Carlo Ierna

The following pages contain a partial edition of Husserl’s manuscript A I 35, pages 1a-28b. The first few pages are dated on May 1927 and are included mostly for completeness’ sake. The bulk of the manuscript convolute, however, is from 1912. Four pages of the convolute, 31a-34b, have been published as Beilage XII (210, 2–216, 2) in Hua XXXII. The manuscript was excluded from the text selection of Husserliana XXI3 based on its much later date of composition. A I 35/24a is mentioned in Husserliana XXII (p. xxi, n. 4) as confirmation for Zermelo’s 1902 “oral report” to Husserl of his own independent discovery of the paradox. The text presented here for the first time has already been the target of at least three extensive commentaries, while still unpublished, by Claire Ortiz Hill and Guillermo Rosado Haddock. These present a good survey in english of the central issues on the text and contain many translated quotations.

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories of space. Husserl’s Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development.

In the article Husserl’s view of the infinite around 1890 is analysed. I give a survey of his mathematical background and other important influences (especially Bolzano). The article contains a short exposition on Husserl's distinction between proper and symbolic presentations in the "Philosophie der Arithmetik" and between finite and infinite symbolic collections. Subsequently Husserl’s conception of surrogate presentations in his treatise "Zur Logik der Zeichen (Semiotik)" is discussed. In this text Husserl gives a detailed account of infinity, using surrogate presentations. The conclusion is that with surrogates we can only operate according to blind psychological rules.

Among the correspondence between Husserl and Brentano kept at the Houghton Library of Harvard University there is a letter from Husserl to Brentano from 29 XII 1889, whose contents were completely unknown until now. The letter is of some significance, both historically as well as systematically for Husserl’s early development, painting a vivid picture of his relation and indebtedness to his teacher Franz Brentano. As in his letter to Stumpf from February 1890, Husserl describes the issues he had encountered during the elaboration of his habilitation work into the Philosophy of Arithmetic, but also announces that he has finally found "clarity" regarding the arithmetica universalis

On July 14, 1866 Franz Brentano stepped up to the pulpit to defend his thesis that “the true method of philosophy is none other than that of the natural sciences”. This thesis bound his first students to him and became the north star of his school, against the complex background of the progress and specialization of the natural sciences as well as the growth and professionalization of universities. I will discuss the project of the renewal of philosophy as science in the School of Brentano and how this aimed to provide a scientific foundation for the humanities independently from the natural sciences, while preserving the unity of science. Through his well-known re-introduction of the concept of intentionality as criterion to distinguish internal and external perception, Brentano was able to supply an empirical foundation for the Geisteswissenschaften. While philosophy would use the method of natural science, its domain would not be nature, but consciousness: a full-blooded science of the mind that did not require a reduction to the physical in order to be scientific. Brentano’s science of consciousness was empirical, but not experimental, and relied on subjective methods, but was not introspective. Brentano’s students Carl Stumpf, Anton Marty, Alexius Meinong, Christian von Ehrenfels, Edmund Husserl and others came to occupy important chairs in philosophy throughout Europe. While they were certainly not all orthodox followers, they adapted and spread his theories far and wide in the schools and movements they founded and influenced: Gestalt psychology, Prague linguistics, phenomenology, etc.. Moreover, the 19th century idea of scientific research as a collaborative and collective achievement led to a division of labor in Brentano’s school. Each of his students was meant to work out a part of the greater whole: Stumpf, the philosophy of sound and music; Marty, language; Meinong the history of philosophy; Husserl, mathematics; etc. Yet all of them also contributed to the shared project of the renewal of philosophy as science and discussed the (foundational) relation of philosophy to other sciences in programmatic works. Though often forgotten and overlooked due to contingent historical circumstances, the scientific paradigm of the School of Brentano was very fruitful and highly influential in philosophy and the human sciences in general, throughout the second half of the 19th and into the 20th centuries. Yet it is relevant then as now to preserve the independent scientific dignity of the humanities.

This present collection of (translations of) reviews is intended to help obtain a more balanced picture of the reception and impact of Edmund Husserl’s first book, the 1891 Philosophy of Arithmetic. One of the insights to be gained from this non-exhaustive collection of reviews is that the Philosophy of Arithmetic had a much more widespread reception than hitherto assumed: in the present collection alone there already are fourteen, all published between 1891 and 1895. Three of the reviews appeared in mathematical journals (Jahrbuch über die Fortschritte der Mathematik, Zeitschrift für Mathematik und Physik, and Zeitschrift für mathema- tischen und naturwissenschaftlichen Unterricht), three were published in English journals (The Philosophical Review, The Monist, Mind), two were written by other members of the School of Brentano (Franz Hillebrand and Alois Höfler). Some of the reviews and notices appear to be very superficial, consisting merely of para- phrases (often without references) and lists of topics taken from the table of con- tents, presenting barely acceptable summaries. Others, among which Höfler might be the most significant, engage much more deeply with the topics and problems that Husserl addresses, analyzing his approach in the context of the mathematics of his time and the School of Brentano.

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories of space. Husserl’s Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development.

In this article I will analyse Husserl’s conception of the infinite as ex- pressed in the paragraph Unendliche Mengen of his Philosophie der Arithmetik (PA). I will give a short exposition on his distinction be- tween proper and symbolic presentations and then proceed to the logi- cal distinctions that Husserl makes between finite and infinite symbolic collections. Subsequently (in section 2.3), I will discuss Husserl’s addition of surrogate presentations as a sub-type of symbolic presentations in his short treatise Zur Logik der Zeichen (Semiotik). In this later text (which was originally intended as an appendix to the never published second volume of the PA) Husserl gives a more detailed account of how we can conceive of the infinite, using surrogate presentations.

Franz Brentano is not usually associated with mathematics. Generally, only Brentano’s discussion of the continuum and his critique of the mathematical accounts of it is treated in the literature. It is this detailed critique which suggests that Brentano had more than a superficial familiarity with mathematics. Indeed, considering the authors and works quoted in his lectures, Brentano appears well-informed and quite interested in the mathematical research of his time. I specifically address his lectures here as there is much less to be found about mathematics in his published works. Besides Brentano’s own work, it is quite remarkable to see that practically all of his better-known students sooner or later produced a work on the philosophy of mathematics. This also encourages the supposition not just of a common interest in the matter, but of a common theoretical core. All this prompts the question: Can we speak of a Brentanist philosophy of mathematics?

Among the various lecture courses that Edmund Husserl held during his time as a Privatdozent at the University of Halle (1887-1901), there was one on "Ausgewählte Fragen aus der Philosophie der Mathematik" (Selected Questions from the Philosophy of Mathematics), which he gave twice, once in the WS 1889/90 and again in WS 1890/91. As Husserl reports in his letter to Carl Stumpf of February 1890, he lectured mainly on “spatial-logical questions” and gave an extensive critique of the Riemann-Helmholtz theories. Indeed, in K I 28 many lectures on this subject can be found, which are for the greater part published in Husserliana XXI. The lecture contained in K I 28/4-12 at the Husserl-Archives Leuven, however, was left out of the selection, because the lecture contained “an analysis of the concept of number” whose content is “already known” from the Philosophie der Arithmetik. Indeed, since the lecture is from the WS 1889/90, the manuscript allows a glimpse of Husserl’s ideas halfway between his Habilitationsschrift (1887) and the Philosophie der Arithmetik (1891). Among other things, in this lecture we find the first documented use of the terms "Gestalt" and "Gestaltmoment" in Husserl.

Among the various lecture courses that Edmund Husserl held during his time as a Privatdozent at the University of Halle (1887-1901), there was one on Ausgewählte Fragen aus der Philosophie der Mathematik (Selected Questions from the Philosophy of Mathematics), which he gave twice, once in the WS 1889/90 and again in WS 1890/91. As Husserl reports in his letter to Carl Stumpf of February 1890, he lectured mainly on “spatial-logical questions” and gave an extensive critique of the Riemann-Helmholtz theories. Indeed, in K I 28 many lectures on this subject can be found, which are for the greater part published in Husserliana XXI. The lecture contained in K I 28/4-12 at the Husserl-Archives Leuven, however, was left out of the selection, because the lecture contained “an analysis of the concept of number” whose content is “already known” from the Philosophie der Arithmetik. Indeed, since the lecture is from the WS 1889/90, the manuscript allows a glimpse of Husserl’s ideas halfway between his Habilitationsschrift (1887) and the Philosophie der Arithmetik (1891). From an exam- ple Husserl gives in the lecture: “Der wievielte Januar ist heute?” (The how-many-th of January is it today?), it can be estimated to be from January 1890, placing it in close relation to the letter to Stumpf and the views and doubts expressed therein. The bundle of papers K I 28 is wrapped in a blue cover, bearing only “Vor- lesungen” (Lectures) as a title. The lecture “On the concept of Number” in K I 28/4-12 is contained again in a separate cover, without any title. As in the case of most of his lectures, Husserl folded the paper in half, using the right half as a margin for annotations. Before the lecture there is a single page (K I 28/3) with some annotations on the treatment of imaginary numbers as assumptions in calculation. This text is published here as an Appendix to the lecture, because, while it is not part of the lecture itself, it is closely related to the subjects discussed therein.