The Commutative Property

In The Joy of x, Steven Strogatz points out that the commutative property (the fact that the order of actions doesn’t matter) isn’t intuitive in multiplication. Forget for a minute that you know multiplication is commutative, that 3 X 7 is equal to 7 X 3. Instead ask yourself: If you add 3 + 3… 7 times, why should it be equal to 7 + 7... 3 times? He elaborates the non-intuitiveness of the problem:

“Try counting by sevens: 7, 14, 21. Now turn it around and count by threes instead: 3, 6, 9… Do you feel the suspense building? So far none of the numbers match those in the sevens list, but keep going… 12, 15, 18, and then, bingo, 21!”

 

Shifting to a different perspective, he says the commutative law is hard to come to terms with, because in daily life, the order of things does matter. Cause before effect. Socks first, shoes second.

 

And then he tells the amusing tale about the physicist, Murray Gell-Mann. He had been rejected by Princeton. Harvard said Yes, but weren’t committing to a scholarship. And so:

“His best option, though he found it depressing, was MIT.”

MIT was depressing? To Gell-Mann, MIT was a “technological institute, beneath his rarified taste”. But he did accept. Years later, he said he’d even considered committing suicide(!) at the time, but didn’t go through. Why not? Because he understood not everything is commutative.

“Attending MIT and killing oneself didn’t commute. He could always go to MIT and commit suicide later if he had to, but not the other way around.”

 

It’s not surprising that Gell-Mann understood non-commutativity, jokes Strogatz. He was a physicist, one of the eventual great’s of quantum mechanics. And so he remembered what Heisenberg had found weird decades back when he was formulating quantum mechanics, that for certain kinds of mathematical entities (matrices), p X q ≠ q X p:

“Without that breakdown of the commutative law, there would be no Heisenberg uncertainty principle, atoms would collapse, and nothing would exist.”

 

The book is endlessly informative, and entertaining.

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