Husserl is preoccupied with the epistemological justification of (mental as well as mechanical) symbol manipulation ever since his very first publications. Already in his review of Schröder he states:

But is calculation deduction? Not at all. Calculation is a blind procedure with symbols, according to mechanically reiterated rules for the transformation and transposition of the signs in the respective algorithm. […] The whole procedure spares us and surrogates for [erspart und ersetzt] a manifold of pure deductions, but it is itself not one. (Hua CW V, 55 f.)

Husserl here distinguishes between the technique of logical and mathematical computation (Logikkalkül) from the justification of such computations (Logik des Kalküls). The latter ultimately requires a fully authentic, proper, conceptual analysis and reflection. The symbolic is founded on the authentic and proper (eigentlich): at the beginning and end signs and concepts must be translatable into one another. Since “machines don’t think for themselves, in the machines no thoughts correspond to the signs.” (Hua Mat I, 247), it falls to us to supply the “thoughtful” part to the “thoughtless” algorithm. This requires a self-reflective awareness of the history of the development of our cognitive tools and the sedimented meanings they embody: awakened reason.

In our own mind, we can do both the uninsightful symbol manipulation as well as the explicating reflection on its result. We can therefore justify our own algorithmic operations on signs and then justify the outsourcing of symbol manipulation to a machine through its parallelism to our own (mental) operations:

To every correct derivation [regelrechten Herleitung] thus corresponds a result that, when interpreted conceptually, yields a correct proposition. This is due to the precise parallelism between mental operations and operations on signs. (Hua Mat I, 247 f.)

When we use the same algorithm either insightfully or uninsightfully, and in the latter case, either in our minds or in machines, the parallelism ensures a correct result everywhere. This is Husserl’s position at least until 1902/03 (see e.g. Hua Mat II, 232). Ultimately, every complex calculation can be reduced to an equation that can be checked by proper conceptual analysis. Once we have grounded the concepts and operations in a domain that is still properly conceivable, we can use the symbol system to reach beyond it. It is in this specific sense that for Husserl and the Brentanists all higher mathematics is founded on the concept of number and can be grounded in the small but inevitable domain of properly conceivable numbers (Husserl 2005, 301. Compare Hua XVIII, § 54). Nevertheless, besides the epistemological justification of the use of symbols, algorithms, and machines, there still remains the question of “Ethically Justifying Mechanical Calculation”, which I discuss in the next section.