In “Husserl on Mechanical Calculation” I show that algorithms were meant to be a major topic in the second volume of Husserl’s Philosophy of Arithmetic, which was never published. Nevertheless, Husserl developed his ideas and did write material on this crucial topic, which likely also influenced the Prolegomena. Specifically regarding mechanical symbol manipulation we see a significant continuity from his early to his late works, even beyond the Logical Investigations up to the Crisis. So: what does Husserl actually mean with “algorithm” in this context?

an arithmetical algorithm, i.e. a system of formal rules by means of which we can solve problems concerning numbers by purely mechanical operations, i.e. from known numbers and number relations it allows us to find unknown ones. (Hua XII, 132; Hua CW V, 139)

One can, namely, conceive of calculation as any rule-governed method of derivation of signs from signs within any algorithmic sign-system according to the “laws”- or better: the conventions – for combination, separation, and transformation peculiar to that system. (Hua XII, 258; Hua CW X, 273)

Mathematical calculation solves problems through “conversion of the initial thoughts into signs – calculation – and conversion of the resulting signs back into thoughts” (Hua XII, 258; Hua CW X, 273) Husserl immediately points out that this is a universal method for any calculation or computation with signs in a rule-based system. The specific concept of number (e.g. cardinal or ordinal) does not matter at all, because they are alle governed by the same algorithm, in fact we could even develop the algorithm independently from any specific conceptual domain: the algorithm does not depend on the domain, but rather the domain on the algorithm (Hua Mat I, 314 f.). Husserl sticks to this position also in his Logic lectures from 1895 and 1896: “Calculating is an operation with signs and not with the concepts themselves, and the result of the calculation is again something purely signitive, a certain combination of signs on paper.” (Hua Mat I, 247) Calculation is not conceptual, but mechanical: algorithmic symbol manipulation. In the next section of my chapter I then go on to discuss Mechanical Symbol Manipulation specifically.