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Single Idea 8661
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
]
Full Idea
The natural numbers are quite primitive, and are what we first learn about. The order of objects (the 'ordinals') is one level of abstraction up from the natural numbers: we impose an order on objects.
Gist of Idea
The natural numbers are primitive, and the ordinals are up one level of abstraction
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.12
A Reaction
Note the talk of 'levels of abstraction'. So is there a first level of abstraction? Dedekind disagrees with Friend (Idea 7524). I would say that natural numbers are abstracted from something, but I'm not sure what. See Structuralism in maths.
Related Idea
Idea 7524
Order, not quantity, is central to defining numbers [Dedekind, by Monk]
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]
|