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Single Idea 18392
[filed under theme 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
]
Full Idea
Classes, because they have a particular cardinality, are essentially a certain number of ones, things that, within the particular class, are each taken as a unit.
Gist of Idea
Classes have cardinalities, so their members must all be treated as units
Source
David M. Armstrong (Truth and Truthmakers [2004], 09.1)
Book Ref
Armstrong,D.M.: 'Truth and Truthmakers' [CUP 2004], p.114
A Reaction
[Singletons are exceptions] So units are basic to set theory, which is the foundations of technical analytic philosophy (as well as, for many, of mathematics). If you can't treat something as a unit, it won't go into set theory. Vagueness...
Related Idea
Idea 18396
The set theory brackets { } assert that the member is a unit [Armstrong]
The
21 ideas
with the same theme
[a series of isolated 'ones' on which counting is built]:
16146
|
Two can't be a self-contained unit, because it would need to be one to do that
[Democritus, by Aristotle]
|
17844
|
The unit is stipulated to be indivisible
[Aristotle]
|
17845
|
If only rectilinear figures existed, then unity would be the triangle
[Aristotle]
|
17859
|
Units came about when the unequals were equalised
[Aristotle]
|
12369
|
A unit is what is quantitatively indivisible
[Aristotle]
|
12273
|
Unit is the starting point of number
[Aristotle]
|
24035
|
Unity is something shared by many things, so in that respect they are equals
[Descartes]
|
24036
|
I can only see the proportion of two to three if there is a common measure - their unity
[Descartes]
|
12956
|
Only whole numbers are multitudes of units
[Leibniz]
|
12920
|
There is no multiplicity without true units
[Leibniz]
|
9147
|
Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction
[Fine,K on Leibniz]
|
9801
|
Numbers must be assumed to have identical units, as horses are equalised in 'horse-power'
[Mill]
|
8641
|
You can abstract concepts from the moon, but the number one is not among them
[Frege]
|
9989
|
Units can be equal without being identical
[Tait on Frege]
|
17429
|
Frege says only concepts which isolate and avoid arbitrary division can give units
[Frege, by Koslicki]
|
7207
|
Counting needs unities, but that doesn't mean they exist; we borrowed it from the concept of 'I'
[Nietzsche]
|
9576
|
Multiplicity in general is just one and one and one, etc.
[Husserl]
|
18392
|
Classes have cardinalities, so their members must all be treated as units
[Armstrong]
|
9895
|
A number is a multitude composed of units
[Dummett]
|
18071
|
A one-operation is the segregation of a single object
[Kitcher]
|
17435
|
Objects do not naturally form countable units
[Koslicki]
|