How do scientists visualize or intuitize math?

[RandyKramer]

Aside: I'm an old guy, it's been 40 years since I took physics and calculus--if 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 ask--there 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:

1. For physicists (or other scientists that use a lot of math) at the leading edge of their fields (doing original research), what kind of understanding and manipulation of math do they have and use--do they apply rote rules, or do they somehow have a gut level understanding of mathematical representations and transformations and do they do their thinking in that intuitive space?
2. If they do their thinking in intuitive space, do you have any insights on how to help others develop such an intuitive / gut level understanding of mathematical representations and transformations?

Thanks!

 

[lavinia]

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

 

[MathAmateur]

"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.

 

[bossman27]

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:

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).

While the experts are more like:

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.

Pure math is probably even more abstract, but the point is that people who are good at math and physics are good because they have the ability to look at problems and recognize patterns in a deeper and more abstract way (and probably do it much faster as well). It surely has partly to do with natural aptitude, but at least for us non-savants, a lot of it obviously has also to do with having enough experience and practice and to build up and become familiar with those "pattern libraries," if you like.

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=Cj4y0EUlU-Y

 

[Aero51]

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.

 

[WannabeNewton]

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_(topology); 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.

 

[DiracPool]

"The Shape of Space" is a good visualization book and so is "Who is Fourier"

First of all, is "intuitize" a legitimate word? Second, I bought that book "Who is Fourier?", and its companion book "What is quantum Mechanics?" by the "transnational college of Lex" or whatever pseudo-anime operation they portended to be, back in the 90's and I don't understand it anymore now than I did then. I also bought "calculus the easy way" back in the 90's and it wasn't so easy. In each of these Fiasco publications, they try to put on you that we're going to make learning the material easy by creating a fantasy world with wizards, and goblins, and magic potions to help you through the learning curve.

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 fable-heuristic device to learn mathematical physics should be, well...marginalized.

 

[ImaLooser]

1. For physicists (or other scientists that use a lot of math) at the leading edge of their fields (doing original research), what kind of understanding and manipulation of math do they have and use--do they apply rote rules, or do they somehow have a gut level understanding of mathematical representations and transformations and do they do their thinking in that intuitive space?
2. If they do their thinking in intuitive space, do you have any insights on how to help others develop such an intuitive / gut level understanding of mathematical representations and transformations?

Thanks!

Math on the leading edge can never be done by applying rote rules. Leading edge (they tell me) comes in two forms. Very complex problems that require great technical skill, and problems where there is no at all clue as to how to proceed other than an intuitive leap (or waiting for someone else to "come up with an idea" in some other area that you can then apply.) Truly leading edge math these days often requires both. Hardly anyone is really at the leading edge of math, fifty people maybe.

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.

 

[RandyKramer]

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 interesting--it 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:

1. I'd tell their parents to watch that video about the importance of a father
2. I'd tell them both (that is parent and child) to learn as much as they possibly can as young as they possibly can, with the hope / intent of developing a knowledge of the math and the science at an intuitive level--as someone here said, learning as many of those things as possible before the brain is fully developed
3. I'd tell the child, especially, that it is important to learn what all those who came before her have already learned to help develop that intuition,
4. and I'd tell her that it is probably at that point she can start to move beyond what is already known, and that at that point she may well be in a trackless wilderness in which she may have to develop her own math, or review some of the (unused?) existing math that may turn out to be useful for whatever she is investigating
5. but, to also pursue a broad education and perspective, as, reasonably often, it seems that knowledge (or events) in some other field may lead to a solution for her (one of the videos I watched in the last few days made the point that Einstein got some insight into gravity by watching the behavior of a ball that he (and others) were playing with on a trampoline, and he saw the circular path that occurred around depressions in the trampoline (like those made by feet). I had never heard this before, so have no idea whether it is true--otoh, I have no reason to believe it is not true...

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 Merriam-Webster.

 

[bossman27]

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.

I wouldn't be so discouraged. Certainly, you're never going to be Paul Dirac (nor will I), but I imagine you didn't have such unrealistically high expectations to begin with. There are definitely some people who simply can't grasp mathematics, but that seems to be a lifelong condition rather than one that comes with age, and it usually coincides with a distase for math rather than a strong desire to learn it.

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 there's 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 :rofl:.

 

[RandyKramer]

bossman27,

Thanks for the reply and encouragement!

 

[collinsmark]

Speaking of Feynman, he has a lot to say on this very topic.

(Sorry for the poor audio.)
https://www.youtube.com/watch?v=Cj4y0EUlU-Y

 

[RandyKramer]

collinsmark,

Thanks!

 

[BobG]

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 ask--there 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.)

(bolding mine)

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.