more from this thinker
|
more from this text
Single Idea 9901
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
]
Full Idea
The fact that Zermelo and Von Neumann disagree on which particular sets the numbers are is fatal to the view that each number is some particular set.
Gist of Idea
Numbers can't be sets if there is no agreement on which sets they are
Source
Paul Benacerraf (What Numbers Could Not Be [1965], II)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.279
A Reaction
I agree. A brilliantly simple argument. There is the possibility that one of the two accounts is correct (I would vote for Zermelo), but it is not actually possible to prove it.
The
26 ideas
with the same theme
[general ideas concerning numbers]:
|
1729
|
We perceive number by the denial of continuity
[Aristotle]
|
|
13273
|
Pluralities divide into discontinous countables; magnitudes divide into continuous things
[Aristotle]
|
|
10958
|
Perhaps numbers are substances?
[Aristotle]
|
|
13152
|
We can talk of 'innumerable number', about the infinite points on a line
[Newton]
|
|
16920
|
Numbers are formed by addition of units in time
[Kant]
|
|
9823
|
Numbers are free creations of the human mind, to understand differences
[Dedekind]
|
|
23186
|
Numbers enable us to manage the world - to the limits of counting
[Nietzsche]
|
|
13412
|
Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order
[Benacerraf]
|
|
13413
|
We must explain how we know so many numbers, and recognise ones we haven't met before
[Benacerraf]
|
|
9901
|
Numbers can't be sets if there is no agreement on which sets they are
[Benacerraf]
|
|
9912
|
There are no such things as numbers
[Benacerraf]
|
|
12334
|
There is no single unified definition of number
[Badiou]
|
|
12335
|
Numbers are for measuring and for calculating (and the two must be consistent)
[Badiou]
|
|
10599
|
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1)))
[Smith,P]
|
|
13861
|
Number theory aims at the essence of natural numbers, giving their nature, and the epistemology
[Wright,C]
|
|
10027
|
Mathematics is higher-order modal logic
[Hodes]
|
|
8958
|
In Field's version of science, space-time points replace real numbers
[Field,H, by Szabó]
|
|
8865
|
If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise?
[Yablo]
|
|
9923
|
We should talk about possible existence, rather than actual existence, of numbers
[Burgess/Rosen]
|
|
9622
|
'There are two apples' can be expressed logically, with no mention of numbers
[Brown,JR]
|
|
17453
|
The meaning of a number isn't just the numerals leading up to it
[Heck]
|
|
12238
|
The Aristotelian view is that numbers depend on (and are abstracted from) other things
[Oderberg]
|
|
9998
|
What is the relation of number words as singular-terms, adjectives/determiners, and symbols?
[Hofweber]
|
|
10002
|
'2 + 2 = 4' can be read as either singular or plural
[Hofweber]
|
|
21644
|
Numbers are used as singular terms, as adjectives, and as symbols
[Hofweber]
|
|
21646
|
The Amazonian Piraha language is said to have no number words
[Hofweber]
|