To my mind all of these things are better suited for learning the power of proof and the mathematical way of analyzing problems.
I think the main thrust of the article was less about the power of mathematics and more about the the habits of close reading and careful attention to detail required to do rigorous mathematics.
I’m not totally sure why, but I think a big part of it is that analysis has a pretty complicated technical foundation that already implicitly uses topology and/or logic (to define limits and stuff), even though you can sort of squint and usually kind of get away with using your intuitive notion of the continuum.
Seems like it’s precisely because of the complicated technical foundation that real analysis was recommended. Theorems have to be read carefully, as even simple ones often have lots of hypotheses. Proofs have to be worked through carefully to make sure that no implicit assumptions are being introduced. Even great mathematicians ran into trouble playing fast and loose with the real numbers. It took them about two hundred years to finally lay rigorous foundations for calculus.
Seems like it’s precisely because of the complicated technical foundation that real analysis was recommended.
What I’m saying is, that’s not a good reason. Even the math with simple foundations has surprising results with complicated proofs that require precise understanding. It’s hard enough as it is, and I am claiming that analysis is too much of a filter. It would be better to start with the most conceptually minimal mathematics.
Even great mathematicians ran into trouble playing fast and loose with the real numbers. It took them about two hundred years to finally lay rigorous foundations for calculus.
...implying that it is actually pretty confusing. There are good reasons for wanting to learn analysis because it is applied so widely. But from the specific perspective of trying to learn lessons about math and rigorous argument in general, it seems like you want a subject that is legitimate math but otherwise as simple as possible. To some extent, trying to do real analysis as a first real math class is like trying to teach physics class in a foreign language. On the one hand, you just want to learn the physics, but at the same time you always have to translate into your native tongue, worrying that you made a subtle mistake in translation. If you want to learn how to prove stuff in general, you don’t also want the objects that you’re proving stuff about to be overcomplicated to the point that it’s a whole chore just to understand what you’re talking about. That is an important but distinct skill from understanding and inventing proofs.
Oh, sure, in expressing agreement with Epictetus I was just saying that I don’t think that you get the full benefits that I was describing from basic discrete math. I agree that some students will find discrete math a better introduction to mathematical proof.
Ok that makes sense. I’m still curious about any specific benefits that you think studying analysis has, relative to other similarly deep areas of math, or whether you meant hard math in general.
I think that analysis is actually the easiest entry point to the kind of mathematical reasoning that I have in mind for people who have learned calculus. Most of the theorems are at least somewhat familiar, so one can focus on the logical rigor without having to simultaneously having to worry about understanding what the high level facts are.
I see. That could be right. I guess I’m thinking about this (this = what to teach/learn and in what order) from the perspective of assuming I get to dictate the whole curriculum. In which case analysis doesn’t look that great, to me.
I think the main thrust of the article was less about the power of mathematics and more about the the habits of close reading and careful attention to detail required to do rigorous mathematics.
Seems like it’s precisely because of the complicated technical foundation that real analysis was recommended. Theorems have to be read carefully, as even simple ones often have lots of hypotheses. Proofs have to be worked through carefully to make sure that no implicit assumptions are being introduced. Even great mathematicians ran into trouble playing fast and loose with the real numbers. It took them about two hundred years to finally lay rigorous foundations for calculus.
What I’m saying is, that’s not a good reason. Even the math with simple foundations has surprising results with complicated proofs that require precise understanding. It’s hard enough as it is, and I am claiming that analysis is too much of a filter. It would be better to start with the most conceptually minimal mathematics.
...implying that it is actually pretty confusing. There are good reasons for wanting to learn analysis because it is applied so widely. But from the specific perspective of trying to learn lessons about math and rigorous argument in general, it seems like you want a subject that is legitimate math but otherwise as simple as possible. To some extent, trying to do real analysis as a first real math class is like trying to teach physics class in a foreign language. On the one hand, you just want to learn the physics, but at the same time you always have to translate into your native tongue, worrying that you made a subtle mistake in translation. If you want to learn how to prove stuff in general, you don’t also want the objects that you’re proving stuff about to be overcomplicated to the point that it’s a whole chore just to understand what you’re talking about. That is an important but distinct skill from understanding and inventing proofs.
Oh, sure, in expressing agreement with Epictetus I was just saying that I don’t think that you get the full benefits that I was describing from basic discrete math. I agree that some students will find discrete math a better introduction to mathematical proof.
Ok that makes sense. I’m still curious about any specific benefits that you think studying analysis has, relative to other similarly deep areas of math, or whether you meant hard math in general.
I think that analysis is actually the easiest entry point to the kind of mathematical reasoning that I have in mind for people who have learned calculus. Most of the theorems are at least somewhat familiar, so one can focus on the logical rigor without having to simultaneously having to worry about understanding what the high level facts are.
I see. That could be right. I guess I’m thinking about this (this = what to teach/learn and in what order) from the perspective of assuming I get to dictate the whole curriculum. In which case analysis doesn’t look that great, to me.