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Axioms/Geometry/Hilbert: can do without real numbers - quantifiers: go beyond regions of the physical space - predicates: among others: "is a point"- "x is between y u z", - "inclusive betweenness": i.e. it is permissible that y = x or y = z.
Segment congruence/congruence: (instead of distance) four-digit predicate -"xy cong zw-" intuitively: "the distance between point x and point y is the same as that from point z to point w" - angle congruence: six-digit predicate "xyz-" W-Comg tuv-": the angle xyz (with y as the tip) has the same size as the angle tuv (with u as a tip) - N.B./Field: Distance and angle size cannot be defined at all because it is not quantified using real numbers.
Addition/multiplication: not possible in Hilbert's geometry - (only with arbitrary zero point and arbitrary 1) solution: intervals instead of points.
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Hilbert/Geometry/Axioms/Field: Multiplication of intervals: not possible because we need an arbitrary "unity-interval" - solution: comparison of products of intervals - Generalization/Field: is then possible on products of space-time intervals with scalar intervals ((s) E.g. temperature difference, pressure difference) - Field: therefore space-time points cannot be regarded as real numbers.
Geometry/Field: a) metric: platonic, quantification via real numbers (> functions) - b) synthetic: without real numbers: E.g. Hilbert, also Euclid (because he had no theory of real numbers) - (also possible without functions) advantage: no external, causally irrelevant entities.
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