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Single Idea 8641

[filed under theme 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units ]

Full Idea

What are we supposed to abstract from to get from the moon to the number 1? We do get certain concepts, such as satellite, but 1 is not to be met with. In the case of 0 we have no objects at all. ..The essence of number must work for 0 and 1.

Gist of Idea

You can abstract concepts from the moon, but the number one is not among them

Source

Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884], §44)

Book Ref

Frege,Gottlob: 'The Foundations of Arithmetic (Austin)', ed/tr. Austin,J.L. [Blackwell 1980], p.57

A Reaction

Note that Frege seems to be conceding psychological abstraction for most other concepts. But why can't you abstract from your abstractions, to reach high-level abstractions? And why should numbers not emerge at those higher levels?

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]
20361 We need 'unities' for reckoning, but that does not mean they exist [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]