The following joke is very old (the following version comes from wikipedia).
An astronomer, a physicist and a mathematician are on a train in Scotland. The astronomer looks out of the window, sees a black sheep standing in a field, and remarks, “How odd. Scottish sheep are black.” “No, no, no!” says the physicist. “Only some Scottish sheep are black.” The mathematician rolls his eyes at his companions’ muddled thinking and says, “In Scotland, there is at least one sheep, at least one side of which looks black.”
One skill that mathematicians are encouraged to develop is the ability to precisely formulate ideas, questions, etc., sometimes overly precisely. We have lots of vocabulary and syntax conventions that helps us to draw very fine distinctions.
This is a common point of contention between mathematicians and non-mathematicians when discussing mathematical ideas. Such as, let me think, teachers and students in a math classroom.
Math teachers, how many times have your students objected, “But you know what I meant!” when you correct something they say in a discussion or write on an exam? Students, how many times have you said that?
(I would say that communication that only works when both parties already know what the other is trying to say is of rather limited use . . . but I digress.)
Mathematics is not just a big collection of mathematical facts. It’s the hidden patterns, symmetries, and structure of mathematical objects, the poetry of interaction among objects and their properties. The advanced branches of mathematics take as their fundamentals the nuances of the simple branches. If we are blurry and vague on the little objects, then the interesting mathematical ideas about their interaction don’t even exist in any meaningful way.
There’s a partial analogy with history classes here. History is a subject with a reputation for being grounded in a large repository of facts to learn. Which is preferable, merely learning myriad historical facts? Or learning myriad (but possibly less) historical facts and the relationships among these facts, the stories told by these events, the ideas the underlie them, etc.? I think we all would prefer the latter. But that doesn’t mean that we go to the other extreme and say the best history class would have no facts at all, just higher-order reasoning. That is arrantly nonsensical; reasoning about what?
Compare the situation with music. I was never very strong in band class (I played the cornet, for some suitably liberal definition of “played”). Wrong notes from me were not uncommon. And the band class teacher insisted, insisted that I and the other wrong-note-players focus on playing the correct notes. We spent time on that that we could have been spending playing beautiful music, or learning about. And it’s not like we didn’t know what note we were supposed to play. It was supposed to be an A and I played an A-flat, but you knew what I meant, right?
And even though she knew what I meant, she wouldn’t just let it slide. Picky, picky, picky, picky! Wouldn’t just ignore it and move on to more interesting topics, more mentally stimulating topics. And she calls herself a teacher!
I hope it’s obvious that I’m being ridiculous.
Imagine a civilization with extraordinarily primitive “musical instruments”, and a universally bad ear for music. Their instruments only play tones “approximately”. If one plays a C and another tries to follow along, he might play a B, or a C-sharp, or a very out-of-tune C, and that’s as close as this civilization would expect. After all, if they’re that close, it will be generally understand that he “meant” to be playing the same note.
Imagine what music would be like in such a scenario. The would be no point in having songs as we know them, because an. In some sense the only aesthetically meaningful aspect their music (to the extent we could stand hearing it at all) would be in the rhythms. (It’s like Victor Borge’s apocryphal one-key piano, but harder on the ears.) Melody, harmony, key signatures, chord progressions, these are all meaningless concepts if our notion of “note” is that blurry. A musicologist travelling in this strange world could explain chord progressions to these people, could explain the nuances of minor versus major keys, but it would be pointless to do so. Both parties would feel like they were wasting their time, and both parties would be right.
And so it is in mathematics: the interesting concepts don’t even come into existence until we can talk about and understand the simple things with precision and nuance.