I will begin summarizing my own contribution “Husserl on Mechanical Calculation: Outsourcing Symbolic Intentionality” in our volume Husserl and Awakened Reason by stating the premises point by point:

• 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”.