Philosophy of Arithmetic

Part of the Synthese Library book series (SYLI, volume 345)


The Philosophy of Arithmetic,1 Husserl’s youthful work dedicated to a philosophical, or better, epistemological foundation of mathematics, shows the shift in his interests from more properly mathematical issues to those regarding the philosophy of mathematics. Husserl strives to understand and clarify what numbers and numerical relations are, a problem that he recasts in terms of the subjective origin 2 of the fundamental concepts of set theory and finite cardinal arithmetic. We will try to show that on the whole this work of Husserl’s does not deserve the criticism and ensuing neglect that it suffered from, ever since Frege published his well-known Review.3 Besides its hotly contested psychologism, we find ideas and conceptualizations that not only were original then, but are still interesting today, such as those concerning the autonomy of the formal-algorithmic aspect of abstract algebra and mathematics. Moreover, it is here that the Husserlian idea of a universal arithmetic receives its first formulation, the full elaboration of which will take at least ten more years, until his research on these topics reaches its stable form in 1901.4


Natural Number Arithmetical Operation Cardinal Number Numerical System Numerical Field 
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  1. Cantor G (1883) Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Teubner, LeipzigGoogle Scholar
  2. Cantor G, Mitteilungen zur Lehre vom Transfiniten, Zeitschrift für Philosophie und philosophische Kritik, 91 (1887), 81–125; 92 (1888), 240–265Google Scholar
  3. Dedekind R (1888) Was sind und was sollen die Zahlen. Vieweg, Braunschweig 1888 (reprint: Vieweg & Sohn, Braunschweig 1969). English translation: Dedekind R (1963) Essays on the theory of numbers (ed & trans: Beman WW), Dover, New York 1963Google Scholar
  4. Frege G (1885) Ueber formale Theorien der Arithmetik. In: Frege 1990, 103–111Google Scholar
  5. Schröder E (1898) Über zwei Definitionen der Endlichkeit und G. Cantorsche Sätze. Nova Acta Academiae Caesareae Leopoldino-Carolinae Germanicae Naturae Curiosum 71Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Universität Hamburg Philosophisches SeminarHamburgGermany

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