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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 R4. - Also the special theory of relativity (SR). - (SR: few changes because of gradients and Laplace equations that involve non-affine Newtonian space-time). - III 64 Field Thesis for AR we can get more general affine structures.
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's possible when the spatio-temporal intervals themselves are comparable, i.e. that they lie on the same line or on parallel in affine space.
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 - old: f is a coordinate function (representation function) attributing points of R4 points on line L - New: fL. Assigns real numbers to points of L - that's "comparable" with the old f in the same sense that for each point x and y on L, I fL (x) - fL (y) I = df(x, y) are represented - ((s) space distance). - The comparison is invariant under choice of orientation.
III 68 f
Product/Equality/Between/Field: we can now define equality and "between" for products with signs.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980