Wednesday, December 07, 2005

thinking physically

What does it mean to "think physically?". When in the midst of a physics discussion, you may well be urged to think physically about the equations in front of you. This will usually mean that you are to form pictures in your mind in which the symbols have properties they may have in some physical situation. For example you may think of E, energy, as a bubbling effervescent mass, or think of it as a ball rolling down a hill trying to reach a minimum, or as a fluid which can flow to different parts of a container but who's volume must stay fixed.

I was reminded of this question which I have pondered occasionally when I read the following post by Jaques Distler. I know very little about the subject which is highly mathematical. After a list of three math heavy constructions, he says:

You might be forgiven if, at first, you don’t see the pattern in this list. But, if you think physically, the answer becomes clear. These topologically-twisted theories...

My point is not that this has nothing to do with the real world. It might. But it might not. So what does it mean to physically interpret some equations that might not have anything to do with the real world? It is to pretend that it does. I think that this activates some clever thinking skills in us. This is how I do math and I remember something Richard Feynman said about how he proved or disproved theorems... each time a new postulate was added he simply added that on to the picture of the object in his head until he could see that this object did or did not have such and such a property. He was "thinking physically" about the theorem.

So people say that string theory has been highly succesful as stimulating new mathematical discoveries. Is it the gut feeling that one's objects of discussion may be "real" in some deep sense that encourages coherent mental pictures to be formed stimulating solid intuition about promising directions? Can one use this lesson to trick oneself into becoming a better mathematician?

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