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Re: Mainstream Mathematics?
Posted:
Jun 12, 2013 2:08 PM
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On Wednesday, June 12, 2013 2:55:58 AM UTC-7, Norbert_Paul wrote:
> Zeit Geist wrote:
>
> > On Tuesday, June 11, 2013 2:22:42 PM UTC-7, Norbert_Paul wrote:
>
> >>
>
> >> Acutally I one question comes into my mind:
>
> >> Why can the fact that
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> >> "Topology covers catergorical statements
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> >> that are true or wrong"
>
> >> make topology unsuitable for spatial
>
> >> data modelling?
>
> >> That quotation is an actual reviewer's
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> >> objection against a research proposal which
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> >> was then rejected.
>
> >
>
> > Does that statement imply that Topology cannot be used
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> > as a foundation for Spatial Data Modelling?
>
>
>
> This statememnt is used as an argument atgainst unsing
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> 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 9-intersection model". So a "categorical
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> 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 Tri-Corder out, scan an area
of the surface of some planet. I then beam
back up to my ship. I plug my Tri-Corder 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/MJEJRH-SDH1990.pdf.
>
>
> In later papers this error is avoided by using other definitions of "region".
>
> The initial "region" definition is on page 808:
>
> Region: * 2-dimensional object in IR^2
>
> * connected point-set
>
> * non-empty 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 = {}
>
> Counter-Example:
>
> IR^2 is a region. It satisfies all above "axioms". As it has
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> 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 = {}
>
> Counter-Example:
>
> 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
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> int B = (0,1) x (0,1)
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> fr B = {0,1} x [0,1] \/ [0,1] x {0,1}
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> 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 =/= {}
>
> Counter-Exapmple:
>
> 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
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> boundary of an object must be arbitrarily close to some point in
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> the interior." This claimed property has been added to the
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> 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 counter-example.
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 9-intersections 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
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