The Death of Philosophy

by Thomas-Fogiel, Isabelle; Lynch, Richard;

  The Kantian theory of the schema, as a necessary construction of the concept in the intuition, cannot account for [irrational] numbers like √2. This is why Kant explains that if this number can be thought, it cannot be represented (that is, depicted in the intuition). I can think it, because I can consider a square as the product of two factors even if I cannot represent any of the factors to myself as a number. Thus, knowledge or manipulation is possible, but I cannot produce this number as given in the intuition. It follows that Kant admits concepts that are neither pure creations (√2 is not the product of a mathematician’s fantasy, it is necessary), nor illusory (propositions about these numbers are of the order of knowledge, not of appearance), nor empty (√2 has a geometrical analogue in the diagonal of a square). We are thus confronted by a concept that is, from the point of view of Kantian theory, imminently paradoxical—a number that can be understood as necessary, that is capable of being known, but that I cannot represent in the intuition as the unity of a multiplicity. To preserve his theory, Kant brutally solves the problem: all this simply proves that √2 is not a number. Only rational numbers will be considered to be numbers. The square root of 2, because I cannot produce it as the unity of a multiplicity, thus negatively confirms the theory that every number must be produced in the intuition. We would be wrong to be ironic about this solution, which apparently consists in drastically reducing the field of concepts to be explained (by eliminating every number that does not belong to the class of rational numbers) to better preserve a theory of arithmetic. Here Kant does not succumb to the ideological argument in which one answers an opponent who has shown that a theory does not account for reality with the retort, “That proves that reality does not exist.” Indeed, in Kant’s era, it was not yet possible to mathematically understand irrational numbers, and we would have to wait for Richard Dedekind, with his idea of the “Dedekind cut” (Schnitt), before these numbers (that Leibniz himself considered to be “amphibious beings”) could be given a mathematical status. Given that, if Kant could not mathematically understand the status of these numbers, the fact nevertheless remains that philosophically he recognized that there are concepts that cannot be represented but are nonetheless neither empty, nor illusory, nor metaphysical. This is thus a curious hole in the proposition that “a concept without intuition is empty.”

  August Wilhelm Rehberg tried to propose a solution that seemed more in conformity with the letter of the Critique. To the question, “Since the understanding has the power to create numbers at will, why is it incapable of thinking √2 in [rational] numbers?”33 Rehberg answers that this impossibility is explained by “the nature—inaccessible to any human capacity of elucidation—of the transcendental faculty of the imagination and its connection with the understanding.”34 This answer is rather strange, because it consists in explaining a fact with a mystery—if we cannot understand a given mathematical fact, that is because, in the final analysis, we know nothing about how the imagination works. This is an explanation that obviously poses a problem for the critical project: philosophy must investigate the conditions of possibility for scientific judgments (mathematics and physics); these conditions of possibility cannot refer to a mystery, incomprehensible for reflection, without causing the entire project to crumble upon its basis. If the entire Critique of Pure Reason should boil down to explaining the facts from which it starts by invoking in fine “a nature, inaccessible to any human capacity of elucidation,” it would have absolutely no value. The quintessence of the Kantian theory being to subordinate all concepts to methods that make them accessible to human consciousness, it cannot do anything by this method with an inexplicable mystery seated in a faculty of incomprehensible function. Consequently, Rehberg, by wanting to be more faithful than Kant himself to the letter of the critical project, reveals the impossibility of this purely verbal solution that appeals to a faculty of imagination without being able to concretely show how it operates. For all that, we would be deceiving ourselves if we were to explain Rehberg’s answer as a simple weakness on his part. This strange solution is rather the index of a real problem in the critical project. Indeed, no matter what explanation is given, certain mathematical concepts transgress its principles. If Rehberg’s solution makes the transcendental method more or less irrational, Kant’s own solution leads to the following paradoxical admission: there are scientifically valid notions for which we cannot show their method of construction. Consequently, the impossibility of constructing a concept in the intuition is no longer proof of its emptiness; which is obviously significant for the Kantian critique of metaphysics.35 Kant’s recognition of knowable but unrepresented concepts points toward an exit from strictly Kantian principles, an exit that is definitively realized when it comes to understanding the properties of numbers.

  The meaning of the phrase “Time … has no influence on the properties of numbers,”36 as such—that is, not in reference to Kantian principles—is easy to understand: when I do ordinary addition, I add 7 to 5. If I reflect not only on the terms but on the operation itself, I immediately grasp that it is commutative, the result remains the same no matter the order of the factors. In a parallel way, if I perform an operation of subtraction, I notice that this does not possess the same property. The commutative and associative properties are properties of the calculation, of the act of addition or multiplication. Given this simple fact, two explanations are possible. Either these properties are reduced to the sole principle of identity—a Leibnizian solution that Kant can obviously only reject—or else the associative, commutative, etc., properties are taken as axioms—a solution proposed by Johann Schultz, but which Kant also rejected: “It is true that arithmetic has no axioms, since its object is actually not any quantum.”37 Indeed, for Kant, axioms are the result of the application of a quantity to the formal intuition and are thus present only in geometry, like propositions such as “Two points determine one and only one straight line,” or “The shortest distance between two points is a straight line.” It follows that if, for Kant, the properties of arithmetic do not depend upon the exclusive application of the principle of identity, they are not analytic; if they do not presuppose recourse to the intuition of time, then they are not synthetic, in the Kantian sense of the term. Nevertheless, they constitute immediately certain knowledge. In such a situation, Kant’s only option is to innovate with respect to his own taxonomy by determining a totally new class of knowledge, “a pure intellectual synthesis that we represent to ourselves in thoughts.”38 We discover the contradiction in terms that contain this strange—in the Kantian context—expression “intellectual representation.” Intellectual representation is indeed the sign of a difficulty, the visible trace of a problem that jeopardizes the very definition of the concept of validity. This is confirmed if we consider the details with which Kant explains this singular expression. If he denies the existence of arithmetic axioms in this letter to Schultz, the philosopher does speak of “postulates, that is, immediately certain practical judgments.”39 What does this term mean in this context? First of all, it is not about postulates in the Euclidian sense, that is, propositions that one is required to accept without being able to demonstrate them, because in Kant strictly Euclidian postulates refer (according to the standard usage of the time) to axioms. Rather, the term “postulates” in this context marks the difference in status between arithmetical properties and mathematical procedures in the Euclidian sense. But nor does “postulate” refer to its use in the Critique of Practical Reason. It is clear here that the term “practical” has absolutely no moral dimension, only concerns speculative reason, and is nevertheless neither a hypothesis nor a demand.40 The properties of arithmetic thus curiously seem to go beyond the division between theoretical and practical, as the term “intellectual” exceeds the theory of representation. The only meaning that can be given to Kant’s expression, it seems to me, is the following: arithmetical operations are operations that establish a relation; properties are the reflection upon this action of esta
blishing a relation. In other words, the mere doing of the action ipso facto produces its rules. This is precisely what is expressed in the definition of postulates as “immediately certain practical judgments.” “Practical,” because the properties of arithmetic describe an action that must be done;41 “certain,” for to do the action is to find its rules; “immediately,” because this does not presuppose any construction in time or space. In sum, the postulate’s goal is to explain the syntagma of “pure intellectual syntheses”—it refers to an action that has to be accomplished and not to an object constructed in time or space; it is given as a certain synthetic proposition; it goes beyond the divide between theoretical and practical reason (arithmetical propositions are certain for speculative reason not—it will be easily agreed—for practical reason); but on the other hand, as “time … has no influence” on these properties, they do not belong to theoretical propositions as the Critique of Pure Reason defines them. In a word, arithmetical properties are called intellectual because they are certain truths that nevertheless do not belong to the sphere of representation. “Intellectual means a concept whose content is an action”—such is the surprising definition that we find in Kant’s Reflections.42

  Before drawing any conclusions relative to this use of the term “intellectual” in the Kantian system, I should respond to the potential objector who, armed with some of the closing sentences of Kant’s letter to Schultz, doesn’t see—from the beginning of these analyses—any tension between the proposition that “time has no influence on the properties of numbers” and the famously Kantian principle that every proposition, in order to be coherent and valid, must be obtained through the intuition. My response to this objection will allow me to even more clearly bring out the difficulty posed by the use of the term “intellectual.”

  Kant, aware of the importance of the concession he has just made, immediately corrects himself:

  But insofar as specific magnitudes (quanta) are to be determined in accordance with this, they must be given to us in such a way that we can apprehend their intuition successively; and thus this apprehension is subject to the condition of time. So that when all is said and done, we cannot subject any object other than an object of a possible sensible intuition to quantitative, numerical assessment, and it thus remains a principle without exception that mathematics applies only to sensibilia.43

  This means quite simply that if the properties of arithmetic are not acquired by the intuition, in general, however, arithmetic has no other use than to be applied to sensible objects—in a word, arithmetic is designed exclusively for geometry, and still more concretely, physics.44 We should note, first of all, that arithmetic applies to the intuition or the sensible indirectly. Indeed, it is not possible to show how a good number of arithmetical propositions are constructed; only when it is useful for geometry or physics can arithmetic as a whole be called intuitive, simply because these disciplines’ objects are. This explanation attests to a considerable weakening of the initial doctrine. The associative property is discovered in a purely “intellectual” way; there is thus a sensible and exact45 discourse about a calculation, without this discourse being subordinated to the methods of intuitive construction. But as the propositions of arithmetic are used only for geometry or for sensible objects, they are intuitive in a mediated way. And yet to be useful for geometry or sensible objects is not entirely synonymous with being constructable in space and time. Besides, this is expressed in the letter’s dual proposition: “It is true that arithmetic has no axioms, since its object is actually not any quantum . . .” and “But insofar as specific magnitudes (quanta) are to be determined in accordance with this . . .”46 Kant’s final specification makes it possible not so much to reintegrate arithmetic in the theory of representation as to distinguish two levels in arithmetic—the first purely intellectual, the second representational because it indirectly aims at objects that, as it happens, are not stricto sensu intuitions but syntheses of a concept and an intuition (objects of geometry or physics). At the first level, certain propositions are true (the commutative property of addition, etc.), independently of any reference to intuition. This assertion remains, even with his final reservation, fundamentally anti-Kantian. As proof, logical empiricism (notably in its purest expression, namely in Rudolf Carnap) can quite definitely say that arithmetical propositions absolutely do not need to have recourse to space and time in order to be grasped and at the same time show that arithmetical propositions have no application except in the real world. To say that arithmetic is used only for geometry or physics thus does not show that arithmetical propositions all need the intuition to be identified as legitimate. Briefly, despite his ultimate denial, the letter to Schultz shows that Kant makes exceptions to his own conception of mathematics. Thus, there is indeed a tension between his general theory (which demands that all synthetic propositions be connected to a concept or an intuition) and the status of arithmetic.47 Therefore, it is clear that the term “intellectual” is the index of a difficulty that jeopardizes the critical project as a whole. At the same time as it indicates the impossibility of the Kantian notion of validity, this term signals in another way the Kantian system’s failure.

  Thus, more than an indication of a problem, the term “intellectual” appears as the moment when the critical edifice cracks and even implodes. In making use of this concept, Kant jeopardizes his own principles and demands the overcoming of his own doctrine. The terms “representation” and “intellectual” could not be more opposed—the first indicates the necessary relation to an object as a marker of valid knowledge; the second calls for overcoming this relation. This contradictio in adjecto and its interpretation as a tension between two mental orientations will become clearer if we consider the second problematical occurrence of the term, namely, at the moment of the utterance of “I” in the Critique of Pure Reason.

  These difficulties have been remarkably illuminated by Michel Henry—who shows, on the one hand that the “transcendental I” signals less the apogee of subjectivity as its liquidation;48 who remarks, on the other hand, that this moment of the transcendental “I” is the moment when the Kantian edifice self-destructs;49 and who notes, finally, that the system’s self-destruction is distinguished by the use of the term “intellectual.”50 I will take up and extend Henry’s demonstration.

  Use of the Term “Intellectual” with Respect to the “I”

  With respect to “I think,” Kant writes:

  For it must be observed, that when I have called the proposition, ‘I think’, an empirical proposition, I do not mean to say thereby, that the ‘I’ in this proposition is an empirical representation. On the contrary, it is purely intellectual, because belonging to thought in general.51

  This characterization of the “I” as an intellectual representation follows from a purely negative definition: “this ‘I’ is as little an intuition as a concept.” It could certainly be said that this is not a matter of a condition of possibility, but then why speak of an “intellectual representation”? The difficulty only increases if we recall Kant’s characterization of consciousness: “in the transcendental synthesis of the manifold of representations in general, and therefore in the synthetic original unity of apperception, I am conscious of myself, not as I appear to myself, nor as I am in myself, but only that I am.”52 We thus have a strange situation: the positing—the existence—of a nonobject, to whose nature consciousness does not have access while being nevertheless capable of defining it and of making it into a certain proposition clearly not at all a hypothesis or a judgment governed by the “everything happens as if.” But what interests me is that Kant’s embarrassment with respect to the grasping of this “nonobject” that is “I am” manifests precisely through his use of the term “intellectual.” This characterization even enjoys a privilege relative to all the other definitions of the “I” that Kant attempted, for as Michel Henry rightly notes, “Wherever Kant attempts to designate the being of ‘I think,’ the only expression that h
e uses without immediately feeling the need to rectify and replace it with another is ‘intellectual representation.’”53 This term, “intellectual representation,” expresses the impossibility of saying the “I” in Kant’s philosophy, for if he keeps to the method he advocates, he does not in fact have the right to say what he says about the “subject.” Indeed, when Kant determines the transcendental apperception as “I” and attributes spontaneity rather than passivity to it, he uses a type of reasoning that gives birth to a mode of certainty that is not at all dependent upon investigation into conditions of possibility. If we had to summarize the steps of the apagogic reasoning that Kant employs, we would get the following development: the unity of experience is real since mathematics and physics exist as real sciences. To be able to account for this fact, it is necessary to postulate, as a condition of possibility, an ultimate synthetic function that Kant calls the synthetic unity of apperception. In other words, in order to know an object, I must link the representations. This linkage requires the unity of consciousness. Subjectivity is thus nothing other than the objective condition of knowledge; it is a requirement of thought, not a fact upon which one reflects. But if we had to respect this reasoning, we would obtain neither the “I” nor (even less) the “I am.” Nor does anything oblige us to define this synthetic function as the spontaneity of a subject rather than as the functioning of a structure. The regressive reasoning does not even allow us to affirm the proposition that “it is necessary that something is thinking” but simply allows the conclusion that “something is thought.” In his book La notion d’a priori, Mikel Dufrenne has remarkably articulated this characteristic of the Kantian doctrine, “An entire part of the analysis is conducted as if the ‘I think’ … were a cogitatum est.”54 I am not insisting here on this aspect of the Kantian doctrine, which has had a paradoxical legacy—both in logical positivism, as in the early Wittgenstein, and in certain developments in the human sciences, like in Claude Lévi-Strauss—an aspect quite aptly described by Jean Hyppolite’s incisive phrase “a transcendental field without a subject.”55 It suffices for my purposes to note that a good number of the statements in the Critique of Pure Reason would be strictly impossible if one held to a mere apagogic reasoning. The critical project’s most important theses exceed the prescribed method and point to another way of investigating. Thus, if we stay with the simple apagogic method, we are forced to conclude either that something is thought or, to put things better, that “something or ‘it’ (Es)56 thinks.” To say that it is a spontaneous “I” that thinks, more is required than what is contained in this reasoning’s conclusion. This is why Kant, in order to characterize how the “I” is grasped, must transgress his own principles and use the adjective “intellectual,” so uncommon to his pen. The term “intellectual” certainly shows the impossibility of saying the “I” but also expresses, at the same time and in the same way, the necessity of saying it. Indeed, the “I” is the foundation of the system, and this foundation must be able to speak itself (at least as “Me”) if critical philosophy as a whole does not want to fall into inconsistency in revealing itself incapable of accounting for its own propositions. Isn’t this the reason that Kant keeps the term “representation,” a term much more familiar to the critical project than the adjective that disfigures it? Representation and intellectual—never were two terms more contradictory. Every representation is defined as the application of a concept to an intuition; on the other hand, intellectual designates what is “as little intuition as concept.” Every representation is “necessarily mediated”; intellectual always specifies what is immediate. A representation is simultaneously active (concept) and passive (intuition); the adjective “intellectual” refers only to spontaneity. With this expression, Kant seems to say—at the same time and in the same way—something and its contrary. Intellectual representation is indeed the sign of a difficulty, which Michel Henry has masterfully summarized, “Kant takes the metaphysics of representivity to the limit, to that extreme point where claiming ultimately to found itself, to subordinate its own condition to representation, it falls into the abyss and self-destructs.”57 The theory of representation, if it wants to claim to be, must make recourse to an authority that denies it. Thus, not only does mathematics—at the origin of the theory of representation—escape, with some of its propositions, from this law that says that knowledge necessarily is the connection between a concept and an intuition, but also the theory of representation cannot account for certain philosophical statements, like the status of the “I” in the Critique.