
Re: Mainstream Mathematics?
Posted:
Jun 12, 2013 5:55 AM


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".
> 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".
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.
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.
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.
I have never seen an Erratum but maybe there is one within the numerous papers on 9intersections an I didn't yet spot it.
NP

