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Single Idea 9189
[filed under theme 18. Thought / E. Abstraction / 3. Abstracta by Ignoring
]
Full Idea
By applying the operation of abstraction to a system of objects isomorphic to the natural numbers, Dedekind believed that we obtained the abstract system of natural numbers, each member having only properties consequent upon its position.
Gist of Idea
Dedekind said numbers were abstracted from systems of objects, leaving only their position
Source
report of Richard Dedekind (Nature and Meaning of Numbers [1888]) by Michael Dummett - The Philosophy of Mathematics
Book Ref
'Philosophy 2: further through the subject', ed/tr. Grayling,A.C. [OUP 1998], p.146
A Reaction
Dummett is scornful of the abstractionism. He cites Benacerraf as a modern non-abstractionist follower of Dedekind's view. There seems to be a suspicion of circularity in it. How many objects will you abstract from to get seven?
The
28 ideas
from Richard Dedekind
17611
|
We want the essence of continuity, by showing its origin in arithmetic
[Dedekind]
|
17612
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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]
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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]
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18094
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Dedekind says each cut matches a real; logicists say the cuts are the reals
[Dedekind, by Bostock]
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13508
|
Dedekind gives a base number which isn't a successor, then adds successors and induction
[Dedekind, by Hart,WD]
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18096
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Zero is a member, and all successors; numbers are the intersection of sets satisfying this
[Dedekind, by Bostock]
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18841
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Categoricity implies that Dedekind has characterised the numbers, because it has one domain
[Rumfitt on Dedekind]
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14130
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Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer
[Dedekind, by Russell]
|
10090
|
Dedekind defined the integers, rationals and reals in terms of just the natural numbers
[Dedekind, by George/Velleman]
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7524
|
Order, not quantity, is central to defining numbers
[Dedekind, by Monk]
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17452
|
Ordinals can define cardinals, as the smallest ordinal that maps the set
[Dedekind, by Heck]
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14131
|
Dedekind's ordinals are just members of any progression whatever
[Dedekind, by Russell]
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22289
|
Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic
[Dedekind, by Potter]
|
9153
|
Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects
[Dedekind, by Fine,K]
|
9189
|
Dedekind said numbers were abstracted from systems of objects, leaving only their position
[Dedekind, by Dummett]
|
9979
|
Dedekind has a conception of abstraction which is not psychologistic
[Dedekind, by Tait]
|
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]
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