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Single Idea 12273

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

Full Idea

They say that the unit [monada] is the starting point of number (and the point the starting-point of a line).

Gist of Idea

Unit is the starting point of number

Source

Aristotle (Topics [c.331 BCE], 108b30)

Book Ref

Aristotle: 'Posterior Analytics and Topica', ed/tr. Tredennick,H/Forster,ES [Harvard 1960], p.329

A Reaction

Yes, despite Frege's objections in the early part of the 'Grundlagen' (1884). I take arithmetic to be rooted in counting, despite all abstract definitions of number by Frege and Dedekind. Identity gives the unit, which is countable. See also Topics 141b9

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]