- #36

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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 reason why I'm concerned with a mathematically consistent definition of spacetime is precisely because of a disputation I had in the past about the derivation of the Lorentz transformations where either part seemed to be short of words to defend their position. It's not just a matter of concept, it's a practical need for communication.

In abstract, yes, one should understand fully the "mainstream" theories before trying to build a new one. For sure my knowledge is limited here to the fundamentals of Special Relativity, in particular to the derivation of the Lorentz transformations. Yet, there are already enough things here that are not clear (to my opinion) to justify some deep dive. If foundations are not well laid down, the building is going to be fragile. This idea of a precise mathematical coordinateless definition for spacetime is something that is floating around in my mind since a few years, and now and again I've been looking for a similar approach. Until I found a seemingly suitable line of thought in the works by Matolcsi (and in more primitive form in the "Neoclassical Physics" by a German physicist whose name I cannot recall right now).

Also, being a theorist willing to say things with his own words is not the only possible reason for developing an alternative foundation.

Last thing. When we define what A vector space is, we are not defining an actual instance of a vector space. We just say when a vector space should be called so. And of course the conclusions we derive from the related axioms are valid for any algebraic object that is entitled to be called a vector space. On the other hand, when we talk about spacetime we are not thinking of a category of things, but of THE mathematical representation for the framework where Physics takes place. Of course such representation may vary from theory to theory. Now spacetime may be one instance in a class of objects sharing some axiomatic properties. I think the best analogy is with affine spaces: they are defined by axioms as a class, but when we want to represent a blackboard by an affine plane, we actually construct the object itself and show that's an affine plane, not just say that "there is a plane and its properties are".

Just to clarify my position. I'm not willing to convince anybody, nor asking for advice on what to study first. I just appreciate hearing everybody's approach and learning new things. In the end it's only for the pleasure of discussion, my research career has already died long ago, even before its birth probably.