On Sun, Sep 23, 2018 at 11:50 PM, Keith F. Lynch <kfl@keithlynch.net> wrote:
One of the most astonishing (to me) calculus theorems is that any smooth curve (in the sense that all of its derivatives are continuous) can only be extended in one way. Any non-zero-length segment of such a curve, no matter how short, uniquely determines the rest of the curve. Also, if you know all the derivatives at any one point, then you know the value at every point.
What's most astonishing to me is the fact that calculus classes, while usually not actually making any false statements, seem designed to mislead people into thinking that this statement is true. It's completely false. Consider the function f(x) = 0, defined on [0, 1]. If your claimed theorem was true, the the only smooth extension of this function to the real numbers would be identically 0. But one can extend it for x > 1 with the function f(x) = e^(- 1 / (x-1)^2) and for x < 1 with the function f(x) = -47 e(- (1/x^2)) and the resulting function is infinitely differentiable everywhere, even at 0 and 1. In fact, given an infinitely differentiable function defined on [0,1], and another infinitely differentiable function on [2,3], there is a way (infinitely many ways, in fact), to interpolate between them to produce an infinitely differentiable function on [0, 3] that extends them both. So rather than knowing exactly what a function can do by knowing in on a non-zero length segment, it can literally do anything whatsoever any short finite distance away from that segment, and remain infinitely differentiable. Why do so many people think the false "theorem" is true? One reason is that when we consider functions from the complex numbers to the complex numbers, rather than from the reals to the reals, the theorem is true. The statement that a function from C to C is differentiable is a much stronger statement than the statement that it is differentiable as a function from R^2 to R^2. The latter statement says that the function is well approximated locally by a function in the family of linear functions from R^2 to R^2, a space with real dimension 4. The former statement says that this function is one of the ones that corresponds to multiplication by a complex number, a linear subspace of dimension 2. The statement that a function is differentiable from C to C doesn't just say it's smooth; it says it's conformal, a much more rigid criterion. The fact that the theorem is true in complex analysis has little bearing on the kind of curve-fitting we are talking about here, since there is no meaningful extension of the functions in question to complex arguments and values, and no reason to expect that the resulting function would be complex-analytic if we did extend it. But I don't think confusion with the complex numbers is the main reason people believe this false theorem; it's a widespread misconception even among those who have never considered the idea that calculus can be done with complex numbers. I think the real reason the false theorem is believed is that calculus texts place great emphasis on the notion of the Taylor series of a function, and want to emphasize its importance. Its importance would seem diminished if the texts let out the dirty secret that even if the Taylor series converges, there is no reason to expect that it converges to the original function, rather than to some completely different function, even on a neighborhood of 0. The expression for the remainder term is given, but nothing is done with it, and the texts quickly move on to other topics, without mention of the fact that even if it converges, this remainder need not converge to 0 as the number of terms goes to infinity, even in a neighborhood of 0.
Since it seems plausible that my own motion is smooth, i.e. I never experience infinite anything (distance, velocity, acceleration, jerk, snap, crackle, or pop (yes, those are the official terms for successive derivatives of position)), that implies that all my past and future travels are predetermined.
Only if you think it plausible that your motion is well-defined and smooth for complex values of t, and that there is reason that the resulting function is differentiable in the strong form of being differentiable as a function from C to C, rather than as a function from R^2 to R^2. Andy