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Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being ad

by ajkjk 3y ago

Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. A simple example is: nobody needs a proof of how differentiation or integration work. They need to know how to do it, think with it, apply it to problems, and generalize it. The proof is a tool for not being wrong, and that is a perfectly fine thing for mathematicians to do and a waste of everybody else's time (unless they really need convincing). It is an absolute travesty how many people enjoy math until they get to their first college class which is taught by a disconnected mathematician that ruins the subject for them. At my school everyone stopped at multivariable because that's the level at which mathematicians systematically ruined math -- by teaching and testing the wrong stuff. Personally, I survived by the skin of my teeth, and then later found physics and actually learned multivariable calculus and learned to love math again. Despite the efforts of the people who were paid to teach it.

anon291 3y ago
> Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. That's great. Not everyone does. Thinking simply via formal systems is not inherently inferior to 'doing' something with it. Both have their uses. In particular, as I said, quantum mechanics would not be where it's at today if it weren't for many pre-quantum mathematicians pushing around symbols on a page.
- ajkjk 3y ago
  I'm very much not arguing that nobody should do pure math. I'm arguing that teaching pure math to undergrads who are trying to learn applied math is a tragedy and ruins math for a lot of people.
Koshkin 3y ago
> nobody needs a proof of how differentiation or integration work This "nobody" includes a lot of industry workers, e.g. those in finance and economics, not only "pure" mathematicians.
- ajkjk 3y ago
  They... don't? The only reason they need proofs is to write papers proving other things. Not to do the actual work of their fields.
nonameiguess 3y ago
I think you're generalizing far too much from limited experience. Not all mathematics, even applied mathematics, corresponds neatly to intuitive physics models you build naturally by being an animal that interacts with the world. Quantum mechanics was already given as an example, but I had the opposite experience with cryptography. So much of the training as a CS student was along the lines of "accept that this works" and then practice by implementing something, and sure, it worked, but at no point did it ever become obvious why any of it should work. Until I finally went back and spent a solid year and a half learning group theory and algebraic number theory in a more rigorous way, figuring out all of the proofs we had skipped over, at least trying it myself, but giving up and reading someone else's proof if I couldn't figure it out within a day or two. And in this case, the abstraction definitely came before the application. This is math that was thought to be compeletely useless for centuries, but people worked on it anyway just because they loved math. Lo and behold, it made the Internet possible, so I'm glad someone did it and loved the tedium.
- ajkjk 3y ago
  My stance in general is not that we shouldn't teach students why things work; just that rigorous proofs aren't at all a good way to do it. Indeed, as far as I could tell, most of the mathematicians who taught math classes I took had at best a rudimentary knowledge of how to use any of the theory they were teaching to do anything. I studied physics in undergrad and the general method was, yes, it involved detailed understanding of all the theory, far better than anyone learned it in the relevant math class (differential equations, multivariable calculus, and linear algebra, in particular --- a 1-semester course in QM has to teach everyone linear algebra because the math course on the subject is garbage; a 1-semester course in E&M has to teach multivariable calculus also because the math course was garbage, etc.) But yeah, absolutely you have to really learn e.g. group theory and algebraic number theory to do cryptography. I actually quit engineering and switched to physics precisely because I was so annoyed that an intro electrical engineering class used the Fourier transform as black magic without bothering with why it worked. In particular many of the exact theorems that are used in formal math are totally irrelevant and historical accidents. Calculus could have been developed in a thousand different ways; we happened to pick one, the particular names of the theorems or what order the results are constructed in is, IMO, totally irrelevant to anything outside pure mathematics (particular examples: which construction of the integers you use; which construction of the real numbers you use; which definition of the derivative; which definition of the integral, stuff like that. In the case of integrals, yes, different constructions have different theoretical properties in terms of which functions can be integrated --- but, in any application, the answers intrinsically can't be affected based on your choice of definition, so the actual definition can't matter.

by ajkjk 3y ago

anon291 3y ago

> Personally I don't care that it's about physical things. But I care that it's about doing things: understanding concepts intuitively, solving problems, being adept at thinking mathematically. The thing that is not useful is tedious definitions and tedious proofs. That's great. Not everyone does. Thinking simply via formal systems is not inherently inferior to 'doing' something with it. Both have their uses. In particular, as I said, quantum mechanics would not be where it's at today if it weren't for many pre-quantum mathematicians pushing around symbols on a page.

ajkjk 3y ago
I'm very much not arguing that nobody should do pure math. I'm arguing that teaching pure math to undergrads who are trying to learn applied math is a tragedy and ruins math for a lot of people.

Koshkin 3y ago

> nobody needs a proof of how differentiation or integration work This "nobody" includes a lot of industry workers, e.g. those in finance and economics, not only "pure" mathematicians.

ajkjk 3y ago
They... don't? The only reason they need proofs is to write papers proving other things. Not to do the actual work of their fields.

nonameiguess 3y ago

I think you're generalizing far too much from limited experience. Not all mathematics, even applied mathematics, corresponds neatly to intuitive physics models you build naturally by being an animal that interacts with the world. Quantum mechanics was already given as an example, but I had the opposite experience with cryptography. So much of the training as a CS student was along the lines of "accept that this works" and then practice by implementing something, and sure, it worked, but at no point did it ever become obvious why any of it should work. Until I finally went back and spent a solid year and a half learning group theory and algebraic number theory in a more rigorous way, figuring out all of the proofs we had skipped over, at least trying it myself, but giving up and reading someone else's proof if I couldn't figure it out within a day or two. And in this case, the abstraction definitely came before the application. This is math that was thought to be compeletely useless for centuries, but people worked on it anyway just because they loved math. Lo and behold, it made the Internet possible, so I'm glad someone did it and loved the tedium.

ajkjk 3y ago
My stance in general is not that we shouldn't teach students why things work; just that rigorous proofs aren't at all a good way to do it. Indeed, as far as I could tell, most of the mathematicians who taught math classes I took had at best a rudimentary knowledge of how to use any of the theory they were teaching to do anything. I studied physics in undergrad and the general method was, yes, it involved detailed understanding of all the theory, far better than anyone learned it in the relevant math class (differential equations, multivariable calculus, and linear algebra, in particular --- a 1-semester course in QM has to teach everyone linear algebra because the math course on the subject is garbage; a 1-semester course in E&M has to teach multivariable calculus also because the math course was garbage, etc.) But yeah, absolutely you have to really learn e.g. group theory and algebraic number theory to do cryptography. I actually quit engineering and switched to physics precisely because I was so annoyed that an intro electrical engineering class used the Fourier transform as black magic without bothering with why it worked. In particular many of the exact theorems that are used in formal math are totally irrelevant and historical accidents. Calculus could have been developed in a thousand different ways; we happened to pick one, the particular names of the theorems or what order the results are constructed in is, IMO, totally irrelevant to anything outside pure mathematics (particular examples: which construction of the integers you use; which construction of the real numbers you use; which definition of the derivative; which definition of the integral, stuff like that. In the case of integrals, yes, different constructions have different theoretical properties in terms of which functions can be integrated --- but, in any application, the answers intrinsically can't be affected based on your choice of definition, so the actual definition can't matter.