Standard interpretations of Frege's logicism take at face value a
drastically oversimplified picture of nineteenth century mathematics.
Against this background, Frege can easily seem to be outside the
mathematical mainstream, and many commentators have recently concluded
exactly this. This paper digs into the historical background to show that
Frege (and nineteenth century foundations more generally) was more
profoundly engaged with ongoing mathematics than has been realized. Among
other things that are relevant to assessing the mathematical thrust of
Frege's work are a contrast between the Riemann - inspired, Lie and Klein
developed "conceptual" style of Göttingen and the arithmetical style of
Weierstrass and Berlin, differences between Riemann and Weierstrass on
definitional practices, and the early applications of (what is now called
in English) the theory of cyclotomic extensions to number theory. This
historical background is not just interesting in its own right, but it
also prompts a revised assessment of what Frege was trying to do in
Grundlagen.
The historical study suggests a broader (meta-) philosophical
moral. Because of Frege's perceived position as an early pillar of what
has come to be called the analytic tradition, the view of Frege's
philosophy as disengaged with mathematical practice both draws
plausibility from, and lends support to a widespread "quietistic" picture
of the relations between foundational philosophy on one hand and
mathematics and the natural sciences on the other. I'll suggest that it is
misguided to hold that philosophy should "leave everything as it was."