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.
III 32 f
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._____________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