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How do scientists visualize or intuitize math? 
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#1
Feb1913, 05:14 PM

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Aside: I'm an old guy, it's been 40 years since I took physics and calculusif this question has been asked and answered, maybe you could point me to the discussion, or give me some good search terms.
I can imagine that people can have different levels of understanding of various math equations, and different approaches to manipulating / solving them. One might solve math equations in a very rote fashion, applying various rules that he has learned along the way (just to name one, integration by parts). Maybe someone else might have a, well, I was going to say very deep, but maybe it's not very deep, but a sort of intuitive / gut level understanding of the equation, and maybe even visualize the equation and the solution without applying various rules in some rote sequence. (I'm not sure I asked exactly what I wanted to askthere are some things you get a gut level understanding of, maybe things like a length times a width gives an area, and times a height gives a volume, so if you think in those terms, the numbers you are manipulating have meaning beyond the numbers themselves in your head.) Maybe, somewhat similarly, in looking at some of Maxwell's equations and seeing a triple integral, you don't focus so much on the triple integral but think of that as representing a volume. My questions include the following:
Thanks! 


#2
Feb1913, 08:45 PM

Sci Advisor
P: 1,722

I would read Feynmann's Lectures on Physics particularly the second book. It completely shows you how to understand Maxwell's equation  and some others as well  in terms of physical phenomena. The book on Quantum Mechanics  the third volume is also good,
There is also geometrical intuition as well as physical intuition. Visualization is common and is one of the bases for mathematical intuition 


#3
Feb1913, 10:42 PM

P: 35

"The Shape of Space" is a good visualization book and so is "Who is Fourier"
But the main way to learn to visualize math is to do a lot of math. It comes with time. 


#4
Feb1913, 11:47 PM

P: 204

How do scientists visualize or intuitize math?
I wouldn't call myself an expert in math by any stretch, but I once read about a study where both "experts" in physics and "beginners" in physics were asked to look at a number of practice problems and group them into categories. Unsurprisingly I suppose, the experts grouped things together in more abstract ways that had to do with the appropriate method of solving the problem, or the laws that would need to be implemented, etc; the beginners tended to group things more superficially (for instance, grouping the "ramp" problems together).
Basically, the beginners solved problems more like this: Of course, learning how to visualize and internalize concepts effectively is a big part in learning efficiently and retaining/retrieving information as well, so I certainly echo the first two posters. I'll leave off with a little Feynman on the subject: http://www.youtube.com/watch?v=Cj4y0EUlUY 


#5
Feb1913, 11:54 PM

P: 546

For me, I reason both intuitively and inductively. In addition, I can reconstruct and manipulate scenarios in my head with great detail. In a sense , when I am scientifically reasoning I recreate the situation in my head and "experience" the mathematics and physics. Its a wonderful ability. The down side is that my memory is terrible.



#6
Feb2013, 12:43 AM

C. Spirit
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One thing I've learned from micromass while learning topology is that it is good to have a sort of reference object you can visualize in your head when doing problems so that you have a plan of attack. For example, if I am trying to prove a particularly non  trivial and highly "geometric" result regarding topological spaces, it is usually second nature to first visualize the result in euclidean 1,2,or 3  space in order to get an idea of why the result would be true and how to go about proving it. I can say without hesitation that this has helped a ton in solving various problem sets. It won't work all the time of course (for example I tried as hard as I could but I couldn't for the life of me visualize the long line, http://en.wikipedia.org/wiki/Long_line_%28topology%29; others probably can but this is an example of the limits of my own imagination) but it is a handy tool nonetheless. Like others above said, it is a matter of practice it would seem.



#7
Feb2013, 02:14 AM

P: 611

In short, it doesn't work. Not only doesn't it work, it makes you feel even more stupid than you should because now you're so dumb you can't even follow a simple child's fairy tale. Who ever came up with this stupid fableheuristic device to learn mathematical physics should be, well...marginalized. 


#8
Feb2113, 01:45 AM

P: 570

In math grad school at age 40 I was very good with geometric visualization but that was all worked out in the 19th century so it didn't count for much. Most algebra isn't geometric. I was terrible at algebra and practice didn't help. I think its one of those things that is so hard you have to learn it before age 16, when your brain is still growing. Special math hardware gets built into the brain. I have that for skiing instead, so an older person learning to ski would have to work damn hard to do what I can do. Those people can think in a purely mathematical way, without reference to the real world. I couldn't. The only way to get that intuition is practice, practice, practice. Feynman worked his *** off for many years. And even that may very well not be enough. It's very competitive. Some of the older professors would never be able get those jobs today. 


#9
Feb2113, 02:53 PM

P: n/a

Thanks to all who have responded so far!
The pointer to Feynman has been very helpful, and I've watched several of his videos since then. One in particular I'd mention is: Feynman: 'Fun to Imagine' 12: Ways of Thinking, Part Two of Two If you're interested, watch that one, then pick others from those that come up on that youtube page. An unstated part of my question was whether I had a chance of doing any of this stuff at my age. It seems rather unlikely, but I won't rule it out entirely. Feynman's video titled "the importance of a father" is very interestingit is interesting to hear about the things his father did for him that probably helped him develop his interest and approach to science. Here it is: The Importance of a Father If I was very young, and wanted to get into this stuff, or if I was trying to encourage some young person to get into it, I'd offer them a few things:
Re: intuitize: although a google search shows a few earlier uses of intuitize, it seems "intuit" would be the current generally accepted (or preferred) "correct" word form, at least according to MerriamWebster. 


#10
Feb2113, 11:05 PM

P: 204

With that in mind, I'm reasonably confident in saying you could still learn plenty of math at your age. The key is to enjoy the learning process, not just the thought of the thought of the end result. Find interesting/fun applications of what you're learning, come up with your own questions about the world around you and see if you can figure them out. And when learning knew concepts, try to grasp why it works that way "intuitively," not just how to calculate answers in a "rote" algorithmic way. Another thing I sometimes do when I hear about an interesting subject, even when it's beyond my current abilities, is to read about it and see if I can piece together at least a qualitative understanding; you'd be surprised how far a solid foundation of integral and derivative calculus (plus maybe some linear algebra and diff EQ thrown in) will get you in the way of qualitative understanding. Even sometimes when theres a gap in my understanding, it prompts me to learn the necessary piece of math, and of course I'm only the better for it. One a side note, I'm glad you liked the Feynman videos. I can certainly empathize with having Feynman video marathons. I used to think it'd be a good idea to watch a few Feynman videos to get excited for studying, but half the time I'd end up clicking on related videos for an hour (or more) before getting anything done . 


#11
Feb2513, 09:44 AM

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bossman27,
Thanks for the reply and encouragement! 


#12
Feb2513, 02:00 PM

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Speaking of Feynman, he has a lot to say on this very topic.
(Sorry for the poor audio.) 


#13
Feb2813, 05:43 PM

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collinsmark,
Thanks! 


#14
Feb2813, 06:43 PM

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If I can't understand the story behind the equation, my eyes tend to glaze over. And understanding that the equation has no real story other than simply modeling patterns seen in the real world counts as understanding  especially if you understand the pattern the equation represents. I guess the average person would say I like math, just because I do have a natural ability for it and do like thinking about numbers, but I wouldn't say I really enjoy math just for math's sake (which kind of puts a limit on just how far one can go in math). But it is funny how things are so relative. I thought Calculus was easy, but that's not very high level math around here. 


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