To quote from today's discussion, "Topology allows us to talk about objects, such as the Klein Bottle, that geometry does not allow." I find it ironic that topology can bestow any system of discussion while it precludes the capacity to talk about the difference between a mug and a donut. We cannot talk about the difference because the only thing we can see, topologically speaking, is the quantity and quality of drawn circles crossing. For the discussion to have gone as far as it did, we need more substance than a few circles crossing. What truly does topology see? Allow us to see? Allow us to talk about?
How does it allow us to talk about the Klein Bottle? The Klein Bottle, topologically, takes the same surface as, say, a plate. You can follow the surface contiguously from the inside and outside, one solid surface. Therefore (borrowing logic from Euclid) a plate is a Klein Bottle. Or rather, topology gives us the tools to talk about the Klein Bottle, but not to distinguish it from a plate!?
Unless topology does give us a tool set, a conversation, a method of describing differences between a donut and a mug, then we cannot talk about the differences between a plate and a Klein bottle and therefore lose all intricacies and any awareness the Klein bottle may have given..
late for class...
Subscribe to:
Post Comments (Atom)
3 comments:
You have misunderstood a great deal; Topology endeavors to distill the basic spatial qualities of objects, both visible to the human eye (3-Dimensional) and extra-sensory (more than 3 Dimensions). For example, we all know that a coffee cup is not EQUAL to a donut - one need only look at them to know this. However, Topology proves that one can be transformed into the other by simple "continuous" operations like pushing, stretching, squashing ... etc.
In technical terms, we say that a coffee cup and a donut are EQUIVALENT spaces, not equal spaces - because it has been proven that there exists a Continuous Mapping Function (a.k.a "homeomorphism") between these two spaces! To use the terminology of the topologist, the sphere and coffee cup are "homeomorphic". In simple terms, this means that they share a fundamental, irrefutable similarities (a.k.a. "topological invariants") in their spatial composition.
In short, I think that it is important to remember that abstraction in thought yields profound understanding... I challenge you to point to a any human discipline which has not benefitted by some infusion of knowledge gleaned from an abstract analysis of that very same discipline.
I hope you're not trying to promote "separate but equal" ideologies..
I think we were just stuck on semantics.. equal, similar, smells the same, they tend bleed together when we get lazy.
m. broderick is both charitable and correct Eric. i just saw the post but had been writing from the bberry since the seminar with follow up points. the plane and the klein bottle are not at all the same. its not semantics, if anything its the logical consistency of a mathematical system (though a mathematician, like Frege, who is also a logician, ha ha, might want to say a lot more about logic here. let me first deposit the notes that i wrote and then follow up with a broom. that way, in fairness to m. moser, it will have the stammerings of a thought and then hopefully the lucidity of a reflection. best, P
Post a Comment