more from Penelope Maddy

Single Idea 17825

[catalogued under 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory]

Full Idea

If you wonder why multiplication is commutative, you could prove it from the Peano postulates, but the proof offers little towards an answer. In set theory Cartesian products match 1-1, and n.m dots when turned on its side has m.n dots, which explains it.


'Commutative' means it works in any order

Gist of Idea

Set theory (unlike the Peano postulates) can explain why multiplication is commutative


Penelope Maddy (Sets and Numbers [1981], II)

Book Reference

'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.347

A Reaction

'Turning on its side' sounds more fundamental than formal set theory. I'm a fan of explanation as taking you to the heart of the problem. I suspect the world, rather than set theory, explains the commutativity.