Being mathematically critical: how should a student approach statements that appear to be obvious?
Very occasionally, I will read or hear a theorem, and think: isn't that obvious?
Not in a contemptuous "I can immediately see how to prove this" way, but rather in a "I would have thought this was self-evident" way. It then turns out that the result is not completely elementary. This always reminds me of how lacking and unexperienced I must be in the area of mathematical rigour. Below are two scenarii, to illustrate what I mean.
- Suppose I do not know of Rolle's theorem. I am in an exam, and I am dealing with an $\mathbb{R}\to\mathbb{R}$, everywhere differentiable function $f$. I write:
Since $f$ is everywhere differentiable and $f(0)=f(1)$, there must be $x\in (0,1)$ s.t. $f'(x)=0$.
This seems blatantly true, so I carry on without further justification.
- Suppose I do not know of the Bernstein-Schröder theorem. A friend of mine presents me with a proof of something $-$ somewhere in the middle of his argument, he or she writes:
Since $|X|\geq |\mathbb{R}|$ and $|X|\leq |\mathbb{R}|$ we have $|X|=|\mathbb{R}|$.
Lacking experience and being slightly lackadaisical, I accept this and carry on reading. At best, I might reflect on the fact that we are not dealing with the usual laws of arithmetic — but I do not linger on it.
To me, this means my way of reading mathematics, or approaching a statement, is still very naive.
I suppose that, given the time, it would be healthy to stop at every statement, and ask: how would I prove this? But in the context of an exam, or skimming over someone else's work, one does not necessarily have the time for this: so what should one be on the look-out for in these seemingly obvious statements? How can one tell when it is necessary to stop and think: this is not clear.
I apologise if this is a bit general, but I would love to hear thoughts/advice.
Solution 1:
I realize that there is a popular style, especially in undergraduate and beginning graduate coursework (e.g., in the U.S.) to behave as though every small detail were equally deserving of attention and worry. I disagree with this attitude, at least because it fails to distinguish important from unimportant details by declaring every one important.
It is subtler to ask about the "obviousness" of things like Rolle's theorem or intermediate value theorem, and such. I agree, these are "obviously" true, ... or should be, and, indeed, if whatever notion of "real numbers" or "continuity" or "differentiability" we had failed to allow us to prove these things (or provided counter-examples), then more likely we'd enhance those notions however it took to make the conclusion true.
Further, in practice (as opposed to exercise sets and exams in school), I think the most productive approach is to first see what interesting conclusions might arise from some intuitive, heuristic approach, and, if sufficiently interesting, go back and try to bulwark things as much as possible, to give an appropriate degree of surety to the conclusion. I do not say "absolute surety", even though that pretense is part of our mythology. (E.g., the supposed surety of Euclidean geometry, apart from parallel postulate issues, was shown to have gaps/problems by Hilbert's careful analysis.)
And I do think it is especially perverse if one gets into the habit of exaggerated self-doubt, or doubt-of-intuition, as a matter of course. Easy to get nowhere from such a viewpoint, both because of paranoia, and because, after all, what guide is there for "what to do next" than some sort of (refined) intuition?
Solution 2:
If I'm reading math I always (at least subconsciously) categorize statements as:
- I know how to prove this.
- I've seen it before but don't remember how to prove it. (Usually means move on)
- I haven't seen it before. (Either look it up, try to prove it, or move on)
For every single statement I ask "why?" (at least subconsciously) and put it into one of these categories. If the answer instead is "It's true, well...just because," that means you don't know how to prove it.