Ideas of Richard Dedekind, by Theme
[German, 1831  1916, Born and died at Brunswick. Taught mathemtics in Zurich and Brunswick.]
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2. Reason / D. Definition / 9. Recursive Definition
22289

Dedekind proved definition by recursion, and thus proved the basic laws of arithmetic [Potter]

4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10183

An infinite set maps into its own proper subset [Reck/Price]

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
22288

We have the idea of self, and an idea of that idea, and so on, so infinite ideas are available [Potter]

4. Formal Logic / G. Formal Mereology / 1. Mereology
10706

Dedekind originally thought more in terms of mereology than of sets [Potter]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
9823

Numbers are free creations of the human mind, to understand differences

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
17452

Ordinals can define cardinals, as the smallest ordinal that maps the set [Heck]

10090

Dedekind defined the integers, rationals and reals in terms of just the natural numbers [George/Velleman]

7524

Order, not quantity, is central to defining numbers [Monk]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
14131

Dedekind's ordinals are just members of any progression whatever [Russell]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17611

We want the essence of continuity, by showing its origin in arithmetic

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
10572

A cut between rational numbers creates and defines an irrational number

14437

Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Russell]

18094

Dedekind says each cut matches a real; logicists say the cuts are the reals [Bostock]

18244

I say the irrational is not the cut itself, but a new creation which corresponds to the cut

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
9824

In counting we see the human ability to relate, correspond and represent

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
17612

Arithmetic is just the consequence of counting, which is the successor operation

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
9826

A system S is said to be infinite when it is similar to a proper part of itself

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18087

If x changes by less and less, it must approach a limit

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
13508

Dedekind gives a base number which isn't a successor, then adds successors and induction [Hart,WD]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18096

Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Bostock]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2ndorder
18841

Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14130

Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Russell]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8924

Dedekind originated the structuralist conception of mathematics [MacBride]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
9153

Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Fine,K]

9. Objects / A. Existence of Objects / 3. Objects in Thought
9825

A thing is completely determined by all that can be thought concerning it

18. Thought / E. Abstraction / 3. Abstracta by Ignoring
9189

Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dummett]

9827

We derive the natural numbers, by neglecting everything of a system except distinctness and order

18. Thought / E. Abstraction / 8. Abstractionism Critique
9979

Dedekind has a conception of abstraction which is not psychologistic [Tait]
