DELEUZE ON KANT’S MATHEMATICS OF THE SENSORY BODY: THE DIFFERENTIAL RELATION IN THE INSTANCE OF LEARNING
HélioRebello
São Paulo State University, Brazil
ABSTRACT
This 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 ofset 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, thathe 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, expectfrom 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 goesfrom axioms to derivative theorems, while inthe problematic thededuction 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 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 problematicsare not the same that interests to the axiomatic model.In short, problematizing constitutes procedures whose accidents and events do not allow beingneither 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 anaxiomatic 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 thatare 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:
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, becausethey 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 (intuitusvelconceptus).” (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 (realitasphaenomenon), 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 aspecific waythe 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])