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Ehrenfels’ dissertation “Über Größenrelationen und Zahlen” is nearly always mentioned in his biographies, but never actually discussed anywhere. The main reason for this is surely that it remains unpublished (the original is preserved at the FDÖPForschungsstelle und Dokumentazionszentrum für Österreichische Philosophie in Graz), and secondarily that the philosophy of mathematics in Ehrenfels and in the School of Brentano in general didn’t garner much interest until recently.

Ehrenfels began his studies in philosophy with Franz Brentano and Alexius Meinong in Vienna during the Winter Semester 1879/80. When Meinong was called to Graz in 1882, Ehrenfels followed him in the winter semester of 1884/1885 and wrote his dissertation on “Relations of Magnitude and Numbers. A psychological Study” under Meinong’s guidance.

Ehrenfels dissertation contains an account of the concept of number that by and large fits in the framweork of the Brentanist philosophy of mathematics, though (unsurprisingly) with Meinongian accents. Ehrenfels asks whether number is something given in sensation, a relation, or neither, and then argues, not unlike early Husserl, that by abstracting from all particular properties of two objects and taking them merely as the foundations of a relation of difference, we obtain the concept of “two-ity” (Zweiheit), which would be the prototype of number. Unity is then obtained by abstracting away one foundation of twoity. Similarly to Husserl in the second chapter of the Philosophy of Arithmetic, Ehrenfels argues that when adding another object, we would need to take into account not only the first-order relations among the foundations, but also the second-order relations of difference among the first-order relations of difference. The rapidly increasing  complication of such higher order relations soon leads beyond our presentational capacities, which prompts the need for indirect ways of presenting numbers. Ehrenfels then points out that bigger numbers can only be obtained by combining presentations of partial numbers (Teilzahlen), e.g. nine through three times three. Of course, again like Husserl, Ehrenfels then claims to have described the essence of the concept of number, not to have defined it, which would be impossible.

In the Stanford Encyclopedia of Philosophy entry on Ehrenfels (forthcoming pending review) I also inserted a brief section on Ehrenfels’ philosophy of mathematics, including a discussion of his 1891 article “Zur Philosophie der Mathematik” (“On the Philosophy of Mathematics”) and his 1922 book “Das Primzahlgesetz” (The Law of Primes).

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