# Stefania Centrone: Logic and Philosophy of Mathematics in the Early Husserl

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DOI: 10.1007/s10743-012-9120-7

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- Ierna, C. Husserl Stud (2013) 29: 251. doi:10.1007/s10743-012-9120-7

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Stefania Centrone’s book provides an in-depth analysis and interpretation of Edmund Husserl’s early work, from the publication of the *Philosophy of Arithmetic* (1891) to that of the *Logical Investigations* (1900/1901). The book consists of three long chapters, respectively discussing Husserl’s first book, the *Philosophy of Arithmetic*, the idea of pure logic in the *Prolegomena*, and the problem of the imaginary in mathematics, which was the topic of Husserl’s 1901 “Double Lecture” for the mathematical society in Göttingen. Each chapter is followed by a number of article-length appendixes that focus in great detail on a specific issue or text, such as the definition of computable functions and Husserl’s 1896 lectures on logic.

As a result the book has a strong systematic-interpretative character, rather than an historical-philological one. Moreover, Centrone avoids reading later themes and terminology from Husserl’s transcendental phenomenology into his earlier works. Instead of seeing them merely as confused sketches and unripe anticipations of his later phenomenology, Centrone’s book takes Husserl’s early contribution to the philosophy of the formal sciences to be a serious endeavor in its own right. While the subject matter calls at times for a technical mathematical analysis, Centrone is generally able to convey Husserl’s theories in a readable and understandable way, without becoming trapped too much in historically conditioned jargon.the articulation of formal logic in logical levels according to a structure that is very close to what, today, is effectively used in standard logical textbooks, the unification of formal logic and mathematics in a most general mathematico-formal science that purports to be the concrete realization of the Leibnizian ideal of a

mathesis universalis, and the explicit conception of abstract mathematics as a theory of structures (p. ix).

*Logical Investigations*is almost entirely bypassed:

In doing so, Centrone stresses that the intrinsic worth of Husserl’s early theories for the philosophy of logic and mathematics is independent of the ultimate answer to the question of Husserl’s alleged psychologism in theDagfinn Føllesdal’s conjectured in 1958 that Frege was an important factor in Husserl’s conversion from the psychologism of this book to the anti-psychologism of the Prolegomena. This claim has been contested by Mohanty and others, but Føllesdal’s defense is very convincing. However, we will approach Husserl’s first book from a perspective that is orthogonal to the psychologism issue (p. xii).

*Philosophy of Arithmetic*and his complex relation to Frege. On the other hand, only by building upon serious theoretical analyses of Husserl’s early works, like the ones Centrone offers in her book, can we hope to work towards a conclusion on this long-standing debate. Instead of writing the

*n*+ 1th episode of the saga of Frege versus Husserl, then, Centrone concentrates on the substantive issues in mathematics and logic that Husserl and Frege debated: the nature of logic, defining a concept by defining its extension, the nature of cardinal numbers, the use of abstraction, and the relation between equinumerosity and bijection.

In a similar respect, while a full-blown account would have certainly gone far beyond the scope of the book, some perspective on the relation between Husserl’s early philosophy of mathematics and logic and his later transcendental idealism would have been very useful. As it stands, “phenomenology” is mentioned so few times (practically only in the introduction and only to declare it outside the scope of the book) that it didn’t even make it into the index. Nevertheless, Centrone’s book shows very clearly that the worth of Husserl’s early theories far exceeds that of mere anticipations of his later transcendental phenomenology. As Peter Simons observes in his foreword to the book: “Stefania Centrone’s thorough and painstaking exposition of Husserl’s early work is a timely reminder that he was a philosopher of insight and stature well before he burst onto the general philosophical scene” (p. v). Indeed, by providing a precise mathematical reconstruction of Husserl’s informal arguments and claims, Centrone can show *inter alia* that it is possible to see Husserl as a pioneer in his research on the computability of arithmetical functions.

Instead of simply revisiting all the old commonplaces of Husserl scholarship, Centrone’s book presents original interpretations and analyses and focuses in detail on a select few sources and inspirations of Husserl’s early position. Hence, we find extensive discussions of Husserl’s relation to Bolzano, while there is hardly any mention of Husserl’s immediate teachers such as Franz Brentano and Carl Stumpf, or even Karl Weierstrass and Leopold Kronecker. It is precisely thanks to this selectivity and focus that Centrone manages to give a very precise analysis of some of the core concepts that drive Husserl’s research in these years: the issues of the extension of the domain of numbers and of imaginary numbers (“without doubt, the guiding thread in Husserl’s reflections on the role of the formal attitude in mathematics”, p. xvi), the notions of manifold and theory of manifolds, and the concept of definiteness or completeness of a theory. Hence, in this book we meet not the pupil of Brentano and the mentor of Heidegger, but the colleague and equal of Cantor, Hilbert, and Frege. Centrone’s book is a big step forward in according Husserl a significant place as a mathematician and logician in the context of the debates and developments of the formal sciences in his time.