
Re: Mainstream Mathematics?
Posted:
Jun 13, 2013 2:11 AM


On Wednesday, June 12, 2013 2:07:46 PM UTC7, Norbert_Paul wrote: > Zeit Geist wrote: > > > On Wednesday, June 12, 2013 2:55:58 AM UTC7, Norbert_Paul wrote: > > >> Zeit Geist wrote: > > > > Any pure idealized mathematical theory usually doesn't > > > address numerical uncertainty. > > > > No. there is even a dedicated theory on errors. > I just meant that pure Mathematical theories, such Set Theory, Topology, Abstract Algebra and even Geometry and the Calculus, do not, at their basic levels do not use or need a theory of measurement and error. Of course, we can then apply those theories and may then need some measurement/error theory.
> > > Why isn't it feasible to use some numerical analysis > > > theory in connection with Topology. > > > There are even researchers (within other communities) that > > suggest using topology to identify such numerical errors. >
I could see that. If we use the measurements to define the topology on some set (space) and then some funtion on those measurements to determine the associated error possibility/probability, where error function varies consistently when the measurement is passed through a continuous function.
Just a bit of free associating there. The point is most Mathematical ideas can be dealt with in a Topological Space. > > > > >> r5: Claim: There do no exist "regions" A, B in IR^2 s.t. > > >> the following intersection pattern holds: > > >> fr A /\ fr B =/= {} > > >> int A /\ int B = {} > > >> fr A /\ int B =/= {} > > >> int A /\ fr B = {} > > >> > > >> CounterExample: > > >> A = [1,0] x [0,1] \/ ([0,1] x ([0,1] /\ Q)) is a "region" > > >> B = [0,1] x [0,1] is a region. > > > > > > Does A satisfy the condition for "region"? > > > Q is NOT connected. > > > > Almost got me. But ([0,1] x ([0,1] /\ Q)) is > > ([0,1] /\ Q) horiziontal strips of > > connected [0,1] x {q} for q in [0,1] /\ Q > > all attached to a vertical line {0} x [0,1]. > > > It is even path connected...
Yes, I think earlier I read it as (0,1) x [0,1] ^ Q. I stand corrected. Your counterexample seems solid.
> > > Actually, don't we need > > > Every point of the Interior must be arbitrarily close to > > > some point of the Boundary. > > > > This cannot be true in R^2 with the natural topology. It is not > > claimed and not needed. > > > > > If you say: > > > Every point of the Boundary must be arb close to > > > some point of the Interior, > > > then then entire space, R^2 still works for the first counterexample. > > > Because the boundary is empty and vacuously satisfies > > > the condition. > > > I dont say this. But the proof uses the statement and needs it. >
Sorry, I should have used the Mathemtical "we" as I did above that.
> > > I don't think he has a solid Topological foundation > > > to do the "topology" presented here. > > > > That is the point. But he claims so. > > Most important. That community compares everything that uses the > > vocabulary "topology" with 9intersections. If it differs it is > > either "wrong" or "too complicated for the average scientific audience".
The theory may be sound, but only for "nice" sets that are "natural". For instance, the set [0,1] ^ Q doesn't appear much in nature. And if we are doing any measurements we can basically can ignore it. After all, if for all real numbers in [0,1], we define
f(x) = 1, when x is rational And f(x) = 0 when x is irrational;
Then the integral from 0 to 1 over f(x) is 0.
But if its supposed to be a Mathematical theory then It should create it own definition on the basis of standard definitions.
It sounds like these people just don't know how Mthematics works. It IS about catagorical statements. When certain conditions A are met, we can also say that conditions B are met.
As far as being "wrong". He may have meant that Topology has results or considers cases that don't correspond to "reality". That does make it useless or improper for your needs. It might just need some refinement and more in depth definitions.
ZG

