|Comparisons, philosophy: here, we are concerned with the conditions under which it is possible to make comparisons. Objects which do not share any properties are not comparable. A comparison always refers to a singled out property among several properties embodied by more than one object. The prerequisite for comparisons is a consistency of language usage. See also analogies, description levels, steps, identification, identity, change, meaning change, ceteris paribus, experiments, observation._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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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. - (Special theory of relativity: few changes because of gradients and Laplace equations that involve non-affine Newtonian space-time).
Field Thesis: for the General Theory of Relativity 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: 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.
Product/Equality/Between/Field: we can now define equality and "between" for products with signs._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
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