Interestingly, I never found this to be a problem with mathematics, although I did find it a problem many times I tried to teach myself physics. In my experience, textbook-level mathematics is almost always a perfect self-contained edifice of logic, whereas textbook-level physics often leaves unclear points that you can clarify only by asking an expert or finding another book that addresses that specific point. (I suppose things might be different if you’re reading bleeding-edge math research papers.)
Out of the large number of mathematical texts I’ve read, I can recall only one occasion when I felt genuinely confused after making the effort to understand the text in-depth. In this case, it turned out that this was indeed a fundamental conceptual error in the text. (I can write down the details if anyone is interested—it provides for a nice case study of what seems “obvious” even in rigorous math.) Otherwise, I’ve always found mathematics to be perfectly clear and understandable with reasonable effort, as long as you can locate all the literature that’s referenced.
The case to which I referred was when I first studied calculus as a teenager. The book I was reading took what I think is the standard approach to handling trigonometric functions, namely first prove that the limit of sin(x)/x is 1 when x->0, and then use this result to derive all kinds of interesting things. However, the proof of this limit, as set forth in the book, used the formula for the length of an arc. But how is this length defined? Clearly, you have to define the Riemann (or some other) integral before it makes sense to talk about lengths of curves, and then an integral must be used to calculate the formula for arc length based on the coordinate equations for a circle—even though that formula is obvious intuitively. But I could not think of a way to integrate the arc length without, somewhere along the way, using some result that depends indirectly on the mentioned limit of sin(x)/x!
All this confused me greatly. Wasn’t it illegitimate to even speak about arc lengths before integrals, and even if this must be done for reasons of convenience—you can’t wait all until integrals are introduced before you let people use derivatives of sine and cosine—shouldn’t it be accompanied by a caveat to this effect? Even worse, it seemed like there was a chicken-and-egg problem between the proofs of lim(sin(x)/x)=1 for x->0 and the formula for arc length.
This was before you could look for answers to questions online, and it was unguided self-study so I had no one to ask, and it took a while before I stumbled onto another book that specifically mentioned this problem and addressed it by showing how arc lengths can be integrated without trigonometric functions. So it turned out that I had identified the problem correctly after all. But considering that I was a complete novice and thus couldn’t trust my own judgment, I had an awfully disturbing feeling that I might be missing some important point spectacularly.
Thanks for this. I guess this goes to show how hard it can be to communicate math well. When I learned the sin(x)/x limit I accepted the “proof” by geometric intuition with no protest and was not alert to any deeper source of confusion here.
Come to think of it, the rigorous treatments of sine that I’ve seen probably all use power series definitions. To see that it’s the same function as the one defined using triangles I expect you have to appeal to derivative properties, so that approach would not skirt the issue.
Interestingly, I never found this to be a problem with mathematics, although I did find it a problem many times I tried to teach myself physics. In my experience, textbook-level mathematics is almost always a perfect self-contained edifice of logic, whereas textbook-level physics often leaves unclear points that you can clarify only by asking an expert or finding another book that addresses that specific point. (I suppose things might be different if you’re reading bleeding-edge math research papers.)
Out of the large number of mathematical texts I’ve read, I can recall only one occasion when I felt genuinely confused after making the effort to understand the text in-depth. In this case, it turned out that this was indeed a fundamental conceptual error in the text. (I can write down the details if anyone is interested—it provides for a nice case study of what seems “obvious” even in rigorous math.) Otherwise, I’ve always found mathematics to be perfectly clear and understandable with reasonable effort, as long as you can locate all the literature that’s referenced.
I agree the problem is even more pronounced in physics.
Also, I am interested in and would appreciate the details of the case study to which you refer.
The case to which I referred was when I first studied calculus as a teenager. The book I was reading took what I think is the standard approach to handling trigonometric functions, namely first prove that the limit of sin(x)/x is 1 when x->0, and then use this result to derive all kinds of interesting things. However, the proof of this limit, as set forth in the book, used the formula for the length of an arc. But how is this length defined? Clearly, you have to define the Riemann (or some other) integral before it makes sense to talk about lengths of curves, and then an integral must be used to calculate the formula for arc length based on the coordinate equations for a circle—even though that formula is obvious intuitively. But I could not think of a way to integrate the arc length without, somewhere along the way, using some result that depends indirectly on the mentioned limit of sin(x)/x!
All this confused me greatly. Wasn’t it illegitimate to even speak about arc lengths before integrals, and even if this must be done for reasons of convenience—you can’t wait all until integrals are introduced before you let people use derivatives of sine and cosine—shouldn’t it be accompanied by a caveat to this effect? Even worse, it seemed like there was a chicken-and-egg problem between the proofs of lim(sin(x)/x)=1 for x->0 and the formula for arc length.
This was before you could look for answers to questions online, and it was unguided self-study so I had no one to ask, and it took a while before I stumbled onto another book that specifically mentioned this problem and addressed it by showing how arc lengths can be integrated without trigonometric functions. So it turned out that I had identified the problem correctly after all. But considering that I was a complete novice and thus couldn’t trust my own judgment, I had an awfully disturbing feeling that I might be missing some important point spectacularly.
Thanks for this. I guess this goes to show how hard it can be to communicate math well. When I learned the sin(x)/x limit I accepted the “proof” by geometric intuition with no protest and was not alert to any deeper source of confusion here.
Come to think of it, the rigorous treatments of sine that I’ve seen probably all use power series definitions. To see that it’s the same function as the one defined using triangles I expect you have to appeal to derivative properties, so that approach would not skirt the issue.