5 ms·

Rigorous = a pain in the ass to learn from, but you gain imaginary points for the pain.

by zorked 11mo ago

godelski 11mo ago
It certainly is more painful, but it is more beneficial. It is also harder to teach, but I stand by my claim. I'll quote Poincare: Math is not about the study of numbers, but the relationships between them. The difficulty and benefit of the rigor is the abstraction. Math is all about abstraction. The abstraction makes it harder to understand how to apply these rules, but if one breaks through this barrier one is able to apply the rules far more broadly. ---- Let's take the Fundamental Theorem of Calculus as an example[0]: f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h} Take a moment here and think about it's form. Are there equivalent ones? What do each of these symbols mean? If you actually study this, you may realize that there are an infinite number of equations that allow us to describe a secant line. So why this one? Is there something special? (hint: yes) Let's call that the "forward derivative". Do you notice that through the secant line explanation that the "backward derivative" also works? That is f'(x) = lim_{h->0} {f(x) - f(x - h)} / {h} You may also find the symmetric derivative too! f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h} In fact, you see these in computational programs all the time! The symmetric derivative even has the added advantage of error converging at an O(n^2) rate instead of O(n)! Yet, are these the same? (hint: no) Or tell me about the general case of f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h} I'm betting that most classes that went through deriving the derivative did not answer these questions for you (or you don't remember). Yet, had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM (Finite-Element Methods) or Computational Methods would be much easier for those who take them. ---- Yet, I still will say that this is much harder to teach. Math is about abstraction, and abstraction is simply not that easy. But abstraction is incredibly powerful, as I hope every programmer can intuitively understand. After all, all we do is deal with abstractions. One can definitely be overly abstract and it will make a program uninterpretable for most, but one also can make a program have too little abstraction, which in that case we end up writing a million variations of the same thing, taking far more lines to write/read, and making the program too complex. There is a balance, but I'd argue that if one is able to understand abstraction that it is far easier to reduce abstraction than it is to abstract. This is just a tiny taste of what rigor holds. You are absolutely right to be frustrated and annoyed, but I hope you understand your conclusion is wrong. Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!). The frustration and pain is quite real! But it is absolutely worth it. [0] Linking an EpsilonDelta video that covers this exact example in more detail https://www.youtube.com/watch?v=oIhdrMh3UJw https://www.youtube.com/watch?v=oIhdrMh3UJw
- zorked 11mo ago
  You are arguing for rigor, not for its didactics. Those are different. > had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM No, you wouldn't. You would also learn things out of order. You would be exposed to things without understanding why you are learning them. People who argue this usually learn things the intuitive way (whether from rigorous material or not - what goes on in their mind isn't rigorous), and then they go back and reassess the rigor in the light of that. Then they pretend that they learned from the rigorous exposition. No, they didn't. It is totally fine to iterate. Learn non-rigorously. Go back to it and iterate on rigor later. As it becomes necessary, and if it ever becomes necessary for your field. > Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall Particularly if you are learning from "rigorous" material. But then you go watch some YouTube videos to make up for the absence of didactics in your textbook. I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes? They are maximally rigorous, therefore maximally great to learn from, right?
  - godelski 11mo ago
    > I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes? The Bourbaki group was quite famous for wanting to restructure math education. Teaching many things that are considered advanced to children. Despite not sticking around we see elements of resurgence and effectiveness. So I'm not sure your argument of "out of order" is accurate. The order is what we make. There's no clear optimal way to teach math. Your argument hinges on that. You might argue that the current status quo is working, so why disrupt it, and I'll point around asking if you really think it's so effective when many demonstrate a lack of understanding all around us. That so many struggle with calculus is evidence itself. We need not even acknowledge that there are many children who learn this (and let's certainly not admit that it's far more common for them to learn it in unconventional ways). > let them sort it out without classes? To suggest I'm arguing for the elimination of educators is beyond silly. I'd hope the caliber of your arguments would match that of your diction.

Rigorous = a pain in the ass to learn from, but you gain imaginary points for the pain.

by zorked 11mo ago

Rigorous = a pain in the ass to learn from, but you gain imaginary points for the pain.

It certainly is more painful, but it is more beneficial. It is also harder to teach, but I stand by my claim. I'll quote Poincare: Math is not about the study of numbers, but the relationships between them. The difficulty and benefit of the rigor is the abstraction. Math is all about abstraction. The abstraction makes it harder to understand how to apply these rules, but if one breaks through this barrier one is able to apply the rules far more broadly. ---- Let's take the Fundamental Theorem of Calculus as an example[0]: f'(x) = lim_{h->0} {f(x + h) - f(x)} / {h} Take a moment here and think about it's form. Are there equivalent ones? What do each of these symbols mean? If you actually study this, you may realize that there are an infinite number of equations that allow us to describe a secant line. So why this one? Is there something special? (hint: yes) Let's call that the "forward derivative". Do you notice that through the secant line explanation that the "backward derivative" also works? That is f'(x) = lim_{h->0} {f(x) - f(x - h)} / {h} You may also find the symmetric derivative too! f'(x) = lim_{h->0} {f(x + h) - f(x - h)} / {2h} In fact, you see these in computational programs all the time! The symmetric derivative even has the added advantage of error converging at an O(n^2) rate instead of O(n)! Yet, are these the same? (hint: no) Or tell me about the general case of f'(x) = lim_{h->0} {f(x + ah) - f(x + bh)}/{(a-b)h} I'm betting that most classes that went through deriving the derivative did not answer these questions for you (or you don't remember). Yet, had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM (Finite-Element Methods) or Computational Methods would be much easier for those who take them. ---- Yet, I still will say that this is much harder to teach. Math is about abstraction, and abstraction is simply not that easy. But abstraction is incredibly powerful, as I hope every programmer can intuitively understand. After all, all we do is deal with abstractions. One can definitely be overly abstract and it will make a program uninterpretable for most, but one also can make a program have too little abstraction, which in that case we end up writing a million variations of the same thing, taking far more lines to write/read, and making the program too complex. There is a balance, but I'd argue that if one is able to understand abstraction that it is far easier to reduce abstraction than it is to abstract. This is just a tiny taste of what rigor holds. You are absolutely right to be frustrated and annoyed, but I hope you understand your conclusion is wrong. Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall (or both!). The frustration and pain is quite real! But it is absolutely worth it. [0] Linking an EpsilonDelta video that covers this exact example in more detail https://www.youtube.com/watch?v=oIhdrMh3UJw https://www.youtube.com/watch?v=oIhdrMh3UJw

zorked 11mo ago
You are arguing for rigor, not for its didactics. Those are different. > had you, you would have instantly known how to do numerical differentiation and understand the limits, pitfalls, and other subjects like FEM No, you wouldn't. You would also learn things out of order. You would be exposed to things without understanding why you are learning them. People who argue this usually learn things the intuitive way (whether from rigorous material or not - what goes on in their mind isn't rigorous), and then they go back and reassess the rigor in the light of that. Then they pretend that they learned from the rigorous exposition. No, they didn't. It is totally fine to iterate. Learn non-rigorously. Go back to it and iterate on rigor later. As it becomes necessary, and if it ever becomes necessary for your field. > Unless you're Ramanujan, every mathematician has spent hours banging their head against a literal or metaphorical wall Particularly if you are learning from "rigorous" material. But then you go watch some YouTube videos to make up for the absence of didactics in your textbook. I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes? They are maximally rigorous, therefore maximally great to learn from, right?
- godelski 11mo ago
  > I mean, why don't we just throw Bourbaki books at freshmen and let them sort it out without classes? The Bourbaki group was quite famous for wanting to restructure math education. Teaching many things that are considered advanced to children. Despite not sticking around we see elements of resurgence and effectiveness. So I'm not sure your argument of "out of order" is accurate. The order is what we make. There's no clear optimal way to teach math. Your argument hinges on that. You might argue that the current status quo is working, so why disrupt it, and I'll point around asking if you really think it's so effective when many demonstrate a lack of understanding all around us. That so many struggle with calculus is evidence itself. We need not even acknowledge that there are many children who learn this (and let's certainly not admit that it's far more common for them to learn it in unconventional ways). > let them sort it out without classes? To suggest I'm arguing for the elimination of educators is beyond silly. I'd hope the caliber of your arguments would match that of your diction.