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Proportion/Relation/Bigelow/Pargetter: in any case, we can assume proportions between relations.
Problem: but not proportions between properties.
Flux/Bigelow/Pargetter: but assumes that speed is a property rather than a relation.
Vector: to explain its nature, we now need something that fills the gap between property and relation.
Solution/Bigelow/Pargetter: for all things with the same property, there is a relation; that of accordance!
Formally: if Fx and Fy, there is a relation RF, so that x RF y.
Properties/Relation/Bigelow/Pargetter: even if two individuals have different properties, there is a relation between them: formal: there is a relation RFG between Fx and Gy...
...so that x RFG y.
At any rate, we assume this in case F and G are vectors of the same kind.
For example, rotating homogeneous disk:
1. points on same radius (same direction): each has a different speed.
Then there are some that are 1m/sec faster than others. etc.
Relation: between properties: because point x has the property (here: speed or location?) it stands in a certain relation to the point y: it is so and so much faster.
Properties/Bigelow/Pargetter: are therefore also in proportions.
2. corresponding to points on the same circumference (same speed, different direction).
Relations/Property/Bigelow/Pargetter: then we have relations between velocities with respect to size (if the points lie on the same radius) e.g. speed of x has r times the size of the velocity of y:
x Pr y.
For example, be a point at the same distance from the center of the borderline, then it has the same speed (size).
z P0 y
The two relations are summarized as follows
x Pr y.
z P0 y,
then we have a derived relation between x and z.
Definition derived relation P*/Bigelow/Pargetter: we define it by saying:
x P* z iff for a y, x pr y and y p0 z,...
Proportion/properties/Bigelow/Pargetter: on the rotating disk, two points will be placed in this "two-step-proportion" of the form P*. Namely, by virtue of their intrinsic properties.
Vectors/Bigelow/Pargetter: the property of instantaneous speed are considered vectors because they are in a family of two-step proportions!
n-step Proportion/Bigelow/Pargetter: this can be generalized to proportions that include n steps. This gives us more general vectors.
Vector/Bigelow/Pargetter: the vector of a speed of a point on a rotating disk can be represented as an ordered pair of real numbers.
General: all ordered n-tuples of real numbers can be understood as vectors. We need some for the flux theory, but not all of them.
Vectors/Bigelow/Pargetter: are useful for representing physical properties, because they can be embedded in a network of proportions.
Ratios/Bigelow/Pargetter: are special cases of real numbers. Conversely however, not all real numbers correspond to ratios.
Proportion/Bigelow/Pargetter: is a more general term than ratio and forms the basis for our system of real numbers. Some proportions in the geometry, for example, do not correspond to ratios.
E.g. pentagon:...._____________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.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990