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Compactification [Split from Have we found enough puzzle pieces to get a big picture]


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20 hours ago, studiot said:

If we wish to base the discussion on set theory a good place to start is Menger's axioms

D1  :  The empty set has dimension -1

D2  :  The dimension of a space is the least integer, n , for which every point has arbitrarily small neighbourhoods whose boundaries have dimension less than n.

Suspecting this may not be relevant to the thread. But out of curiosity: what are the simplest meanings of 'small neighbourhood' and 'boundary' that you can come up with, which are based on set theory?

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Dimension Theory by Hurewicz and Wallman devotes the entire book to establishing the answer to this and developing topology as a result.

Remember you are the one who first noted (in this thread) that topology may have, but does not require a metric (or norm).

My thesis is that there are several different approaches as a result of different starting points and application goals.


 

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Sooo.
I gather from this discussion that both the Mathematical version and the Physical version will produce similar results.
The difference is the Mathematical version has to necessarily be more generalized, to encompass more cases, while the Physical version is more specific, and usually pertains to the case of 'limiting dimensions' by examining boundary conditions.

Or am I still missing something ?
( IOW dumb it down a bit, for the rest of us )

Edited by MigL
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1 hour ago, MigL said:

The difference is the Mathematical version has to necessarily be more generalized, to encompass more cases, while the Physical version is more specific, and usually pertains to the case of 'limiting dimensions' by examining boundary conditions.

That sounds fair to me. At least I hope you are right. In which case a mathematician will accept some transformations to be eligible for the status of being compactifications that a physicist would not, and maybe that would be the only difference. I hope that there are no transformations that physicists will deem to be compactifications but mathematicians would not agree to. That would make communication difficult.

 I get the sentiment that physicist mentality says "do not come here to our two horse town with your law from the big city; we have six-gun justice here, and this town may be too small for the both of us (sound of click)." Once you hear it, you try to adjust. Just kidding. 

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3 hours ago, taeto said:

At least I hope you are right. In which case a mathematician will accept some transformations to be eligible for the status of being compactifications that a physicist would not, and maybe that would be the only difference.

Just because a Physicist finds some aspects of Euclidian geometry uninteresting it doesn't mean he doesn't accept the things Mathematicians find interesting such as Napoleon's theorem.

But you never know, it might become useful someday for something other than bothering victorian schoolboys. Then all physicists around the world will instantly adopt it.

I believe that is called 'action at a distance'; which of course has long been acceptable in Maths, but no longer in Physics.

:)

Edited by studiot
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36 minutes ago, Mordred said:

Lol good luck tracking every math theory on compactification.... You can nearly get  PH.D on the topic

Maybe nowadays. In my time it was just a trick that we learned in a first course on topology. Actually the textbook for the course was probably on mathematical analysis. I would not have expected that this topic would come to any notoriety in the mean time. If a student would come to me and suggest to write a PhD on compactification I would try to dissuade, since it does not seem a serious research topic. Maybe I am just too out of touch.

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9 hours ago, MigL said:

Oh great...

Now I had to look up Napoleon's Theorem also.

It is difficult to find a Maths theorem that is not of interest to Physicists these days.

Napoleon's is the first that came to mind and is interesting because it is independent of any coordinate system.

David Wells' Dictionary of Curious and Interesting Geometry contains some fascinating entries, many of which are actually useful.

Edited by studiot
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  • 2 weeks later...

Some musings from reading you, brethren in curiosity. If I understand Taeto and Studiot correctly, yeah, dropping properties that seem but commonsense to physicists, chemists and engineers, and delving into more abstract mathematics is a healthy thing to do for someone at some point. Science needs some valiant people to go down those dark alleyways. Drop basic assumptions about compactness, drop space-time itself, what have you.

Easier said than done, though. It's really dark down those alleyways. We've lost too many brave ones to the depths of maths, whence they never came back.

Another thought I'm pondering is that physics has always taken a quantum leap when very deep mathematics has percolated to the more 'math-dummies' like me, at least next-door, which is theoretical physics AFAIK. But the possibility that it's some kind of dice throwing game doesn't bear thinking. I like to think there's going to be another Planck, some day. Then the Paulis, Diracs, and Heisenbergs will appear who clarify the mathematical rules of the game.

I first heard of Napoleon's theorem from a friend mathematician. I'm not sure anybody has used it in physics.

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