Hi urpflanzen:

Coffa recounts how one mathematical result that Bolzano attained helps to explain how he was able to overcome the mistake that Kant made, namely that Kant's definition analytic judgments does not exhaust the conceptual resources of our language, but employs only a tiny fraction of them.

Bolzano proved a theorem that lies at the foundation of the calculus: the intermediate value theorem. This theorem, you might recall if you had a calculus course in high school or in the university, states that a continuous real-valued function that takes values below and above zero must somewhere in between take the value of zero. To a Kantian, this would seem to be intuitively obvious; if the function is continuous, and has no gaps in its range, it must coincide somewhere with the x-axis, i.e. y = 0 [cf. Coffa, 'Emergence of Logicism,' pp. 36-37].

But, as Coffa points out, Bolzano defined continuity not in terms of intuition, but in terms analogous to our standard epsilon-delta formulation. He also provided a notion of the convergence of a series and supplied the criterion now attributed (wrongly, it seems) to Cauchy [Coffa, p. 37]. Then, Bolzano proved the intermediate value theorem. It's a case of not relying upon intuition, but relying on fundamental, mathematical definitions of basic concepts, and deriving what previously had appeared to be defensible only through "intuition" by way of a rigorous demonstration. It seems that analyticity could be described now not it terms of the inclusion of the predicate concept within the subject concept, but in terms of logical derivation from a cluster of concepts.

Coffa remarks: "It was the first time anyone had introduced these concepts and proofs. If Kant had known about Bolzano's paper there can be little doubt that he would have regarded it as a philosophically incoherent effort to prove the obvious. The paper was, instead, one of the landmarks of nineteenth-century mathematics" [Coffa, p. 37].

Thanks, --Ron