(published on 10/22/2007)
|For sure the graphite from the pencil is grainy. You deposit a
finite number of grains. The question is whether the space in which you
deposit them is really grainy. I don't see any way an informal verbal
argument could settle that question. |
You may want to look up "Zeno's paradox" in a calculus textbook. Zeno asked the question that if a runner runs a finite distance in a finite time, you can think of him running a series of small distances, all added together. If you slice up his racetrack into infinitely many small pieces (and the example was one of cutting each piece in half -- no matter how close the runner got to the end, he would have to travel halfway there and then the other half, which could itself be broken into halves and so forth). Zeno reasoned, erroneously, that to travel across an infinite set of these intervals "must" take an infinite amount of time. For the last few hundred years, we realize that the sum of Zeno's geometric series converges to a finite number which is easily calculated to be the length of the original racetrack.
Relativitity and quantum mechanics when interpreted together (and we are not good at doing this) seme to indicate that space and time may not be continuous and flat all the way down to very very very tiny distance scales. On very short distance and time scales, quantum fluctuations can bend space and time around. Ordinary calculations of quantum corrections to observable quantities involve adding up all kinds of processes involving virtual particle exchange. If you add up all of these without upper bound on their momentum (that is, without a lower bound on their wavelength), you sometimes get absurd, infinite answers. So people stop their integrals at some small distance scale and say that the theory breaks down, hoping that we will discover eventually what kinds of interactions take place between particles on these small distance scales.
(republished on 07/18/06)