God, I wish real analysis was at least half as elegant as any other math subject — way too much pathological examples that I can’t care less about. I’ve heard some good things about constructivism though, hopefully analysis is done better there.
Yeah, real analysis sucks. But you have to go through it to get to delightful stuff— I particularly love harmonic and functional analysis. Real analysis is just a bunch of pathological cases and technical persnicketiness that you need to have to keep you from steering over a cliff when you get to the more advanced stuff. I’ve encountered some other subjects that have the same feeling to them. For example, measure-theoretic probability is a dry technical subject that you need to get through before you get the fun of stochastic differential equations. Same with commutative algebra and algebraic geometry, or point-set topology and differential geometry.
Constructivism, in my experience, makes real analysis more mind blowing, but also harder to reason about. My brain uses non-constructive methods subconsciously, so it’s hard for me to notice when I’ve transgressed the rules of constructivism.
As a general reflection on undergraduate mathematics imho there is way too much emphasis on real analysis. Yes, knowing how to be rigorous is important, being aware of pathological counterexample is importanting, and real analysis is used all over the place. But there is so much more to learn in mathematics than real analysis and the focus on minor technical issues here is often a distraction to developing a broad & deep mathematical background.
For most mathematicians (and scientists using serious math) real analysis is a only a small part of the toolkit. Understanding well the different kinds of limits can ofc be crucial in functional analysis, stochastic processes and various parts of physics. But there are so many topics that are important to know and learn here!
The reason it is so prominent in the undergraduate curriculum seems to be more tied to institutional inertia, its prominence on centralized exams, relation with calculus, etc
Really, what’s going on is that in the general case, as mathematics is asked to be more and more general, you will start encountering pathological examples more, and paying attention to detail more is a valuable skill in both math and real life.
And while being technical about the pathological cases is kind of annoying, it’s also one that actually matters in real life, as you aren’t guaranteed to have an elegant solution to your problems.
Update: huh, nonstandard analysis is really cool. Not only are things much more intuitive (by using infinitesimals from hyperreals instead of using epsilon-delta formulation for everything), by the transfer principle all first order statements are equivalent between standard and nonstandard analysis!
God, I wish real analysis was at least half as elegant as any other math subject — way too much pathological examples that I can’t care less about. I’ve heard some good things about constructivism though, hopefully analysis is done better there.
Yeah, real analysis sucks. But you have to go through it to get to delightful stuff— I particularly love harmonic and functional analysis. Real analysis is just a bunch of pathological cases and technical persnicketiness that you need to have to keep you from steering over a cliff when you get to the more advanced stuff. I’ve encountered some other subjects that have the same feeling to them. For example, measure-theoretic probability is a dry technical subject that you need to get through before you get the fun of stochastic differential equations. Same with commutative algebra and algebraic geometry, or point-set topology and differential geometry.
Constructivism, in my experience, makes real analysis more mind blowing, but also harder to reason about. My brain uses non-constructive methods subconsciously, so it’s hard for me to notice when I’ve transgressed the rules of constructivism.
As a general reflection on undergraduate mathematics imho there is way too much emphasis on real analysis. Yes, knowing how to be rigorous is important, being aware of pathological counterexample is importanting, and real analysis is used all over the place. But there is so much more to learn in mathematics than real analysis and the focus on minor technical issues here is often a distraction to developing a broad & deep mathematical background.
For most mathematicians (and scientists using serious math) real analysis is a only a small part of the toolkit. Understanding well the different kinds of limits can ofc be crucial in functional analysis, stochastic processes and various parts of physics. But there are so many topics that are important to know and learn here!
The reason it is so prominent in the undergraduate curriculum seems to be more tied to institutional inertia, its prominence on centralized exams, relation with calculus, etc
Really, what’s going on is that in the general case, as mathematics is asked to be more and more general, you will start encountering pathological examples more, and paying attention to detail more is a valuable skill in both math and real life.
And while being technical about the pathological cases is kind of annoying, it’s also one that actually matters in real life, as you aren’t guaranteed to have an elegant solution to your problems.
Update: huh, nonstandard analysis is really cool. Not only are things much more intuitive (by using infinitesimals from hyperreals instead of using epsilon-delta formulation for everything), by the transfer principle all first order statements are equivalent between standard and nonstandard analysis!