Philosophy of Arithmetic

Philosophy of Arithmetic: Psychological and Logical Investigations (Philosophie der Arithmetik. Psychologische und logische Untersuchungen) is an 1891 book about the philosophy of mathematics by the philosopher Edmund Husserl. Husserl's first published book, it is a synthesis of his studies in mathematics, under Karl Weierstrass, with his studies in philosophy and psychology, under Franz Brentano, to whom it is dedicated, and Carl Stumpf.

Structure
The Philosophy of Arithmetic constitutes the first volume of a work which Husserl intended to comprise two volumes, of which the second was never published. Comprehensively it would have encompassed four parts and an Appendix.

The first volume is divided in two parts, in the first of which Husserl purports to analyse the "Proper concepts of multiplicity, unity and amount" (Die eigentliche Begriffe von Vielheit, Einheit und Anzahl) and in the second "The symbolic amount-concepts and the logical sources of amount-arithmetic" (Die symbolischen Anzahlbegrife und die logischen Quellen der Anzahlen-Arithmetik).

Content
The basic issue of the book is a philosophical analysis of the concept of number, which is the most basic concept on which the entire edifice of arithmetic and mathematics can be founded. In order to proceed with this analysis, Husserl, following Brentano and Stumpf, uses the tools of psychology to look for the "origin and content" of the concept of number. He begins with the classical definition, already given by Euclid, Thomas Hobbes and Gottfried Wilhelm Leibniz, that "number is a multiplicity of unities" and then asks himself: what is multiplicity and what is unity? Anything that we can think of, anything we can present, can be considered at its most basic level to be "something". Multiplicity is then the "collective connection" of "something and something and something etc." In order to get a number instead of a mere quantity, we can also think of these featureless, abstract "somethings" as "ones" and then get "one and one and one etc." as basic definition of number in abstracto. However, these are just the proper numbers, i.e. number which we can conceive of properly, without the help of instruments or symbols. Psychologically we are limited to just the very first few numbers if we want to conceive of them properly, with higher numbers our short-term memory is not enough to think of them all together, but still as identical to themselves and different from all others. Husserl contends that as a result, we must proceed to the analysis of symbolically conceived numbers, which are in essence the numbers used in mathematics.

History
The book is a product of Husserl's years of study with Weierstrass (in Berlin) and his student Leo Königsberger (in Vienna) on the mathematical side and his studies with Brentano (in Vienna) and Stumpf (in Halle) on the psychological/philosophical side. The book is mostly based on his habilitationsschrift of 1887 "On the Concept of Number" (Über den Begriff der Zahl). Husserl also lectured on the concept of number between 1889 and 1891, much in the same vein. He continued working on the second volume up to at least 1894.

Gottlob Frege was critical of Philosophy of Arithmetic, and accused Husserl of relying too much on the metaphysical and not enough on the logical aspects of mathematics. Frege's criticisms influenced negatively the young mathematician's career as a professor. Husserl's Logical Investigations secured his reputation ten years later, but Frege and others never accepted Husserl as a practitioner of true logic.

Editions
The original edition:

Philosophie der Arithmetik Psychologische und logische untersuchungen, von Dr. E. G. Husserl. 1. bd. 1891, xvi, 324 p. 23 cm.   LC Classification: QA9 .H8 Dewey Class No.: 510.1 Other System No.: 

Husserliana edition:

Philosophie der Arithmetik Psychologische und logische untersuchungen - mit ergänzenden Texten (1890-1901) Series: Husserliana - Edmund Husserl Gesammelte werke, Vol. XII Husserl, Edmund Ed. Eley, Lothar 1970, 585 p., Hardcover ISBN 90-247-0230-5

Official English translation of the Husserliana edition:

Philosophy of Arithmetic Psychological and Logical Investigations - with Supplementary Texts from 1887-1901 Series: Edmund Husserl Collected Works, Vol. X Husserl, Edmund Tr. Willard, Dallas 2003, 580 p., Hardcover ISBN 1-4020-1546-1