On Kant’s Schema of Reality

2.1 Is the schema of reality a transcendental schema?

Let us revisit the transcendental role Kant assigns to schemata, defined as the mediator between categories and possible objects of experience. Kant generally posits that a transcendental schema – that is, a transcendental determination of time – can resolve the heterogeneity between the concepts of understanding and objects of sensibility, “insofar as time is contained in every empirical representation of the manifold” (KrV, A138/B177). This general condition is essential to Kant’s argument of the schematism. Without this general condition, Kant’s doctrine of schematism would be futile.

Keeping in mind this general condition of how a schema in general would solve the problem Kant wishes to address, one can notice that a particular problem arises with respect to the empirical representation of sensation in the schema of reality. “Sensation in itself,” as Kant explains, “is not an objective representation,” by which Kant means that, “neither the intuition of space nor that of time is to be encountered in it” (KrV, A165/B208). It follows that the object of sensation – the real having an intensive magnitude – cannot be intuited over an extended time. This appears to be at odds with the condition that time must be generally contained in the empirical representation of the manifold for objects to be representable through a schema (KrV, A138/B177 above).

The object of sensation is purportedly perceived as a unity rather than a manifold to represent time. In the first edition of the TD, Kant explains how a category can rightfully represent an object through a successive apprehension. That is to say, in order to apprehend a unity in the manifold of representations, “it is necessary first to run through and then take together this manifoldness” (KrV, A99). In the AP, however, we learn that sensations are apprehended not successively but in a way that fills only an instant (KrV, A167/B209).

On Kant’s account, objective representations are those determined in extensive manifold of intuition (time or space). “Without intuition there can be no object” (MAN, AA 04:475n). Kant calls form “that which allows the manifold of appearance to be intuited” (KrV, A20/B34).^[7] In contrast, Kant explicitly distinguishes between the pure form of intuition and the real that fills it. The real in appearance is precisely the material aspect that fills the same intuition and cannot itself be intuited. The real does not constitute the intuition but comes about in addition to it (KrV, A166/B207). Kant’s use of intuition in this context is narrow and technical. As an a priori principle, “all intuition is extensive magnitude” (KrV, B201). The object of sensation, however, denotes a non-intuitive character as having not extensive but intensive magnitude (KrV, B207). If the schema’s primary goal is to demonstrate how a category can represent objects in general (KrV, A147/B187), how can the schema of reality in particular represent an object when its corresponding representation is purportedly not objective? The schema of reality appears to be tasked with delivering an outcome that it cannot promise in the first place. This difficulty leads to the apparent inconsistency of attempting to represent an object (KrV, A147/B187) that cannot have an objective representation (KrV, A165/B208). In other words, the schema of that which corresponds to sensation appears to be a contradiction in terms.

2.2 Is intensive magnitude a magnitude?

Kant presents a specific conception of magnitude in general. Intensive magnitude, in particular, poses a difficulty when put against that background. Indeed, it is unclear if intensive magnitude qualifies as a magnitude within the framework of Kant’s own definitions. At times, Kant suggests that intensive magnitude is not a ‘genuine’ [eigentliche] magnitude (Refl, AA 18:322).^[8] In the Axioms of Intuition, Kant writes,

The consciousness of the homogeneous manifold in intuition in general, insofar as through it the representation of an object first becomes possible is the concept of a magnitude. (KrV, A161/B203)

Kant’s concept of intensive magnitude poses a challenge to the abovementioned assumption that magnitudes in general are defined on the ground of intuitions. How can a non-extensive magnitude be represented as extensive and still be considered a magnitude if magnitudes by definition depend on intuition? Moreover, in light of the abovementioned definition, what is the manifold of intuition in the case of intensive magnitude, and what are the homogeneous parts in its synthesis? While Kant initially defines magnitude in terms of homogeneous parts in intuition, he later characterizes intensive magnitude as a degree that lacks such parts,

There are objects in which we distinguish no multitude of homogeneous parts; this is intensive magnitude. This magnitude is a degree. (V-Met-L2/Pölitz, AA 28:562)

Such an apparent inconsistency leads us to a crucial question regarding Kant’s philosophy of mathematics: can Kant’s concept of intensive magnitude be captured through his own account of mathematical cognition? Kant’s definition of mathematical cognition entails exhibiting a priori the intuition corresponding to the concept, or what Kant terms the construction of concepts (KrV, A711/B739). However, it appears that the notion of intensive magnitude does not align with this definition. It invites the question: how can a mathematical concept be constructed when it allegedly lacks spatiotemporal intuition?

It remains unclear whether Kant’s system can accommodate the mathematical cognition of intensive magnitudes in a consistent manner. While Kant identifies the principle of AP as mathematical in consideration of the fact that “the real in the perception of appearances could be generated in accordance with rules of a mathematical synthesis” (KrV, A178/B221), there remains a difficulty in constructing intensive magnitudes mathematically in accordance with Kant’s own definitions.^[9]

2.3 How to connect the degrees?

It is still unclear how one should consider the a priori synthesis that fills time in degrees as a determination of time, let alone the claim that there is a necessary connection between different degrees in the presumed generation of that degree.

We can illustrate this difficulty through an empirical example. Let us consider a generative process for a quality. When attempting to visualize various shades of blue with different degrees of the same quality <blueness>, I might conjure images of Azure (light blue), Sapphire, and Cobalt (dark blue) as emerging from one another. While I can imagine the emergence of different shades in this quality, I am still unable to determine the degree of darkness between them or the temporal sequence through which this generative process unfolds. All these shades appear as distinct fragments of blueness, seemingly unrelated except for their comparative degree of that quality (darker or lighter). Is the distinction in degrees between Azure and Sapphire the same as that between Sapphire and Cobalt? Have I overlooked various shades between them, such as, say, Cerulean? How does one shade necessarily follow from another? There seems to be no rule of continuity that can explain how I have jumped from one shade to another, even if I am able to imagine a generative process like the one described above.

Now contrast such a process with how I can envisage the construction of an extensive magnitude. In the case of drawing a line in thought (KrV, A102, B154, A161 – 2/B203), Kant could argue that parts necessarily follow each other, because one could be conscious of the act through which the manifold was being produced in every moment. When drawing a line, I could see in my spatial intuition how parts emerge in connection with each other through a motion in time. In the case of the schema of reality, however, the synthesis of the degree which is attributed to the real in sensation seems to be merely based on the possibility of its analysis ad infinitum – for instance, I can in principle posit an increasing number of shades of blue between any two given shades but not a motion whose product is the continuous spectrum of those shades. Hence, it seems that although I can first imagine different degrees of a quality, the unifying rule by means of which I connect those degrees cannot be properly conceived. The connection between lesser or greater degrees must also be explained through a rule-governed mathematical synthesis for those degrees to be necessarily connected.

Thus, as we move forward, it is crucial to bear in mind the characteristics of a viable resolution to Kant’s schema of reality.

Firstly, the principle of “apprehension in an instant” must be understood in light of both the TD and the schematism. This implies that the assertion that the real must have intensive magnitude is intrinsically connected to the continuous and uniform generation of a degree, which is a transcendental determination of time under the category of reality. The nature of this connection must be expounded upon.

Secondly, any coherent interpretation of “apprehension in an instant” must be mathematical in a way that “the real could be generated in accordance with rules of mathematical synthesis” (KrV, A178/B221). It is insufficient to merely interpret the degree of the real in an analytical fashion, coming in comparative forms of “more or less”. This would be a bottom-up discursive approach, which Kant rejects as an empirical deduction. In contrast, Kant’s TD instructs us that the proof of the AP must be read in the opposite direction: It is only through the synthetic act of continuously and uniformly generating a quantity that the forms of comparison, such as “lesser and greater,” first become possible. Against the background of the TD, the central question of the AP remains the question of a law-governed procedure of a generation of a degree over time [quaestio juris] not the question that concerns what Kant takes as a fact; namely that sensations come in degrees [quaestio facti] (KrV, A84-A88/B117 – 120). The proof of the AP concerns the former rather than the latter.

To arrive at a coherent solution that meets these requirements, it will prove useful to examine some of the existing literature on the subject, recognize what is lacking, and build upon previous approaches.

4.1 What is Uniformity?

But what is a uniform generation in the first place? One way to imagine uniformity is through a spatial analogy: let us contemplate a scenario where we aim to distribute a certain number of the same entity across different spatial locations. We can deem this distribution uniform if, and only if, in every arbitrarily chosen but equally sized portion of space, the same number of that entity can be found. Consider, for instance, a hypothetical situation where all living Kant scholars were uniformly dispersed among the universities worldwide. For the sake of this argument, let us take universities as units, assuming they are all of equal size. If the quantity of all living Kant scholars is uniformly distributed across all such equally sized universities, then irrespective of any particular university we select, the number of Kant scholars in that university would remain the same. In simpler terms, if we assume a uniform distribution, the specific university I choose to go to would not affect the number of Kant scholars I encounter; indeed, the number of Kant scholars will remain constant across all universities. You could expect exactly the same number of scholars in each university simply by knowing first, the total number of all universities and second, the total number of Kant scholars worldwide.

Given the uniform distribution you could write the following relation,

The left-hand side of the equation simply yields the notion of an average. It tells us how many Kant scholars, on average, are in each university. In addition, the assumption of uniformity informs us that this average, no matter how big or small, is precisely the number of Kant scholars we will find in each university. If one university held even one more or less than the average, the distribution could not be called uniform. Also, if I average over two, three or rather a hundred universities, the average, being equal to the number of scholars in each university, will not change. The constant number of Kant scholars in each university indicates that such a number is not a function of, or in any way dependent on, the specific university that we might deliberately choose.

Assuming that such a uniform distribution is in place, let us now see how a uniform generation of a quantity – in this case, Kant scholars – would look. I go visit all universities successively one at a time and record the sum total of Kant scholars in my survey. Upon going to the first university, I encounter, say K number of Kant scholars. Upon visiting the second university, the cumulative number of Kant scholars will add up to 2K. Since the distribution is uniform, this pattern continues so that by visiting the nth university, the total number will accumulate to nK.

To put it in Kant’s terms, by traversing the universities worldwide and summing up the count of Kant scholars, I have engaged in a synthetic act of apprehension that runs through the manifold of universities, resulting in the emergence of a uniformly generated total number of scholars. Guided by the condition of uniformity, the spatially extensive dimension governing one quantity (the number of universities) gives rise to another dimension wherein the other quantity can be constructed (the cumulative count of scholars). In this manner, the condition of uniformity establishes a relationship between these distinct quantities; the distinct quantity of universities on one hand, and that of Kant scholars, on the other, can be related through the assumption of uniformity.

The above hypothetical scenario aims to illustrate how similarly in Kant’s schema of reality, through an assumption of uniformity, two distinct quantities – intensive (the degree of the real) and extensive (time) – can be related in a sense that one contains the other. However, the notion of uniformity stipulated by the schema of reality pertains to a distribution, (or more precisely, a generation), which takes place not in space but in time. We aim to comprehend the transition from nothingness to a certain degree of a perceived quality within a generative process where, at equal time intervals, the same constituent units of that degree are purportedly generated.

To establish a better connection to the illustrative example mentioned above, we must recognize that equal time intervals serve as units and replace the universities that could contain more or less Kant scholars. By encompassing the entirety of a time interval, symbolized by going through all the universities in our analogy, we ultimately arrive at a cumulative quantity—an intensive magnitude that represents the degree of the real. This degree serves as the counterpart to the number of Kant scholars in the earlier scenario.^[13]

In this assumed process, if ΔD represents the difference in degrees at two separate moments within a time interval of ΔT, the condition of uniformity will necessitate that for any two arbitrarily chosen time intervals, such as ΔT1 and ΔT2, the average degree generated during those intervals must consistently be equal. In a similar fashion, we can formulate the uniform generation of a degree in time as follows:

To clarify, Kant establishes this relationship as a prerequisite for the schema of reality: Under the category of reality, the construction of a degree must exhibit a uniform generation in the transcendental determination of time. When Kant asserts that the generation of a degree must adhere to uniformity, he is, in effect, articulating the equation delineated above.

However, a uniform generation alone cannot encompass the intensive magnitude Kant ascribes to ‘the real’ in sensation. The condition of continuity must also be introduced, allowing Kant to discuss an apprehension that fills an instant.

By incorporating the continuity condition, Kant can establish a mathematical analogy between the generation of a quantity over a duration of time and when that duration becomes infinitely small, approaching zero. It is the continuity thesis that enables Kant to consider such infinitesimal time intervals. If I define dD/dT as the apprehension of a degree (dD) in nearly an instantaneous moment (dT) yielding the “quantity of something insofar as it fills time”, this is to be sure some representation that cannot be intuitive or objective. The extension is infinitely small to yield any magnitude or evoke ‘the consciousness of a homogeneous manifold’. Nevertheless, through a mathematical analogy that factors in the uniformity condition, I can envision a uniform and continuous generation over an extended period of time (ΔD/ΔT) that yields an equivalent apprehension in an instant. Since the time intervals in the uniform model of generating a quantity can be taken arbitrarily, I can now relate the rising of a degree in an instant (/dT) to the rising of that degree in any time interval (ΔD/ΔT). This will read as follows:

When Kant endorses both conditions of uniformity and continuity, he is, in effect articulating this equation above. To quote once again the definition (e) discussed earlier: “the schema of a reality, as the quantity of something insofar as it fills time [dD/dT], is just this continuous and uniform generation of that quantity in time [ΔD/ΔT]” (KrV, A143/B183). Through this mathematical analogy, Kant can establish a connection between the transcendental synthesis of a quantity over a specific duration of time on the one hand, and the degree of an empirical sensation that purportedly fills only an instant on the other. Kant employs this analogy to ground the latter on the former.

By applying both the continuity and uniformity conditions to the synthesis of reality, Kant can define intensive magnitude as “that magnitude which can only be apprehended as a unity, and in which multiplicity can only be represented through approximation to negation=0” (KrV, A168/B210). This definition finds coherence solely within the context of the aforementioned mathematical analogy, drawing upon the neglected background of the infinitesimal method utilized in differential and integral calculus.

The definition also highlights the importance of making a distinction between intensive magnitude and degree when interpreting Kant. It is not enough to consider intensive magnitude and degree as synonymous terms. If an intensive magnitude can be called a degree, not every degree can yield intensive magnitude. The term approximation to zero refers to the use of the infinitesimal method, which eventually leads to the assumption of a continuum. While a degree can exhibit discrete variations, it is inherent in the concept of intensive magnitude that if it is conceived as a degree, it must necessarily exhibit continuous variations. Without the continuous variation, the concept of intensive magnitude cannot be properly defined.

In his Das Princip der Infinitesimal-Methode und seine Geschichte, Hermann Cohen comments that in the AP, Kant has systematically exhibited the concept of differential by his principle of intensive magnitude: “Kant could not feel the need to emphasize the identity of differentials and intensive magnitudes”, since such an identity was already a general assumption in Kant’s time (Cohen 1883, 14). Cohen’s observation holds significant merit. In my proposed interpretation, I aim to demonstrate that comprehending why Kant formulates the schema of reality as he does requires framing his discussion within the presumed mathematical context. While Cohen is generally correct in recognizing the connection between intensive magnitude and differential calculus, he is mistaken in equating differentials with intensive magnitudes. A more precise formulation of the relationship between the proof of the AP and the calculus of differentials would be to identify intensive magnitude as a relation between differentials. As we have seen, the notion of “apprehension in an instant,” which yields an intensive magnitude, signifies a relation between two differentials: an infinitesimal apprehension of the generation of a degree (dD) within an infinitely small time interval (dT), which can be linked to and grounded upon a successive apprehension of a quantity (ΔD) over a period of time (ΔT).

On my reading, Kant’s schema of reality and henceforth his principle of intensive magnitude in the AP is based on such a central mathematical analogy that enables Kant to render intuitive what is first deemed non-intuitive. Kant construes mathematical analogies as formulas that assert the identity of two relations of magnitude so that if two members of the proportion are given, the third can also be constructed (KrV, A179/B222). The mathematical analogy is constitutive of the principle of AP.

Furthermore, we can recall that Kant entertains the use of analogies for an objective representation of time. In the Transcendental Aesthetic, he has a similar difficulty in representing the inner intuition as that which lacks an objective representation. By means of an analogy, there he appeals to the representation of space maintaining, “because this inner intuition yields no shape, we also attempt to remedy this lack through analogies” (KrV, A13/B50).

The secondary literature often fails to recognize the presence and significance of the mathematical analogy within Kant’s schema of reality and the proof of AP. When, in the same passage (KrV, A179/B222), Kant distinguishes between philosophical and mathematical analogies to discuss categories of relation under ‘analogies of experience,’ he is also implying that the latter, being constitutive, is already at work in the mathematical principle of intensive magnitude (AP). The mathematical analogy, which underlies Kant’s reasoning and conceptual framework, is overlooked or not given due attention by Kant commentators.

To provide a more comprehensive insight into Kant’s use of this mathematical analogy in the schema of reality, I will now further explore the foundations of Kant’s thought. In the next segment, I illustrate how Kant’s transcendental philosophy has implicitly appropriated the Newtonian method in viewing motion as a unifying synthetic act that can describe a manifold.

4.2 The Newtonian Context Underlying Kant’s Schema of Reality

At the heart of the TD, specifically from B151 onward, Kant endorses a theory of motion that serves as the a priori condition for the reception of form and matter. Kant draws on Newton’s insight in taking motion as the description of space. If in Newtonian physics motion can describe geometrical spaces, Kant’s transcendental philosophy adopts the Newtonian model and translates motion to a synthetic act of understanding that can unify the manifold of pure intuition. According to Kant’s TD, thinking of objects, especially geometrical figures, presupposes such a synthetic act: “We cannot think of a line,” writes Kant, “without drawing it in thought”. The synthetic act of drawing, the process, comes before the outcome. Kant believes that if we abstract from the manifold in space and still attend to the same synthetic act (KrV, B155), such a motion as description of space, can also “represent,” or rather, transcendentally determine, time:

We cannot even represent time without, in drawing a straight line… attending merely to the action of the synthesis of the manifold we successively determine the inner sense [time].” (ibid)

Kant identifies such an attention to the act of synthesis with “motion as action of the subject”. The motion that in Newton’s thought could describe a geometrical space, now, in transcendental philosophy, determines time. Kant enunciates this line of thought in a footnote, writing:

Motion as description of space, is pure act of the successive synthesis of the manifold in outer intuition in general through productive imagination and belongs not only to geometry but even to transcendental philosophy. (KrV, B155n)

In Kant’s transcendental philosophy, pure acts of successive synthesis generally yield pure concepts of understanding (KrV, B104). To obtain objective validity, these pure concepts must transcendentally determine the intuition of time (cf., KrV, B128). This is in general part of Kant’s transcendental logic that extends to the doctrine of schematism. Our concern, in particular, is how reality, among other pure concepts, can possibly determine time. As we saw above, Kant would eventually tell us how a transcendental determination of time must be thought of in connection to the category of reality. He defines the schema of reality as the continuous and uniform generation of a degree we can associate with sensation. Now, if Kant’s doctrine of transcendental time-determination in general hinges on the synthetic act of successively connecting together a manifold through the Newtonian doctrine of “motion as the description of space”, then the schema of reality – the transcendental determination of time under the category of reality – must also be read against the Newtonian background of motion as description of space on which Kant builds his transcendental philosophy. Such a crucial link between Kant’s teachings in the TD and the schema of reality which inherently links transcendental philosophy to the Newtonian doctrine of motion is, nevertheless, vastly underdiscussed.

It remains unclear how, without considering the mathematical foundation of Kant’s argument in specific sections of the TD, the Schematism, and the AP, one can make coherent sense of crucial passages such as B155n. Nonetheless, for the purpose of our present discussion, I will argue that these passages imply how the mathematical science of motion undeniably does contribute to Kant’s Transcendental philosophy as a whole, and more specifically, to the elucidation of the schema of reality.^[14]

A close examination of Newton’s seminal work, The Method of Fluxions (1736), especially paragraphs 55 and beyond within the chapter Transition to the Method of Fluxions, would reveal Kant’s embrace of Newton’s science of motion in postulating a transcendental schema for reality.

“Motion as description of space” carries the distinctive mark of Newtonian terminology. Within §55, Newton commences by presenting his methodological proposal centered around “a Space described by local Motion” (Newton 1736, 19). Newton sets the goal of finding the velocity of motion at any given time while continuously accounting for the length of the “space described”.

He introduces two distinct genres of quantities: “those quantities which I consider as gradually and indefinitely increasing, I shall hereafter call Fluents, or Flowing Quantities” (ibid, 20). Additionally, he defines Fluxions as “the Velocities by which every Fluent is increased by its generating Motion” (§60). Motion coming in different velocities is characterized as the Fluxion that continuously generates the Fluent, namely a space that it locally describes. In this way, Newton’s inquiry into velocity and the continuous description of space becomes a general problem involving the relation between Fluxions and Fluents: “the Relations of the Flowing Quantities to one another being given, to determine the relations of their Fluxions” (ibid, 21). Newton’s method of Fluxions postulates that every continuous magnitude can be conceived as a product of local motion.

Although Kant, in the AP, presents a synthetic account of continuous magnitude as flowing (KrV, A170/B211 – 2), he does not exclusively explore this underlying Newtonian framework. The relation between intensive magnitude and that which he alludes to as flowing or elapsing remains underdiscussed. Nonetheless, as Cohen observes, the fact that Kant only alluded to this was likely due to the common assumption, in Kant’s time, that this relationship held between intensive magnitude and differentials. In other words, Kant’s notion of intensive magnitude aligns with Newton’s concept of Fluxion, enabling the continuous description of a spatiotemporally extensive intuition as a Fluent.^[15]

Through the lens of the uniformity condition, we can discern how Kant establishes a connection between the successive apprehension of a quantity over time and the instantaneous apprehension of a certain degree supposedly in sensation. On one hand, time extends and a degree is continuously and uniformly generated, while on the other hand, there is an intensity in sensation that corresponds to that same degree. These two aspects are bridged and grounded through a mathematical analogy that juxtaposes two distinct types of quantities: extensive and intensive. It is important to note that this relationship can only be conceived when extension is predicated upon a synthetic act, as emphasized by Kant in the TD. In Newton’s terms, this implies that we can conceive any continuous quantity as a Fluent, namely as described by a motion throughout space. To describe a space by a continuous motion is the core of Newton’s insight of a science of Fluxions that Kant has implicitly integrated into his transcendental philosophy.

Kant appropriates the Newtonian relationship between Fluxions and Fluents, in order to define the concept of intensive magnitude within the mathematical framework of the AP. The intricate construction of this concept, or more precisely, the a priori exhibition of its corresponding intuition, if such a concept is to be cognized mathematically according to Kant’s definitions, becomes plausible when viewed in light of the mathematical analogy borrowed from Newton’s method of Fluxions. Thus, Kant’s schema of reality, which stipulates the transcendental determination of time in terms of the continuous and uniform generation of a quantity, can be understood as a similar endeavor to establish a relation between a Fluxion (the degree of the real in sensation) and a corresponding Fluent (time).^[16] When we put Kant’s arguments from the AP in the context of his overall teachings in the TD and the schematism, we can recognize a philosophical project with striking affinity to Newton’s mathematical project in Method of Fluxions. This is a foundational thought that underpins Kant’s philosophical project, and nevertheless, it is largely overlooked: The Newtonian calculus of relating Fluxions to Fluents has become the Kantian project of explaining the instantaneously apprehended degree of sensation as a case of time-determination. In other words, Kant’s schema of reality utilizes Newtonian mathematics to tackle a philosophical problem – namely, how “the real in the perception of appearances could be generated in accordance with rules of a mathematical synthesis” (KrV, A178/B221).

In the mathematical framework Kant has adopted in thinking the schema of reality, there is still one last aspect that requires consideration—a crucial question that lingers: why, within the framework of the Newtonian thought, does Kant stipulate that the schema of reality adhere to the uniformity condition?

4.3 Why Uniformity?

So far, we have observed how the uniformity condition can assist Kant in establishing a connection between the instantaneous apprehension of sensation in degrees and the successive apprehension of those degrees over time. To gain a deeper understanding of why Kant employs the uniformity condition, we can turn, I suggest, to the Phoronomy chapter of Metaphysical Foundations of Natural Science (MAN), specifically Explication 3.

In that section, Kant explores the notion of uniformity to address a similar challenge. There, Kant discusses the motion of a body undergoing acceleration and deceleration due to gravity, raising the question of how one can assert that the body is at rest at its turning point. The concept of rest, Kant posits, can in no way be constructed if we merely define rest as lack of motion. However, the construction can be rendered possible “through the representation of a motion with infinitely small speed through a finite time and can therefore be used for ensuring application of mathematics to natural science” (MAN AA, 4:486).

Similarly, in the context of the first Critique, “sensation in itself” does not contain the manifold of time, its apprehension is instantaneous; nevertheless, it is still crucial for Kant’s doctrine of time-determination to ground the possibility of apprehension in an instant on a unifying synthetic act over time. It is true that the degree of the real is given to sensation as a unity, but that degree, Kant would insist, presupposes a synthesis through which it can be constructed. As Cohen has stressed, “if the unity of a manifold must be thought, that unity itself must before anything else be thought of” (Cohen 1871, 428).^[17]

Returning to Kant’s example of an ascending and descending body influenced by gravity, the rising and falling as well as the state of rest that separates them can be united and necessarily connected through a motion that exhibits ongoing continuity under a uniformity condition. Kant describes such a motion as “first uniformly decelerated and thereafter uniformly accelerated” (MAN, AA 4:485; my emphasis). The entire process of the velocity of a body decreasing and increasing is governed by a constant factor, namely “the continual influence” of gravity. Assuming that the influence of gravity remains unaltered over time, we can observe a continuous and uniform generation – and diminution – of an intensive magnitude (velocity) as the body in question would rise and fall.

If we recognize that the same mathematical model of thinking is applicable to both the schema of reality and the case of continuous and uniform generation of velocity under the influence of gravity, the following obvious question arises: What does this example from mathematical physics of motion have to do with the schema of reality, which is meant to provide a transcendental grounding for the degree of influence in our senses? The answer, I contend, lies in Kant’s systematic presentation of transcendental philosophy. A closer reading of the proof of AP can reveal such systematic considerations.

Kant’s schema of reality is not merely about the possibility of perceiving a degree in an instant, but the continuous and uniform generation of that degree in time as that which represents a real in time. The real in time can be considered a cause influencing our senses in degrees. Although this is not a task for transcendental philosophy per se to fully account for the causality of the real as the affecting object of sensation, Kant still wants to make the AP open to the possibility of regarding reality as the cause of sensation, i. e., as the “moment” of a force being sensed (KrV, A168 – 9/B210). To this end, I take Kant to invite us to ask the following: If there is a physical body before me which I can perceive through my senses, how is it possible that I can perceive this object influencing my senses more or less as having a degree? The possibility of a fact is questioned in a transcendental fashion.

Against the general background of his transcendental argument, but for systematic reasons, Kant still aims to distinguish between the transcendental grounding of reality and causality. It is noteworthy to recognize that Kant deliberately sets forth a condition to establish this systematic distinction. He enunciates that in the proof of the AP, “I do not take into consideration the succession of many sensations” (KrV, A167/B209).

This is a highly crucial condition in the proof of the AP from which the uniformity condition can be derived. If I had a manifold of successive sensations before myself, I would then have to explain how such a succession can be objectively posited. This is the task Kant assigns to the Analogies of Experience where the categories of relation make possible experience defined in terms of a necessary connection between perceptions (KrV, A176/B218). In the AP, however, Kant is investigating the condition of the possibility of perception per se. If we recognize that perception can vary in degrees, it becomes essential for the synthetic unity to account for the necessary connection within the manifold of a single perception. The degree of one unchanging perception must be grounded on a mathematical construction not the relation between different perceptions.

Here, Kant is looking for an a priori synthesis that grounds the fact that reality has a constant degree of influence on our senses. The schema of reality abstracts away from alterations of that reality in the succession of time. Since alteration in sensation is not the concern of the schema of reality and the main aim of the AP is to ground the fact that the same sensation can have different degrees, Kant has to abstract away from the alteration of the real in time and regard the cause of sensation (i. e., the real) as a ‘moment’. The uniformity thesis is thus inherently linked to the possibility of accounting for the cause of sensation as an unalterable ‘moment:’

All alteration is therefore possible only through a continuous action of causality, which insofar as it is uniform, is called a moment. (KrV, A208/B254; my emphasis)

The question now takes the following form: If I want to disregard all the alteration in sensation but still consider sensation, “under the continuous influence” of an unchanging real, what is the corresponding synthetic act of determining that degree in time? In this case, I am not dealing with the categories of relation since the real or the corresponding sensation does not undergo any change. My focus is solely on the degree of influence that may be caused by that very same real which I assume as the object of my sensation.^[18] In such a scenario which exclusively concerns Kant’s constitutive and mathematical principle of reality and not his regulative and dynamical principle of causality, the a priori synthesis I am seeking must necessarily involve a uniform generation.

In a similar manner to how the constant influence of gravity can be translated into uniform acceleration in a falling body, an unchanging cause of sensation can be translated into a uniform generation of a degree. If in the former, each moment of time corresponds to a cumulative velocity, in the latter, Kant would say, time can also receive certain degrees of reality, at each moment. As mentioned earlier, Kant is not concerned with the succession of sensations (KrV, A167/B209) and abstracts from the categories of relation and, consequently, from alteration. This means that the intensive magnitude of the real remains constant over time. Taking this into account, we can establish a mathematical analogy between the generation of that magnitude within any arbitrary time interval ΔT and its instantaneous apprehension within dt. This allows us to generate the degree of a time-independent intensive magnitude in time, expressed as:

This uniformity “through a continuous action of causality” aligns with the previous notion we discussed and is now derived under the systematic consideration of the AP, imposed by the conditional “if I do not take into consideration the succession of many sensations” (KrV, A167/B209).

Kant is making an analogy between a transcendental generation of a degree over time and its empirical apprehension in an instant. For this analogy to work, it is absolutely necessary that the a priori synthesis of reality be both continuous and uniform. The uniformity condition serves a dual purpose. Firstly, it establishes a relationship between two different types of magnitudes within a synthetic act, a degree in each moment of time. On the transcendental level, the generated degree can be understood as a measure of depth or an additional dimension of time that represents the varying degree of perception. Furthermore, the uniformity condition assists Kant in discerning between the categories of reality and causality. This systematic consideration imposes a condition on the mathematical construction of a quantity that fills only an instant. The condition is none other than uniformity.