Books on Amazon
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.
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