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Single Idea 13510

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

Full Idea

Russell toyed with the idea that there is nothing to being a natural number beyond occurring in a progression

Gist of Idea

Could a number just be something which occurs in a progression?

Source

report of Bertrand Russell (Introduction to Mathematical Philosophy [1919], p.8) by William D. Hart - The Evolution of Logic 5

Book Ref

Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.148

A Reaction

How could you define a progression, without a prior access to numbers? - Arrange all the objects in the universe in ascending order of mass. Use scales to make the selection. Hence a finite progression, with no numbers!

The 19 ideas with the same theme [which type of numbers is the most fundamental?]:

11044 One is prior to two, because its existence is implied by two [Aristotle]
10091 God made the integers, all the rest is the work of man [Kronecker]
10090 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
17452 Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
9983 Cantor took the ordinal numbers to be primary [Cantor, by Tait]
18256 Quantity is inconceivable without the idea of addition [Frege]
13510 Could a number just be something which occurs in a progression? [Russell, by Hart,WD]
14128 Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell]
14129 Ordinals presuppose two relations, where cardinals only presuppose one [Russell]
14132 Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
13489 Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD]
18255 Addition of quantities is prior to ordering, as shown in cyclic domains like angles [Dummett]
9191 Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett]
13411 If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
9151 Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K]
18102 A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
13892 One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
8661 The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]