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
17453

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
17452

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

6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
17457

A basic grasp of cardinal numbers needs an understanding of equinumerosity

6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
17448

In counting, numerals are used, not mentioned (as objects that have to correlated)

17450

Understanding 'just as many' needn't involve grasping oneone correspondence

17451

We can know 'just as many' without the concepts of equinumerosity or numbers

17455

Is counting basically mindless, and independent of the cardinality involved?

17456

Counting is the assignment of successively larger cardinal numbers to collections

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
17459

Frege's Theorem explains why the numbers satisfy the Peano axioms

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
17454

Children can use numbers, without a concept of them as countable objects

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
17458

Equinumerosity is not the same concept as oneone correspondence

17449

We can understand cardinality without the idea of oneone correspondence
