People bitched about non-euclidean geometry but they're also perfectly consistent and, oh, suprise! it turns out hyperbolic geometry is a better model of the real world at cosmological scales, oops. It's "absurd" as a real-world interpretation, but it's hardly a contradiction of the sort that would invalidate AoC or even the first incompatible-with-human-experience mathematical thing. I don't know when those hit schools but given that Cauchy's big book came out in 1821 I'm going to guess it might have been a bit before WWI.īanach-Tarski, everyone's favorite nonconstructive construction, has nothing to do with infinitesimals, it's an axiom of choice thing. They're conflating the foundational crisis and the various philosophical schools that came out of that with the increase in rigor in analysis (dropping infinitesimals, epsilon-delta limits, all that stuff) a century before. Thankfully my roommate set me down (I was really embarrassed to seem this dense :) ) over video chat and finally explained it in a way that made sense directly to me! The walls I'd built up about Limits were impossible to climb by myself, and I was confused because I'd never had trouble with a concept in any field taught from a book (history, writing, genetics, etc)! But his explanation was tailored to me, and he got a big kick out of it because he'd been in a private school and had understood limits from a really good teacher explanation since he was 15. I couldn't figure out not just the concept, but the usefulness and even the point!
I followed the books, bought new "For Dummies" and "Learn in 24 hours" books, and derivatives weren't bad, and I could actually do the type of Limit examples directly from the text. I got some books from my old college roommate (bio/math double major), and had the absolute hardest time with limits. I later (after my son was born) thought it through, and figured I should at least be familiar with some of the easy parts and definitions of Calc1 in case my boy needed help in school. My physics class was for non-majors, too, and the most complicated the professor got (and complained about every period) was trig-based. When I got to college I filled my out-of-major-but-necessary classes (one year of math for Bio) with college algebra and trig, both with As. I "finished" math available at Trig, and only one other person wanted to learn Calculus with me but the teacher didn't want such a small class. I grew up in a very rural small school system in Tennessee. Of course, actually "constructing" a hyperreal number line using ultrafilters is pretty complex, so it's not necessarily a great alternative to standard analysis. I have a feeling my high school teacher defined continuity at a point with almost that exact sentence.
It's also pretty similar to how you might intuitively think of continuity. However, in nonstandard analysis, the statement "f is continuous at a" is defined as "for all x, if x is really (read: infinitesimally) close to a, then f(x) is really close to f(a)." This only has a single quantifier, so the statement itself is potentially easier to digest. If you define continuity with limits, then continuity will also be a statement involving 3 alternating quantifiers. Typically, statements involving higher numbers of alternating quantifiers are thought of as more complex, and the statement "the limit as x approaches a of f(x) is L" requires 3 alternating quantifiers. If you mean the definitions of limits and continuity, I think it likely has to do with the quantifier complexity of the statements.
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