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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.
Clarification
'Commutative' means it works in any order
Gist of Idea
Set theory (unlike the Peano postulates) can explain why multiplication is commutative
Source
Penelope Maddy (Sets and Numbers [1981], II)
Book Ref
'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.
| 17823 | If mathematical objects exist, how can we know them, and which objects are they? [Maddy] |
| 17824 | The master science is physical objects divided into sets [Maddy] |
| 17825 | Set theory (unlike the Peano postulates) can explain why multiplication is commutative [Maddy] |
| 17826 | Standardly, numbers are said to be sets, which is neat ontology and epistemology [Maddy] |
| 17828 | Numbers are properties of sets, just as lengths are properties of physical objects [Maddy] |
| 17827 | Sets exist where their elements are, but numbers are more like universals [Maddy] |
| 17829 | Number words are unusual as adjectives; we don't say 'is five', and numbers always come first [Maddy] |
| 17830 | Number theory doesn't 'reduce' to set theory, because sets have number properties [Maddy] |