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Single Idea 9989
[filed under theme 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
]
Full Idea
The fact that units are equal does not mean that they are identical. The units can be equal just in the sense that once can be substituted for any other without altering the name assigned, i.e. the number.
Gist of Idea
Units can be equal without being identical
Source
comment on Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §54) by William W. Tait - Frege versus Cantor and Dedekind XI
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.59
A Reaction
[this is in reference to Thomae 1880] Presumably this might mean that units have type-identity, rather than token-dentity. 'This' unit might be a token, but 'a' unit would be a type. I am extremely reluctant to ditch the old concept of a unit.
The
21 ideas
with the same theme
[a series of isolated 'ones' on which counting is built]:
16146
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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
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If only rectilinear figures existed, then unity would be the triangle
[Aristotle]
|
17859
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Units came about when the unequals were equalised
[Aristotle]
|
12369
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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
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There is no multiplicity without true units
[Leibniz]
|
9147
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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
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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
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Counting needs unities, but that doesn't mean they exist; we borrowed it from the concept of 'I'
[Nietzsche]
|
9576
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Multiplicity in general is just one and one and one, etc.
[Husserl]
|
18392
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Classes have cardinalities, so their members must all be treated as units
[Armstrong]
|
9895
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A number is a multitude composed of units
[Dummett]
|
18071
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A one-operation is the segregation of a single object
[Kitcher]
|
17435
|
Objects do not naturally form countable units
[Koslicki]
|