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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

[which type of numbers is the most fundamental?]

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