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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.88 Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Potter]
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
no. 66 p.83 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 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