On Math

From Jonathan Gardner's Physics Notebook
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I love mathematicians. I rely on their work, I appreciate the way they see the world, and so on and so forth.

But they make terrible physicists.

Before going further in this essay, let me first say this: I believe physics is harder than math, and great physicists are much brighter than great mathematicians. This depends, of course, how I define a physicist and a mathematician.

To me, the line is the same as that between Physics and Philosophy: Pure mathematicians refuse to experiment or observe nature, while pure physicists do nothing but experiment and measure.

How should a physicist approach math? In the most un-mathematical and brutal way possible.

Take a carpenter, the worker of wood. In his toolbox, you'll find a variety of tools. He is familiar with how all of them work, and very good at the function of some of them. He does not care where the tools came from or why they work, only that they do a certain thing and the exact parameters of their operations.

That is how a physicist should treat math. Ignore all the theories except for the assumptions and conclusion. Memorize the assumptions and conclusions, and maybe some alternate forms, so that you can recognize problems that might benefit from them.

Physics could ignore, altogether, the mathematical process of creating a new proof. However, they cannot ignore basic logic, even though they may decide to "skip a few steps" and rely on their intuition.

How do you get good at math? You have to put on the mathematicians hat from time to time, and learn how the proofs work. You have to be able to assault an equation with heartless cruelty. You have to say, at the moment you have arrived at the conclusion, that all the work you did was wrong, and then go find all the mistakes you made.

Don't beat yourself up if you didn't see how to solve a particular problem. Memorize how it was solved, and then keep those tools in your tool box.

Notice that physicists often work backwards. They start with the conclusion, and then ask, "How can I get here?" That's just like solving a maze by starting at the exit. It seems a lot of problems are easier to solve that way. That means sometimes it's a good idea to guess what the solution is, and work backwards and see if that was right.

Intuition is a powerful thing. Rely on it, tune it, love it. Your brain is the greatest supercomputer ever, and its principle power is pattern recognition. If you can't see the answer to a problem, look at lots and lots of data, and let your brain work out the patterns.

If you're really, really stuck, pull out the "kitchen sink". Keep a notebook of ALL the solutions to all the problems you know, and try each one out, one at a time.

Keep your mind tuned. If it's been a while since you've worked on a subject, go back and learn it again. Never, never, never write down your solutions in the book. Make yourself work to get the answer again.

Finally, let me say this. Train yourself to love solving problems. A simple way is this: reward yourself for solving a problem, with a small candy or a few minutes of playing games or watching tv. You CAN train your body and mind to love solving problems. You should get to a point where every solution is full of anticipation and then the final endorphin rush when you get it right.