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Topic: Mainstream Mathematics?
Replies: 32   Last Post: Jun 16, 2013 2:07 PM

 Messages: [ Previous | Next ]
 Norbert_Paul Posts: 48 Registered: 3/23/10
Re: Mainstream Mathematics?
Posted: Jun 12, 2013 5:55 AM

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
>> "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".

> 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
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".

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
an empty boundary its boundary is connected.
Hence A = B = IR^2 has this pattern.

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.

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 =/= {}
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
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.

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.

NP

Date Subject Author
6/11/13 Norbert_Paul
6/11/13 LudovicoVan
6/11/13 Norbert_Paul
6/11/13 Tucsondrew@me.com
6/12/13 Norbert_Paul
6/12/13 Tucsondrew@me.com
6/12/13 ross.finlayson@gmail.com
6/13/13 Norbert_Paul
6/13/13 fom
6/14/13 Norbert_Paul
6/14/13 LudovicoVan
6/14/13 fom
6/15/13 ross.finlayson@gmail.com
6/15/13 ross.finlayson@gmail.com
6/15/13 FredJeffries@gmail.com
6/16/13 ross.finlayson@gmail.com
6/11/13 Tucsondrew@me.com
6/11/13 Norbert_Paul
6/11/13 Tucsondrew@me.com
6/12/13 Norbert_Paul
6/12/13 Tucsondrew@me.com
6/12/13 Norbert_Paul
6/13/13 Tucsondrew@me.com
6/13/13 Peter Percival
6/13/13 Norbert_Paul
6/11/13 Peter Percival
6/11/13 Rick Decker
6/13/13 Dan Christensen
6/11/13 William Elliot
6/12/13 Norbert_Paul
6/12/13 amzoti
6/12/13 David Bernier
6/13/13 Stephen Wynn