Kant on Mathematical Construction and Quantity of Matter
	
	Kant's reasoning in his special metaphysics of nature is often opaque, and the character of his a priori foundation for Newtonian science is the subject of some controversy. Recent literature on the Metaphysical Foundations of Natural Science has fallen well short of consensus on the aims and reasoning in the work. Kant claims that his work grounds Newtonian science. However, various of the doctrines enunciated in the Metaphysical Foundations and even the character of the reasoning in the work have been taken to present insuperable obstacles to accepting this claim. As Kathleen Okruhlik has argued, the Metaphysical Foundations has far more of a Leibnizean than Newtonian character, since the doctrine of body in the Dynamics is characteristically Leibnizean. Okruhlik’s position is that Kant's claims to defend Newtonian science are not to be taken at face value, and that prior ontological commitments play a hidden but central role in Kant's special metaphysics. Okruhlik points, for instance, to Kant's long held interest in a force ontology, and suggests that it may underwrite his desire to ground Newtonian mechanics with a dynamical concept of matter. Kant's desire to retain some Newtonian concepts -- absolute space, for example -- is held to be compatible with the overarching framework of relativistic motion and force ontology, since all fall under a methodological goal of reason to achieve explanatory completeness. On this reading, the Metaphysical Foundations reiterates the dynamical conception of matter first announced in Kant’s pre-critical Physical Monadology (1756), according to which the impenetrability of bodies is based on a fundamental force of repulsion. Since Newton writes, in Query 31 of the Opticks, that original matter was likely composed of ‘solid, massy, hard, impenetrable, moveable particles’, it looks as if a serious disagreement between Kant and Newton has been established.
	Standing in contrast to this is the view that Kant's stated aim of providing an a priori foundation for Newtonian science can be taken seriously, but only by holding Kant’s rather peculiar ruminations on matter theory to be marginal concerns in the Metaphysical Foundations. Michael Friedman thinks it unlikely that the theory of matter plays an important role in the Metaphysical Foundations, and holds ‘the idea that Kant's central disagreement with Newton is located here, at the level of matter theory...to be profoundly misleading’. Friedman's suggestion is that the central issue between Kant and Newton is the latter's assumption of absolute space and motion. Absolute space presents a difficulty, since Kant denies that it can be an object of experience. But if Newtonian science is to act as a case in concreto for the critical metaphysics, Kant must in some way show that absolute space is amenable to critical treatment. It turns out that Kant's foundation for natural science does include a foundation for absolute space, since it leads up to the defence of absolute space as ‘an idea that is to serve as a rule for considering all motion therein only as relative.’ [IV 560]
	My own position mitigates the dispute between Okruhlik and Friedman. I maintain that it is possible to give a strict and consistent reading of Kant’s stated aims and his doctrine of body. On the one hand, Kant’s theory of matter is prominently featured in the Metaphysical Foundations, so no one is more responsible for erecting the ‘mistaken’ impression that matter theory is a central issue there than Kant himself. On the other hand, Kant’s discussion of matter in the Dynamics is centred on the very same concepts of density and bulk used by Newton to define quantity of matter in the Principia. Moreover, as Friedman points out, ‘Kant disputes the “common opinion” that Newton is able to do without the assumption of an immediate attraction at a distance as an essential property of matter...’. So there is no compelling reason to suppose that Kant's Dynamics stands at odds with anything essential in Newton. The view taken in this paper is that the mathematical construction of the Newtonian concept of quantity of matter is a key step in Kant’s Metaphysical Foundations, and that Kant’s criteria of construction, quite apart from any Leibnizean commitments, are enough to generate the peculiar doctrine of matter contained therein.
	Kant concludes the Preface to the Metaphysical Foundations by suggesting that his a priori foundation for Newtonian science is incomplete. He claims only partial success in ‘following the mathematical method’ and suggests that it may yet be possible to bring a ‘metaphysical portion’ of his treatise into ‘unison with the mathematical doctrine of motion’ [IV 478] -- that is, the part of the treatise with which he was satisfied. His mathematical doctrine of motion, or kinematics, is given in the first chapter of the Metaphysical Foundations, the Phoronomy. The arguments there characterise the sense in which matter's motion can be constructed in relation to categories of quantity. Such constructions are mathematical representations created in intuition that correspond to the concept of ‘matter as the moveable in space’ [IV 470]. Three criteria of construction are explicit in the Phoronomy. Matter as moveable is constructible insofar as features of its motions are: a) given in intuition in space and time [IV 487]; b) subject to a rule for the composition of motion [IV 486]; and c) mathematically tractable in terms of addition and subtraction [IV 489]. In discussing these criteria, Kant rejects candidate representations inconsistent with the sense of mathematical tractability so defined.  For example, with respect to a), Kant says that the concept of an object brought under the category of causality but not given in intuition is non-constructible. In relation to b), Kant insists that all concepts of motion, including rest, must be represented through the composition of motion. This ensures that motion is represented in a manner distinct from relative space itself.  In connection with c), Kant says that Phoronomy is limited to representations of quantity of motion through the addition and subtraction of homogeneous parts.
	The above criteria define the sense in which motion is constructible as an extensive magnitude. In constructing concepts, we construct ‘objects viewed as quanta’ through a homogeneous synthesis which we ‘create for ourselves’ in space and time [B 751]. The construction itself goes beyond the empirical concept of matter as moveable and passes to the intuition that exhibits the concept of motion in concreto [A 722n]. Hence, the concept of quantity of motion is constructed a priori in a determinate fashion, without the help of empirical data [B 752].  
	It is important to remember that, according to Kant, the mathematically tractable features of concepts of quantity invoked in Newton’s laws must be a counterpart to some construction in intuition a priori. Given this understanding of Kant's natural philosophy and the evidence of the Phoronomy, Kant’s motive for constructing concepts of quantity is clear. But the Phoronomy, which deals with pure kinematic properties of motion, is the only chapter in which Kant carries out anything like rigorous constructions. What remaining portion of Kant’s special doctrine of body -- the metaphysical part of which in the Preface he claims to have exhausted -- can be, but has not yet been mathematically treated, and hence, brought into unison with his constructions in the Phoronomy? I suggest that Kant hoped to show the dynamical concept of quantity of matter to be constructible, thereby uniting the remaining ‘metaphysical portion’ of the Metaphysical Foundations with his mathematical treatment of matter’s motion. Moreover, the latter construction supplies an a priori foundation for the concept of quantity of matter implicit in Newton’s laws, which is defined in terms of bulk and density. It turns out that this appears to land Kant with a peculiar conjunction of mechanistic and dynamical concepts of matter.
	To begin, consider the textual support for my claim that Kant wanted to construct the dynamical concept of quantity of matter. The first half of the Dynamics shows that the possibility of matter’s filling space, and in particular, its capacity to resist other bodies, must be explained in terms of two fundamental forces; one, the force of repulsion, the other, the force of attraction. Much of the focus in this section is on how such a concept of matter can be given a mathematical treatment. In a recent article, Friedman has interpreted the argument in this part of Kant’s Dynamics. According to Friedman, the epistemological question of matter’s possibility is to be kept separate from the issue of matter as a possible object of sense experience. As an object of experience, matter is solid and resistant to the penetration of other bodies. Thus, Kant rejects the dogmatic claim that ‘matter consists of an aggregate of parts external to one another – parts that are always divisible into further smaller parts to infinity’. At the same time, Friedman argues, a dynamical concept of matter is to be represented a priori as (potentially) infinitely divisible, where infinite divisibility depends only on the character of the space.  But on this interpretation, Kant’s Dynamics has a serious failing. For it fails to establish that it is actually Newton’s concept of quantity of matter, a concept defined in terms of bulk and density, that meets the conditions of schematism and mathematical construction. My reading of Kant’s Dynamics aims to succeed where Friedman’s account fails.
	We can begin with the construction of matter’s bulk. On the basis of the criteria of mathematical construction described in the Phoronomy, it is possible to represent matter’s extensive quantity in terms of spatial extension. First of all, the representation in question is of ‘the various distances of matter and its parts from one another’, and a body and its parts are given in sensible intuition. Secondly, the representation would reflect the distinction between the moving parts in a body and the moving body as a whole, so the representation is subject to the rule of composition of motion. Finally, the representation of matter’s extensive quantity is conceivable through subtraction and addition of the homogeneous parts of a body. Hence, it is possible to provide a general determination of matter’s bulk through a priori construction. But more than this is required if Newton’s definition of quantity of matter as a measure of both bulk and density is to be mathematically constructed. Specifically, Kant must show that the concept of matter’s density, is also determinable a priori. This is a problem that preoccupies Kant in the second half of his Dynamics [IV 517-535]. 
	Beginning at Proposition 8, Kant discusses the possibility of constructing the dynamical concept of matter in connection with a ‘law of the relation of forces’, which is itself a function of the direction of the original forces and the distances between matters. Kant speaks as though a law describing the limitation of matter by opposing forces is really possible, but says that the magnitude of this law is unknown [IV 517]. He envisions a method for constructing the dynamical concept of matter through the ‘action and reaction of fundamental forces’, and thereby establishing a priori the possibility of a body's filling of space in a determinate degree. Later in the chapter, the discussion is crowned by the claim that the dynamical explanation of the interaction of repulsive and attractive forces would provide an account of matter’s density or ‘specific variety’. Indeed, what Kant appears to maintain is that density can be explained if the attractive force is equal in all materials, while the repulsion of different materials differs [IV 534]. According to Kant, the envisioned law of the relation of forces would express a formula for determining the quantity of matter associated with the dynamical concept of matter, and thus enable him to complete the mathematical construction of quantity of matter.
	Although the text makes it quite clear that Kant had in mind some sort of construction relating to density, Kant concedes that there is no ‘law of the relation of forces’, so it is difficult to imagine exactly what he thought such a construction might be like. In terms of Phoronomy, Kant's sense of mathematical construction is reasonably clear. The empirical intuition of moving things in space and time is given in the synthesis of a manifold of parts, and general features of extensive magnitude associated with matter’s motion can be represented a priori under categories of quantity in a manner consistent with Kant’s criteria of construction [B 202].  Moreover, it is easy to see that the criteria for constructing extensive magnitude described in the Phoronomy can be carried over for the construction of matter’s bulk. But the situation with regard to density is rather less straightforward, since density is conceptually linked to intensive magnitude. Nevertheless, if the dynamical concept of quantity of matter is to be meaningful, or ‘really possible’, it must be possible to construct matter’s density due to fundamental forces.  
	The specific problem facing a construction of matter’s density, is that intensive magnitudes such as forces are not themselves constructible in the sense Kant defines in the Phoronomy. First of all, we do not sensibly perceive determinate degrees of intensive magnitude [B 210]. In addition, intensive magnitudes are not necessarily summed in the same way as extensive magnitudes. So the sense of mathematical tractability Kant defines for the construction of extensive magnitude cannot be straightforwardly applied to intensive magnitude. Indeed, the criteria of construction described in the Phoronomy seem intended only for the representation of extensive magnitude, and the dynamical concept of matter is the concept of an extensive magnitude that also has intensive magnitude. So the question arises, ‘Does it even make sense to think of the dynamical concept of matter as constructible?’ 
	In the first Critique, Kant claims that categories of quantity and quality are both mathematical and constitutive of knowledge of objects a priori [B 110]. So evidently, his settled view is that there is some sense in which concepts brought under categories of quality are capable of mathematical treatment. Indeed, it suggests that he believed it to be possible to ‘go beyond the empirical concept’ which is understood purely qualitatively and ‘pass to the intuition which exhibits the concept’. If so, it might be possible to mathematically construct not only matter’s bulk, but also its density [A 722]. Still, the question of how quantity of matter conceived in terms of its quality of ‘filling space due to forces’ is to be given a mathematical treatment is far from clear. Kant certainly does not give a complete account, and there is no prima facie ground for thinking that such an account is even possible. There are hints as to how Kant thought such an account might proceed, which I will try to lay out in the next few paragraphs; but I shall have to reserve a complete treatment of this topic for a later date.
	The primary clue to the puzzle surrounding the construction of extensive magnitudes that are also intensive magnitudes may lie in the Doctrine of Method. There, Kant says that there are actually two types of mathematical construction: 1) geometrical constructions and, 2) symbolic constructions [B 745]. The former, Kant writes, are constructions of magnitudes (quanta) ‘as in geometry’, and these have to do with the intuition of objects in space. The latter are constructions of magnitudes (quantitas) ‘as in algebra’, and these are abstracted from the constitution of objects. While constructions of extensive features descriptive of quantity of motion and matter are clearly of the former, geometrical kind, Kant may have thought that concepts associated with intensive magnitude can be mathematically constructed in the latter, symbolic sense.   Newton’s force law, the law of gravity, is expressed algebraically. And Kant does say in his Doctrine of Method that algebraic magnitudes are to be symbolically constructed.  Presumably the law of the relation of forces, like the law of gravity, would be expressed algebraically, so it may be that intensive quantities relating to force (e.g., density) are to be symbolically constructed. In order to assess this suggestion, we will look at Kant's view of symbolic construction. The idea here is that a fuller account of Kant’s view of mathematical construction, one that includes both geometrical and symbolic construction, might help to explain the sense in which Kant thought about the mathematical construction of ‘matter filling space due to forces’.  
	All mathematical constructions, whether geometrical or symbolic, must ‘exhibit the a priori content of a concept in intuition’. Indeed, mathematical construction is defined as a method for exhibiting in intuition the a priori content of a concept. In geometrical construction, a successive iteration of points generates an ideal representation of a given extensive magnitude. Hence, geometrical construction is a method for the intuitive representation of an extensive magnitude. Symbolic constructions are also characterised by the repetition of an operation, although the operation that is iterated is not the iteration of points. Symbolic constructions link algebra and time, and represent algebraically described magnitudes in so far as they exhibit the successive progression underlying the number series or the operation of a calculus. So although symbolic constructions do not exhibit the intuition in space corresponding to a concept, they are linked to intuition in time. Time, Kant says, is ‘a rule for the synthesis in imagination for the production of images’ [B 182]. As the most general form of synthesis in intuition, time is the basis for all representations. Kant writes; ‘The pure image of all magnitudes of the outer sense is space, of the objects of the senses in general, however, is time.’ [B 182]  
	In relation to mathematical construction, what this means is that magnitudes with external determinations can be exhibited a priori in space; but, that every magnitude, whether extensive or intensive, can be exhibited a priori in time. Since the dynamical concept of matter is conceptually linked to both extensive and intensive magnitude, it is a potential candidate for a mathematical construction that exhibits magnitude in relation to both space and time. Specifically, Kant might have had in mind a construction exhibiting the relationship between certain intensive magnitudes that are ‘symbolically constructed’ and certain extensive magnitudes that are ‘geometrically constructed’. Such a construction would have to conform to what Kant says about symbolic representation in determinate judgement. 
	Kant says that for determinate judgements involving symbols, it is the rule or procedure of the determinate concept that determines the agreement between a concept and its symbol. Such a link presupposes that the concept symbolised is ‘supplied with an intuition’ so that ‘what agrees with the [symbolic] concept is merely the rule of this procedure, not the intuition itself.’ [V 352]. The implication for the argument in the Dynamics is as follows: An algebraic ‘law of the relation of forces’ would justify the mathematical construction of the dynamical concept of quantity of matter if the symbolic construction of density expressed agreement with the rule for the geometrical construction of matter’s bulk. Such a construction would not involve a direct representation of degrees of intensive magnitude through the successive iteration of points; rather, it would involve a symbolic linking of degrees of intensive magnitude to marks exhibited in the geometrical construction of matter’s bulk. In this way, the mathematical construction of the dynamical concept of quantity of matter might link degrees of force with the representation of matter’s bulk. Thus, the dynamical concept of matter would have a two-step mathematical construction. In the first step, a geometrical construction of the extensive magnitude of a body would exhibit the distances between, and directions of, the moving parts of a body. This would amount to the construction of matter’s bulk. In the second step, a symbolic construction of density would justify the assumption that degrees of intensive magnitude are meaningfully linked to the geometrical construction of matter’s bulk. This step would complete the construction of the dynamical concept of matter.
	A number of Kant’s interpreters have suggested that the construction of forces would be impossible even in principle, since forces are not directly observed, and therefore not fit subjects for geometrical construction. Kant does indicate that a construction connected to a ‘law of the relation of forces’ is a viable project, but he offers no more than clues as to how such a construction might proceed. The clues gathered here suggest that a mathematical construction of quantity of matter would be meaningful only if it expressed some relationship between intensive magnitudes associated with matter’s density and the representation of the extensive magnitude of a body. The difficulty facing the construction of the dynamical concept of quantity of matter is that a geometrical construction of matter’s bulk could not be meaningfully related to the symbolic construction of matter’s density in the absence of the appropriate ‘law of the relation of forces’. Thus, Kant’s proposed construction in the Dynamics fails not on principle, but for lack of the requisite scientific discovery of the ‘law of the relation of forces’.
	In conclusion, let me briefly review my argument. Kant's special metaphysics is intended to provide the a priori foundation for Newtonian science, which is to be achieved by exhibiting the a priori content of Newtonian concepts and laws. Kant envisions a two-step mathematical construction of the dynamical concept of matter involving a geometrical construction of matter’s bulk and a symbolic construction of matter’s density. Since Newton himself defines quantity of matter in terms of bulk and density, there is no reason why we shouldn’t interpret Kant’s Dynamics as a defence of a Newtonian concept of matter. When Kant’s reasoning is understood in relation to his criteria for mathematical construction, it is possible to maintain that matter theory is central to the Metaphysical Foundations, but that this does not undermine Kant’s stated aim of giving an a priori foundation for Newtonian science. 

Jennifer McRobert 
Acadia University