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Single Idea 16146

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

Full Idea

Democritus claimed that one substance could not be composed from two nor two from one. …The same will clearly go for number, on the popular assumption that number is a combination of units. Unless two is one, it cannot contain a unit in actuality.

Gist of Idea

Two can't be a self-contained unit, because it would need to be one to do that

Source

report of Democritus (fragments/reports [c.431 BCE]) by Aristotle - Metaphysics 1039a15

Book Ref

Aristotle: 'Metaphysics', ed/tr. Lawson-Tancred,Hugh [Penguin 1998], p.218

A Reaction

Chrysippus followed this up the first part with the memorable example of Dion and Theon. The problem with the second part is that 2, 3 and 4 are three numbers, so they can count as meta-units.

Related Ideas

Idea 16058 Dion and Theon coexist, but Theon lacks a foot. If Dion loses a foot, he ousts Theon? [Chrysippus, by Philo of Alexandria]

Idea 16059 Change of matter doesn't destroy identity - in Dion and Theon change is a condition of identity [Chrysippus, by Long/Sedley]

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]