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← Mathematics: stupid and clever questions for people who understand

Jos Gibbons's Avatar Jump to comment 25 by Jos Gibbons

I don’t know whether septiman444 knows the answer to his/her own question, but I’m sure some people won’t, so I’ll answer it. Its answer boils down the same was as “what’s 1/0?” (except that, as far as I’m aware, no non-standard theory solves this one in the way wheel theory solves that).

What about (1/x).sin(1/x) when x ---> 0 ?

As with division by 0, the “what is it?” (in this case, what is the limit) question shouldn’t be worded so as to assume the existence of a quantity that, it turns out, doesn’t exist. In this case, it would mean, “for which a is it true that, for any δ>0, there’s an ε such that, if |x-0|<δ |(1/x) sin (1/x)-a|<ε?” (The wording is different when a=∞, but that’s not “the limit” here either.) Needless to say, the problem is no a has that property. For the special case x = 1/(2n+½)π sets (1/x)sin(1/x) to (2n+1/2)π, which diverges as n goes to ∞ (equivalent to x going to 0), whereas by contrast if x = 1/nπ we get 0 instead.

Although it may be tangent to the actual discussion topic

Stealth pun!

I would be very interested in your take on how Isaac Newton discovered the derivative. For instance, what was the mathematical world like just prior to the year 1665, with respect to problems that are now child's play for the Calculus? Legend has it that an apple inspired the discovery of gravity, was there a similar "Eureka" moment that led to the birth of "fluxions"? And please consider this request a compliment to your astounding erudition in a wide range of fields. I don't think many could attempt to divine the mind of an acknowledged genius like Newton, if you'll pardon the expression.

I see you pick and choose which puns to point out. Anyway, I’ll give it a shot.
The first thing to understand is that, when mathematicians solve a new maths problem, historically it’s usually because a new practical problem requires it. The second thing is to understand that, while the derivatives of a handful of specific functions were needed for one approach to calculating the areas and volumes of certain shapes, which is why even the ancient world did some very rudimentary “calculus” as we would now identify it, to motivate a differentiation algorithm for arbitrary functions required mathematicians to care about how quickly arbitrary functions change, which basically meant it had to wait for Newton to care about motion. By contrast, Leibniz was a polymath who wanted to one day see calculations solve every problem regardless of field, so essentially he invented calculus because he noticed it didn’t exist yet. (Well, technically Newton beat him by 9 years; there’s a big controversy regarding whether Leibniz really knew this, but at any rate his emphasis differed from that of Newton, because Leibniz was all about the integrals.)
The next thing to realise is that calculus has quite a detailed history. As I said, developments came because we wanted them. Eureka moments are so named because Archimedes allegedly yelled Eureka when, upon noticing his body’s displacement of water, he worked out how to solve a problem that had been assigned to him. I don’t think we know, for any of the specific developments in the history of calculus, that there’s an associated “He worked it out like Dr House in such and such a coincidental moment” story. And, of course, both the bath story and the apple story are now considered myths. A weaker requirement, which Wikipedia calls the Eureka effect, is just that you “suddenly work it out”, which it seems is quite common in mathematics. Or so we think. In my experience, it’s more like, “we realise we could solve the problem step by step in this way, and one hard part (possibly the only one) is here”, and that one hard part has a Eureka moment, or possibly gets broken down again etc. I suppose there has to be something discrete in how the problems are solved, but a gradual breakdown might seem unworthy of the term “Eureka moment”.

Wed, 09 May 2012 06:05:37 UTC | #940680