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Identity/Identification/Field: in many areas, there is the problem of the continuous arbitrariness of identifications. - In mathematics, however, it is stronger than with physical objects. I 181
Intensity relations between pairs or triples, etc. of points. - Advantage: that avoids attributing intensities to points and thus an arbitrary choice of a numerical scale for intensities.
Addition/Multiplication: not possible in Hilbert's geometry. - (Only with arbitrary zero and arbitrary 1) - Solution: intervals instead of points.
Non-Classical Degrees of Belief/DB/Uncertainty/Field: E.g. that every "decision" about the power of the continuum is arbitrary is a good reason to not assume classical degrees of belief. - (Moderate non-classical logic: That some instances of the sentence cannot be asserted by the excluded third party).
Figure/Points/Field: no Platonist will identify real numbers with points on a physical line. - That would be too arbitrary ( "what line?"). - What should be zero - what is supposed to be 1?
III 32 f
Hilbert/Geometry/Axioms/Field: multiplication of intervals: not possible, because for that we would need an arbitrary "standard interval". - Solution: Comparing products of intervals. - generalization/Field: is then possible on products of spacetime intervals with scalar intervals. ((s) E.g. temperature difference, pressure difference). - Field: therefore, spacetime points must not be regarded as real numbers.
FieldVsTensor: arbitrary. - Solution/Field: simultaneity.
Def Equally Divided Region/Equally Split/Evenly Divided Evenly/Equidistance/Field: (all distances within the region equal: R: is a spacetime region all of whose points lie on a single line, and that for each point x of R the strict st-between (between in relation to spacetime) two points of R lies, there are points y and z of R, such that a) is exactly one point of R strictly st-between y and z, and that is x, and -b) xy P-Cong xz (Cong = congruent) - ((s) this avoids any arbitrary (length) units - E.g. "fewer" points in the corresponding interval or "the same number." - ((s) but not between temperature and space units (Which common measure?) - But definitely in mixed products - Then: "the mixed product... is smaller than the mixed product..." - equidistance in each separate region: scalar/spatio-temporal.
Arbitrariness/Arbitrary/Scales Types/Scalar/Mass Density/Field: mass density is a very special scalar field which, due to its logarithmic structure, is "less arbitrary" than the scale for the gravitational potential. - ((s) >objectivity, >logarithm) - Logarithmic structures are less arbitrary. - Mass density: needs more fundamental concepts than other scalar fields. - Scalar field: E.g. height._____________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