Science, according to its own conception, is as infinite as the world and can only be realised in the finitude of a scientific community in stages and by using symbolic aids [Hilfsmittel der Symbolik], by sharing the lived insights [lebendigen Einsicht] among a communicative scientific community. (Hua XLII, 273, n. 1)

Symbolic aids do not only enhance our capacity for (scientific) knowledge, but also our capacity for (ethical) action. If we have the obligation to choose the greater good, then we have an obligation to use our symbolic aids. Even if we do not have the required “autonomous insight” as individuals, we have it collectively, as a community. In the scientific community, “autonomous insight” cannot mean that everyone needs to know everything. Division of labor is unavoidable and is based on trust in the respective specialization (Hua XLII, 274, n. 1. Also see Hua XVIII, § 71). This division of labor is enabled again by the use of symbolic aids for obtaining, storing, and communicating knowledge. However, Husserl warns us in the Crisis about this progressive technicization of science:

Thinking becomes a thinking with surrogates, that has its own evidence in the context of this technique, its own goals, its own tasks and solutions, which diverge from the evidence of goals, tasks, solutions, that were originally indicated by such words and symbols. (Hua XXIX, 35)

Husserl’s terminology from the 1890s for symbolic intentionality in the context of mathematics is now used in the same way for science at large. When we convert concepts into signs, manipulate the signs according to an algorithm, and then convert signs into concepts, we risk forgetting our original aims and values for which the whole symbolic technology was invented in the first place (see i.a. Hua XLII, 430, 436). There is a risk of loss of meaning in the “thoughtless” intermediate phase without awakened reason as a historical-critical reflection (see e.g. Hua XXIX, 190).

If science and rationality are to guide humanity and help it flourish with the tools it develops, the trust and division of labor in science and society need to be well-founded on historical-critical thinking that can provide the ultimate proper, insightful justification. This requires a thorough analysis of symbolic intentionality and symbolic technologies: how humans and machines use signs. Husserl’s theory of symbolic computation can still help us today: in order to understand the world and act ethically in it, we need to understand the tools to which we have outsourced our thinking and the theories on which this very possibility is based.

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.

In his works throughout the 1880s and 1890s, in the Prolegomena, and up to the Crisis, Husserl explicitly connects the use of mechanical tools for thought and our scientific and technological achievements. Likewise the issue of founding, clarifying, and justifying the use of symbol systems (whether mental or mechanical), remains a central topic from the beginnings of Husserl’s philosophy through to its very end. Husserl starts by adopting and adapting the Brentanist philosophy of mathematics: mathematics and symbolic methods are indispensable for science. Symbolic intentionality and symbolic technologies pervade nearly all aspects of our mental life and scientific practice. This seems not only possible and useful, but even necessary. Besides Husserl also others in the School of Brentano, like Ehrenfels, were concerned with the consequences of outsourcing mechanical symbol manipulation: if we can do complex mathematics only thanks to our tools, what does this mean for the status of mathematics as analytic, deductive, and a priori? What changes if we have to use memory aides (“Hülfsmittel für das Gedächtniss”, Ehrenfels 1891, 326) or outsource our cognitive labor to machines, acknowledging their results as correct, while we never carried out any deductions ourselves (Ehrenfels 1891, 329)? How can we rely on the (mental) labor done by (or made superfluous by) mechanical devices? What would justify this? The question of “Justifying Mechanical Calculation” is then addressed in the next section of my article.

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.

• there was a shared philosophy of mathematics in the School of Brentano
• mathematics is analytic, deductive, and a priori
• mathematics is the foundational science and the model science
• mathematics is the exemplar of symbolic intentionality

I have discussed the Brentanist philosophy of mathematics in more detail elsewhere (Ierna 2011, 2017, 2021, 2022), here I focus on symbolic intentionality and symbolic technologies. Symbolic intentionality occurs when our mental acts use signs to present their content or object indirectly, inauthentically, improperly. We can only think some objects (god, other minds, large numbers) by proxy: pointing at them without being able to fully grasp them. This view was transmitted from Brentano through Stumpf to Husserl. In his early writings, Husserl presents a detailed elaboration of the Brentanist philosophy of mathematics. But he was not the only one: all of Brentano’s students wrote about the philosophy of mathematics in these terms, and generally used mathematics as the prime example to introduce symbolic presentations and symbolic intentionality. For Husserl mathematics was hence a game of signs: translation of concepts into signs, (physical) symbol manipulation, translation of signs into concepts. In fact, notwithstanding (sub-)titles such as “the concept of number” or “psychological analyses” Husserl very strongly argued against the view that mathematicians would actually work with the authentic, proper concepts of number. What is of paramount importance for mathematics, considered as a “general theory of operations” (Hua XII, 283; Hua CW X, 299), is the algorithm with which we manipulate mathematical signs. I further elaborate what this entails in the second section: “Husserl on Algorithmic Symbol Manipulation“.