Mathematics and Nature

Is any area of mathematics sacred?
That is to say are there any areas of mathematics which are pure abstractions.
(Or to put it more bluntly are the Pure Mathematicians really doing applied mathematics and they just don't know it yet?)As a possible avenue of this discussion, what are we doing when we say we are doing mathematics.addendum:I am asking is there any area of current pure mathematical research which will escape being subsumed into physical theory.
Rather than asking is Maths a Science, I am asking is there any area of Pure Mathematics which is of no used to Science, especially physics.Over the last century we have seen areas we thought useless to physics, such a noncommutative geometry actually come into play in the physical sciences.Does Pure Mathematics mean no more than Mathematics with no current use.

This thread was actually inspired by work from John Baez and a few others.
They are currently working on Quantum Mathematics, which is basically the study of mathematics which is important to Quantum mechanics (Not yet field theory though).Weird results have come up such as using measurements of quantum systems to study the Riemann zeta function and possibly solve the Riemann zeta hypothesis experimentally.A very exciting field and I can't wait until they get to QFT.My surprise isn't so much that Number theory has important applications to Quantum Mechanics, but that very obscure "curiosities" in number theory which only the number theorists knew about (Not even other Mathematicians) turn up all the time in Quantum Systems apparently and we might be seeing Quantum Number Theory as a research field in the next few years.Bizarre!!

The way I understand it, the Mobius strip is a topology, just like a Klein bottle.
Topology is used to examine connectness in mathematical entities independant of their geometry or other specifics, just like differential geometry dicusses geometry without the need for embedding.So something can have the property of a mobius strip topology even if it is not a surface, simply by having that connectivity. Examples would be mobius energy in knots and junction ladders in conformal field theory have boundaries with mobius topology.So even if we cannot embed a 2D figure in 3D space which has the mobius topology, that does not mean that the mobius strip does not exist, as it is a topology not a shape.I'm not the best on topology so there might be some mistakes.This message has been edited by Son Goku, 12-17-2005 01:55 PM

To quote John Listing himself, the inventor/discoverer of the mobius strip: In this manner the Mobius strip was a connectivity he discovered in oriented three-dimensional polyhedra, as it came up again and again when discussing connectivity it became a hot topic in topology.August Mbius then invented the one sided representation of this topology to aid him in thinking about it, as it is often easier to understand connectivity when it is related to shapes.The Mobius strip is real, in that many things posses that connectivity. However no sequence of atoms can arrange themselves into the shape of August Mobius.

No, that’s my point. It doesn't matter, the Mobius strip is a topological property which something can either posses or not.
I can't make a piece of paper which posses this property, only one that comes asymptotically close to the property.
However I can have energy levels and boundaries which posses this topology.

Yes, it is.
And?

When we do Fluid Dynamics, to pick a random area of physics, nobody thinks that there literally is a Stress Tensor sitting at every point in the fluid, a 3 x 3 matrix that literally sits there and takes values.The Stress Tensor can describe the fluid. It is a series of nine values and Stress on the fluid is a combination of nine independant "directions" of Stress.Thats it, we know they aren't "real". What does this mean?
To be honest it sounds like you have a vague idea of what we do and it amounts to us sitting around all day appreciating mathematics and laughing at laymen.
We study physics, just once more that word is physics, the science that studies the laws governing the natural world, the real world.
We do it because we love reality and how it works. If we liked mathematics so much we would have been mathematicians.
However when you start digging nature behaves more and more like our most abstract mathematical systems. This is why we talk in terms of mathematics, not to make ourselves seem smart or aloof, but because we need it. Nature behaves mathematically.
End of story, that is it.

The mobius concept is a topology! The concept is a topological one.
How can compare a topological concept to topology itself? What reality? We aren't applying it yet.
If you want me to compare it to reality, then the boundaries of junction ladders in conformal field theory can be compared to the concept of a mobius strip topology and found to behave the same.
So the boundaries of junction ladders in conformal field theory have a mobius strip topology.

The paper loop has the mobius strip as a topology.
Of course the paper loop is not a mobius strip.
Just as an apple isn't the colour green, it is coloured green.
The paper has a mobius strip topology, it is not the mobius strip.The boundaries of junction ladders in conformal field theory also have a mobius strip topology.