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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
p.116 Order, not quantity, is central to defining numbers
p.124 Dedekind gives a base number which isn't a successor, then adds successors and induction
p.146 Dedekind said numbers were abstracted from systems of objects, leaving only their position
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
64 p.376 An infinite set maps into its own proper subset
2-3 p.23 Dedekind originally thought more in terms of mereology than of sets
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