
Re: Mainstream Mathematics?
Posted:
Jun 12, 2013 2:08 PM


On Wednesday, June 12, 2013 2:55:58 AM UTC7, Norbert_Paul wrote: > Zeit Geist wrote: > > > On Tuesday, June 11, 2013 2:22:42 PM UTC7, Norbert_Paul wrote: > > >> > > >> Acutally I one question comes into my mind: > > >> Why can the fact that > > >> "Topology covers catergorical statements > > >> that are true or wrong" > > >> make topology unsuitable for spatial > > >> data modelling? > > >> That quotation is an actual reviewer's > > >> objection against a research proposal which > > >> was then rejected. > > > > > > Does that statement imply that Topology cannot be used > > > as a foundation for Spatial Data Modelling? > > > > This statememnt is used as an argument atgainst unsing > > topology that way. > > > > > If so, why? > > > > He says that topology does not respect the "inherent uncertainty > > of numerical data". >
Any pure idealized mathematical theory usually doesn't address numerical uncertainty. Why isn't it feasible to use some numerical analysis theory in connection with Topology.
> > > Because it "makes categorical statements that are > > > true or wrong"? > > > > The context of the statement shows that "topology" is confounded > > with Max Egenhofer's "4 or 9intersection model". So a "categorical > > statement" is if two point sets in a space "meet", "overlap", etc. > > > > > True in what context? > > > What does it mean for a statement to be wrong? > > > > For example: "The intervals [1,2] and [2,3] overlap". > > Actually the reviewer is confounding "statement" with "predicate". >
I'm unfamiliar with the context of this discussion. I have, however, been looking at sites on Spatial Data Modeling and Engenhofer.
Please correct me if I way off base, but SDM seems to address the problem of taking spatial position of objects in a space, the data, and produce a representation of the objects in that space, the model.
So, I take my TriCorder out, scan an area of the surface of some planet. I then beam back up to my ship. I plug my TriCorder into The Holodeck computer. Then, using it SDM program creates a hologram of the planet.
I won't go on from here, just in case I'm incorrect. However, some questions?
Is that book of Egenhofer a Mathematical text or a Computer Science book on SDM?
Was the Reviewer you speak of Math or CS?
And are a Mathematician and/or a Computer Scientist? > > But Egenhofer's initial work, in fact, contains mathematically wrong > > statements (together with wrong proofs). > > See Lemmas 1 and 2 in http://www.spatial.maine.edu/~max/MJEJRHSDH1990.pdf. > > > In later papers this error is avoided by using other definitions of "region". > > The initial "region" definition is on page 808: > > Region: * 2dimensional object in IR^2 > > * connected pointset > > * nonempty interior > > * connected boundary > > > > Against the two "proven" Lemmas the intersections named r2: r5, r9, and r14 DO exist: > > r2: 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: > > IR^2 is a region. It satisfies all above "axioms". As it has > > an empty boundary its boundary is connected. > > Hence A = B = IR^2 has this pattern. > Does he later put bounds on "region" so that the entire space is not allowed?
> > 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.
A subset X of the real line that contains at least two distinct points is connected if and only if it is an interval.
Theorem 3.4 Bert Mendelson, "Introduction To Topology", Third Edition
> > The sets > > int A = (1,0) x (0,1) > > fr A = ({1,0} x [0,1]) \/ ([1,0] x {0,1}) \/ B > > int B = (0,1) x (0,1) > > fr B = {0,1} x [0,1] \/ [0,1] x {0,1} > > have exactly this pattern of intersections. > > > > r9: is symmetric to r5. > > > > r14: 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 =/= {} > > CounterExapmple: > > A = R^2 \ {(0,0)} = int A > > B = R^2 \ {(1,1)} = int B > > fr A = {(0,0)} > > fr B = {(1,1)} > > > > Note: The wrong argument in proof 1 is that "any point on the > > boundary of an object must be arbitrarily close to some point in > > the interior." This claimed property has been added to the > > defintion of "region" in later publications. >
Actually, don't we need
Every point of the Interior must be arbitrarily close to some point of the Boundary.
This bounds the object in all dimensions.
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 have never seen an Erratum but maybe there is one within the > > numerous papers on 9intersections an I didn't yet spot it. >
I don't think he has a solid Topological foundation to do the "topology" presented here.
> > NP
Thanx for your time, ZG

