Ideas from 'Cardinality, Counting and Equinumerosity' by Richard G. Heck [2000], by Theme Structure

[found in 'Notre Dame Journal of Formal Logic' (ed/tr -) [- ,]].

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6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
The meaning of a number isn't just the numerals leading up to it
6. Mathematics / A. Nature of Mathematics / 3. Numbers / c. Priority of numbers
Ordinals can define cardinals, as the smallest ordinal that maps the set
6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
A basic grasp of cardinal numbers needs an understanding of equinumerosity
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
In counting, numerals are used, not mentioned (as objects that have to correlated)
Understanding 'just as many' needn't involve grasping one-one correspondence
We can know 'just as many' without the concepts of equinumerosity or numbers
Is counting basically mindless, and independent of the cardinality involved?
Counting is the assignment of successively larger cardinal numbers to collections
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
Frege's Theorem explains why the numbers satisfy the Peano axioms
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Children can use numbers, without a concept of them as countable objects
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
We can understand cardinality without the idea of one-one correspondence
Equinumerosity is not the same concept as one-one correspondence