|Numbers: whether numbers are objects or concepts, has been controversial in the philosophical discussion for millennia. The most widely accepted definition today is given by G. Frege (G. Frege, Grundlagen der Arithmetik 1987, p. 79ff). Frege-inspired notions represent numbers as classes of classes, or as second-level terms, or as that with one measure the size of sets. Up until today, there is an ambiguity between concept and object in the discussion of numbers. See also counting, sets, measurements, mathematics, abstract objects, mathematical entities, theoretical entities, number, platonism._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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Real Numbers/Bigelow/Pargetter: Thesis: Real numbers are universals of higher level.
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).
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.
Instantiation/Bigelow/Pargetter: Quantities such as length, mass, speed are in turn instantiated by individuals such as photons, electrons, macroscopic objects.
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.
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.
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".
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.
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.
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 1941,1961).
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.
Definition grandparents/Frege/Quine/Bigelow/Pargetter: x GE y iff x E² y
Definition great-grandparents: x UGE y iff x E³ y
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.
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,...
...and one between S and Q, then there is a derived relation between R and S.
Wiener: (1912) 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._____________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