III 121
Nominalization/Field: we use two topologies on the same set (the amount of space-time points) instead of topologies on two different sets, which are connected by a function. - Therefore, we do not have to quantify on functions.
a) temperature-based region (warmer, colder or similar to) (region as a set of points)
b) the amount of space-time points - thus we get temperature continuity. - Here: purely affine geometry. I.e. only intermediate relation without simultaneity relation or spatial congruence relation.
This then applies for all physical theories that have no Newtonian space-time, but a space-time with flat four-dimensional space R
4. - Also the special theory of relativity. - (Special theory of relativity: few changes because of gradients and Laplace equations that involve non-affine Newtonian space-time).
III 64
Field Thesis: for the General Theory of Relativity we can get more general affine structures.
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Relativity Theory.
Product/Field/(s): Products of differences: = distances between = points = distance. - Pairs of intervals can only be multiplied if they are of the same kind (scalar or spatial-temporal).
Solution: with "mixed multiplication" we can still say that a result is greater than the result of another multiplication with the same components. - That is possible when the spatio-temporal intervals themselves are comparable, i.e. that they lie on the same line or on parallel in affine space.
III 68
Product/Comparison/Field: so far we have only spoken of products of absolute values.
New: now we also want products with signs.
Platonist: this is easy: with new representation functions. Suppose we only have points on a single line L.
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Platonism.
Old: φ is a coordinate function (representation function) attributing points of R4 (four dimensional space) points on line L.
New: φL assigns real numbers to points of L - that's "comparable" with the old φ in the same sense that for each point x and y on L, I φL (x) - φL (y) I = dφ(x, y) are represented. - ((s) Space distance). - The comparison is invariant under choice of orientation.
III 68f
Product/Equality/Between/Field: we can now define equality and "between" for products with signs.
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Definability, >
Spacetime, >
Spacetime points.