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Single Idea 7524

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers ]

Full Idea

Dedekind said that the notion of order, rather than that of quantity, is the central notion in the definition of number.

Gist of Idea

Order, not quantity, is central to defining numbers

Source

report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4

Book Ref

Monk,Ray: 'Bertrand Russell: Spirit of Solitude' [Vintage 1997], p.116

A Reaction

Compare Aristotle's nice question in Idea 646. My intuition is that quantity comes first, because I'm not sure HOW you could count, if you didn't think you were changing the quantity each time. Why does counting go in THAT particular order? Cf. Idea 8661.

Related Ideas

Idea 646 When we count, are we adding, or naming numbers? [Aristotle]

Idea 8661 The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]

The 23 ideas from 'Nature and Meaning of Numbers'

22289 Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Dedekind, by Potter]
10090 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
17452 Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
14437 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
18094 Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
18841 Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
13508 Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
18096 Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
14130 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
14131 Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
9153 Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]
9189 Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett]
9979 Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait]
9823 Numbers are free creations of the human mind, to understand differences [Dedekind]
9824 In counting we see the human ability to relate, correspond and represent [Dedekind]
8924 Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride]
10183 An infinite set maps into its own proper subset [Dedekind, by Reck/Price]
10706 Dedekind originally thought more in terms of mereology than of sets [Dedekind, by Potter]
9825 A thing is completely determined by all that can be thought concerning it [Dedekind]
22288 We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Dedekind, by Potter]
9826 A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
9827 We derive the natural numbers, by neglecting everything of a system except distinctness and order [Dedekind]