Thread by @wtgowers

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/

(but not right at the extreme) I feel I ought to comment.

So first, as many have pointed out, the truth of a mathematical sentence depends on the definitions involved and on the system to which that sentence belongs. For example, there is a very important context, ... 2/

arithmetic base 2, in which the sentence '1+1=10' is true.

Secondly, which definitions and rules we adopt depends a lot on what we happen to find useful, either for mathematical purposes or for the purposes of modelling the world. I can imagine a (slightly contrived) ... 3/

situation in which people might find it useful to say that 2+2=5, namely the following. Suppose you are an event organiser and the event splits up naturally into short presentations with breaks between them. Maybe you'd get into the habit of talking about sessions ... 4/

in terms of the number of breaks. So a 2-break session would consist of three presentations separated by two breaks. A 2-break session followed by a 2-break session would then be a 5-break session, and it might be convenient to abbreviate that to 2+2=5. 5/

Maybe some people would take that as an Orwellian assault on the notion of objective truth. Such people might insist on pointing out that your definition of "plus" in this context is not the usual one. They might ask you please to use a different symbol, $, say, ... 6/

which would be defined by m$n=m+n+1. But you'd be within your rights to say that no confusion has arisen, so you're going to stick with using the + symbol thank you very much. Of course, you would also cheerfully admit that this was a different use of the symbol. 7/

As I understand it, the reason this debate has generated so much heat is that an analogy is drawn between the use of the word "woman" and the use of the word "plus". The idea is, I think, that calling trans women women is a bit like saying that 2+2=5. 8/

Here are a few reasons that this is a terrible analogy.

The first, and by far the most important, is that definitions in pure mathematics are, by and large, politically neutral. (There may be counterexamples, and I'll be happy to hear about them, but in general ... 9/

if we say "Let's define X to be Y" that's just a straightforward mathematical convenience with no political consequences. Of course, the moment mathematics is used to model the world, there may well be political consequences.) By contrast, ... 10/

the way we decide to use the word "woman" has huge consequences. You can't say to a trans woman, "What are you worried about? I'm using the word 'woman' in a way that excludes anybody who was born with a penis, but don't worry - we agree about all the facts ... 11/

of the situation even if we use different language to describe them." That's because most of the time the word "woman" is used by people who aren't particularly thinking about whether trans women are or are not included, so to refuse to use the word is to imply ... 12/

that they are *not* included in all those other contexts. And that is a serious and very negative consequence for trans women.

A second reason that the analogy is bad is that while I had to make quite an effort to come up with a context in which it might be reasonable to ... 13/

use the mathematical sentence '2+2=5', I don't have to make any effort at all to come up with a context in which it is reasonable to use the word "woman" to refer to trans women, namely when one wants to include trans women in general statements about women. 14/

An analogy that I read somewhere and do find quite helpful is that of adoptive parenthood. To say "You are not that child's mother" to an adoptive mother would be offensive, for a similar reason, so we use the word "mother" for adoptive mothers. And yet we don't see anyone ...15/

complaining that the notion of objective truth is under threat, or that the nature of parenthood is being undermined.

I do admit to finding the highly charged nature of the debate concerning trans women difficult, because it goes so strongly against my mathematical habits. 16/

One side says "Trans women ARE women" and the other side says "Trans women ARE NOT women" where a mathematician might say "It is better to define 'woman' in a way that includes trans women" or "It is better to define 'woman' in a way that excludes trans women." 17/

But it seems that temporarily adopting a neutral linguistic stance in this debate is taken as entertaining the possibility that the other side has a point that's worth listening to (even if one ends up disagreeing with it), and that one must not do. 18/

A third reason the analogy is bad is that out of the two sentences '2+2=4' and '2+2=5' it's clear that the first is by far the more likely to be useful. So by implicitly linking '2+2=4' to "Trans women are not women" and '2+2=5' to "Trans women are women", 19/

the suggestion is that it is far more natural and sensible to say that trans women are not women, and that ridiculous mental contortions are needed if one wants to call them women. But it isn't hard to use the word "woman" in that way, and there are good reasons to do so. 20/20