Your comment made me think, and I’ll look up some of the recommendations. I like the analogy with musicians and also the part where you talked about how the analogy breaks down.
However, I’d like to offer a bit of a different perspective to the original poster on this part of what you said.
To summarize: the math I think you’re looking to learn is proofy, not computational, in nature.
Your advice is good, given this assumption. But this assumption may or may not be true. Given that the post says:
I don’t care what field it is.
I think there’s the possibility that the original poster would be interested in computational mathematics.
Also, it’s not either or. It’s a false dichotomy. Learning both is possible and useful. You likely know this already, and perhaps the original poster does as well, but since the original poster is not familiar with much math, I thought I’d point that out in case it’s something that wasn’t obvious. It’s hard to tell, writing on the computer and imagining a person at the other end.
If the word “computational” is being used to mean following instructions by rote without really understanding why, or doing the same thing over and over with no creativity or insight, then it does not seem to be what the original poster is looking for. However, if it is used to mean creatively understanding real world problems, and formulating them well enough into math that computer algorithms can help give insights about them, then I didn’t see anything in the post that would make me warn them to steer clear of it.
There are whole fields of human endeavor that use math and include the term “computational” and I wouldn’t want the original poster to miss out on them because of not realizing that the word may mean something else in a different context, or to think that it’s something that professional mathematicians or scientists or engineers don’t do much. Some mathematicians do proofs most of the time, but others spend time on computation, or even proofs about computation.
Fields include computational fluid dynamics, computational biology, computational geometry...the list goes on.
Speaking of words meaning different things in different contexts, that’s one thing that tripped me up when I was first learning some engineering and math beyond high school. When I read more advanced books, I knew when I was looking at an unfamiliar word that I had to look it up, but I hadn’t realized that some words that I already was familiar with had been redefined to mean something else, given the context, or that the notation had symbols that meant one thing in one context and another thing in another context. For example, vertical bars on either side of something could mean “the absolute value of” or it could mean “the determinant of this matrix”, and “normal forces” meant “forces perpendicular to the contact surface”. Textbooks are generally terribly written and often leave out a lot.
In other words, the jargon can be sneaky and sound exactly like words that you already know. It’s part of why mathematical books seem so nonsensical to outsiders.
Excellent points; “rigorous” would have been a better choice. I haven’t yet had the time to study any computational fields, but I’m assuming the ones you list aren’t built on the “fuzzy notions, and hand-waving” that Tao talks about.
I should also add I don’t necessarily agree 100% with every in Lockhart’s Lament; I do think, however, that he does an excellent job of identifying problems in how secondary school math is taught and does a better job than I could of contrasting “follow the instructions” math with “real” math to a lay person.
Your comment made me think, and I’ll look up some of the recommendations. I like the analogy with musicians and also the part where you talked about how the analogy breaks down.
However, I’d like to offer a bit of a different perspective to the original poster on this part of what you said.
Your advice is good, given this assumption. But this assumption may or may not be true. Given that the post says:
I think there’s the possibility that the original poster would be interested in computational mathematics.
Also, it’s not either or. It’s a false dichotomy. Learning both is possible and useful. You likely know this already, and perhaps the original poster does as well, but since the original poster is not familiar with much math, I thought I’d point that out in case it’s something that wasn’t obvious. It’s hard to tell, writing on the computer and imagining a person at the other end.
If the word “computational” is being used to mean following instructions by rote without really understanding why, or doing the same thing over and over with no creativity or insight, then it does not seem to be what the original poster is looking for. However, if it is used to mean creatively understanding real world problems, and formulating them well enough into math that computer algorithms can help give insights about them, then I didn’t see anything in the post that would make me warn them to steer clear of it.
There are whole fields of human endeavor that use math and include the term “computational” and I wouldn’t want the original poster to miss out on them because of not realizing that the word may mean something else in a different context, or to think that it’s something that professional mathematicians or scientists or engineers don’t do much. Some mathematicians do proofs most of the time, but others spend time on computation, or even proofs about computation.
Fields include computational fluid dynamics, computational biology, computational geometry...the list goes on.
Speaking of words meaning different things in different contexts, that’s one thing that tripped me up when I was first learning some engineering and math beyond high school. When I read more advanced books, I knew when I was looking at an unfamiliar word that I had to look it up, but I hadn’t realized that some words that I already was familiar with had been redefined to mean something else, given the context, or that the notation had symbols that meant one thing in one context and another thing in another context. For example, vertical bars on either side of something could mean “the absolute value of” or it could mean “the determinant of this matrix”, and “normal forces” meant “forces perpendicular to the contact surface”. Textbooks are generally terribly written and often leave out a lot.
In other words, the jargon can be sneaky and sound exactly like words that you already know. It’s part of why mathematical books seem so nonsensical to outsiders.
Excellent points; “rigorous” would have been a better choice. I haven’t yet had the time to study any computational fields, but I’m assuming the ones you list aren’t built on the “fuzzy notions, and hand-waving” that Tao talks about.
I should also add I don’t necessarily agree 100% with every in Lockhart’s Lament; I do think, however, that he does an excellent job of identifying problems in how secondary school math is taught and does a better job than I could of contrasting “follow the instructions” math with “real” math to a lay person.