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Single Idea 8266
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
]
Full Idea
That one-to-one correlated sets of objects are equinumerous is a more sophisticated achievement than the simple ability to count sets of objects.
Gist of Idea
Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous
Source
E.J. Lowe (The Possibility of Metaphysics [1998], 2.9)
Book Ref
Lowe,E.J.: 'The Possibility of Metaphysics' [OUP 2001], p.49
A Reaction
This is an objection to Frege's way of defining numbers, in terms of equinumerous sets. I take pattern-recognition to be the foundation of number, and so spotting a pattern would have to precede spotting that two patterns were identical.
The
19 ideas
with the same theme
[view that one-one correspondence is basis of numbers]:
8649
|
Two numbers are equal if all of their units correspond to one another
[Hume]
|
9956
|
'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F'
[Frege, by George/Velleman]
|
13527
|
Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC
[Frege, by Wolf,RS]
|
22292
|
Hume's Principle fails to implicitly define numbers, because of the Julius Caesar
[Frege, by Potter]
|
17442
|
Frege thinks number is fundamentally bound up with one-one correspondence
[Frege, by Heck]
|
14425
|
A number is something which characterises collections of the same size
[Russell]
|
18145
|
Many things will satisfy Hume's Principle, so there are many interpretations of it
[Bostock]
|
18148
|
Hume's Principle is a definition with existential claims, and won't explain numbers
[Bostock]
|
18149
|
There are many criteria for the identity of numbers
[Bostock]
|
10140
|
We derive Hume's Law from Law V, then discard the latter in deriving arithmetic
[Wright,C, by Fine,K]
|
8692
|
Frege has a good system if his 'number principle' replaces his basic law V
[Wright,C, by Friend]
|
17440
|
Wright says Hume's Principle is analytic of cardinal numbers, like a definition
[Wright,C, by Heck]
|
13893
|
It is 1-1 correlation of concepts, and not progression, which distinguishes natural number
[Wright,C]
|
8784
|
Neo-logicism founds arithmetic on Hume's Principle along with second-order logic
[Hale/Wright]
|
10529
|
If Hume's Principle can define numbers, we needn't worry about its truth
[Fine,K]
|
10530
|
Hume's Principle is either adequate for number but fails to define properly, or vice versa
[Fine,K]
|
8266
|
Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous
[Lowe]
|
8302
|
Fs and Gs are identical in number if they one-to-one correlate with one another
[Lowe]
|
10133
|
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle
[George/Velleman]
|