13027
|
Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
|
|
Full Idea:
Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine.
|
|
From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8
|
|
A reaction:
Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed.
|
9627
|
Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
|
|
Full Idea:
In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures.
|
|
From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4
|
|
A reaction:
This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another.
|
16150
|
One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
|
|
Full Idea:
If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
|
|
From:
Plato (Parmenides [c.364 BCE], 144a)
|
|
A reaction:
This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
|