On the Origin of the “Language” of Formal Mathematics

Husserl’s early, pre-phenomenological researches into the “logic” of symbolic mathematics and the symbolic calculus generally remain, in an important aspect, definitive for his later, explicitly phenomenological researches into the experiential basis of both formal mathematics (formal ontology) and the mathematization of the life world. The aspect in question concerns his characterization of the non-conceptual nature of the symbolic algorithms employed by “formal” mathematics. This view of the matter initially emerged in the wake of his recognition of the need to abandon both Weierstrass’s thesis regarding the foundational role proper to the concept of cardinal number (Anzahl) for universal analysis (arithmetica universalis) and, connected with this, Brentano’s thesis regarding the logical equivalence of the conceptual contents proper to authentic (eigentlich) and symbolic presentations. As a consequence of this, Husserl came to understand the symbolic algorithms operative in formal mathematics as a calculational technique, composed of the signs and “rules of the game” that surrogate for genuine deductive thinking. As he puts it in his review of Ernst Schröder’s Vorlesungen über die Algebra der Logik, “calculation is no deduction. Rather, it is an external (äuβerliches) surrogate for deduction.”