|
> The concept of "compactness" is the technical solution that lets you formulate calculus without them -- it's typically only taught to math majors because infinitesimals are less awkward to do algebra with: we can now prove that the shortcut works, so why bother with the long way unless you have good reason? I don't know where you got the idea that compactness is in any way relevant to the formulation of calculus. Compactness is a property of topological spaces that, to an approximation, is a generalization of sets being finite or infinite. For example, with the discrete topology, a set is compact iff it is finite. There are many related notions of compactness. In R^n, a set is compact (and sequentially compact) iff it is closed and bounded. Compactness is important for some ideas related to calculus, but it's not related to formulating calculus. For example, if a continuous function maps from a compact space to R, then it achieves a maximum/minimum (this can be seen of a generalization that there is always a maximum/minimum of a finite set of real numbers, but not necessarily for an infinite set). 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. This is not what is taught in calculus or analysis classes, and is only relevant for non-standard developments of calculus. The winner in the modern formulation in calculus is the "epsilon-delta" formulation of limits; that's what's taught in both calculus classes (at least to an extent) and analysis classes. The weird thing is that calculus is stuck with Leibniz's notation, which does, in a sense, refer to infinitesimals. I think that's what you're really thinking of (rather than compactness) as how you can formulate calculus without infinitesimals. The thing is that, save notation, this is how calculus is taught today. |
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?