You are standing in a room against one wall facing the opposite wall. The goal is to cross the room in stages, each one being half the distance that remains between the last step and the far wall. You keep this up until you bump into the far wall. Lets switch to an example we can better deal with, a line 2 inches long that we can mark our progress points on. Our first move is half the distance, so we move 1", the next is half the distance remaining, so we move 1/2" for a total of 1.5", our next move is half again or 1/4" for a total of 1.75", and so it goes half again, and again.
Q1) how many points can be marked on the line?
Q2) what is the total of your addition, the total distance traveled along the line?
Q3) this exercise sparked wild debate for hundreds of years, which calmed down (some) in the 1700's. There is a conclusion to be drawn from it that once stated sounds stupidly obvious and simple but had not been solved or agreed upon until then (1700's). Any ideas what it was?
(Why the rewrite? NAC pointed out that my quiz from yesterday may not be clear, so I have added both perspicuity and some run on sentences in order to make sure it is not clear. Sarge then made a smart ass remark about my formative entertainment and religious experiences that I find inappropriate due to the fact that I hauled his ass across an international border in the rain in a $400 VW to his first epiphany.)
I have no idea what problem I'm being asked to solve here, but the most obvious thing to me is that a person would never actually reach a destination if he only moved halfway to it each time he moved. By definition, moving halfway toward an object implies a remaining distance, and always will.
ReplyDeleteHow am I doing?
I think it was Anaxagoras who used such an example to "prove" that motion was impossible. You would like to move across the room but before you can you must first cross the half-way point. And before THAT you first must cross the one quarter point. And BEFORE THAT ... Therefore, motion is impossible.
ReplyDeleteI'll stick with my previous answer to a). A line is an infinity of points.
Mathematically, you can claim the full two inches, because as you edge infinitely closer the distance between you and the end of the line becomes infinitely small. If the distance is infinitely small, expressed as the fraction 1/infinity sign then the distance is 2 because there is no number that you can name that fits between your location on the line and 2.
Maybe someone else wants that last question.
Squatlo is correct and NAC is correct. From a number finally broke through the idea of infinity, and the mathematical formulas for motion and time and gravity. The industrial revolution could not occur until these things were solved, the formula set. It is said at this time there were 100 men who if they could have all been murdered, we would still live in the late middle ages, with streets heaped with shit and our own dead.
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