Modes of continuity in diagram for Intermediate Value Theorem
Piotr BÅ‚aszczyk and Marlena Fila
In the ongoing debate over the role of a diagram in the proof of the Intermediate Value Theorem (IVT), Brownâ€™s takes a clear position: a diagramÂ does constitute proof of IVT. Giaquintoâ€™s points out that a real continuous but nowhere differentiable function lacks a curve, therefore diagrammaticÂ evidence must be restricted to smooth functions. By applying newly-shaped concepts such as pencil-continuity and crossing x-axis to rational and real maps, f : Q â†’ Q, f : R â†’ R, he comes to the conclusion that the same diagram can represent either a false or true statement, depending on the interpretation in terms of the domain of f . We analyze Brownâ€™s and Giaquintoâ€™s arguments in mathematical, philosophical and historical contexts. Our basic observation is the equivalence of IVT and the Dedekind Cut Principle. While Brown does not address the foundational issues at all, Giaquinto seeks to characterize them by the non-mathematical concept of â€˜desideratumâ€™. As for philosophy, contrary to Giaquinto, we show that the diagram itself constitutes the mathematical context rather than needs an interpretation; yet, contrary to Brown, diagram for IVT does not prove anything, since it represents the axiom (completeness) of real numbers. We adopt a historical perspective to show that both Brownâ€™s and Giaquitoâ€™s arguments involve concepts that take us back to the pre-Bolzano era of non-analytic proofs of IVT.