Lexicon of Arguments

 

III 25
Axioms/geometry/Hilbert: geometry 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.
>Quantifiers.
III 26
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.
III 32
Addition/multiplication: Addition and multiplication is 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.
>Spacetime points, >Real numbers.
III 42
Geometry/Field:
a) metric: platonistic, quantification via real numbers (> functions)
b) synthetic: without real numbers: E.g. Hilbert, also Euclid (because he had no theory of real numbers). (This is also possible without functions).
Advantage: no external, causally irrelevant entities.
>Mathematical entities, >Theoretical entities.