Philosophy of Arithmetic

Philosophy of Arithmetic (PA; German: Philosophie der Arithmetik. Psychologische und logische untersuchungen) is an 1891 book by 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 is dedicated, and Carl Stumpf.

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).

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, Hobbes and 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. Hence, Husserl says, we have to move on to the analysis of symbolically conceived numbers, which are effectively those that are used in mathematics.