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Why is multiplication commutative?
Posted:
Aug 28, 2012 1:57 AM


The main point of this page is that the above question can be interpreted in many different ways and that different interpretations lead to different answers. A secondary goal is to discuss whether the fact that we can `just see' that multiplication is commutative means that we have some ability that computers could never have (as, for example, Penrose has suggested).
mn=nm for any pair of positive integers m and n.
Why does this statement seem obvious? After all, if you decide to multiply together two numbers such as 395 and 428 using long multiplication, then the calculations you do will depend very much on whether you work out 395 428s or 428 395s. In the first case you will find yourself adding 128400, 38520 and 2140 while in the second you will add 158000, 7900 and 3160. Why isn't it a miracle that both triples of numbers add to 169060?
Most people would probably say that 395x428 and 428x395 both count the number of points in a 395by428 rectangular grid. If you rotate such a grid through 90 degrees then you turn it into a 428by395 rectangular grid but you clearly leave the number of points unchanged. (They might not express the argument in exactly this form, but this would be the rough line of reasoning.)
This argument, compelling as it is, doesn't quite qualify as a mathematical proof. Can we turn it into one? Here is an attempt.
Attempted Proof
If A is a set of cardinality m and B is a set of cardinality n, then the Cartesian product AxB has cardinality mn. But the map (a,b)>(b,a) is easily seen to be a bijection between AxB and BxA, from which it follows that BxA has cardinality mn. But we already know that it has cardinality nm, so mn=nm.



