Recent content by Enrico

Actually this is a multilinear application. This definition is as saying that a matrix is a linear application. Just as a matrix represents a linear application, I'd say that a tensor represents a multilinear application.

Obviously I'm not willing to start explicitly from there, but actually when I think of fundamentals the proof that ##1+1=2## is what comes to my mind (I had seen that from the Peano axioms. Thanks for quoting the PM, I'll put that into the infinite pipeline of things I  could read in my life).

At this point let me ask a question. Just out of curiosity. What is everybody's favorite definition of a tensor?

First you have to derive the equation then. How do you do that if you still have to define what spacetime is?

It was. With more than one. In particular, I had a paper about this very subject. Submitted to three different reviews. Two of them stated "it's correct but not suitable for us". The third stated "it's suitable for us but wrong". And the reason why they thought it was wrong was impossible to...

I have manually created a link with a specific visible text. Is that the only way?

@PeterDonis @Dale About the blackboard thing: I don't feel comfortable by pointing to the real physical object. If we want to do maths, I'd first define an algebraic object that represents the blackboard and then work on that. For this, I'd use what I was calling a "constructive" approach and...

@strangerep Thanks, I know what all this is, although not fresh in my mind.

Please have a look at post #24.

Something of the kind, yes. But what if one wants to define spacetime before deriving the (special) relativistic metric?

I have read all replies here. Many things to say, I'll try to outline some of them. First, I have always disagreed with the usual habit of Physicists of not being mathematically precise. That always gives space to some kind of subjectivity and as a consequence to endless disputations. The...

Same as for foundations of Maths? I'm not an expert in that in any way, but my understanding is that if you give the Naturals and the concept of Set as primitive concepts, you may build up everything from there. Once you get to the Reals, you should proceed from there, without introducing...

I've been trying to do so. That was just an example to clarify what kind of procedure I have in my mind. In Special Relativity I'd think of something like "a point in space is the equivalence class of all its possible coordinatisations in inertial frames". Just a naive idea, nothing else. Or...

I have only read this after my last reply. Here you get exactly my point. (And from what you say I understand that I won't find the definition I'm looking for in Carroll's notes.)

I'm trying to expose my thoughts, sometimes I cannot find the right words. Let me try again. If we talk about ##\mathbb{R}^4##, the concept is clear: set of 4-ples of real numbers (i.e. applications from {1,2,3,4} to ##\mathbb{R})##, where ##\mathbb{R}## is defined by (if I recall right, it's...