## Wednesday, May 30, 2012

### Thinking like a physicist

@mathematicsprof on Twitter recently tweeted a link to a page asking what it's like to understand advanced mathematics. There are a number of very interesting answers there, but one interested me particularly. I can't figure out if there's a way to link to it directly, but I'll quote it here:

A two part question to determine if you "think like a mathematician," from Prof. Eugene Luks, Bucknell University, circa 1979.

Part I: You're in a room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on the floor.  How do you make hot water for tea?
Answer to Part I: Put tea kettle on stove, turn on burner, heat until water boils.

Part II: Next, you're in another room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on a table.  How do you make hot water for tea?
Non-mathematician's answer to Part II: Put tea kettle on stove, turn on burner, heat until water boils.
Mathematician's answer to Part II: Put the tea kettle on the floor.

Why?  Because a solution to any new problem is elegantly complete when it can be reduced to a previously demonstrated case.
This might be why I'm studying physics more than maths. I can see why putting the kettle on the floor solves the problem rather elegantly - I think it's a nicer solution than the "non-mathematician's answer" up there - but it's not how I would solve the problem. Isn't it obvious that the table is negligible in this situation, so that Part II is reduced to Part I?

Mathematicians aren't, I think, supposed to say things are negligible. Assuming that the table is negligible isn't rigourous. It does, however, get to the right solution without (explicitly, at least) going via the floor. It's still elegant (if you can get over the idea of neglecting the table) and it takes less effort.

Perhaps it's related to the idea that physics is not so much about working out how to describe some given bit of the universe as it is about working out which bits of the universe we can describe and doing so. This is usually expressed in terms of finding symmetries, from what I've seen and heard. Here, I think the system is invariant under the introduction of the table, which is a symmetry.

There are probably other ways of solving the problem, too. I think it's a very interesting exercise in how people think!

1. How does putting the kettle on the floor give you the hot water? I don't follow.

1. Well, putting the kettle on the floor won't actually give you hot water, but you've already shown how to get hot water if there's a functioning stove and a kettle on the floor. So there's an implication that you would then put the kettle on the stove etc., but since that's already been said in part I, it "doesn't need to be said again".

If what you're really interested in is getting a cup of tea, the best answer is probably just to put the kettle on the stove. If you're interested in solving a problem, it's neater to show that the two problems are really the same thing by putting the kettle on the floor. In a more complicated situation, there might be some reason that the kettle couldn't be moved directly from the table to the stove, but the mathematician's answer would still work. So there is a somewhat practical reason that the mathematical way might be superior. But nobody said maths had to be useful!

In most of what I'll vaguely call "theoretical science" a problem is considered solved as soon as you can show it's the same as some other problem you've already solved. To actually use the solution in an application, you'd have to solve both parts, but, eh, that's what engineers are for. (What engineers do is obviously important -- and, to many people, interesting -- but I dropped out of my engineering degree precisely because I find that part of science relatively boring.)

Sorry about the massively long answer! I hope it makes some sense now.