DELEUZE ON KANT’S MATHEMATICS OF THE SENSORY BODY: THE DIFFERENTIAL RELATION IN THE INSTANCE OF LEARNING

Hélio Rebello

São Paulo State University, Brazilherebell @ hotmail.com

ABSTRACTThis article discusses three consequences of Deleuze’s approach to Kant’s philosophy of mathematics. First, that Deleuze develops Kant’s concept of intensive magnitude to propose an ontology for the sensory body. Second, that the Deleuzian approach shifts the conditioning role that the intensive plays in regards to the quantitative in the “possible experience in general”. Third, that the Deleuzian approach enrolls Kant under the circle of the so-called problematic (pedagogy of) mathematics. These consequences are illustrated through the instantiation of Kant’s and Deleuze’s mathematical-philosophical issues in some pedagogical settings related to the process of learning (how to swim and to develop a mathematical theorem). The subjects hereby discussed converge to the mathematics of the sensory body.

INTRODUCTION: Deleuze’s mathematical main idea (axiomatics vs. problematics) and Kant’s “application of mathematics to nature”

Deleuze (1925-1995) understands that Kant’s (1724-1804) mathematical ideas presents a problematic profile. What does Deleuze mean when he approaches Kant’s philosophy of mathematics emphasizing its problematic feature?

Deleuze criticizes the mathematics that became prey to axiomatic-based systems in terms of set theory (Duffy 2006a: 2-4). It does not follow from this disagreement, though, a total disapproval toward the calculus of axioms or against the set theory as such. Deleuze regrets, indeed, the axiomatization in terms of sets that tends to reduce the axioms and their theorems toward a single notion or primitive function as to suffocate the internal differentiation in the axiomatic system. It happens, whenever:

… ‘equivalents’ in cyclical alternation [prevent] difference from displacing itself in these cycles […], rendering repetition imperative but offering only the bare to the eyes of the external observer who believes that the variants are not the essential and have little effect upon that which they nevertheless constitute from within. (Deleuze 1994: 290)

The output of the logic-mathematical operation that Deleuze criticizes, namely, is the creation of general equivalents that corrupt the involved mathematical entities. This

criticism addresses the most basic procedure in the theories of axiomatization, such as Zermelo-Fraenkel’s, whose main tool is the cumulative hierarchy based on the recursive equivalence of the empty set:

The universe of set theory is built up in stages, with one stage for each ordinal number. At stage 0 there are no sets yet. At each following stage, a set is added to the universe if all of its elements have been added at previous stages. Thus the empty set is added at stage 1, and the set containing the empty set is added at stage 2. (Zermelo-Fraenkel theory September 2016)

Deleuze and Guattari call the cumulative hierarchy in set-theory “axiomatic” to emphasize one main defect, which he describes as the disorder of the mathematical entities that are stifled under the blockage of “variants” for the sake of “equivalents”: “the axiomatic blocks all lines, subordinates them to a punctual system, and halts the geometric and algebraic writing systems that had begun to run off in all directions.” (Deleuze & Guattari 2005: 143) The criticism of Deleuze to the axiomatic is not, however, just a philosophical insurrection against the set-theoretically oriented trend in the contemporary logic, since there are dissenting voices in the history of mathematics that echo the Deleuzian claim. According to Duffy:

The relations between the canonical history of mathematics and the alternative lineages that Deleuze extracts from it are most clearly exemplified in the difference between what can be described as the axiomatized set theoretical explanations of mathematics and those developments or research programs that fall outside of the parameters of such an axiomatic, for example, algebraic topology, functional analysis, and differential geometry, to name but a few. (Duffy 2013: 1)

In fact, the closeness between Deleuze and these “alternative lineages” demonstrates, on the one hand, that he was not a naive reader of contemporary mathematics. On the other hand, his possible relationship with some alternative lineages in the contemporary logical-mathematics assures that Deleuze did not practice with respect to mathematics an “intellectual imposture” as Sokal and Bricmont claimed (1998). Indeed, Deleuze’s disagreement regarding the axiomatization can be illustrated by contemporary philosophers interested in developing formal ontologies, which broadly involve the relationship of philosophy with mathematics regarding the study of material things, as Smith explains “formal ontology deals with properties of objects which are formal in the sense that they can be exemplified, in principle, by objects in all material spheres or domains of reality.” (Smith 1998: 19) Husserl, who launched the idea of formal ontology, believed that certain branches of mathematics, such as Riemann’s theory of multiplicities, can be applied to domains of actual objects to formalize the philosophical approach to reality. The axiomatic grants that mathematical entities properly fit ontological entities to be formalized as a general category:

Manifolds are thus in themselves compassable totalities of objects in general, which are thought of as distinct only in empty, formal generality and are conceived of as

defined by determinate modalities of the something-in-general. Among these totalities the so-called definite manifolds are distinctive. Their definition through a complete axiomatic system gives a special sort of totality in all deductive determinations to the formal substrate-objects contained in them. With this sort of totality, one can say, the formal-logical idea of a world-in-general is constructed. The theory of manifolds in this special sense is the universal science of the definite manifolds. (Husserl 1970: 45-46) 

What can formal ontologies, nevertheless, expect from non-set-theoretically axiomatizable mathematics as Deleuze claims?

Smith (1996) and Varzi (1997: 31-32) argue that the formal ontology called mereotopology, that studies the relationship between whole and part and of part with another part, experiences a deficit when it seeks to gather all its axioms under one single primitive definition. It happens because the required cumulative hierarchy decreases the descriptive power of the ontological entities framed in set-theoretic terms. The defect of hanging formalization to ultimate axioms is that the definitions and the formal stability of the derived axioms or theorems lose their realistic character. Smith sums up this issue:

More recent experience in the construction of formal-ontological systems, for example for the purposes of naive physics, has suggested that systems capable of describing real-world phenomena will require large numbers of non-logical primitives, no group of which will be capable of being eliminated formally in favor of any other group (Smith 1996: 288)

Nef adds that there is a basic problem concerning the formalization of the connection between a whole and its parts in terms of set theory, because:

… the set theory […] defines them [sets] either as lists (extensional definition) or as collections of elements endowed with the same property (extensional definition or comprehension). A list is not a whole, and a collection of attributions either [...] and therefore one can understand [...] that the identity of philosophical ontology with set theory is doomed to failure (Nef s/d: 13; my translation)1

Deleuze, in turn, asserts that the lineages that evade the axiomatic based on set theory require a problematic mathematics, since “in the axiomatic, the deduction goes from axioms to derivative theorems, while in the problematic the deduction goes from the issue to the ideas, accidents and events that determine the problem and make the cases that can solve it.” (Smith 2006: 145) Smith provides two examples from mathematics education that clearly distinguish axiomatic from problematic mathematics. If a teacher asks the students to draw a triangle to which the sum of the angles is 180 degrees, they would draw different 1 ...la théorie des ensembles […] les [ensembles] définit soit comme des listes (définition extensionnelle) ou comme des collections d’éléments dotés de la même propriété (défintion extensionnelle ou en compréhension). Une liste n’est pas un tout, et une collection d’attributions non plus […] et on comprend dès lors […] que l’identification de l’ontologie philosophique à la théorie des ensembles est vouée à l’échec.

triangles and, by measuring their angles they would demonstrate the axiom that the sum of the interior angles of every triangle is 180 degrees. However, if the teacher asks them to draw an equilateral triangle it is not enough to know the axiom, for the student must do an experiment to draw a triangle with three equal sides (Smith 2006: 148). Drawing an equilateral triangle launches a problem because the students must deal with the possibility of constructing triangles with unequal sides. In this case, they are introduced to basic topology, i.e., exercises with accidents and events that define that kind of triangle and no other. They must deal with these topological events and accidents before developing the single axiom of the angles summation. The former procedure is an axiomatic pedagogy; whereas the latter is a problematic one. What is at stake in these pedagogies is that the formalizing resources that are inherent to problematics are not the same that interests to the axiomatic model. In short, problematizing constitutes procedures whose accidents and events do not allow being neither reduced nor estimated by the cumulative axiomatic. A more generic example of the problematic knowledge is the science that Deleuze and Guattari call “nomadic”. The nomadic science emphasizes the importance of intuition (topological accidents and events), of sensation, and perception, in contrast with the “royal science” which holds an axiomatic character: “what is proper to royal science, to its theorem or axiomatic power, is to isolate all operations from the conditions of intuition, making them true intrinsic concepts, or ‘categories.’” (Deleuze & Guattari 2005: 373). Moreover, Deleuze and Guattari extract from the point of view that interconnects classic philosophical and mathematical issues the political character of mathematics in contemporary societies:

Mathematical writing systems were axiomatized, in other words, restratified, resemiotized, and material flows were rephysicalized. It is a political affair as much as a scientific one: science must not go crazy. Hilbert and de Broglie were as much politicians as they were scientists: they reestablished order (Deleuze & Guattari 2005: 143)

Indeed, for Deleuze and Guattari set-theoretically based axiomatic is not just a harmless and abstract mathematical operation with which mathematicians deal in their offices or which philosophers apply to solve ontological challenges. The axiomatic can be understood instead as the basic procedure of the contemporary capitalism - the “worldwide axiomatic” - that regulates relations of production and consumption:

Capitalism marks a mutation in worldwide or ecumenical organizations, which now take on a consistency of their own: the worldwide axiomatic, instead of resulting from heterogeneous social formations and their relations, for the most part distributes these formations, determines their relations, while organizing an international division of labor (Deleuze & Guattari 2005: 454)

This historical-social axiomatic additionally operates at the level of subjective processes and constitutes the prevailing digital semiotic system based on “omnidirectional images”. (Deleuze 1989: 265-266) These images are organized in the confluence between social and technical machines that are elements connected by an axiomatizable relationship.

Consequently, the “omnidirectional images” are like sets of numbers that can be recombined – hierarchized - whenever the axiomatic principle is threatened by elements that depart from being axiomatized (Cardoso Jr. 2012).

Even though related to the social and historical consequences of the axiomatics, this article focus on the application of some mathematical entities to ontological entities such as intuitions, sensations and perceptions as they are produced in the body that learns. It claims that Deleuze strived to reinforce the problematic point of view in the Kantian application of mathematics the learning capacity of the body, so that his efforts may be understood as the development of a formal ontology within the Kantian tradition. In short, the approach to Deleuze’s problematic mathematics is better understood as an approximation and a confrontation between Kant’s “application of mathematics to nature” (Kant 2014: 61 [Prol, AA 04: 309.19])2 and Deleuze’s “mathematization of Nature” (Smith 2012b: 83), as both has as object the sensory body and its capacity of learning (De Freitas & Sinclair 2014: 156-157). Kant’s and Deleuze applications of mathematics will be understood henceforth as related philosophical endeavors, since both build their philosophy of mathematics on the “mathematical doctrine of nature” (Kant 2004: 9 [MAN, AA 02: 473]) and the “general doctrine of body” (Kant 2004: 13 [MAN, AA 02: 478.6]). These interconnected issues outline the field of the mathematics of the sensory body.

KANT AND THE MATHEMATICS OF THE SENSORY BODY: the intensive magnitude between the conditioning and the “genetic instance”

Kant considers the mathematical method as a tool to understand how the sensory body elaborates outer intuitions in inner sensations (Friedman 2013: 573). The mathematical approach to this bodily process instantiates metaphysics with concrete examples without which the latter would be meaningless:

It is also indeed very remarkable (but cannot be expounded in detail here) that general metaphysics, in all instances where it requires examples (intuitions) in order to provide meaning for its pure concepts of the understanding, must always take them from the general doctrine of body, and thus from the form and principles of outer intuition; and, if these are not exhibited completely, it gropes uncertainly and unsteadily among mere meaningless concepts... (Kant 2004: 13 [MAN, AA 02: 478.3-9])

Kant defines the sensory process involved in the body and opens the way to its formalization, as he explains this bodily process through the mathematical concept of magnitude and the mathematical function of the infinitesimal.First of all, the sensory body is defined through its ability to capture stimuli from the outside world in a way that simple intuitions are related to sensations to form perceptions:

2 English Edition followed by Kant’s Akademie-Ausgabe (in brackets).

Perception is empirical consciousness, i.e., one in which there is at the same time sensation. Appearances, as objects of perception, are not pure (merely formal) intuitions, like space and time (for these cannot be perceived in themselves). They therefore also contain in addition to the intuition the materials for some object in general (through which something existing in space or time is represented), i.e., the real of the sensation, as merely subjective representation, by which one can only be conscious that the subject is affected, and which one relates to an object in general (Kant 1998: 290 [KrV, B: 207.10-18)

The external stimulus are empirical or sensible intuitions, thus

… even the perception of an object, as appearance, is possible only through the same synthetic unity of the manifold of given sensible intuition which the unity of the composition of the homogeneous is thought in the concept of a magnitude, i.e., the appearances are all magnitudes, and indeed extensive magnitudes, since as intuitions in space or time they must be represented through the same synthesis as that through which space and time in general are determined. (Kant 1998: 287 [KrV, A: 203.4-11]).

Sensible intuitions are understood as extensive magnitudes, because they can be measured as they appear in space and time. Sensations are not equal to the empirical intuitions, because they are not extensive data in space and time. Sensations hold a subjective character which adds some quality to the corresponding extensive magnitude (empirical intuition). Kant provides the definition: “Sensation […] expresses the merely subjective aspect of our representations of things outside us, but strictly speaking it expresses the material (the real) in them (through which something existing is given).” (Kant 2002: 75 [KU, A 05: 189.10-13]) The subject assigns qualitative sensations to the extensive magnitudes of the empirical intuitions that the sense organs receive. The qualitative sensations give uniqueness to the difference in magnitude to each corresponding intuition because the sensation is “the quality of empirical intuition with respect to which the sensation differs specifically from other sensations.” (Kant 2014: 61 [Prol, AA 04: 409.17-18)

According to Kant: perception is the “…representation with consciousness (perceptio). A perception which relates solely to the subject as the modification of its state is sensation (sensatio), an objective perception is knowledge (cognitio). This is either intuition or concept (intuitus vel conceptus).” (Kant 1998: 398-399 [KrV, A: 320.1-5]) The knowledge purport of the sensory body is based on a categorical relationship from which the perception (relation) assigns to the sensation (quality) the objective reality of the intuition (extensive magnitude). In fact, the perception combines the extensive magnitude of an intuition with the corresponding qualitative sensation to compose an intensive magnitude. Kant affirms in the Critique of Pure Reason’s first version: “In all appearances the sensation, and the real that corresponds to it in the object (realitas phaenomenon), has an intensive magnitude, i.e., a degree.” (Kant 1998: 290 [KrV, A: 166.13-15]) According to Jankowaik, “the intensive magnitude is a measure of how an object ‘fills’ space or time”, so

that “realities in objects have intensive magnitudes because sensations do, and it is this dependency relation that grounds Kant’s inference” (Jankowaik 2013: 387, 389), as a degree of brightness or heat, for instance. The intensive magnitude for which the perception stands for is the highest outcome of the sensory body. It provides knowledge as it is “representation with consciousness”. It will be worthwhile, therefore, observing closer the relationship between intuitions, sensations and perceptions as a process that the sensory body promotes to translate an extensive into an intensive magnitude.

The intensive magnitude arises from an operation that places the perception as the term of the relation that interrelates in a specific way the empirical intuition and the corresponding sensation. This process makes it possible, Kant explains “the application of mathematics to nature, with respect to the sensory intuition whereby nature is given to us, is first made possible and determined.” (Kant 2014: 61 [Prol, AA 04: 309.19-21]) The application of mathematics to nature concerns the formalization of the relation between empirical intuitions, quality sensations and intensive perceptions. This operation, though, faces the main problem that the three-termed relationship happens along an instantaneous and infinitesimal process in the sensory body. Therefore, Kant reframes the sensory relationship that the body undergoes in terms of a mathematical function saying that:

…because the real in the appearances must have a degree [intensive magnitude], insofar as perception contains, beyond intuition, sensation as well, between which and nothing, i.e., the complete disappearance of sensation, a transition always occurs by diminution, insofar, that is, as sensation itself fills no part of space and time… (Kant 2014: 60-61 [Prol. AA 04: 309.8-12])

In the automatic process that the sensory body undergoes, the qualitative sensation becomes insensible as it tends to disappear, given that the corresponding intuition goes empty or tends to zero in extensive magnitude related to space and time. This progressive decrease tends to the limit that the intuition that occupies the space-time defined coordinates becomes the pure form of space and time, as it is emptied of its former extensive magnitude. Nevertheless, the form of space and time remains related to an evanescent sensory quality (“between which and nothing, i.e., the complete disappearance of sensation, a transition always occurs by diminution”). The perception, thus, results from the infinitesimal relation that mediates the intuition and the sensation. This relation, though, must observe a condition to be extracted: it is the purely formal magnitude of intuition when the quality of sensation tends to disappear toward the degree zero of magnitude. In short, as the intuition becomes the form of space and time, the intensive magnitude of perception stands for the extensive magnitude of the intuition in correspondence to the quality of the sensation. Kant provides an example to explain this instantaneous and infinitesimal relation between the extensive magnitude, the sensation and the intensive magnitude. He emphasizes the independence of the intensive with regard to the intensive:

…an expansion that fills a space, e.g. warmth, and likewise every other reality (in appearance) can, without in the least leaving the smallest part of this space empty, decrease in degree infinitely, and nonetheless fill the space with this smaller degree

just as wen as another appearance does with a larger one. (Kant 1998: 294 [KrV, A/B: 174.13-17/216.8-12])

The mathematical relationship between different types of magnitude explains what happens when the ontological entities work together in the sensory body. Speaking in more general terms, the Kantian mathematization of nature explains the determination of the magnitudes (extensive or intensive) involved in the ontological issue of the relation between the external and internal instances. The perception as an intensive magnitude, in the one hand, make the external extensive intuition to correspond to the internal qualitative sensation of the body that feels it. In this case, the relation between intuition and sensation has the intensive magnitude of perception as the conditioning term in the “possible experience in general:

…although sensation […] can never be cognized a priori, it nonetheless can, in a possible experience in general, as the magnitude of perception, be distinguished intensively from every other sensation of the same kind. (Kant 2004: 61 [Prol, AA 04: 309.14-18])

On the other hand, according to Simont (1997) in a book that contrasts Kant’s and Deleuze’s respective concept of relation, Kant changed his position regarding the role that the intensive magnitude plays in the relationship between the external and the internal sensory body. The intensive magnitude ceases to be only the conditioning term in the relation of correspondence between intuitions and sensations in the “possible experience in general” to become the “genetic instance” that brings about the empirical experience (Simont 1997: 187-188). Simont provides evidence about this oscillation in Kant’s ontological position by comparing the First Critique first edition (A) passage on intensive magnitudes: “In all appearances the sensation, and the real that corresponds to it in the object (realitas phaenomenon), has an intensive magnitude, i.e., a degree.” (Kant 1998: 290 [KrV, A: 166.13-15]) with the corresponding passage in the second edition (B): “In all appearances the real, which is an object of sensation, has intensive magnitude , i.e., a degree.” (Kant 1998: 290 [KrV, B: 207.9-10]) Simont concludes after this comparison made:

...the mutual exteriority of sensation and its object disappears, the correspondence of the separated terms vanishes, reality and sensation are thought of as one and the same; the intensive magnitude [is] the intelligibility of sensation and reality, indissolubly. And if the production of magnitudes touches the real, the degree may be understood as a genetic instance. (Simont 1997: 188; my translation)3

It is important to detail the kind of generation that the Critique of Pure Reason’s second edition displays. On the one hand, the genetic instance of sensation does not apply to the 3 …l’extériorité mutuelle de la sensation et de son objet disparaît, la correspondance des termes séparés se évanouit, le réel et la sensation son pensés comme un seul et même surgissement ; la grandeur intensive [est] l’intelligibilité de la sensation et du réel, indissolublement. Et si la production de grandeurs touche le réel, le degré peut êtrê compris comme instance génétique.

thing in itself, for the “… pure sensible intuition) [is an] a priori concept, hence independent of experience”, so that “…(the assertion of an empirical origin would be a sort of generatio aequivoca).” (Kant 1998: 264 [KrV B: 167.1-4]). Thus, both editions (A and B) agrees in regards to the transcendental grounds of the a priori. On the other hand, the second edition goes as far as to inspect a special kind of generation that do not deny the aprioristic ground of transcendental philosophy. As the empirical sensation corresponds to an experience in time its generation can be conceived as the continuous transformation of degrees of the sensation from the experience (full elapsed time) to nothing (empty time):

…the quantity of something insofar as it fills time, is this continuous uniform generation of that quantity in time, as one descends in time from the sensation that has a certain degree to its disappearance or gradually ascends from negation to its magnitude. (Kant 1998: 275 [KrV B: 183.2-5])

The generation runs continuously in two senses. By decreasing, the empirical intuition reaches empty time (zero) and it becomes pure intuition, which is the intensive magnitude to that experienced extensive magnitude and its corresponding quality sensation. At this level, the generation of time meets its genetic instance because it is now possible to explain how the intensive magnitude (zero) or pure intuition gradually increases to display the generation of the extensive magnitude: “…thus there is also possible a synthesis of the generation of the magnitude of a sensation from its beginning, the pure intuition == 0, to any arbitrary magnitude.” (Kant 1998: 290 [KrV B: 208.4-6])

By contrast, the Critique of Pure Reason’s first edition also inspects the generation issue, but it is restricted to an operation with representations related to the possible experience in general, for instance, addition in space or time:

I cannot represent to myself any line, no matter how small it may be, without drawing it in thought, i.e., successively generating all its parts from one point, and thereby first sketching this intuition. It is exactly the same even with the smallest time. I think therein only the successive progress from one moment to another, where through all parts of time and their addition a determinate magnitude of time is finally generated. (Kant 1998: 287-288 [KrV A: 162.3-6, 163.1-4])

The intensive magnitude changes its scope from the conditioning role in the possible experience in general to the “genetic instance” in the relationship between intuitions and sensations. Consequently, the functional involvement of the mathematical entities that explain the ontological process that goes on in the sensory body shall change by their turn. That is what Deleuze takes for granted to propose his mathematics of the sensory body.

DELEUZE’S MATHEMATICS OF THE SENSORY BODY and the development of Kant’s philosophy of mathematics: application of mathematics to nature and the intensive magnitude as differential relation

Deleuze also appeals to the resources of mathematics to formalize to a certain extent his philosophy (Salanskis 2006: 51). He starts from Kant’s alleged fluctuation regarding the role of the intensive magnitudes. The scholarly literature has been providing opposite ways of approaching the relationship of Deleuze to Kant in regards to the intensive: “…although many scholars have focused on the differences between the transcendental philosophies of Kant and Deleuze, others have argued that Deleuze's reformulation of the problem of intensity is designed to extend and correct Kant's account, not replace it.” (Ables n/d) This article follows the second path and assumes its main consequence: “…Deleuze's work extends Kant's critical project and Copernican revolution.” (Lord 2012: 83) Deleuze’s thesis on Kant’s intensive magnitude, in fact, unifies the two poles of the alleged Kant’s alternative between the intensive being conditioning or generative to the bodily entities (Deleuze 1994: 222). Simont explains Deleuze’s solution in a nutshell: “Genesis is conditioned by what it is supposed to generate, just as the condition can only be engendered by what it conditions.”4 (Simont 1997: 194) Deleuze, moreover, details this implication as coming up from the differentiating process that the sensory body promotes: “Intensity is the form of difference in so far as this is the reason of the sensible. Every intensity is differential…” The new portray of the intensive magnitude, therefore, extends the mathematics of the sensory body towards the investigation the differentiating process through mathematical and philosophical resources, which is one of Deleuze’s engagements with mathematics according to Dufy: “the deployment of mathematical problematics as philosophical problematics is one of the strategies that Deleuze employs with respect to the history of philosophy.” (Duffy 2006: 3) In fact, Kant’s transcendental philosophy explains how the body produces perceptions from the relationship between intuitions and sensations. As explained previously, he even details this process by mapping out those bodily entities through the infinitesimal function that relates mathematical entities - the quantitative and the intensive magnitudes – and the qualities of feeling. Deleuze in turn develops this Kantian lineage in applying resources from the differential calculus and geometry – differentials, differential relation and derivatives; composite curve-function, straight-tangent and tangent series - to scrutinize the relationship that Kant stablishes between quantitative and intensive magnitudes.

It is useful to move along with an example.

A sound is heard. Hearing a sound is an experience whose occurrence can be measured by a stopwatch, since the sound as an empirical intuition occupies space-time coordinates. One can measure the sound since it starts until its extinction in cycles or oscillations per unit of time (see Figures 1.a and 1.b.). The intuition of the sound is then an extensive magnitude. The sound thereby can be divided into smaller and smaller intervals tending to zero (see Figure 2). The differential of this sound is the smallest possible difference between two consecutive minimum intervals of the captured sound. The sensation of this micro-sound corresponds to the differential (quality) of that minimal intuition (extensive magnitude in micro-periods of time). The perception of the sound, by its turn, arises if at least two

4 La genèse est conditionnée par ce qu’elle devrait engendrer, tout autant que la condition ne peut êtrê qu’engendrée par ce qu’elle conditionne.

differentials (dx and dy), the one relative to the sound cycle and the other to time unit are related to one another. The quotient (dy/dx) between two differentials is called differential relation. The values that this relationship assumes are intensive magnitudes. Deleuze puts it bluntly that the association between the differential relation and the intensive is neglected in the history of philosophy: “The affinity between intensive magnitudes and differentials has often been denied. Such criticism, however, bears only upon a misconception of this affinity.” (Deleuze 1994: 244)

In the example of the captured sound, the differential relation defines the lowest audible sound to an ear. This differential relation between two sounds intervals tending to zero depends on the materiality of the hearing system, capable of responding to the infinitely decreasing of the composite extensive magnitude. This means that not all sensory intuition becomes perception for a certain ear. The differential relation characteristic of the human ear, for example, is different from the dogs’ ear, so that the canine audio spectrum is different from the human. The human and the canine hearing systems take different perceptual values due to their sensory differentials. These differential perceptive values are called derivatives, i.e., the values that the differential relation takes in each case. Deleuze assures: “The intensive factor is a partial derivative or the differential of a composite function.” (Deleuze 1994: 222) If the composite functions are the human and the canine ear they will take diverse derivatives due to their respective physiological structures.

One might claim that the consideration of physiological processes violates the Kantian transcendental clause. Deleuze, nevertheless, does not separate the empirical consideration of senses from the a priori conditions of the sensory experience. Kant also understands that the “physiology of pure reason” combines the “transcendental philosophy” to form the “metaphysics of nature”: “Immanent physiology, on contrary, considers nature as the sum total of all objects the senses, thus considers it as it is given to us, but in accordance with a priori conditions, under which it can be given to us in general.” (Kant 1998: 6978 [KrV AB: 846/874.7-10]) The Deleuzian approach to the “immanent physiology” does not objects Kant. It indicates that the metaphysics of nature can be extended through the differential relation by understanding that the experience of hearing in general has diverse profiles, for instance, in the human and in the dog because the conditions of hearing as experience involves the generation of the hearing as a differentiating process.

In order to make the Deleuze’s approach the philosophy of mathematics closer to the Kantian lineage, it is important, hence, to understand how the intensive magnitude becomes conditioning and generative regarding the relationship that intuition-sensation-perception entertains. On the one hand, the intensive magnitude is conditioning because without the differential relation no perception would be extracted from the external intuition of the captured sound and no quality sensation could be assigned to it. On the other hand, the intensive is not a general condition that holds good regardless of the ontological entities for which the intensive magnitude is conditioning in the empirical experience. As the derivative of the differential relation, the intensive magnitude determines the perception as generated by the physiological features physiology of the given sensory body – canine or human ear in the previous example.

The conditioning and the generative associated roles in the Deleuzian sensory body’s mathematics share as their common characteristic the independence of the intensive magnitude. The differential relation grants this independence: the dx and dy differentials depend on the amplitude of sound in the measured time, but the dy/dx ratio does not depend upon these values. It depends instead on the physiological capacity that this body disposes of hearing that sound the way it does. The differential relation, therefore, can be expressed through another variable, z, which is neither the sound cycle in different amplitudes, nor the time that this cycle lasts. When no ear accurately knows if the sound is perceived or not, then the differential relation is the derivative or intensive magnitude extracted from the starting sound. This relation characterizes the perceptive capacity of the ear as a sieve that regulates all the ear can hear independently of what it actually hears.

This is the point at which Deleuze departs from the Kantian mathematics of the sensory body. Kant, at least in the Critique of Pure Reason first version, does not admit the independence of the intensive magnitude because it is only the degree that fills the extensive magnitude of the intuition and relates it to the quality of the sensation: “In all appearances the sensation, and the real that corresponds to it in the object (realitas phaenomenon), has an intensive magnitude, i.e., a degree.” (Kant 1998: 290 [KrV A: 166.13-15]) Deleuze does not miss pointing out that “Kant's mistake is […] to reserve intensive magnitude for the matter which fills a given extensity to some degree or other.” (Deleuze 1994: 222) From the Deleuzian point of view, Kant holds back the full independence of the intensive and eventually downgrades the conditioning role that it plays. As a degree of the extensive magnitude, the intensive of the perception ends up to be the small-scale image of the intuition-sensation that it conditions. As image or degree, thus, the intensive magnitude has a fainter reality than the intuition-sensation.

“The complete disappearance of sensation”, as Kant says (Kant 2014: 60-61 [Prol. AA 04: 309.10-11]), in the differential relation is not a denumerable limit for Deleuze, it is not a degree in a decreasing scale, but the limit that sets the ontological significance of the intensive magnitude as the condition of the sensory body’s capacities. At the same time, when the sensation reaches zero, the form of the intuition does not remain only as condition of all possible experience. The differential relation, indeed, provides the intensive magnitude that precedes the intuitions, sensations and perceptions because the zero in the differential relation as Deleuze understands it holds a generative character. In the order of the things, the perception occurs only if it is triggered by an intuition. In the order of the reasons, however, the intensive magnitude must perform the genetic instance in the experience for the intuition to exist in space and time and for the quality sensation to correspond to the quantitative magnitude.

In the following section the operation that Deleuze makes over Kant’s mathematics of the sensory will be instantiated and detailed in the example of the sensory body that learns in pedagogical settings.

DELEUZE’S MATHEMATICS OF THE SENSORY BODY EXEMPLIFIED: the differential relation graphically depicted and learning defined as sensory problematic field

To clarify the conditioning and generative role that intensive magnitudes accomplishes regarding the sensory body’s ontological entities, it is useful to present an exercise based on the previous example that combines acoustics and some resources of the differential geometry. First, the Cartesian graphic (Figure 1) describes the sound cycle relating amplitude (X axis) and time (Y axis) of two different sounds. These two figures display graphically the composite functions of these sounds:

Figure 1.a. and 1.b Graphical representation of sound according to frequency (width X vs. time Y)

The higher (Fig. 1.a) or lower (Fig. 1.b) frequency of a sound clearly depends on the axes y (amplitude) and x (time). The relationship between any point at this curve, say Y/X applies for the entire curve as the frequency function of the sound. Being the frequency of the sound graphically represented a sinusoidal curve, one section may be cut off this curve and some random point of it chosen. The point (A) at the curve may be associated with a straight line, called straight tangent (Figure 2):

Figure 2. Graphic representation of differential relation (dy/dx) or curve-tangent relation over point A

At the point A two differentials are found, one related to Y (amplitude) and another to X (time), respectively, dy and dx. The differential relation (dy/dx) is the point that belongs to a tangent-line for the curve considered. Thus, it is observed that dx and dy are ordered pairs of the sound curve. The dy/dx ratio, though, is independent of the curve it comes from. The differential relation belongs to the straight line, so that it indicates that at point A the gradient curvature – slope - of the curve is given by the tangent-line. One can say, therefore, that the differential relation conditions the curve from which it derives. This function is what was previously called the conditioning role of the intensive magnitude for the differential relation between the quantitative magnitude (dy and dx at point A in the curve) and the corresponding quality (gradient of curvature). The conditioning intensive magnitude is determined by the function related to the differential relation (straight-tangent) that differs from the composite function (curve): “When the primitive function expresses the curve, dy/dx = - (x/y) expresses the trigonometric tangent of the angle made by the tangent or the curve and the axis of the abscissae.” (Deleuze 1994: 172)

Moreover, the differential relation is a derivative, because each point of the curve can be analyzed by the series of straight lines that pass by that point. The derivative, therefore, is determined by the series of straight-tangents that expands from the differential relation, so that the derivative indicates the rate of change that the curve undergoes over the considered gradient-point (A): “Where the differential relation gives the value of the gradient at the

given point, the value of the derivative of the differential relation [...] indicates the rate at which the gradient is changing at that point.” (Duffy 2013: 163) The series of straight-tangents that determines the derivative generates the straight-tangent that corresponds to the considered differential relation in the derivative point. This function plays the generative role of the intensive magnitude regarding the differential relation between the quantitative magnitude and the corresponding quality. From this exercise, it is important henceforth to instantiate Deleuze’s mathematics of the sensory body in order to compare it with Kant’s. The learning provides an example that illustrates the sensory body’s differentiating process.

To begin with, Deleuze’s philosophy must show how it can be useful in regards to educational and pedagogical issues. On the one hand, Deleuze is recognized as pedagogue and philosopher of education, both for his concerning about teaching and for some themes that his philosophy promotes. The pedagogy and education topic, though not constituting a clearly defined sector in Deleuze’s philosophy, was explicitly treated in his work. (Cardoso Jr. 2006 37-38) According to Boudinet: “Deleuze can be considered as part of the lineage of the great philosophers of education […]. Deleuze emphasizes educational, formative and pedagogical dimensions…” (Boudinet 2012: 10; my translation). On the other hand, the scholarly literature dedicated to Deleuze’s mathematical ideas clearly indicates that the sensory body as one proper domain for their application (Smith 2012a: 52-54; De Freitas & Sinclair 2014: 156-158). Especially regarding Deleuze’s ideas on education and pedagogy, there are prolific theoretical and practical scholarly literature about the role that the sensory body plays for learning (Cardoso Jr. 2016). For example, recent scholars highlight issues such as “body mind learning” (Semetsky 2013), “embodied pedagogies” (Perry 2013), and “corporeal theorizing” that involve the “physical rationality”. (Knight 2013: 17) The Deleuze studies dedicated to education and pedagogy, nevertheless, has not focused on the sensory body’s mathematics implicated in learning.

The very beginning of the learning process takes place in the problematic field that is established between the object to be learnt and the sensory body that grasps this or that object. Deleuze provides an example: “To learn to swim is to conjugate the distinctive points of our bodies with the singular points of the objective idea in order to form a problematic field.” (Deleuze 1994: 165) The learning problem arises as the single intuition of the water that touches the body of the learner (“distinctive points of our bodies”). The “objective idea” is the idea that emerges to immediately correspond to the “distinctive points of our bodies”. The correspondence between both composes the problematic field of learning to swim: swimming is to be immersed in the sea having nothing but the body. The swimming at sea becomes learnable as when the distinctive points of the swimmer’s body - the movement of waves, tides and currents, together with the wind – are brought together to correspond to the objective ideas as an operation that the sensory body organizes. Practically speaking, learning to swim takes place when one adjusts the movements of a body to the waves and the other implicated circumstances. To this effect, a more elaborated idea than the previous objective idea arises from the relationship of the body with the sea. It is the problematic idea, which engages the movements and actions that perform swimming at a given problematic field. According to the previous established difference between problematic and axiomatic, one can now understand that learning to swim begins in

problematic field because the body of the swimmer deals with accidents and events of being immersed in water. Learning to swim is not like applying an axiom.

The “distinctive points of our bodies” and the “objective idea”, however, can be extremely variable. Evidently, learning how to swim in the sea, in the middle of the waves, and learning to swim in a pool is quite different. The former presents a problematic field closer to that of dancing than to that of learning how to swim in a pool. The reason for this is simple: the water of the sea has a movement of its own; while the water in pool tends to be motionless. According to a swimming coach: “At sea the priority is the direction and not the speed.” (Lima s/d) This sensorial difference that qualifies several pedagogical bodies in the practice of swimming explains why the best swimmers in the pool may not perform well in open sea. The sensory body of the sea swimmer and the pool swimmer differs, in turn, from a swimmer who competes in lake races. Lakes are quieter than sea and more agitated than swimming pools. They also have different salinity than the sea and the pool, which intervenes on the buoyancy and thus on the speed of swimming. These different settings that learning to swim involves shows that the problematic fields of learning cannot be reduced to a general experienced field, since different distinctive points of the body and objective ideas are involved in diverse learning situations. Accordingly, the problematic ideas, which makes the swimming becomes a learned action, can be quite different.

Other examples are possible. The sensory body of language students can be determined in several problematic fields, depending on the language they learn. For example, teaching a dead language like Latin does not involve the same conjugations between points of the body (vocal tract) and the objective ideas (phonemes) that are present when learning English nowadays. Even learning a mathematical theorem involves a problematic pedagogy. Learning a mathematical theorem, unlike learning a language, would be properly portrayed as the problematic field formed by the body when faced with a mathematical entity. Learning math does not involve a concrete object like the sea, and it does not involve either the ability to try new skills of the vocal tract as in the case of learning a language (Cardoso Jr. 2007). In the math class, the learning is inherent to the student’s ability to apply or deduce the theorem on focus. From the sensory point of view of the body that learns the theorem, the problematic field would be different if the involved mathematical entities were differently conceived. According to Deleuze, the cases of math learning differ for being based either on the axiomatic or on the problematic math pedagogy (De Freitas 2013). If the theorem was defined according to operations based on axiomatic set theory, the learning process would proceed on the basis of visual entities that linguistically represent translated operations (union, intersection etc.) as with Venn diagrams. Diversely, if learning the theorem started from Peirce’s “diagrammatic observation” or reasoning the visual entities would be the theorematic operations themselves (Havenel 2016). De Freitas and Sinclair show that Deleuze warns the teachers of mathematics about the difference between learning mathematics through perceptions based on sensitive organs and on linguistic forms (2014: 141-147).

How can the whole process of learning in diverse problematic fields be explained through the mathematics of the sensory-learning body?

THE INSTANTIATION OF DELEUZE’S MATHEMATICS OF THE SENSORY BODY in different pedagogical settings and the expansion of the Kantian mathematics of the sensory

The problematic field of learning is defined according to its three interconnected features (the distinctive points of the body, the objective ideas and the problematic idea). The problematic field is organized around the differentiation process that takes place in the sensory body as the intensive is produced from differential relations. Therefore, the problematic is both conditioning and genetic regarding the sensory entities involved. Deleuze claims that: “Problems are the differential elements in thought, the genetic elements in the truth. We can therefore substitute for the simple point of view of conditioning a point of view of effective genesis.” (Deleuze 1994: 162). One can only understand how to learn from diverse problematic fields, which is located at, or is composed from, the sensory body as it intensively differentiates intuitions. The participation of the body in the learning process generates events and accidents that relate to the corporeal learning (sensations, intuitions and perceptions). This is an important condition for the type of formalization that Deleuze requires. It follows, then, that it is possible to observe the sensory output of the learning process through the sensory body’s mathematics as it maps out the problematic field of learning.

The learning body that learns to swim receives a sensorial stimulus: it feels the water as a material intuition in some of its distinctive points. The intuition corresponds to one sensation that specifies materially the water: the moving sea water or the standing water in the pool. The resultant perceptions related with learning to swim are quite different depending on the environment considered (sea, pool or lake). Hence, one can say that the differential relation conditions each intuition in distinct points on the body to provide a correlative sensation as the objective idea of the sea. This idea, by its turn, becomes perception when these intuitions-sensations are differentiated as ideas that put learning into action as the problematic idea. The differential relation that allows learning to swim is also generative regarding the intuition-sensation involved in the derivative, which specifies the differential relation presupposed in the series of different swimming environments (pool, lake, river, etc.).

The overlapping conditioning-genetic role of the differential relation depends on the expanded series of derivatives in which the bodily differentiating process is involved. The Deleuzian application of mathematics also details the infinitesimal function involved in the intensive bodily process. The differential relation (perception) is stablished between two differentials (tending to zero) of extensive magnitudes (intuitions) connected to some quality (sensation) which are about to disappear or become insensible. Swimming in the sea, swimming in a pool and swimming in a lake, differs from each other based on the events and accidents that come up between the body of the swimmer and each liquid environment in order to generate the perceptions of swimming. As each of these perceptual functions are differentiable, each case of swimming will have its own set of values

(derivatives) from the differential relation that corresponds to it. Therefore, swimming in the pool includes the perception that favors speed, which is the first derivative of the differential relation. The second derivative of this same differential relation (swimming in the pool) is acceleration. As for swimming in the sea, differently, the derived perception is the direction, the first derivative. The sense is its second differential, because the swimmer at open sea must deal with the direction of the ocean currents to calculate the sense of the swimming, for instance, the official course in a competition. In the case of the sea, thus, the speed and the acceleration would eventually generate the third and the forth derivatives as they are differentiated from direction and sense as the first and the second derivatives, and so on. Deleuze explains the expansive nature that determines differential relations, which unlimitedly conditions the primitive or composite function differentiated at a given tangent-point:

… in so far as it expresses another quality, the differential relation remains tied to the individual values or to the quantitative variations corresponding to that quality (for example, tangent). It [this tangent-point or quality] is therefore differentiable in turn… (Deleuze 1994: 172)

Deleuze’s approach to the dialects of the pure reason takes Kant’s problematic idea to understand it in terms of recursive derivative process as it conditions experience:

Ideas are themselves problematic or problematizing […] Ideas have legitimate uses only in relation to concepts of the understanding; but conversely, the concepts of the understanding find the ground of their (maximum) full experimental use only in the degree to which they are related to problematic Ideas… (Deleuze 1994: 169)

For Kant, there is also a recursive differentiation related to problematic ideas as they regulate the pure concepts of understanding. The possible experience in general as an idea conditions or limits the concepts of the understanding in a way that this idea plays the unconditioned (universal conditions of all empirical experience) for them. The idea of experience in general is problematic because it is the unconditioned that makes all concept of understanding determinable:

The absolute whole of the series of conditions for a given conditioned is always unconditioned, because outside it there are no more conditions regarding which it could be conditioned. But the absolute whole of such a series is only an idea, or rather a problematic concept, whose possibility has to be investigated, particularly in reference to the way in which the unconditioned may be contained in it as the properly transcendental idea that is at issue. (Kant 1998: 465 [KrV AB: 417-418.1-2/445.1-6])

This relationship that the sensory body promotes made the learning an action as it implies the problematic field that condition the learning to swim. It means that learning is an experience in general and that everyone is able to know and practice something relying on the sensory activity only. Had not, however, the perception a problematic intensive

magnitude that generates the intuition-sensation related to swimming, the derivatives (e.g. speed and acceleration in a pool) would be ineffective regarding the special swimming environment. It might happen if the sensory body’s mathematics does not develop the intensive beyond the conditioning role. If the intensive does not observe the genetic role regarding the learning body, the differentiation would remain incomplete and it would miss the differentiable problematic field that the variegated learning situations demand. It would be, in short, the learning to swim as the “possible experience in general” only. The learning to swim, though, is made possible because the intuitions-sensations become of intensive magnitude as the sensitive points of the body adjust to the water perception (objective idea) in specific perceptive situations (sea, pool, lake). The differentiating process that the body undergoes should not remain unspecific and undifferentiated.

FINAL REMARKS

This article has emphasized that Kant and Deleuze provide an important topic in the philosophy of mathematics as they explain how intuitions correspond to sensations to compose perceptions as ontological sensory entities. Sensory body’s mathematics is the approach to this transforming process. The philosophical development of mathematical ideas allows to precise and, to some extent, formalize this corporeal accomplishment. The sensory entities are consistent with mathematical formal entities as far as their correspondence is defined as the problematic field in which intuitions, sensations and perceptions are produced under the differentiation process which relates extensive magnitudes, qualities and intensive magnitudes. The process of differentiation is the starting point for both the Kantian and the Deleuzian sensory body’s mathematics, since they propose a formal ontology that envisages the body as organized by infinitesimal-differential relations. However, Deleuze redeploys Kant’s mathematics of the sensory body on a basic feature. For Deleuze, the intensive magnitude is called derivative and can be summarized as the differential relation (perception) between two differentials (tending to zero) of an extensive magnitude (intuition) corresponding to a sensation that becomes insensible in qualitative terms in the process of infinitely diminution of its quantitative magnitude (intuition). Kant understands that the sensory body differentiates through a similar differential relation, but he conceives of the derivative as an infinitesimal closure that halts the differentiating process. Deleuze instead claims that the derivative conditioning function establishes the recursive nature of the differential relation by which the differential relation becomes indefinitely derivable, until the limit that such derivatives by the expansion of their series reach functions that differs from the straight-tangent function differentiable at the considered point.

Furthermore, the aim of understanding Deleuze’s starting point from Kant’s mathematics of the body shows the extent to which the Deleuzian formal ontology may fairly be included in the Kantian lineage. Consequently, he bets on a dip in the mathematical-philosophical tradition, interconnecting diverse branches such as the differential calculus, the infinitesimal calculus and the differential geometry, purposely aiming to avoid the set-theoretic choice (DeLanda 2006: 236-237). Nevertheless, the Deleuzian way of addressing

the formal ontology is not merely the recovering of a minor branch in the tradition to propagate Kant’s heritage, because the use of mathematics to support and formalize the ontology divides the contemporary philosophical scene, as Smith and Duffy show to qualify the differences between Badiou and Deleuze (Smith 2012c; Duffy 2013: 155-159).This scenario, though, is not restricted to the French contemporary ontologies. Deleuze believes that the philosopher can develop formal instruments through a problematic, not axiomatic, mathematics. This claim particularly applies to an approach to differential calculus that resists set-theoretic formalization. The axiomatic would eventually make the problematic field of the involved variables be determined according to the truthful cases of solution to differentials equations. This operation suffocates the differentiating relation under the axiomatic equivalence circuit. In this case, the differentials lose their generative role. They are defined only as the support of the set theory, mischaracterizing the differential relation along with its problematic feature (Cardoso Jr. 1996a: 201-203). In fact,

It is historically inaccurate to reduce all modern interpretations of the calculus to set theoretical ones. Deleuze is quite aware of this, and the history that he charts in a number of places of his work traces the development of the calculus that continue to elude this reduction. (Duffy 2013: 161)

If it is in general true that the set axiomatic approach may present the damage of eluding the differentiating process of the sensory body, reducing the concept of set to the axiomatic equivalence is still a rough approach, though. In fact, the opposition between problematic and axiomatic presents challenges for further Deleuzian studies. Deleuze and Guattari were sensitive to this possibility, because they consider the logic of the “fuzzy sets”. They warn that when traditional set theory tries to hierarchize these sets, they immediately transform their nature. In ‘fuzzy sets’, connections between its elements are distributed in a completely different manner, in a way that the cumulative hierarchy no longer applies to this set conception. Fuzzy sets forbid that operation, for “The axiomatic manipulates only denumerable sets, even infinite ones, whereas the minorities constitute” fuzzy, “nondenumerable, nonaxiomatizable sets, in short, masses, multiplicities of escape and flux.” (Deleuze & Guattari 2005: 551 n.8) In fact, what fuzzes a set are specific logical operations of consolidation and consistency applied to the membership function in an ordinary set (Deleuze & Guattari 2005: 470). By hypothesis, these “logical operations of consolidation and consistency applied to the membership function” may develop a fuzzy logic to cope with intensive quantities, since the fuzzy set membership function manages recurrent derivatives as to consolidate the membership functions of an ordinary denumerable set. More recent developments in set theory indicate such operations can be developed without taking the risks that Deleuze points to. This is possible once variants of the set as a formal entity are explored to allow more realistic ontological approaches, less abstractive and, therefore, more intuitive and operative (Cardoso Jr. 1996b: 309). For example, in the conception of set that allows the primacy for the property definition that ascribes membership and not the contrary (Nef 2006: 201-202). This same attitude is observed in the paraconsistent set theory (Weber & Cotnoir 2014) and is developed by the

theory of “non-well-founded sets” based on the “anti-foundation axiom” (Aczel 1996: 6-10).

On the one hand, one can understand these appeals to an alternative concept of set as pointing to a formal ontology that is not based on axiomatic in the sense defined by Deleuze. On the other hand, the call upon a new concept of set claim for a new type of axiomatization as Kant’s and Deleuze’s mathematics of sensory body suggest.

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