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I 193
Def Problem of Quantities/S/R/Field: the representational theorems used for the generation of many numerical functors in physics (e.g. distance, relative velocity, acceleration). - They are not available for relativism because they depend on structural regularities of the space time which are lost when one discards those parts of the space time that are not completely occupied by matter as the space is. - Definition of distance without numbers by congruence and "between".
>
Spacetime , >
Representation theorem .
III 84f
Law of movement/Nominalisation/Field: therefore we need the concepts trajectory and differentiation of the vector field.
Cf. >
Nominalism .
Derivation: of scalars can be equated with differences of scalars - so also derivations of vectors with differences of vectors.
Problem: differences of vectors are themselves vectors; spacetime can be assumed to be infinite, but not temperature.
III 88
Law of movement/Nominalisation/Field: with the concept of the tangent on a trajectory. - The trajectory can be differentiated if the tangent is not purely spatial. - The accelerations of points (on one or more trajectories) are compared with the gradients of the gravitational potential at the points.
Def Law of movement/Newtonian gravitational theory/Field: (if only gravitational forces are effective): for every such T, T',z,z',S,S', y and y': there is a positive real number k so that
a) the second directional derivative of the spatial separation of S from T to z in relation to zy> is taken twice is k-times the gradient of the gravitational potential on z.
b) the second directional derivative of the spatial trajectory from S' from T' to z' corresponding to the other coated symbols.
Nominalistic: one only has to use the second directional derivative in (12').