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Single Idea 9153

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism ]

Full Idea

Dedekindian abstraction says mathematical objects are 'positions' in a model, while Cantorian abstraction says they are the result of abstracting on structurally similar objects.

Gist of Idea

Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects

Source

report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Kit Fine - Cantorian Abstraction: Recon. and Defence §6

Book Ref

-: 'Journal of Philosophy' [-], p.21

A Reaction

The key debate among structuralists seems to be whether or not they are committed to 'objects'. Fine rejects the 'austere' version, which says that objects have no properties. Either version of structuralism can have abstraction as its basis.

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]
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]
14131 Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
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]
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]
18841 Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
14130 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
9153 Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]
9979 Dedekind has a conception of abstraction which is not psychologistic [Dedekind, by Tait]
9189 Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dedekind, by Dummett]
9824 In counting we see the human ability to relate, correspond and represent [Dedekind]
9823 Numbers are free creations of the human mind, to understand differences [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]