Is the matheme transparent and exhausted by its phenomenality?
Derrida states in L’origine de la géométrie that:
L’objet mathématique semble être l’exemple privilégié et le fil conducteurle plus permanent de la réflexion husserlienne. C’est que l’objet mathématique est idéal. Son être s’épuise et transparaît de part en part dans sa phénoménalité. Absolument objectif, c’est-à-dire totalement délivré de la subjectivité empirique, il n’est pourtant que ce qu’il apparaît. Il est donc toujours déjà réduit à son sens phénoménal et son être est d’entrée de jeu être-objet pour une conscience pure. (O., 6.)
For a discussion as to whether for Husserl the constitution of any object in general has as its mode the mathematical object, I refer you to Walter Biemel, Eugen Fink, and Roman Ingarden as they discuss Biemer’s lecture on “Les phases décisives dans le développement de la philosophie de Husserl.” Be that as it may. For now. Here I will only focus the relation between the constitution of the mathematical object and ideal objectivity in general, such as operated by Husserl. As my ulterior research interest is Derrida’s revolutionary import for science studies and environmental studies alike, I will read Husserl by the lenses of Derrida.
Husserl will contend that the mathematical object—or the matheme as already defined elsewhere—is ideal, ideal through and through, having no share in matter and sensibility, to wit, the so-called ‘real world.’ No share with material bindings and its irreducibly reciprocal meshwork of ever differential implications, it is thoroughly transparent, lifted away from the Real, exhausted by its properly own phenomenality. Linguistics, e.g., partakes of bounded idealities; the matheme is the one non-thing that partakes of free idealities. The matheme is thus freed from any particular subjectivity, empiricity, and is simply what it appears, or phenomenalizes, to be. It is always already reduced to its own phenomenal appearing as such, neither more nor less. If it appears at all, it appears for a certain something, and this certain something can only be a pure and transcendental consciousness. If one wants a Husserlian phenomenology to be coherent, it is necessary that there are such free, purely ideal, entities. Which is why Husserl often stress that the matheme is irreal.
Let me first position the strategical operation of L’Origine (originally published Die Frage nach dem Ursprung der Geometrie als intentional-historisches Problem). It is distinguishable from the Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie, not because of its descriptive novelty since nearly all motifs are already present in other earlier and contemporaneous investigations. L’Origine still addresses the status of the ideal objects of science (of which geometrical objects are but examples), the production of such objects, and the constitution of exactitude. Also addressed is the interrelated conditions of possibility of such ideal objectivity, viz. language, intersubjectivity, temporality, and the world as the unity of ground and horizon. Last, reductions of various sorts are still employed. As Derrida points out, L’origine’s originality is neither distinguishable by way of its double critique of, one the one hand, a certain technicist and objectivist irresponsibility in the practice of science and philosophy, and on the other hand, a certain “historicisme aveuglé par le culte empiriste du fait et la présomption causaliste” (O., 4). This so since the first criticism was already operative in works such as Formale and transzendentale Logik, Cartesianische Meditationen, and Die Krisis der europäischen Wissenschaften und die transzendentale Phänomenologie, whereas the second criticism had already been articulated in works such as Logische Untersuchungen, Ideen 1, and “Philosophie als strenge Wissenschaft.” “Mais jamais les deux dénonciations de l’historicisme et de l’objectivisme n’avaient été si organiquement unies que dans L’Origine de la Géométrie” (O., 4). A new scheme is created: it brings forth a new type of historicity, and it articulates the new tools and original direction of historical reflection. It concerns the historicity of ideal objectivities, which is to say their origin and their tradition, the latter in its ambiguous sense of the term including the movement of transmission as well as the perdurance of the heritage. Therefore:
La naissance et le devenir de la science doivent donc être accessibles à une intution historique d’un style inuoï, où la réactivation intentionnelle du sens devrait précéder et conditionner—en droit—la détermination empirique du fait.
If there is to a birth and a development of science at all, then empirical determinations of fact in general must be preceded and conditioned by a certain intentional reactivation. Any science must, to be constituted as science and beholder of exactitude, if there ever be such, be such constituted that it for any posteriority is perfectly subsumed under an absolute intentional reactivation of its very sense. The empirical world is, by itself, without sense, asensical; only a transcendental subject can bestow sense upon empirical nature, and thus only a transcendental phenomenology can properly ground the sciences. Husserl contends that for empirical facts and events to have sense at all, they must first be commanded by the sense of ideal objectivity, which as such can only be established, constituted, and understood by a pure consciousness. It is only sense as a transcendentally constituted ideal object that at all allow us to bestow meaning and sense upon empirical entities and events. Or such is the hypothesis. And this is what Derrida undertakes to investigate into. What I find disconcerting with Derrida’s text, however, is that the programmatic focus on the very Husserlian conception of scientificity seems to gradually fade and become preoccupied with a predominant focus on philosophy. Which is also one of the reasons I find it acutely pertinent to take up where Derrida left out, to follow up on what he nonetheless had already explicitly announced. If we continue our reading from the passage above we will find:
Dans leur irréductible originalité, l’historicité de la science et la réflexion qu’elle appelle, la Geschichtlichkeit et l’Historie, ont des conditions aprioriques communes. Aux yeux de Husserl, leur dévoilement est principiellement possible et devrait nous amener à reconsiderér dans leur plus large extension les problèmes de l’historicité universelle. Autrement dit, la possibilité de quelque chose comme une histoire de la science impose une relecture et un réveil du “sens” de l’histoire en général: son sens phénoménologique se confondra en dernière instance avec son sens téléologique.
Derrida is after the historicity of science and the reflection that it invites; if, for Husserl, their disclosure is possible in principle, opening up also for a wider reflection on universal historicity in general as well as the “sense” of history in general, they will in Derrida’s reading appear essentially aporetic: in this reading the phenomenological sense of historicity will not simply “merge” with its teleological sense, since the very conceptions of phenomenological and teleological sense will be demonstrated as essentially problematic. And here it is that a systematic investigation into the historicity of scientificity in general fades away, somewhat obscurely. Instead, Derrida undertakes to demonstrate the revolutionary import on philosophy: the deconstruction of the conditions of possibility of the constitution of ideal objects of which the mathematical objects are posited as the most pure and exemplary seems, in Derrida’s reading, to bypass science altogether and goes head-on onto philosophy. Which is somewhat unjustified since Husserl himself is stressing the import of transcendental phenomenology for science no less than philosophy; in fact, Husserl is in this particular work L’Origine more into scientific than philosophic implications. If now Derrida’s deconstructive investigation of Husserl’s transcendental phenomenology irreducibly problematizes the latter, then there is all the more reason to investigate into the implications for science, its conceptions, and scientificity in general. In any case, it seems that Derrida, somewhat against the grain of L’Origine itself, stakes out a course that singles out the bearings on philosophy in general.