Recent content by Ravid

\sqrt 2 is an exact expression of the square root of two. What I suggest you mean is that rationals can be expressed exactly as finite combinations of integers with finitary (binary) algebraic operations, e.g. division, as opposed to as solutions of an equation or a limit. But then again, \sqrt...

The distinction you make here is arbitrary. \sqrt{2} is obtained from 2. The process needed to obtain it is different from that to obtain 1/2, but the difference is not fundamental. People found it hard to accept irrationals, but they were a conscious extension of the rational number counting...

Perhaps I could mount a serious argument if you actually explained why you think the rationals are derived but the irrationals not. Both arose as a "conscious extension of something more basic." Both have examples arising in nature, as I indicated. In post #23 you gave historical reasons for...

That's a pretty specious argument. Both involve choices - we can choose to take the ratio of anything we want. For example, we might have taken the ratio of the circle's radius to its diameter. Now take two atoms (where by atom I mean indivisible thing). How many ways can you 'choose' to...

I was using 1^{-1}\cdot 1v=v but now that you mention it I suppose that requires the axiom.

The expression v1 is not defined in a typical (i.e. left) vector space. The fact that 1v=v follows from the properties of 1 in the field, since 1v=w\Rightarrow v=1w=1v. The real question is, why do you care? You know that it's true, so whether or not you have to take it as an axiom is a triviality.

Lie algebras are obtained by differentiating Lie groups at the identity. A formal introduction is not straightforward, but I can briefly explain this example: any element of SU(3) has determinant one and thus preserves e_1\wedge e_2\wedge e_3. Differentiating along a path X_t we get...

I can see roughly what you are trying to achieve here, and I think it might even be an instructive exercise (even if it is, as I believe, doomed to eventual failure) but I find your particular choice of numbers baffling. How can you claim the division of a whole into two equal parts is in some...

I am used to a co-ordinate free approach to the subject, and hence I didn't really understand much of what you just wrote. Never mind, though, I figured out the relationships myself by studying Kobayashi/Nomizu. Thanks anyway, especially for the recommandations.

Yes, the definition I am talking about is in terms of the structural equation, which is why I have a hard time understanding where it comes from. How are the connection forms defined by the Levi-Civita connection, and how does the curvature form agree with the Riemannian curvature tensor (I...

I 'know about' the Levi-Civita connection in that I know what it is and the fundamental lemma, but I have not studied them in any particular depth. I learned about Cartan connections via Klein geometry, i.e. homogeneous spaces and Lie groups/algebras, and a principal bundle with a Cartan...

Why is the definition of the curvature form for a Cartan connection the correct definition, and what does it actually tell you? I've read that it "measures the failure of the structural equation" but I suppose I don't really understand the structural equation (of a Maurer-Cartan form) anyway...