The American physicist J Willard Gibbs thought so, defining the difference as “a mathematician can say what he likes… a physicist has to be at least partly sane”. We should perhaps take what he says with a grain of salt – as a mathematician, Gibbs invented a whole new branch of the discipline – modern vector calculus – so maybe he was at least a little bit crazy too.
Inspired by Gibbs, Marcus Wilson, a lecturer at New Zealand’s University of Waikato, takes a light-hearted look at the differences between the disciplines and concludes:
What is it that makes a physicist sane (if only in part)? Everything has to be related back to the ‘real world’, or the ‘real universe’. That is, a physicist has to talk about how things work in the world or universe in which we live, not some hypothetical universe. That’s how I think of it…
And that means giving things dimensions and working in units.
Here’s a question which illustrates the point. What is the length of a side of a cube whose volume is equal to its surface area? The over-zealous mathematics student blunders straight in there: Let the length be x. Then volume is x^3, and surface area is 6 x^2 (the area of a face is x^2, and there are six on a cube). So x^3 = 6 x^2 ; cancelling x^2 from both sides, we have x=6. Six what? centimetres, inches, furlongs, parsecs? The point is that the volume of a cube can never be equal to its surface area. Volume and area are fundamentally different things.
Part of the blame, he says, rests with computer programmers.
In writing a computer programme to do a physics calculation, we almost always don’t have explicit record of the units or dimensions in our calculations. Our variables are just numbers. It’s left to us to keep track of what units each of these numbers is in. Strictly speaking, I’d say it’s rather slack.