Ideas of Richard Dedekind, by Theme

[German, 1831 - 1916, Born and died at Brunswick. Taught mathemtics in Zurich and Brunswick.]

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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
An infinite set maps into its own proper subset [Reck/Price]
4. Formal Logic / G. Formal Mereology / 1. Mereology
Dedekind originally thought more in terms of mereology than of sets [Potter]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
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
Dedekind defined the integers, rationals and reals in terms of just the natural numbers [George/Velleman]
Order, not quantity, is central to defining numbers [Monk]
Ordinals can define cardinals, as the smallest ordinal that maps the set [Heck]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Dedekind's ordinals are just members of any progression whatever [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
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
A cut between rational numbers creates and defines an irrational number
Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Russell]
Dedekind says each cut matches a real; logicists say the cuts are the reals [Bostock]
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
In counting we see the human ability to relate, correspond and represent
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
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
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
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
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
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 2nd-order
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
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
Dedekind originated the structuralist conception of mathematics [MacBride]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Fine,K]
9. Objects / A. Existence of Objects / 3. Objects in Thought
A thing is completely determined by all that can be thought concerning it
18. Thought / E. Abstraction / 3. Abstracta by Ignoring
Dedekind said numbers were abstracted from systems of objects, leaving only their position [Dummett]
We derive the natural numbers, by neglecting everything of a system except distinctness and order
18. Thought / E. Abstraction / 8. Abstractionism Critique
Dedekind has a conception of abstraction which is not psychologistic [Tait]