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I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it. I really

by lmartel 13y ago

I've read this article several times at this point (it does tend to pop up everywhere) and it resonates with me but I'm not sure what to do about it. I really want to experience the kind of math the author writes about; can anyone recommend a place to start as someone who has only ever done "fake" high school math? I'm in college now and I'm halfway through a computer science degree; I've tried a few times to break into theoretical math classes but I've found the bar for entry pretty high (especially when I only have room for one or two courses), with most classes and even peers asking for years of experience and "mathematical maturity." Have any of you ever succeeded in learning some math outside formal curricula?

abecedarius 13y ago
http://www.amazon.com/Measurement-Paul-Lockhart/dp/0674057554 http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755... (by guess who) looks like the best starting point since this resonated with you. (I've dipped into it but not seriously tackled it yet.) It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.) More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.
westoncb 13y ago
Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll actually get somewhere if you put effort into it. If you haven't had much experience with proofs, it's worth focusing briefly on the process of proving explicitly as a preliminary. That's actually a good thing to do with another person or in a class -- can be pretty difficult to get some of the subtleties involved, and to know when you've done things correctly or not. I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.
- abecedarius 13y ago
  Courant and Robbins is really great. I'd recommend checking it out after the Lockhart book (or together with it), as it's more textbookish; there's more danger of bouncing off.

by lmartel 13y ago

abecedarius 13y ago

http://www.amazon.com/Measurement-Paul-Lockhart/dp/0674057554 http://www.amazon.com/Measurement-Paul-Lockhart/dp/067405755... (by guess who) looks like the best starting point since this resonated with you. (I've dipped into it but not seriously tackled it yet.) It's hard to keep at it, learning on your own. I sometimes find problems where the usual solutions or explanations feel kind of ugly, and try to make them cleaner, like rewriting someone's code. (A couple days ago it was Snell's law: this optics tutorial http://www.bigshotcamera.com/learn/imaging-lens/refraction http://www.bigshotcamera.com/learn/imaging-lens/refraction linked from HN just dropped this formula down, and to most kids it's going to be magic. Can you formulate the law of refraction in a more elementary way and derive it from some simple assumptions? I did come up with a version that never mentions sines, but I'm not really satisfied and it's gone back on the to-do list to try to take it further. See: hard to keep at it.) More generally, this kind of work can come up all the time when programming if you say, "No, I'm not going to look up the algorithm, I'll work one out for myself and then see what's been done." Occasionally you find something kind of new that way, besides often deepening your appreciation of the usual solutions. For example, last week I found a new way to avoid the epsilon-loops in Thompson's regular-expression search algorithm -- new to me, at least. This has minor significance and came out of a ridiculous amount of work rediscovering things taught in automata-theory classes, but Lockhart wasn't kidding: it's a rush when you figure it out.

westoncb 13y ago

Hey Imartel -- I've been in a similar situation. I think my first starting point was "Mathematics and the Imagination" or "Gödel's Proof." Mathematics and the Imagination is a good high level overview, and would provide some foundational notions that'll reappear repeatedly -- but, it won't given you any practice in mathematical methods. For that I would recommend "What is Mathematics?" by Courant and Robbins. Can be pretty challenging, but you'll actually get somewhere if you put effort into it. If you haven't had much experience with proofs, it's worth focusing briefly on the process of proving explicitly as a preliminary. That's actually a good thing to do with another person or in a class -- can be pretty difficult to get some of the subtleties involved, and to know when you've done things correctly or not. I mention "Gödel's Proof" because it's the first thing I came across that informed me I am in fact interested in mathematical systems. I'm more interested in architectures than problem solving though. If you suspect that might be the case for yourself, might check it out -- it's about 100 pages.

abecedarius 13y ago
Courant and Robbins is really great. I'd recommend checking it out after the Lockhart book (or together with it), as it's more textbookish; there's more danger of bouncing off.