I first learned problem solving skills studying physics, but because my school wasn't satisfying my "justified" criteria for physics, I started studying math. Math for me had always been for physics, which in turn, was for understanding and forming beliefs about the world. Hence, I didn't do math for math's sake.
While studying I was usually very stressed, and took the material very seriously. When trying to make some sort of headway on finding a thesis topic in mathematical physics, the professor mentoring me was trying to get me to slow down.
I've been studying from Rudin's Principles of Mathematical Analysis, in my Analysis course. It is by far my favorite math book yet.
I took what my professor said seriously and read Rudin, making sure I wouldn't move to the next sentence until I understood the current. I breathed deeply, relaxed, calmly concentrated, and waited patiently for the understanding.
So here's the strange part(to me). My face started to feel numb, my hands began shivering slightly, and the concepts flowed much more easily. It's like I was manipulating logical mathematical constructs, without relying on the symmetry of symbolic representations or words. I could see the ideas, then describe them, not the other way around. Now when I read mathematical writing, If I am patient, I can see a much deeper logic and relationship between the concepts. The closest analogy I have is the feeling one gets when trying to describe music with notation as one hears it.
Does this sound strange? I've always found it fun to solve problems, but I've never had a deep emotional response to it. At least not in this way. I'm a little nervous talking about this, so I wanted to see if this sounded somewhat familiar to anybody here...
Mathematical thinking
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Re: Mathematical thinking
Feeling mad with power is perfectly normal when studying math. Is that what you are describing?
Re: Mathematical thinking
I don't recall ever experiencing numbness or shivering when working with math, but in a more general sense, yes, I often find math deeply enjoyable, even in a totally abstract way, separate from all its usefulness or application.
I've been thinking about this a lot in the last few days. If someone asked me why I like math, I might talk about how important it is to science and understanding the world around us, or how useful it is even in daily life. I imagine that if I asked an athlete why they like sports, they might tell me about the many health benefits of exercise, but that's "why I should agree that exercise is important", not "why I should enjoy exercise". Similarly, I can't give much of a reason for why I enjoy math, except to say that I do enjoy it, and the fact that it's also very useful is just a happy... I was going to write "coincidence", which doesn't quite convey what I want to say, but can't think of a better word. Fermat's Last Theorem is just as useful with the really difficult proof we have as it would have been with some hypothetical simple proof like Fermat claimed to have, but the first is much less satisfying than the latter.
To paraphrase a (possibly fabricated or misattributed?) Richard Feynman quote: "Math is like sex. Sure it has practical results, but that's not why we do it."
I've been thinking about this a lot in the last few days. If someone asked me why I like math, I might talk about how important it is to science and understanding the world around us, or how useful it is even in daily life. I imagine that if I asked an athlete why they like sports, they might tell me about the many health benefits of exercise, but that's "why I should agree that exercise is important", not "why I should enjoy exercise". Similarly, I can't give much of a reason for why I enjoy math, except to say that I do enjoy it, and the fact that it's also very useful is just a happy... I was going to write "coincidence", which doesn't quite convey what I want to say, but can't think of a better word. Fermat's Last Theorem is just as useful with the really difficult proof we have as it would have been with some hypothetical simple proof like Fermat claimed to have, but the first is much less satisfying than the latter.
To paraphrase a (possibly fabricated or misattributed?) Richard Feynman quote: "Math is like sex. Sure it has practical results, but that's not why we do it."
No, even in theory, you cannot build a rocket more massive than the visible universe.
Re: Mathematical thinking
I don't think I've ever had a religious experience while doing math, partly because at that time, I am too busy doing math.
 skeptical scientist
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Re: Mathematical thinking
I sometimes have a sense of understanding a concept/argument/problem on a deep level that I can't put into words right away, which might be similar to what you're talking about. I think it's one aspect of mathematical "intuition". However, I've learned not to entirely trust this sense. Sometimes it works great and I can translate the intuition into words and get a new proof out of it, but sometimes when I do the analysis to make my intuition precise, it turns out that my unconscious mind was matching patterns incorrectly.
I'm not telling you not to enjoy the sensation you describe, but don't go all wobblyheaded over it.
I'm not telling you not to enjoy the sensation you describe, but don't go all wobblyheaded over it.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
 Cleverbeans
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Re: Mathematical thinking
I've long held the believe that mathematicians are epiphany chasers, constantly seeking the emotional thrill of sudden insight into a previously intractable problem. With that being said, about a year and a half ago I thought I had solved a problem presented on these forums only to discover despite a very strong emotional sense that I had it right and a relatively articulate "explanation" for it I was terribly wrong and had vastly oversimplified the problem. It was an emotional blow, but it taught me not to trust my intuition to much and the importance of peerreview in mathematics. That's why I'd second Skeptical's comment about learning to not trust the feeling to much, but enjoy it as well when it comes.
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Re: Mathematical thinking
Religious experiences are induced in a particular portion of the brain, and they can be induced at will with TMS. Based on the descriptions I've heard religious people give, I'm pretty sure I've experience them as well at times, mainly when I'm either thinking about space (and get even a minor handle on the scale of it) or when I get into a good flow with math. I used to have your same physical symptoms after finishing good, hard math tests where I knew the subject matter well and could crank through it. I similar occasionally have it when doing programming, though I rarely know what I'm doing well enough to flow like that. ^_^
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