The above proposition is occasionally useful.

Fascinating.

A bizarre discussion is going on on Twitter at the moment, concerning whether 2+2=5. Apparently, it's "woke" to try to undermine the truth we all know, that 2+2=4. As a mathematician who fondly imagines that he is towards the woke end of the spectrum ... 1/

— Timothy Gowers (@wtgowers) August 3, 2020

This all started a couple of weeks ago when Critical Theory/Social Justice researcher James Lindsay mocked the insistence of such specimens that mathematics was a social construct and needed to be decolonized. This is most frequently synopsized in the claim that 2+2=5.

Of course it is all a matter of definitions. Anything can be anything depending on your agreed definitions. The technique of Critical Theory/Social Justice (CTSJ) devotees is make a claim outside reality and then back into it by changing the accepted definitions of words. It is a petty indulgence, witless and counterproductive, but it is what they do.

Twitter has been filled in the past week or so with CTSJ trying to justify their arbitrary position, providing example after example that they are true believers rather than real mathematicians. There are some reasonably good faith efforts by others trying to be even handed and show that under just the right narrow circumstances, the CTSJ faithful can be correct. And then, far outweighing them, there is the avalanche of mockery.

And while worthy of mockery for their dangerous totalitarian beliefs it has inadvertently driven a useful stress test. There is much knowledge we accept because it is usefully true but which we do not examine in detail or necessarily understand why it is usefully true.

Imagine being called upon to prove that the world was indeed round. We so broadly and consistently believe it to be round that the challenge is absurd. And yet when you sit down and try and formulate the argument and evidence that would compellingly demonstrate the truth of the proposition, it is surprisingly challenging. Probably as few as one in a hundred could do so without resort to Google.

The CTSJ are wrong (on just about everything) but their desperate and amateurish efforts to prove they are right serve as good foil for everyone else to dust-off their knowledge of mathematics and rediscover its beauty, precision, sophistication and utility. And its challenges.

This is not dissimilar to the experience with the Intelligent Design people back in 1980s or so. They are wrong but proving them wrong requires a lot more careful thinking than you would imagine. And the CTSJ are not nearly as good at making their argument as the Intelligent Design people are with theirs.

All of this calls to mind a footnote from long ago. It was from Principia Mathematica by Alfred North Whitehead and Bertrand Russell. Published in 1910, it was their attempt to put mathematics on a firm and logic-based foundation, an effort which extended to three volumes. It is famous for the fact that not till page 379 in the first edition of Volume I do they find it safe to conclude:

From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2.

379 pages is a long way to get to one plus one equals two. But well worth it.

And in fact they were not completely done. They didn't actually complete the proof until page 86 of Volume II. And when having finally proved it to their satisfaction, they observe, tongue in cheek "The above proposition is occasionally useful."

What would they have made of the CTSJ nonsense? Mincemeat, if it were Russell responding.