One of my favourite bits of Isaac Asimov’s science writing was an appendix in which he derived E=mc**2 from the Michaelson-Morley experiment, using only simple geometry & math.

Even your title makes me laugh – the neccessity of mathematics.

I was teaching math to a friend’s twelve-year-old. She’s smart but oriented to history and argument (and will probably become a lawyer). So she said to me – why should I learn this stuff? I’ll never use it again. And I had no answer for her, because she’s probably right. Somehow, “the beauty of it (math)” seemed a lame answer.

Again, nice blog.

Admiringly…Scott Hatfield

I think you are a little too optimistic.

I did say “the basics”, i.e. giving people a concrete sense of what the Riemann tensor and the Einstein equations mean, that goes beyond the usual waffly “space-time is curved” that frustrates everyone with its vagueness.

And stop making tensors sound scary! The basics of differential geometry are incredibly easy: a tangent vector v is like a railway line, a scalar function f is like air pressure, a 1-form df is like the contours of constant air pressure, and you can calculate a number <v,df> by asking how fast someone on the train measures the air pressure to be changing. (Yes, there’s a little bit of fine print to add, but nothing conceptually much more difficult.) Tensors of various kinds arise simply from multiplying and adding together these basic objects.

IMHO, it would actually be much easier to understand multivariable calculus if it was taught at the same time as these concepts. I never really, really became confident with manipulating partial derivatives until I realised I could draw pictures of everything and treat them all as tangent vectors.

I think anyone who did high school calculus and physics could learn the basics of SR and GR pretty easily.

I think you are a little too optimistic. Speaking as an undergrad physics student, I have had great difficulty finding a good explanation of GR beyond “space-time is curved”. It may be because I haven’t found the right source (thanks for the link, btw) but I think I’d need to take an entire course to begin to understand. Anyways, not everyone takes calculus in high school, and they certainly don’t cover multivariable calculus, much less vector calculus, and tensors. Where I am, we don’t learn about tensors until well into the upper div physics classes.

It is a very nice piece of science writing, with real, but not phenomenally complicated mathematics.

Maybe your point should be directed at not stripping the math from science classes, rather than at how we teach math?

(AFAIK, nobody in the science-blogging business is making a living exclusively off blog profits. The whole whirl is a study in what kind of content gets written by volunteer labor.)

Some people just honestly don’t want to know about these things and are really never, ever going to use them.

Which is why I talk about being accessible to the “generally interested lay audience”, rather than to everybody. I agree that people should be encouraged to remedy where they’re weak on the basics, but if I’m a good communicator it’s to people who are already interested.

I mean, explaining the math of Kepler’s laws is one thing, but I can’t imagine the same happening for General Relativity

I think anyone who did high school calculus and physics could learn the basics of SR and GR pretty easily.

In terms of public understanding of mathematically-based science, the hardest audience is going to be those who never coped with algebra and calculus in high school. Some people just honestly don’t want to know about these things and are really never, ever going to use them, and that’s fair enough, but people who finished high school without reaching that level of mathematical literacy and then later feel locked out of something important need to be encouraged to remedy their problem, rather than being scared off with “mathematics is difficult, boring and nerdy” messages from people who should know better.

Um, all right. Now, what’s a gauge theory?

“You remember how we’ve pointed out that the laws of physics have certain symmetries? Like if I do an experiment it doesn’t matter whether I do it today or tomorrow, or whether my lab bench faces north or southeast, or whether my lab is in New York or Shanghai? Good. Let’s consider the orientation just to be specific.

“So: if I take my entire experimental setup and turn the whole thing by some angle, it’s not going to make a difference in the results I get. In fact, for a large number of experiments (e.g. polarization measurements) where the apparatus comes in a number of different pieces, you can rotate each piece by the same angle and get the same results.

“Now here’s where the insight comes in: if the parts are all separated from each other by large distances, they obviously can’t tell instantly when another part of them has been turned. That would be information, and relativity tells us that information travels at a finite speed. So the old rotational symmetry has to be ‘gauged’. That is, we need to think of rotating by a different angle at each point in spacetime. Gauge theories are mathematical models physicists have developed to cope with this sort of ‘gauged’ symmetry.”