
Re: Mainstream Mathematics?
Posted:
Jun 12, 2013 5:07 PM


Zeit Geist wrote: > 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.
No. there is even a dedicated theory on errors.
> 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.
Here you must ask the reviewer. But he is anonymous. Actually I think that the argument is forged in the attempt to take down a proposal. I cannot justify that here but reading the whole review suggests that.
>>> 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? Article (not book) on Computer Science
> Was the Reviewer you speak of Math or CS? CS but obviously from the GeoInformation field.
> And are a Mathematician and/or a Computer Scientist? ^ "you?"+ Computer Science but coworking with a Mathematician
>> 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? Yes.
>> 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 and looks like this:  1  .  . < the horizontal lines [0,1]x{q}  .  q2  q1  0 0 1 ^  + the vertical line {0} x [0,1]
> > 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.
> 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".

