Last week there was a debate about "2 + 2 = 5". Math ppl pointed out it's false in *standard* math, but could be true in other maths. Which—yes, BUT it happens 2 + 2 = 5 produces the null ring, an exceptionally *boring* math.

Now, 2 + 2 = 0 on the other hand— THAT'S interesting.

The null ring suffers from trivialism. There's only one element and not much to say about it.

2 + 2 = 0, on the other hand, means you're in the modular field of order 2, which there's LOTS to say about, and even more to say about if you consider the ring of polynomials over it.

Polynomials over Z_2 (the field where 2 + 2 = 0) are actually *useful*. AES (widely considered the current gold standard for encryption) works by doing math on degree 8 polynomials on Z_2, encoded as 8-bit binary numbers where each bit is a coefficient.

2 + 2 = 0 is useful!

Hm, yeah, I suppose so.

Z_4 is less interesting to me than Z_2 *makes a smug group theory hipster face* https://twitter.com/porglezomp/status/1294497307602411521

Anyway, here's where I'm gonna go off the rails. The 2 + 2 = 5 argument last week was really about relativism. Some shitty conservative was saying you can't just stand up and say 2 + 2 = 5, cuz what he was REALLY trying to say is you can't stand up and say trans women are women.

Shitty conservative wanted to push back against relativism— the idea that different value systems should be simultaneously considered and treated, at least for purposes of politeness, as valid— because he wanted only his value system to be treated as correct.

He considers his values "facts". He was trying to say, you can't stand up and say 2 + 2 = 5—cuz to him denying a "fact" like that is equivalent to disagreeing with him about politics. But he chose a bad example, because mathematicians stood up and said— well, why *not* 2 + 2 = 5?

Mathematicians deal with tautologies. Everything in math is true by definition. But mathematicians are used to being relativistic about their tautologies. They're used to considering lots of different systems, each true by definition, but mostly incompatible with each other.

To a mathematician, the response to an outlandish statement is "well, what would the consequences of that be?"

But, of course, 2 + 2 = 5 isn't a very good example of *that* idea either. Cuz the consequences of that (unless u have a construction I didn't think of) aren't useful.

So: 2 + 2 = 5 is the null ring. It's valid, but not useful.

…what's that a metaphor for? What's it say about relativism? Anything?

Well, here's one angle. I mentioned "trivialism". This is the idea a system can prove "too much". It comes up in logical systems w/contradictions.

If you define some axioms, but you explore the axioms and discover they lead to even one contradiction, this starts a cascade effect where you can prove *anything* is true. If every statement is true, then no statement is meaningful.

This is one reason mathematicians actually do proofs by starting with an premise and if it leads to a contradiction, saying you've disproved the premise. (You can get around this with paraconsistent logics with resolution processes, but that's adding rules, so let's ignore this.)

If we want an analogy to relativism, maybe one lesson math teaches us is it's OK to be relativistic, but not to overlook bullshit. People can come at a thing with different starting assumptions (different backgrounds). But if they start contradicting themselves, they can buzz off

Maybe we can accept there is more than one valid system out there. But then maybe there are also systems where people have taken enough messy assumptions that they can just prove any statement true they can think of, and maybe at that point nothing they have to say is useful.

…or maybe you can just remember human belief systems and political values are not mathematical objects you can form unambiguous logical sentences about, they're something more vague and mushy, and stop trying to treat human interaction like a computer program.

That works too.

CORRECTION: AES doesn't use degree 8 polynomials. It uses degree 7 polynomials. I got confused about the definition of "degree". What I was trying to say is AES uses polynomials with 8 terms