Ideas from 'Nature and Meaning of Numbers' by Richard Dedekind [1888], by Theme Structure

[found in 'Essays on the Theory of Numbers' by Dedekind,Richard [Dover 1963,0-486-21010-3]].

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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 / p. Counting
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 / 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 / D. Concepts / 6. Abstract Concepts / c. 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