/sci/ - Science & Math

Is the dimensionality of a real vector space only expressible as a natural number?

If so, why?

Can the higher dimensions be visualized as a curvilinear coordinate system?

Or are these just transformations in <span class="math">\mathbf{\mathbb{R}}^{3}[/spoiler]?

I can't help but feel like the full, true, all-encompassing and immaculate nature of reality is something supremely weird and totally beyond the bounds of human intellect.

The string theorists are starting to tell us that models of the universe should be based on higher dimensions... the 4 or even 5 dimensional models of relativity were not enough. In 2010, some astronomical data indicated that there were 3 times as many stars as previously thought. New subatomic particles are theorized and discovered with every generation... Current theories suggest the accelerating universe is caused by a mysterious dark energy in far corners of the universe...

I see an increasingly complicated big picture, and an increasingly limited ability to observe the universe.

I think science is going to dead end at some point in the future-- it will discover a problem to which it postulates a theory for which it is unable to experiment and therefore unsure of being true.

tl;dr The universe is of an intricate and incomprehensible complexity, and we can only observe a very, very, very tiny piece of the whole.

Do you agree science probably has a limit? If not, what makes you think we have the intellectual capacity, technology or observational capability to figure the universe out?

Question:

Is differential geometry usually studied at the graduate level (for physics/math majors)?

When does one begin (academically speaking) to encounter the mathematics used in relativity?

tensors, Ricci flow, non-Euclidean space and all that jazz.