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Real Numbers/Bigelow/Pargetter: Thesis: Real numbers are universals of higher level.
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Universals, >
Real numbers, >
Levels/order, >
Description levels
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They are relations between relations (or between properties).
They are precisely the relations of higher levels or proportions with which we had compared quantities (see above 2.5).
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Relations.
Proportions/Bigelow/Pargetter: should be identified with real numbers.
Real numbers/Bigelow/Pargetter: are then themselves physical! Like other proportions and relations. They are instantiated by physical quantities such as length.
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Proportions.
Instantiation/Bigelow/Pargetter: Quantities such as length, mass, speed are in turn instantiated by individuals such as photons, electrons, macroscopic objects.
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Instantiation.
Instantiation/Bigelow/Pargetter: being instantiated makes a causal difference. They are then abstract as universals, but not abstract in the sense that they would be causally inactive.
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Abstraction, >
Abstractness.
Abstraction/Bigelow/Pargetter: is only a process of drawing attention to one or the other universal that are instantiated around us. But this does not create a new thing.
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Numbers/Bigelow/Pargetter: there is a strong tendency to assume that they are objects that instantiate relations and properties, but are not themselves properties or relations. They seem to be "abstract objects".
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Numbers/Frege.
Bigelow/Pargetter: pro: they can be that without ceasing to be universals.
Numbers/Frege/Bigelow/Pargetter: the theory we are discussing here is about relations of relations. This probably also applies to relations between properties. For example: length comparisons etc.
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Properties, >
Measurements.
Properties/Bigelow/Pargetter: if we want to avoid them, we can also compare the endpoints instead of the lengths of two objects.
Relation/Bigelow/Pargetter: we can generally come from properties to relations by saying that there is a relation between objects by virtue of a shared property (e.g. length). For example "smaller than" etc. that is a derived relation.
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Definitions/Frege.
Derived relation/Bigelow/Pargetter: will then exist between the properties that generate these relations.
Frege/Bigelow/Pargetter: his theory is now based on relations between relations. For example, parent relation and grandparent relation. (Lit. Quine 1941
(1), 1961
(2)).
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Relations/Frege.
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Parents/Grandparents/Bigelow/Pargetter: the relations are different, but closely related, if two things are connected by the grandparent relation, the same two things will be connected by a chain involving two instances of the parent relation.
If a is grandparent of b, there is a c so that a is a parent of c and c is a parent of b.
Notation (see above 2.6): Rn: n-fold relation: e.g.
(s) Grandparents-R = (parents-R)².
X Rn y
Means that we get from x to y through n applications of the relation R
x R x1
x1 R x2
xn-1 R y.
Grandparents/formal/spelling/Bigelow/Pargetter: if x is grandparent of y then x is parent² of y.
Ancestor/Ancestor Relation/Bigelow/Pargetter: is just a generalization of it.
Descent/predecessors/predecessor relation/ancestor/nominalism/Bigelow/Pargetter: the predecessor relation or ancestor relation was one of the biggest problems for nominalism.
Problem: you have to have a realistic attitude towards relations, there must be relations here.
Frege/Whitehead/Bigelow/Pargetter: get much more out of the parent relation than one could have predicted.
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Def grandparents/Frege/Quine/Bigelow/Pargetter: x GE y iff x E² y
Def great-grandparents: x UGE y iff x E³ y
etc.
N.b.: because grandparent relation and great-grandparent relation are connected in different ways with the same basic relation (parents), there is now automatically a relation between these:
If x UGE² y then x GE³ y.
In general: given are two relations R and S, we can have a relation between them, by virtue of the
x Rn y iff x Sm y.
Ratio/Proportion/logical form/Bigelow/Pargetter: these relations of relations are called ratios or proportions. For example, in the above case, R to S is m:n.
Negative ratios/Bigelow/Pargetter: we obtain by changing the variables x and y:
x Rn y iff y Sm x.
For example, grandchild-relation: has the ratio -2:1 ((s) inverse relation of the grandparents-relation)
x grandchild y iff y E² x.
Recursive rule/relationship/ratio/Bigelow/Pargetter: if R and S have a proportion (ratio) with respect to another relation Q:
If there's a relationship between R and Q,...
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...and one between S and Q, then there is a derived relation between R and S.
Wiener: (1912)
(3) varies the approach of Whitehead: when
The ratio of R to Q is n:1
If the ratio of S to Q is m:1
Then we conclude
the ratio of R to S is n:m.
N.b.: this allows us to set up the ratio n:m between R and S, even if it is not possible to iterate R or S.
For example, your relation to Eva and your mother's relation to Eva. The ratio of these two relations will then be n:(n+1)
N.b.: We cannot simply get such relationships through iteration! For example, because no one stands in relation to them as you stand to Eve (you do not have so many successors).
Solution/Wiener/Bigelow/Pargetter: no iteration of the relation to Eva, but iteration of the basic unit: here the parent relation.
Rational numbers/Bigelow/Pargetter: in order to receive them in their full complexity, we must assume that the given relation has the correct patterns of instances. Problem: the parent relation may not have enough instances to generate an infinite number of rational numbers.
((s) Parent relation: is linear).
Ratio/ratios/proportions/rational numbers/solution/Bigelow/Pargetter: set theory.
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Sets, >
Set theory.
1. Quine, W.V.O. (1941). Whitehead and the rise of modern logic. In: The philosophy of Alfred North Whitehead (ed. P.A. Schilpp). pp.125-63. La Salle, Ill. Open Court.
2. Quine, W.V.O. (1961). From a logical point of view. Logico-philosophical essays 2d ed. New York, Harper & Row.
3. Wiener, N. (1912). A simplification of the logic of relations. Proceedings of the Cambridge Philosophical Society 17 (1912-14), pp.387-90.
