I 42
Order/Universals/Antisymmetry/Bigelow/Pargetter: the antisymmetry can then establish an order (hierarchy) between infinitely many different universals.
Order/Hierarchy:
1. individuals: Definition individual/Bigelow/Pargetter: what is not instantiated by anything.
2. rule: the rest is obtained by the following rule:
If t1, t2,.... tn are types, then also (t1, t2... tn) is a type...
>
Individuals, >
Type/Token, >
Universals.
((s) that is, summaries of types are also types).
Def type/Bigelow/Pargetter: is then a set of universals, which can consist of one to infinitely many.
Domain/Bigelow/Pargetter: the union of all types, each type is a subset of the domain. There may also be empty subsets.
>
Domains, >
Sets.
I 362
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.
>Real numbers.
Order/Relation/Bigelow/Pargetter: for this theory we need to extend the natural order created by relatios.
>
Proportions.
Geometry: shows proportions that cannot be displayed in whole numbers.
>
Geometry.
Proportion/terminology/Bigelow/Pargetter: we call proportions ratios that cannot be expressed in whole numbers.
Realism/Bigelow/Pargetter: pleads for the assumption that there are objects with the proportions of the golden section rather than claiming there is no golden section.
>
Realism.
Real numbers/Bigelow/Pargetter: Assuming that there is no golden section, would there be no real numbers?
I 363
Is the existence of real numbers contingent on the existence of quantities?
>
Quantity, >
Quantities/Physics.
Aristotle/Bigelow/Pargetter: demands that every quantity must be instantiated to exist.
>
Instantiation, >
Ontology, >
Existence.
VsAristotle: this seems to make mathematical facts dependent on empirical facts.
>
Mathematical entities, >
Empiricism.
Platonism/Bigelow/Pargetter: all quantities exist for him, regardless of whether they are instantiated or not. This guarantees pure mathematics.
>
Platonism, >
Mathematics.
