Husserl and Mathematics by Mirja Hartimo (review)

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• Husserl and Mathematics by Mirja Hartimo
• Andrea Staiti

Mirja Hartimo. Husserl and Mathematics. Cambridge: Cambridge University Press, 2021. Pp. 214. Hardback, $99.99.

Mirja Hartimo has written the first book-length study of Husserl's evolving views on mathematics that takes his intellectual context into full consideration. Most importantly, Hartimo's historically informed approach to the topic benefits from her extensive knowledge of Husserl's library. Throughout the book, she provides references to texts and articles that Husserl read, marked, and annotated, although regrettably, very few of these sources are quoted explicitly in his published works. The overall picture that emerges from Hartimo's book shows that Husserl continued to read and engage with mathematics throughout his career, and, even though extensive discussions of mathematical issues are not frequent in his later works, some of the ideas he develops, for instance, in The Crisis of the European Sciences and Transcendental Phenomenology, can be read against the backdrop of the developments of mathematics in his time.

Chapter 1 sets the method for the rest of the book. Hartimo emphasizes the importance of a methodological device that takes center stage in Husserl's work from the 1920s onward: Besinnung. Hartimo claims that Besinnung is a method to reactivate the "goals and purposes" (21) that originally motivated scientists engaging in a particular line of research. According to Husserl, as specialized research progresses and becomes standard practice, such original goals and purposes might be blurred by other concerns. When applied to mathematics, Besinnung thus amounts to a reflection "on what mathematicians should do, on what the genuine goal of their activities should be" (21–22). Hartimo argues that this kind of approach to mathematics comes close to Penelope Maddy's naturalism, which takes its departure from what mathematicians are actually doing and then proceeds to inquire into the ontological status of mathematical objects and theories. Hartimo identifies particularly in David Hilbert, a colleague of Husserl's at Göttingen, a vivid embodiment of what Husserl considered to be exemplary work oriented toward the ideal goal of mathematics.

In chapters 2 and 3, Hartimo expands upon Husserl's conception of what the true goal of mathematics should be. She does so, first, by addressing the classical topic of Husserl's psychologism in his first (and only) work entirely devoted to mathematics, The Philosophy of Arithmetic, and Frege's well-known critical review of it. In Hartimo's rendition, Husserl criticizes Frege for his attempt to provide formal definitions of basic notions, such as equality. For Husserl, such basic notions are intuitively available and require no definition—rather, they should be put to work to clarify vague notions. Frege, in return, attacks what he takes to be Husserl's wishy-washy psychological account of numbers, which revolves around the concrete act of collecting items and abstracting from their specific contents. Ultimately, Hartimo argues, Husserl takes Frege's criticism to heart and in his later Prolegomena no longer pursues a psychological foundation of number, but rather emphasizes the distinction between the subjective and the objective sides of mathematics, with philosophers studying the former and mathematicians focusing on the latter. While Hartimo's verdict on the philosophical yield of the debate is in line with the standard interpretation of Frege's review as prompting Husserl's antipsychologistic turn, she makes a remark that might be worthy of further discussion: "While for Husserl the debate was about logicism, Frege shifts the debate to be about psychologism and here commentators have followed Frege's example" (50). It would be interesting to imagine what a Fregean reply to Husserl's view about the uselessness of formal definitions for basic notions like equality might look like. In any case, Hartimo's reconstruction makes it clear that, however motivated, Husserl's rejection of psychologism does not amount to an acceptance of logicism.

In chapter 3, Hartimo puts forward two of her central claims: (1) Husserl was a structuralist about mathematics; and (2) for Husserl, the goal of modern mathematics is encapsulated in the concept of definiteness. According to (1), mathematics is about the formal structures of systems, which can be isolated and compared. Definiteness, in turn, is the property of systems...