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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 We want the essence of continuity, by showing its origin in arithmetic
1 p.4 Arithmetic is just the consequence of counting, which is the successor operation
4 p.15 A cut between rational numbers creates and defines an irrational number
p.27 p.263 If x changes by less and less, it must approach a limit
1888 Letter to Weber
1888 Jan p.173 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 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [George/Velleman]
p.21 Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Fine,K]
p.45 Dedekind has a conception of abstraction which is not psychologistic [Tait]
p.71 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Russell]
p.99 Dedekind says each cut matches a real; logicists say the cuts are the reals [Bostock]
p.101 Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Bostock]
p.116 Order, not quantity, is central to defining numbers [Monk]
p.124 Dedekind gives a base number which isn't a successor, then adds successors and induction [Hart,WD]
p.146 Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dummett]
p.200 Ordinals can define cardinals, as the smallest ordinal that maps the set [Heck]
p.248 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Russell]
p.251 Dedekind's ordinals are just members of any progression whatever [Russell]
p.267 Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt]
Pref p.31 Numbers are free creations of the human mind, to understand differences
Pref p.32 In counting we see the human ability to relate, correspond and represent
3 n13 p.581 Dedekind originated the structuralist conception of mathematics [MacBride]
64 p.376 An infinite set maps into its own proper subset [Reck/Price]
2-3 p.23 Dedekind originally thought more in terms of mereology than of sets [Potter]
I.1 p.44 A thing is completely determined by all that can be thought concerning it
V.64 p.63 A system S is said to be infinite when it is similar to a proper part of itself
VI.73 p.68 We derive the natural numbers, by neglecting everything of a system except distinctness and order