The Dialecticity of Mathematical Concepts and the Problem of the Origin of Geometry: Cavaillès avec Derrida

Comment tenir dans le même champ (ce champ où se meut dans une ubiquité fondamentale le regard philosophique) le sens inaugurale où s’instaure l’déalité, et l’autorité apodictique des systèmes d’énoncés qui l’expriment?

(Desanti 1968, 125)

I

In the long and brilliant introduction to his translation of Husserl’s posthumously published text on the origin of geometry, Jacques Derrida systematically develops the principles, methods, and tools of a historical phenomenology. According to Derrida, the appendix of the Krisis inaugurated a new kind of historical questioning, unprecedented so far in Husserl and, in fact, in the history of philosophy. “The singularity of our text,” he explains, depends on its articulation of “a new scheme [un dessein inédit]: on the one hand, it brings to light a new type or profundity of historicity; on the other hand, and correlatively, it determines the new tools and original direction of historic reflection” (1989, 26; 1962, 4).¹ Derrida understands the novelty of this conception to result from the problem that Husserl addressed in the text, namely, the origin of ideality, and in particular that of geometrical (and mathematical) objective ideality. The terms that are combined in the formulation of the problem—the origin of ideality—lead us to leave behind Platonism (for which there could be no origin, hence construction and constitution, but only revelation, of idealities) and transcendental phenomenology (insofar as we are questioning a historical becoming, not the constitution of an object of conscious experience), as well as, needless to say, historical empiricism and psychologism (because unless ideality is denied, the essence of ideality cannot be reduced to mere facts, whether natural or psychological).

Idealities are not here seized as mere illusions or mistakes and will not die. How did the objective, necessary, and universal truths of atemporal idealities arise in history, that is, in time and contingency? How could geometry, and science in general, and the history of science emerge in the first place? That is, how are we to think the necessarily historical genesis of ostensibly atemporal idealities? Out of the work upon this formidable problem, in the wake of Husserl’s late reflections on the origin of geometry, one of the major philosophies of the second half the twentieth century arises. It is the program of understanding ideality as constituted by historical becoming itself, which certainly would then be “historical” in an unexpected sense (historique en un sens insolite) (1989, 1; 1962, 4). One has to find the means of considering the factuality of becoming and temporality as preceding and conditioning the production of ideality. All sciences, including and above all those that concern mathematical idealities and their application (exact and natural sciences), are produced and constituted in their historical movement. History, therefore, is not the inevitable and implacable human degradation of ideal entities, which are or subsist by themselves apart from their historical decay. On the contrary, such ideal entities, their substantiality and ideality are a production of historical becoming. “The historicity of ideal objectivities, i.e., their origin and tradition…, obeys very unusual rules [règles insolites], which are neither the factual interconnections of empirical history, nor an ideal and ahistoric adding on. The birth and development of science must then be accessible to an unheard-of style of historical intuition in which the intentional reactivation of sense should—de jure—precede and condition the empirical determination of fact” (1989, 26; 1962, 4–5). As is well known by anyone acquainted with Derrida’s subsequent work, language, and specifically “writing” (écriture or archi-écriture), will be the means to reach such unusual rules and such an unheard-of style of historical intuition, since no ideality is thinkable beyond its inscription in the space and time of an originary communication.

And yet it is uncertain that a historical phenomenology, even one rebuilt and radicalized by Derrida, actually clarifies the historical structure of geometrical idealities, let alone solves the problem—at the heart of any phenomenological program—of the foundation of mathematical knowledge...