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
4. Formal Logic / G. Formal Mereology / 1. Mereology
Dedekind originally thought more in terms of mereology than of sets
6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
Numbers are free creations of the human mind, to understand differences
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
Dedekind defined the integers, rationals and reals in terms of just the natural numbers
Order, not quantity, is central to defining numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
We want the essence of continuity, by showing its origin in arithmetic
6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
I say the irrational is not the cut itself, but a new creation which corresponds to the cut
A cut between rational numbers creates and defines an irrational number
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
Arithmetic is just the consequence of counting, which is the successor operation
In counting we see the human ability to relate, correspond and represent
6. Mathematics / A. Nature of Mathematics / 4. 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 / 4. The Infinite / m. Limits
If x changes by less and less, it must approach a limit
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
Dedekind gives a base number which isn't a successor, then adds successors and induction
6. Mathematics / B. Foundations for Mathematics / 6. Mathematical Structuralism / a. Structuralism
Dedekind originated the structuralist conception of mathematics
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
We derive the natural numbers, by neglecting everything of a system except distinctness and order