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. Nature of 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. Nature of Numbers / f. Cardinal numbers
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A basic grasp of cardinal numbers needs an understanding of equinumerosity

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
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In counting, numerals are used, not mentioned (as objects that have to correlated)

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 / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
17450

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

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

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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Frege's Theorem explains why the numbers satisfy the Peano axioms

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
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Children can use numbers, without a concept of them as countable objects

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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We can understand cardinality without the idea of oneone correspondence

17458

Equinumerosity is not the same concept as oneone correspondence
