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Single Idea 8924

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism ]

Full Idea

Dedekind is the philosopher-mathematician with whom the structuralist conception originates.

Gist of Idea

Dedekind originated the structuralist conception of mathematics

Source

report of Richard Dedekind (Nature and Meaning of Numbers [1888], §3 n13) by Fraser MacBride - Structuralism Reconsidered

Book Ref

'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.581

A Reaction

Hellman says the idea grew naturally out of modern mathematics, and cites Hilbert's belief that furniture would do as mathematical objects.

The 28 ideas from Richard Dedekind

17611 We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
17612 Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
10572 A cut between rational numbers creates and defines an irrational number [Dedekind]
18087 If x changes by less and less, it must approach a limit [Dedekind]
18244 I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
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]
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]
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]
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]