Regarding your first point, this quote by Gian-Carlo Rota perhaps says what I want to say best:
“Most people, even some scientists, think that mathematics applies because you learn Theorem Three and Theorem Three somehow explains the laws of nature. This does not happen even in science fiction novels; it is pure fantasy. The results of mathematics are seldom directly applied; it is the definitions that are really useful. Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do. This you cannot see unless you take a course in abstract differential equations. What applies is the cultural background you get from a course in differential equation, not the specific theorems. If you want to learn French, you have to live the life of France, not just memorize thousands of words. If you want to apply differential equations, you have to live the life of differential equations. When you live this life, you can go back to molecular biology with a new set of eyes that will see things you could not otherwise see.”
Regarding your second point, I think one of the principles behind democracy is that even if each individual voter is not an expert, their collective decisions (ideally) will be similar to those of ‘experts’. Think of Boosting in AI—you can often combine many weak learners to form a strong learner. So it is reasonable to expect politicians to be more qualified (restricted to policy making, of course) than the voters.
Yes the concepts are the main thing useful you learn when you learn math. But not all math is equally useful in all areas. Economists hardly ever use differential equations; that is not a math that helps much there. So in learning econ Congressfolks should learn some math, yes, but probably not differential equations.
That’s an unfortunate example: to the extent which economics is quantifiable, it’s all differential equations… heck, marginal utility is a differential equation!
Maybe they weren’t explicitly stated as differential equations but:
Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do.
Speaking of differential equations in economics, a friend of mine has had an idea that there should be an economics textbook for mathematicians, because it annoyed him so much that they seem to dance around mathematical concepts—for example, marginal anything is clearly a derivative, although normal econ textbooks never call it that.
That’s odd—if anything, econ usually gets accused of being way too much about mathematical formalism. This, for example, might as well be “an economics textbook for mathematicians”; maybe your friend will find it helpful.
Regarding your first point, this quote by Gian-Carlo Rota perhaps says what I want to say best:
“Most people, even some scientists, think that mathematics applies because you learn Theorem Three and Theorem Three somehow explains the laws of nature. This does not happen even in science fiction novels; it is pure fantasy. The results of mathematics are seldom directly applied; it is the definitions that are really useful. Once you learn the concept of a differential equation, you see differential equations all over, no matter what you do. This you cannot see unless you take a course in abstract differential equations. What applies is the cultural background you get from a course in differential equation, not the specific theorems. If you want to learn French, you have to live the life of France, not just memorize thousands of words. If you want to apply differential equations, you have to live the life of differential equations. When you live this life, you can go back to molecular biology with a new set of eyes that will see things you could not otherwise see.”
Regarding your second point, I think one of the principles behind democracy is that even if each individual voter is not an expert, their collective decisions (ideally) will be similar to those of ‘experts’. Think of Boosting in AI—you can often combine many weak learners to form a strong learner. So it is reasonable to expect politicians to be more qualified (restricted to policy making, of course) than the voters.
Yes the concepts are the main thing useful you learn when you learn math. But not all math is equally useful in all areas. Economists hardly ever use differential equations; that is not a math that helps much there. So in learning econ Congressfolks should learn some math, yes, but probably not differential equations.
That’s an unfortunate example: to the extent which economics is quantifiable, it’s all differential equations… heck, marginal utility is a differential equation!
Maybe they weren’t explicitly stated as differential equations but:
so it can’t be helped.
Speaking of differential equations in economics, a friend of mine has had an idea that there should be an economics textbook for mathematicians, because it annoyed him so much that they seem to dance around mathematical concepts—for example, marginal anything is clearly a derivative, although normal econ textbooks never call it that.
Not in the discrete case.
That’s odd—if anything, econ usually gets accused of being way too much about mathematical formalism. This, for example, might as well be “an economics textbook for mathematicians”; maybe your friend will find it helpful.
Do you mean a math text for economists?