This article has two parts. In the first part, I trace the development of Gottlob Frege's views on the foundations of arithmetic, from the conception of logicism to the attempt to lay geometrical foundations for arithmetic. I trace Frege's path from 1902, when he learned of Russell's paradox in the system of The Basic Laws of Arithmetic, to the late manuscripts "Sources of Knowledge of Mathematics and Mathematical Sciences", "Numbers and Arithmetic" and "A New Attempt at the Foundations of Arithmetic", written in 1924/25, but not published until 1969. Between 1910 and 1924/25, Frege abandoned important theses of logicism.

In the second part of the article, I discuss the philosophical sources of mathematics. There were fundamental philosophical influences on Frege's new position. The views on the philosophy of mathematics, especially on the concept of geometric intuition, of the neo-Kantian philosophers Bruno Bauch and Richard Hönigswald provided the intellectual background to Frege's ideas. Bauch, who was a good friend of Hönigswald, was Frege's colleague at Jena University between 1911 and 1918. He had very good relations with Frege during this period.

I want to examine Bauch's and Hönigswald's influences on Frege. Therefore I analyse the papers of these philosophers and present the philosophical arguments in Frege's late essays. These arguments were included in the neo-Kantian context of discussions about the philosophy of mathematics. This fact is an important cause for Frege's having disregarded the contemporary discussion of the paradoxes of set theory.

This article scrutinizes some of T. Parsons' arguments in favor of assuming Meinongian fictional objects. Parsons justifies his Meinongian approach by showing with respect to carefully chosen example sentences that the special paraphrases commonly used in order to circumvent the assumption of non-existent objects are not always adequate. In particular, he explores a Fregean approach that amounts to conceiving of fictional objects as intensional objects and rejects it. The present author reanalyzes Parsons' examples on the basis of Frege's original distinctions. He shows that an adequate account of fictional objects requires neither their interpretation as intensional objects nor the assumption of non-existing objects.

The apparent absence, in reference or allusion, of Peirce to Frege and of Frege to Peirce, to work of the one in work of the other, and vice versa, would suggest — probably or even plausibly that neither Peirce nor Frege were aware of the other's work. It nevertheless seems almost inconceivable that Peirce and Frege could have been unaware of the the other's work, with circumstantial evidence available that would favour some likelihood of such awareness. This paper collates and speculates about the circumstantial evidence that is available.