 The differences concerning the resources available to logic revolve around a key issue, namely, whether the additional resources Frege assigns to logic require an appeal to non-logical constructions, specifically to a faculty of ‘intuition’, that is, an extralogical source which presents our minds with phenomena about which judgments can be formed.  (Recall the discussion above about Frege’s early interest in appeals to intuition.)  The debate over which resources do and do not require an appeal to intuition is an important one. Frege continued a trend started by Bolzano (1817), who eliminated the appeal to intuition in the proof of the Intermediate Value Theorem in the calculus (which in its simplest form asserts that a continuous function having both positive and negative values must cross the origin). Bolzano proved this theorem from the definition of continuity, which had recently been given in terms similar to the definition of a limit (see Coffa 1991, 27). A Kantian might simply draw a graph of a continuous function which takes values above and below the origin, and thereby ‘demonstrate’ that such a function must cross the origin.  But appeal to a graph involves an appeal to intuition, and both Bolzano and Frege saw such appeals to intuition as potentially introducing logical gaps into a proof. There are reasons to be suspicious about such appeals: (1) there are functions which we can’t graph or otherwise construct for presentation to our intuitive faculty, e.g., the function \(f\) which maps rational numbers to \(0\) and irrational numbers to \(1\), or the functions noted by Weierstrass, which are everywhere continuous but nowhere differentiable; (2) once we take certain intuitive notions and formalize them in terms of explicit definitions, the formal definition might imply counterintuitive results; and (3) the rules of inference from statements to constructions and back are not always clear.