Ideas from 'Logicism, Some Considerations (PhD)' by Paul Benacerraf [1960], by Theme Structure

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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

13412

Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order

13413

We must explain how we know so many numbers, and recognise ones we haven't met before

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

13411

If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

13415

An adequate account of a number must relate it to its series