
Re: Visual Presentation of Real Number System
Posted:
Nov 29, 2013 1:56 AM



On Nov 28, 2013, at 9:19 PM, Joe Niederberger <niederberger@comcast.net> wrote:
> Happening, selecting, picking are either physics, or computation. To paraphrase Yoda, "The real number > just is, there is no picking". > > > My problem is not with infinite digits. You cannot actually pick a random real (or at least ever finish picking it.)
I don't know why you are having a problem with this. You make it sound like Zeno's paradoxes are actual paradoxes. And it isn't infinite digits, it is an infinite *sequence* of digits. You are right, we never finish picking it, like we never finish 1/2 + 1/4 + 1/8 + 1/(2^n)..., but we know the sum is 1. You started with the reasonable argument that a ray of light would never hit a particular number. All I did is realize that the ray of light must hit a non particular number, which we know realize is an uncomputable number.
> This business of saying "it happens all the time" is pure H.S. That is neither logical, or mathematical. For an example that is easier to unwind, look up the formal definition of the axiom of choice. There is no choosing involved.
I didn't see this discussion as being so odd that we needed to unwind it.
> > > The metaphors of choosing, or "picking a random real" are crutches, much like those pictures you are always railing against. I think those metaphors though, are more insidious than pictures, because while its easy for a mathematically mature person to distinguish between a picture and the mathematics that may map to that picture, the metaphors surrounding probability are more ingrained and unconscious.
Using physics metaphorically is not a problem, when, as you stated, it is done by a mathematically mature person. Even using pictures is not a problem under the same circumstances. When I draw a line and say it is a mathematical line I am speaking metaphorically. Actual physics gets around this because it uses constants which are uncomputable numbers. The speed of light is an uncomputable number. Plank's constant is an uncomputable number. In every sense of this discussion.
I still am not sure what your issue is. You started with the argument of the impossibility of a ray of light hitting a particular number, which I found reasonable. I just realized, along the same line of reasoning, the ray of light couldn't hit any particular number. It seems that it hits an uncomputable number. Do you have a problem with the idea of an uncomputable number? Or is it with my infinite sequence argument concluding with the comparison of the size of the set of uncomputable numbers and computable numbers?
Bob Hansen

