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)

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Counting is the assignment of successively larger cardinal numbers to collections

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

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
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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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Equinumerosity is not the same concept as oneone correspondence

17449

We can understand cardinality without the idea of oneone correspondence
