more from this thinker | more from this text
Full Idea
Hume's Principle can't tell us what a cardinal number is (this is one lesson of Benacerraf's well-known problem). An infinity of pairs of sets could actually be the number two (not just the simplest sets).
Clarification
See Idea 8302 for Hume's Principle
Gist of Idea
No particular pair of sets can tell us what 'two' is, just by one-to-one correlation
Source
report of Paul Benacerraf (What Numbers Could Not Be [1965]) by E.J. Lowe - The Possibility of Metaphysics 10.3
Book Ref
Lowe,E.J.: 'The Possibility of Metaphysics' [OUP 2001], p.215
A Reaction
The drift here is for numbers to end up as being basic, axiomatic, indefinable, universal entities. Since I favour patterns as the basis of numbers, I think the basis might be in a pre-verbal experience, which even a bird might have, viewing its eggs.
Related Idea
Idea 8302 Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe]
| 16896 | If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge] |
| 18161 | The theory of classes is superfluous in mathematics [Wittgenstein] |
| 8697 | Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them [Benacerraf, by Friend] |
| 8304 | No particular pair of sets can tell us what 'two' is, just by one-to-one correlation [Benacerraf, by Lowe] |
| 9906 | If ordinal numbers are 'reducible to' some set-theory, then which is which? [Benacerraf] |
| 17821 | You can ask all sorts of numerical questions about any one given set [Yourgrau] |
| 17815 | We can't use sets as foundations for mathematics if we must await results from the upper reaches [Yourgrau] |
| 10560 | Set-theoretic imperialists think sets can represent every mathematical object [Fine,K] |
| 10222 | Mathematical foundations may not be sets; categories are a popular rival [Shapiro] |
| 17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
| 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |
| 9643 | Set theory may represent all of mathematics, without actually being mathematics [Brown,JR] |
| 9644 | When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR] |
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |