Thursday, September 15, 2022

Math for scientists?

In Quillette, Thornett says most STEM majors should skip calculus and learn statistics. I was in physics, which of course needs calculus and differential equations for the models the field uses--and more than just those. But most week-to-week research work didn't use anything beefier than trig--there were months where all I was doing was programming, or hardware work. On the other hand, there were (much shorter) spells when derivations were all I did.

More statistics would have been useful, especially if better targeted to hypothesis rejection, etc. Simple statistics should be taught as soon as possible, starting before high school. How to Lie with Statistics is good.

But I don't think calculus is intrinsically all that hard. There are only two main concepts to learn--what's a derivative, and what's an integral. The rest is just algebra--and the clever changes of variables and whatnot that nobody actually uses in the field. We look them up, or if we need numerical answers, use programs designed to do integrations in stable ways. (Just trying to translate your textbook equations into computer code is fraught with pitfalls thanks to the finite resolution of numbers in a computer.)

I think we can do calculus better if we divide it. One short course would be calculus basics (differentiation and integration), another would be calculus methods (how to play the fancy games), and then you go on to advanced calculus (with the fancy rigor). The first is the one you require of non-math majors.

True, a field geologist may not use much of it, but what happens when he wants to model crustal pressures? Even having just a passing knowledge of rates of change beats just accepting a black box result. I wrote earlier of different levels in math: arithmetic, understanding the abstractions when explained, able to use the abstractions yourself, and able to do research. I think a minimum for most scientists would be "understanding the abstractions when explained." The field geologist needs to know enough to know the tools to use, and have an approximate notion of what sort of results the computer program should give him and why. But how much more depends on what he'll be doing.

There's a story that Terry Pratchett was addressing some physicists, and explaining that he'd wanted to become one, but had problems with calculus and wound up a writer instead--and met with laughter, as people explained that they rarely used anything more than algebra. They were wrong and Pratchett right. If he wanted to understand the field, he needed to know the language. And in the event he probably contributed more as a writer than he ever would have as a physicist.

From "Physicists continue to laugh", translated by Lorraine Kapitanoff:

Group QuestionedTotalKnewDon't KnowAnswer
Writer-Realists 11 74 They argue until hoarse in smoke filled rooms. It is not known why they set up unintelligible dangerous experiments using huge apparatus.
Writer-Visionaries 58 58 0They work on enormous electronic machines called electronic brains. They work primarily in the cosmos.
First year college students65 65 0They speculate a lot. They make discoveries no less than once a month.
Graduate students 30 10 20They solder circuits. They ask the older ones to find the leak. They write articles.
Young scientific staff members, experimenters 19 190 They run to the equipment department. They scrub rotary vacuum pumps. They flap their ears at seminars.
Young scientific staff member, theoreticians 19 19 0They converse in corridors helping to make great discoveries. They write formulae, mostly incorrect.
Older scientific Staff members761They attend meetings. They help younger scientific staff members to find the leak.
Members of the personnel department550 Experimenters must arrive at 8.25 so that at 8.30 they can sit silently next to apparatus which is running. Theoreticians do not work at all.
Members of the guard force6 6 0They walk back and forth. They present passes upside down.
Representatives of the Ministry of Finance18180They spend money to no purpose


David Foster said...

There are very few real-world problems that can be solved developing a general solution formula via symbol manipulation, as is done in calculus classes. Much more commonly, they require numerical, step-by-step solutions, which means computer solutions unless one is a masochist.

What is important for the non-mathematician is to learn how to translate the real-world problem into equation form.

james said...

Quite true. And the specialist needs to know approximately what the solution should look like.