| > I don't know where you got the idea that compactness is in any way relevant to the formulation of calculus. I'm referring to the existence proofs for simple integrals. While you can certainly formulate the proofs without literal compactness, I have yet to see a proof that accomplishes this without invoking a strategy with such a degree of conceptual similarity to those using compactness that I cannot, in good faith, call it a fundamentally different approach. > Compactness is important for some ideas related to calculus, but it's not related to formulating calculus. I'd consider existence proofs for integrals pretty darn important to the formulation of calculus. > The word infinitesimals is also a tricky word to use. To a mathematician, an infinitesimal would probably mean an algebraic object that formalizes the idea of a number smaller than any positive real number. Yes... > [The use of formal infinitesimals] is not what is taught in calculus or analysis classes, and is only relevant for non-standard developments of calculus. Yes, that's why I said it was ridiculous for the author to claim that infinitesimals were fundamental to the development of calculus. > The winner in the modern formulation in calculus is the "epsilon-delta" formulation of limits Prove that continuous functions on [0,1] are Riemann integrable using epsilon-delta limits but without using compactness (or anything that I could reasonably point to and claim "that's compactness, you just called it something different"). > I think that's what you're really thinking of [is Leibniz's notation] Why would you think that? By "infinitesimal" I mean, to use your words, "an algebraic object that formalizes the idea of a number smaller than any positive real number". You assumed that I meant something different, even though you were able to define precisely what the word meant. Why? |