I always lived in physical space and got great support from E.~Cartan's book ``Geometry of Riemannian Spaces''. As I got it, I gradually drifted apart from other physicists. My equations of motion of rotating objects in curved spaces exposed obvious non-conservation of momentum that was absolutely inadmissible for them. I realized that there are two distinct momentum conservation laws, one global, never observed, which requires that physical space is either $\mathbb{E}^3$ or $\mathbb{V}^3$ and one local observed in collisions of bodies in mechanics. I denied the earlier, not the latter, but had no idea how to explain this to physicists which live in $\mathbb{E}^3$ or $\mathbb{V}^3$ and use only Cartesian coordinates.
Later on, I noticed that I cannot explain them much simpler things. Analyzing this phenomenon, I found that we use different ideas of what is function on the space. To my idea, it is function of a point which takes a certain value in given point of the space, whereas for others it is function of three variables.
-- What variables? -- I ask.
-- Coordinates.
-- What are coordinates?
-- Everybody knows what are coordinates.
-- Really? What is one coordinate?
-- One of them, of course.
-- Each coordinate is a function of point. You have three functions of point and call them coordinates, then define other functions as that of these three functions. How would you introduce a function on the space if there are no coordinates?
-- How can it be?
-- I mean, how to introduce a coordinate in the space itself, which is not endowed with any coordinates?
-- How can it be?
Indeed, how can it be in the framework of generally accepted ideology in which coordinates appear before the space? According physicists, there is no space at all, only coordinates exist. By ``space'' they mean span of coordinates, nothing more. Physicists do not need the space, they need only variables to calculate something. Well, how physical space be the main physical object, if it does not exist?
Later on, I noticed that I cannot explain them much simpler things. Analyzing this phenomenon, I found that we use different ideas of what is function on the space. To my idea, it is function of a point which takes a certain value in given point of the space, whereas for others it is function of three variables.
-- What variables? -- I ask.
-- Coordinates.
-- What are coordinates?
-- Everybody knows what are coordinates.
-- Really? What is one coordinate?
-- One of them, of course.
-- Each coordinate is a function of point. You have three functions of point and call them coordinates, then define other functions as that of these three functions. How would you introduce a function on the space if there are no coordinates?
-- How can it be?
-- I mean, how to introduce a coordinate in the space itself, which is not endowed with any coordinates?
-- How can it be?
Indeed, how can it be in the framework of generally accepted ideology in which coordinates appear before the space? According physicists, there is no space at all, only coordinates exist. By ``space'' they mean span of coordinates, nothing more. Physicists do not need the space, they need only variables to calculate something. Well, how physical space be the main physical object, if it does not exist?
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