green numbers give full details.
|
back to list of philosophers
|
expand these ideas
Ideas of Richard Dedekind, by Text
[German, 1831 - 1916, Born and died at Brunswick. Taught mathemtics in Zurich and Brunswick.]
1872
|
Continuity and Irrational Numbers
|
Intro
|
p.2
|
17611
|
We want the essence of continuity, by showing its origin in arithmetic
|
§1
|
p.4
|
17612
|
Arithmetic is just the consequence of counting, which is the successor operation
|
§4
|
p.15
|
10572
|
A cut between rational numbers creates and defines an irrational number
|
p.27
|
p.263
|
18087
|
If x changes by less and less, it must approach a limit
|
1888 Jan
|
p.173
|
18244
|
I say the irrational is not the cut itself, but a new creation which corresponds to the cut
|
1888
|
Nature and Meaning of Numbers
|
|
p.13
|
10090
|
Dedekind defined the integers, rationals and reals in terms of just the natural numbers [George/Velleman]
|
|
p.21
|
9153
|
Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Fine,K]
|
|
p.45
|
9979
|
Dedekind has a conception of abstraction which is not psychologistic [Tait]
|
|
p.71
|
14437
|
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Russell]
|
|
p.88
|
22289
|
Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Potter]
|
|
p.99
|
18094
|
Dedekind says each cut matches a real; logicists say the cuts are the reals [Bostock]
|
|
p.101
|
18096
|
Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Bostock]
|
|
p.116
|
7524
|
Order, not quantity, is central to defining numbers [Monk]
|
|
p.124
|
13508
|
Dedekind gives a base number which isn't a successor, then adds successors and induction [Hart,WD]
|
|
p.146
|
9189
|
Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dummett]
|
|
p.200
|
17452
|
Ordinals can define cardinals, as the smallest ordinal that maps the set [Heck]
|
|
p.248
|
14130
|
Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Russell]
|
|
p.251
|
14131
|
Dedekind's ordinals are just members of any progression whatever [Russell]
|
|
p.267
|
18841
|
Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt]
|
Pref
|
p.31
|
9823
|
Numbers are free creations of the human mind, to understand differences
|
Pref
|
p.32
|
9824
|
In counting we see the human ability to relate, correspond and represent
|
§3 n13
|
p.581
|
8924
|
Dedekind originated the structuralist conception of mathematics [MacBride]
|
§64
|
p.376
|
10183
|
An infinite set maps into its own proper subset [Reck/Price]
|
2-3
|
p.23
|
10706
|
Dedekind originally thought more in terms of mereology than of sets [Potter]
|
I.1
|
p.44
|
9825
|
A thing is completely determined by all that can be thought concerning it
|
no. 66
|
p.83
|
22288
|
We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Potter]
|
V.64
|
p.63
|
9826
|
A system S is said to be infinite when it is similar to a proper part of itself
|
VI.73
|
p.68
|
9827
|
We derive the natural numbers, by neglecting everything of a system except distinctness and order
|