Hm turning a sphere inside out what's the point?

To you it is of no importance at all. To mathematicians it shows that things are not quite as simple as we might expect and that problems in 3 dimensions can be much different from problems in 2 dimensions (where turning a circle inside out in exactly the same way is impossible).

 

Here's the full video:

http://video.google.com/videoplay?docid=-6626464599825291409 [Broken]

 

Sorry for the novice-ness, but what realm of mathematics is that? It is topology?

 

horrors, does pure math have a point?

 

Before taking an advanced mathematical logic course, I would have said that pure math should be studied for its own sake. Now I'm not so sure.

 

Check this video out... Its way out there... dont know what to make of it...
http://www.youtube.com/watch?v=dfDIVJXkfCU&feature=related

 

Not everything needs a purpose applicable to the real world. If you can not appreciate this, you will never appreciate pure mathematics.

 

Yes. Modern topology was founded by Poincare and Brouwer in their study of the behavior of solutions of ordinary differential equations. Theorems of topology have unbelievably powerful applications in many areas of pure and applied mathematics. Ie., the Brouwer fixed point theorem, the Poincare index theorem, the generalized Jordan curve theorem/Mazur's theorem, the Hopf fibration, knot theory, the study of cohomology and the discovery of category theory. As in many areas of pure mathematics, the results far outweigh the initial seed of studying ordinary differential equations.

 

we have all asked them from time to time, but a general question of "what is that good for?" risks being a bit like a parent wondering why his kids listen to rock and roll, it misses the point and shows a rigidity and lack of imagination or intellectual curiosity.

obviously, this theorem is interesting because it reveals a surprizing aspect of geometry, one that contradicts ordinary intuition. the point is not that one can imagine a sphere turning inside out, but rather that it can be done without ever introducing any wrinkle in the sphere as it passes through itself.

the intuitive operation is to push inward on the sphere at each pole, and continue pushing the sphere through itself and inside out. but this leaves a circular wrinkle at the last stage at the equator. amazingly this can be avoided.

any fact about the geometry of manifolds may or may not have applications in engineering and physics or cosmology.

topologically it was a question of the connectedness of a certain set of deformations of a smooth manifold, and was proved by Smale.

It is also an interesting story of mathematical imagination and communication, since the original expository article in scientific american reports that the chain of explanation from the man who envisaged the transformations to the one who drew the pictures passed through one link of explanation by a blind person, I believe it was bernard morin.

 

does anyone ask what is the point of a picasso? nevermind the quite cliche comparison of art to math but why do people necessarily assume math needs a purpose? damn our primary education system.

 

A concept in pure maths can have a use that is not apparent until several decades after it's been discovered.

 

There are two justifications that I see being offered:

1. Some topics developed by pure math have later been applied to the sciences.

2. Not everything has to be useful.

Of these I would say that (2) is more honest. I think a study should be under taken with respect to (1), since as pure math has flourished in recent times I think (1) is less true than in the past. Never mind anecdotal reports, we need to count how much of the total mathematical content generated in the 19th century is in use today.

As for (2), I often use this line for students and I am tempted to agree with it in general, especially when presented with something like the video in this thread. But getting students to study math is one thing, and paying professionals to dedicate their efforts to it is another. Personally, I don't think (2) is sufficient justification for a lifetime career in pure mathematics. Look at all the scientific efforts that could help the world e.g. better solar panels, recycling methods, fusion power, agriculture, birth control, communications networking, medicine, transportation, etc.

 

I dont know about this not being useful. Im just in my first year of engineering, but I read this book on chaos which focused on the qualitative aspects of the subject and it partially introduced string theory and what its trying to accomplish.

As far as I know, string theory is based on topology and if it really is the next big breakthrough, then procedures such as inverting this sphere could have real practical applications. In any case, I think this particular method might even be applicable in fluid dynamics or even in electromagnetics. For example, the earths magnetic field is something akin to a sphere, and perhaps this method of inverting the sphere could describe some phenomenon where the magnetic field varies as such. The practical applications of any theoretical concept are only limited by the imagination of the practitioner. Isnt that true?

 

Does anything have a point?

 

the cartesian plane has points?

 

One thing that people often forget when it comes to "useless" proofs is that although the fact itself may not have any immediate consequences, sometimes other results that depend on the proof can be useful. Even more often, although the fact might not be very useful, the techniques used to prove it are often very useful to prove other things.

As for why people care about this one example specifically: eh, it's just something that mathematicians think is cool, unexpected, etc.

 

Euclidean geometry has points. Are you calling Euclid a liar?