
Re: Visual Presentation of Real Number System
Posted:
Nov 29, 2013 11:27 AM


>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
That is fine for sequences series that converge to a computable number. Any real can be viewed as the limit of a sequence of of rationals. While each of those rationals is computable, the sequence itself, generally, is not.
The problem is "sloppy talk that fails to distinguish the mathematical from the heuristic, the picturesque, the realm of applied from the realm of the math itself."
The problem here is "choose" or "pick"  no one can do that for a random real. To pick or choose means to distinguish one from the rest. A description of what such a number would sort of look like, or what some of its properties would have to be, as you gave, even if completely formal, does not distinguish one random real from all the rest.
There is no mathematization of the "dart" or the actions performed with the dart, that can do the job either. Its an illformed statement, a fantasy that exists neither in the real world in in the world of mathematics.
R Hansen says: >Using physics metaphorically is not a problem, when, as you stated, it is done by a mathematically mature person. & >I still am not sure what your issue is.
My issue is, that in the world of probability, there is a "echo" of the confusion between realworld stuff similar to the way some students may be confused between pictures, and the math that lies behind the picture. With probability though, it also has to do with notions of time and action. My claim, such as it is, is I think its harder for many writers on probability matters to keep those two worlds separate and clear in their writing, in their teaching.
I wasn't accusing you of being similarly confused. As far as the author of that link I posted, while he may not be internally confused, his writing style doesn't help others see the truth. He persists with the "zero probability events happen all the time" meme.
I think greater efforts to mentally separate those two worlds would help the world of mathematical teaching. I wouldn't expunge those real world metaphors any more than I would expunge pictures, just make sure everyone knows which is which.
And, as I said, I was having some fun here. If you came up with your ideas about particular (computable) numbers and nonparticular numbers on the spot in this discussion, then congratulations, and I hope you had some fun too.
Cheers, Joe NM

