Over the course of my childhood, I remember asking exactly one intelligent question. Unfortunately, I couldn’t make my parents understand what I was asking. Perhaps it was that frustration that deterred me from ever formulating an intelligent question again.
I was, I think, six years old at the time, and my question was this: If you’re traveling at 50 miles an hour at 1:00, and you’re traveling at 70 miles an hour at 2:00, must there be a time in between when you’re traveling exactly 60 miles an hour?
What made this question intelligent—and probably what made it incomprehensible to my parents—was that I was very keen to distinguish it from the question of whether your speedometer would have to pass through the 60-mile-an-hour mark. It seemed clear to me that the answer to that one was yes—that even if your true velocity could somehow skip directly from 50 to 70, the speedometer needle, in the course of whipping around from one reading to the other, would have to pass through the midpoint.
I quite vividly remember worrying that my question about your speed would be misinterpreted as a question about your speedometer, a question to which I thought the answer was obvious and therefore could only be asked by a very stupid person—a very stupid person for whom I did not wish to be mistaken. Therefore I prefaced the question with a long discourse on how it was thoroughly obvious to me that if your speedometer reads 50 miles an hour at one time and 70 miles an hour at another, then surely it must pass through 60 on the way, but that this was not not not not not the question I was about to ask, which concerned your actual speed and not the measurement thereof. By the time I got around to formulating the question itself, my parents (or at least my father; I don’t remember whether my mother was present) had quite understandably given up on figuring out what I was trying to get at.
In retrospect, though, what I was trying to get at was the distinction between the Intermediate Value Theorem, which applies to continuous functions (like your speedometer reading) and Darboux’s Theorem, which applies to derivatives (like your velocity). And I had exactly the right intuition, which is that the Intermediate Value Theorem is easy but Darboux’s Theorem is (comparatively) difficult. In other words, it’s pretty much obvious that the speedometer has to pass through 60 but not so obvious that your actual speed has to pass through 60, although in fact it’s true.
Perhaps a much more intelligent (or obstreperous) child would have questioned the continuity of the speedometer reading, or the continuity of space and/or time itself.
What is the most intelligent question you can remember asking as a child? Did you get a good answer?