuna in Kant’s dynamics.Footnote ²⁷

Our understanding of Kant’s project allows us to recognize that Kant need not and cannot a priori coordinate the fundamental attractive and repulsive forces with mathematical constructions in MAN but, further and crucially, that MAN lays the groundwork for the possibility of their coordination.Footnote ²⁸ As we observed above, in order for matter to be determinable by the categorial heading of quality, the filling of space needs to be reducible to the composition of motions. This implies that the fundamental forces can be coordinated with mathematical constructions, but does not require any particular coordination. The transcendental role of the filling of space (with respect to the concept of matter) entails only certain constraints on the coordination of these forces. For instance, the laws governing the diffusion of the two forces cannot be equivalent, for then the filling of space could not have degrees and thus would fail to be a quality. Nonetheless, Kant presents a possible coordination – that is, one that accords with the transcendental requirements – in the second remark to proposition 8 of the Dynamics (Kant Reference Kant2004: 57–9; MAN, 4: 518–22). But the actual coordination principles for these forces, whatever they may be, is beyond the bounds of metaphysics, for the possibility of matter as an object of experience does not require any particular coordination (Kant Reference Kant2004: 55–7; MAN, 4: 517–18).Footnote ²⁹

Indeed, the grounds for the coordination of the fundamental forces will ultimately be empirical.

Nevertheless, the science of the fundamental attractive and repulsive forces – dynamics – is not consequently an improper science. The particular coordination principles for these forces lie in the empirical part of the proper science of physics.Footnote ³⁰ Physics remains a proper science, for it still has a pure core – the doctrine of motion – that makes possible the coordination of the fundamental attractive and repulsive forces with mathematical constructions. And since there must be coordinations for the fundamental forces of attraction and repulsion, whatever they may be, in order for matter to be an object of possible experience, the doctrine of dynamics belongs to the empirical part of the proper science of physics.Footnote ³¹

There are thus a priori principles that make possible the coordination of the concepts of physics with mathematical constructions. These principles include the pure doctrine of motion and those validating the reduction of the further physical determinations of matter to composite motions. Mathematical coordination in physics, even when it requires empirical evidence, as it does in dynamics, depends for its possibility on the pure, coordinated part of the proper science.

4. Why Chemistry is Not a Proper Science

At this point, we have concluded that the application of mathematics to a proper science consists in the coordination of mathematically constructible concepts with those of the science. Additionally, this coordination is made possible by a priori principles. We now move to a consideration of chemistry’s status as an improper science, an issue that has not been explained in the existing literature. Few commentators have been interested in chemistry’s impropriety. Those that have directly considered the issue have applied inappropriate conceptions of mathematical application.Footnote ³² In this section, I argue that there can be no a priori principles of coordination in chemistry. The science of chemistry, for Kant, concerns particular forces of matter. These forces, however, do not belong to the possibility of matter, and hence cannot be coordinated a priori. Furthermore, chemical forces cannot be reduced to the pure doctrine of motion, meaning that chemistry is not even in the empirical part of a proper science. These forces could only be represented by composite motions if their activity were reduced to the interaction of mechanical corpuscles. But such reduction would be metaphysically incoherent. Thus chemistry is improper, and furthermore its impropriety is directly connected with Kant’s rejection of mechanical philosophy.

Before we account for chemistry’s impropriety in detail, it is crucial to come to an understanding of what chemistry is for Kant. Perhaps the clearest definition of chemistry comes from early loose leaves of OP. In these leaves, in particular leaf 23, Kant contemplates forces beyond the fundamental forces of attraction and repulsion handled in the Dynamics of MAN. In this context, Kant describes chemistry as ‘the science of the inner forces of matter’ (Kant Reference Kant1993: 5; OP, 21: 453). Chemistry, like physics or psychology, is a special natural science. As such, at its foundation is an empirical concept. Where physics takes the concept of matter as its basis, chemistry takes the concept of an inner force of matter.

But what are inner forces of matter? In DP, Kant separates forces (as well as changes of matter) into three kinds: mechanical, chemical and organic (DP, 29: 116–17). He has the following to say of the chemical forces and changes of matter.

A similar sentiment appears in the General Remark to the Dynamics where Kant distinguishes chemical action of bodies from the mechanical action of bodies. Bodies mechanically act on each other when they externally communicate motion; whereas bodies chemically act ‘insofar as they mutually change, even at rest, the combination of their parts through their inherent (eigene) forces’ (Kant Reference Kant2004: 69; MAN, 4: 530). Chemical action is then divided into dissolution (Auflösung) and decomposition (Scheiden).Footnote ³³ Kant then explains that by dissolving forces, two matters combine such that every part of the solution contains the same proportion of solvent and solute. The characteristics of such a combination are then different from the reagents, as the sulphur has different properties from the oils it came from in the above passage. The decomposing forces, then, reverse the combination of the dissolving forces, leaving two distinct matters.Footnote ³⁴

So according to Kant chemistry is the science of dissolving and decomposing forces; the chemist is to discover laws governing such forces. Yet this science is not conducted mathematically.

The chemist must experiment and look to experience to discover her science’s laws. We cannot know a priori whether the inner forces of matter are really possible, and thus the project of chemistry is merely to reduce the various forces of dissolution and decomposition as much as possible.

Why must chemistry proceed non-mathematically? This is because there are two ways of applying mathematics to chemistry, both of which fail. First, coordination principles for the inner forces of matter could belong to the possibility of matter. Second, the activity of inner forces could be reduced to effects on motions, making chemistry an empirical part of physics. These options will be examined in turn.

That chemical forces can be coordinated with mathematical constructions does not belong to the possibility of matter (as does the coordination of e.g. motion). That is, the forces of dissolution and decomposition do not play any categorial role for the concept of matter; they play no substantive role in the propositions and explications of Kant’s phoronomy, dynamics, mechanics, and phenomenology. Furthermore, Kant is clear that these doctrines of physics exhaust the a priori principles regarding matter; ‘all that may be either thought a priori in this concept [of matter], or presented in mathematical construction, or given as a determinate object’ must be given by the categorial determinations of the concept of matter (Kant Reference Kant2004: 11–12; MAN, 4: 475–6). So, as the only ground for a priori coordination consists in the categorial determination of the concept of matter, and the chemical forces play no categorial role for the concept of matter, they are not a priori constructible.

Nevertheless, chemistry could be part of a proper science were the action of its forces reducible to the doctrine of motion (as are the fundamental attractive and repulsive forces). However, in Kant’s mind, there are two possible conceptions of the ultimate explanatory grounds in natural philosophy (and two associated methodologies), neither of which can legitimate the reduction of the inner forces of matter to the doctrine of motion.

The physical mode of explanation comprises two modes under itself:

1. 1. The mechanical as one explains something through already available forces. So Descartes explained the dissolution of crab stones [calcium carbonate] by vinegar [acetic acid] mechanically when he supposes the parts of vinegar to be sharp, which therefore penetrate the crab stones as the heat drives the corpuscles into the crab stone as with a warm blow.

2. 2. Dynamical as one lays especially still not available forces at the foundation. In chemistry one explains nearly everything dynamically. The mechanical and dynamical modes of explanation taken together make the physico-mechanical [mode of explanation]. (DP, 29: 105)Footnote ³⁵

Here Kant describes the two physical modes of explanation – the mechanical and the dynamical – and notes that chemistry proceeds dynamically.Footnote ³⁶ This particular division of modes of explanation is echoed in the General Remark to the Dynamics of MAN. There Kant presents these two mutually exclusive (and apparently exhaustive) approaches to the science of matter. The mathematical-mechanical mode of explanation takes absolute impenetrability to be the fundamental property of matter. The mechanist thinks the world is made of absolutely impenetrable corpuscles separated by absolutely empty interstices. The opposed material mode of explanation, metaphysical-dynamism, rejects the mechanical explanans of absolute impenetrability and empty space. Rather, according to the metaphysical-dynamist, space is a plenum, where all its parts are relatively impenetrable to some degree or another. The relative impenetrability of a space is the result of the relative intensity of the fundamental repulsive force in that space. In sum: mathematical-mechanism takes absolute impenetrability at its basis where metaphysical-dynamism takes moving forces at its basis.

Both modes of explanation seek to explain phenomena beyond impenetrability and resistance to motion. The mechanist explains other material phenomena by appeal to the shape, size and impact of absolutely impenetrable corpuscles against a background empty space. For a paradigm of mechanical explanation, take Descartes’s description of magnetism.Footnote ³⁷ A magnet emits tiny screw-shaped parts of matter. It attracts or repels metal by the action of these tiny screws. If the screws that enter the pores of the metal are (say) right-handed, they will push the metal away; if they are left-handed, they will draw metal closer. Kant mentions another example of mechanical explanation in his passage on modes of physical explanation from DP. There he presents a Cartesian explanation of the dissolution of calcium carbonate by acetic acid. The parts of matter that make up the acetic acid are sharp and thus can penetrate and separate the parts of the calcium carbonate. So in chemistry, the mechanist would look to explain the dissolution and decomposition of matters by appeal to the size, shape and impact of corpuscles in an analogous fashion.

On the other hand, forces constitute the ultimate explanatory basis for the metaphysical-dynamist: ‘The general principle of the dynamics of material nature is that everything real in the objects of the outer senses, which is not merely the determination of space (place, extension, and figure), must be viewed as moving force’ (Kant Reference Kant2004: 62; MAN, 4: 523). So the dynamist would, for instance, explain magnetic phenomena by appeal to attractive and repulsive magnetic forces. Her explanation of the dissolution of calcium carbonate by vinegar would similarly appeal to forces of dissolution that combine the parts of the reagents.

It is important to note that the mathematical-mechanist’s explanations are amenable to mathematical treatment. Mathematical-mechanism refers only to the shape, size and impact of parts of matter and explains all material phenomena by reference to these properties. However, all such properties can be coordinated with mathematical constructions. The shape and size of corpuscles can be simply described mathematically, for these parts of matter are just (impenetrable) parts of space. For instance, the shape of the screw-shaped or sharp corpuscles can be constructed just as any figure can be. Similarly the motions and impacts of these parts of space will be constructible via the doctrine of motion and the mechanical coordination of material collision. So, it appears that the mathematical-mechanical mode of explanation makes possible the application of mathematics to material phenomena, generally, and the science of dissolution and decomposition, in particular. This would mean that chemistry, mechanically conducted, is reducible to the coordination principles of MAN, making it part of the proper science of physics.

But the mathematical-mechanical mode of explanation cannot realize its promise of mathematical adequacy in virtue of its metaphysical deficiency. The concepts making up the mechanists’ explanatory basis – absolute impenetrability and empty space – are empty. Kant deems absolute impenetrability to be ‘in fact nothing more nor less than an occult quality’ (Kant Reference Kant2004: 39; MAN, 4: 502).Footnote ³⁸ Kant also claims that empty space is beyond the possibility of perception; this point is made both in the Anticipations of Perception (Kant Reference Kant1998: 193; KrV, A172–3/B213–14) and the Third Analogy (Kant Reference Kant1998: 319; KrV, A214/B261). In Kant’s mind, the concepts of absolute impenetrability and empty space have a status analogous to Newton’s absolute space: all are Undinge (Kant Reference Kant1998: 166; KrV, A39/B56).Footnote ³⁹ Furthermore, Kant takes mechanism to entail a rejection of ‘all forces inherent in matter’ (Kant Reference Kant2004: 64; MAN, 4: 525). By this, he means that mechanism must give up gravity as a universal force of matter, an unacceptable implication.Footnote ⁴⁰

Yet the metaphysical-dynamic approach to chemistry also fails to make possible the construction of the chemical forces, though its deficiency is distinct from mechanism’s.

The metaphysical-dynamical explanations of chemistry, though coherent, cannot be reduced to the pure doctrine of motion. To reduce the inner forces of matter to mathematical physics would require that chemical action be represented by the communication of motion between spaces. Now, for Kant, chemical dissolutions are absolute; that is, every part of a chemical solution is homogeneous (Kant Reference Kant2004: 69; MAN, 4: 530). So, in a complete chemical dissolution, every single part of the solvent needs to combine with a proportional part of the solute in a shared space. But this is to require that during dissolution the reagents’ extensions are absolutely reduced. That is, chemical dissolution requires material penetration. Yet, as Kant shows in proposition 3 of his Dynamics, the communication of motion that may be mathematically represented in physics can only effect a finite compression of matter, never an absolute penetration (Kant Reference Kant2004: 37–8; MAN, 4: 501). So the mathematical coordinations belonging to physics are inadequate for the mathematical representation of chemical forces.Footnote ⁴²

To summarize up to this point, chemistry is an improper science because the concepts of chemical forces cannot be constructed. First, chemical forces do not belong to the possibility of experiencing matter, so there cannot be a priori principles coordinating them with mathematical constructions. Second, chemical forces cannot be reduced to the doctrine of motion. The mathematical-mechanical approach would so-reduce chemical forces, allowing for their mathematical representation, were mechanical philosophy not intellectually bankrupt. The metaphysical-dynamical description of chemistry, though metaphysically legitimate, utilizes forces that outstrip the expressive power of the mathematical doctrines of physics. Therefore chemistry is neither a proper science nor a part of the proper science of physics.

There is nevertheless a crucial difference between mechanical and dynamical chemistry, for Kant. The concepts of dynamical chemistry (of decomposing and dissolving forces) are not incoherent. Though the forces of the dynamic approach to chemistry cannot be mathematically represented, there is nevertheless a legitimate science of these forces.Footnote ⁴³ We can observe and experiment with substances in order to systematize and unify our knowledge of chemical forces even in the absence of mathematical foundations. Take the following example that was well-known in Kant’s time.Footnote ⁴⁴ Metallic silver dissolves in aqua fortis; when copper is added to this solution, silver is precipitated out. According to Kant’s framework of chemical forces, this observation is explained by the judgement that the dissolving force that acts to combine silver and aqua fortis is weaker than the dissolving force between copper and aqua fortis.Footnote ⁴⁵ Here we have a genuine judgement of dynamic chemistry; one that is legitimate despite our inability to a priori coordinate chemical forces with mathematical constructions. Thus, the metaphysical-dynamical methodology of postulating forces opens the possibility of a systematic, chemical science, though a science that is improper. The mathematical-mechanical approach, on the contrary, due to its postulation of empty concepts at its basis, cannot give rise to any coherent science of chemistry.Footnote ⁴⁶

And, despite the lack of a priori coordination principles between the chemical forces and mathematical constructions, this science is yet quantifiable and capable of quantitative regularities. The principle of the Anticipations of Perception guarantees that we can ascribe degrees to the qualities of experience (Kant Reference Kant1998: 290–5; KrV, A166–76/B207–18). So we are able to assign quantitative measurements to chemically relevant properties, such as reagent weights, specific gravities and degrees of heat.Footnote ⁴⁷ Such quantitative measurability makes possible quantitative regularities in chemistry. For example, we may find in experience that it always takes the same volume of warm water to dissolve one millilitre of saltpetre.Footnote ⁴⁸ Now this does not mean that dynamical chemical concepts have thereby been coordinated with mathematical constructions; there is still no mathematical object that represents the dissolving force of the water. Rather, the principle of the Anticipations merely makes possible the quantitative measurements and judgements that Kant observed in contemporary chemical practice, while the science itself remains improper.Footnote ⁴⁹