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Golden Section/Bigelow/Pargetter: this relation is all too real. Nevertheless, it is not a ratio in our sense.
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Ontology, >
Existence, >
Relations.
For example, if we create lines by stringing DF together we will never get a matching point with multiples of DC.
Logical form/General/Incommensurability/Bigelow/Pargetter: n times DF will never be = m times DC.
This also applies to the solution of Wiener (see above).
Proportion: here is 2: (1 + √5), therefore it cannot be represented as ratio a:b for integers a and b.
Incommensurability/proof: can be proven by raa: assuming DF and DC would be commensurable, i.e. there is a distance d that divides both DF and DC. Let's take a look at the rectangle (in the graphic above) FDC, d divides DF and DF equals EC. This divides both DC and EC. Therefore, it must also divide DE. Then the same size must divide both the larger and the smaller rectangle, which is not possible. d would then also have to divide the sides of the third rectangle in the drawing etc. ad infinitum.
Therefore, no finite length can divide both sides of a golden rectangle.
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VsBigelow: incommensurability seems to be against our theory.
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Irrational numbers.
BigelowVsVs: Solution: we redefine "ratio" a little bit: we need a third relation:
Definition Incommensurability/logical form/Bigelow/Pargetter: if two relations R and S are income-survable, then whenever
x Rn y,
follows that
not: x Sm y,
for whichever values of n and m are used. Repetition of n applications of R will never result in a match with m applications of S.
N.B.: nevertheless, we can determine that the results of repeated applications of R and S are in a certain relation to each other. They are arranged in a linear order "<" ("smaller"). I.e. it can be, for an n and an m
If x Rn y and x Sm z, then y < z.
Golden Section/Bigelow/Pargetter: is clearly defined by the list of numbers n and m for which the above scheme applies.
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General: each proportion between two relations R and S can be unambiguously characterized by a list of natural numbers n and m for which the scheme applies.
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Proportions.
Proportion/Bigelow/Pargetter: this theory of proportions is based on Eudoxo's contribution to Euclid's Elements (Book 5 Def 5).
Real Numbers/Bigelow/Pargetter: this theory of proportions as a theory of real numbers was developed by Dedekind and others at the end of the 19th century.
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Real numbers.
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Geometry/Bigelow/Pargetter: geometry has to do with spatially instantiated universals. Therefore, it is vulnerable through empirical discoveries about space. It could be that we discover that space does not instantiate the geometric shapes that we had previously assumed to be instantiated like this.
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Discoveries, >
Space.
Aristotle/Bigelow/Pargetter: according to him the forms would then be discarded.
Plato/Bigelow/Pargetter: he allows first the acceptance of a non-Euclidean space. ((s) But if it is not directly perceptible to us and if it is instantiated in the universe, for example, it is not a problem for Aristotle either.)
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Universals/Platonism/Bigelow/Pargetter: actually he doesn't believe in uninstantiated universals either, but he will find them or invent them. Above all, he will say that pure mathematics is autonomous.
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Ideas/Plato.
