How thinking mathematically might help us all argue better

The Third Discourseman

‘Trans-women are women’ – this is a statement often spouted by people wanting to fight for transgender rights. For many in our day and age, its truth is not negotiable. It’s blindingly obvious, and even an attempt to debate it is backwards and outdated. Yet some remain who might want to deny it- in fact a reasonably broad range of people from TERFs (trans exclusionary radical feminists) to social conservatives and many religious groups. And disagreements over a statement like this one at best cause outrage, offense and argument, and at worse could leave you ‘cancelled’ and out of a job.

Thinking about it logically (by which, as a mathematician, I mean in a set-theoretic sense) it is a pretty simple proposition. ‘A are B’ could mean A=B, but this is probably not what this statement means. After all, I doubt that even the most outspoken transgender activist would claim that the set of all trans-women and the set of all women are equal. Probably what is meant is ‘the set of all transgender women is a subset of the set of all women’ which, in English, means ‘any human that happens to be a trans-woman is also a woman’. Once phrased this way, it becomes clear that the truth of this statement depends entirely on how you define these two sets. In other words this statement is either true by definition, or false by definition.

In other words, if you define a woman as someone who self-identifies as a woman (though you’d have to go further than this since, as it stands, the word is poorly defined since it uses the word ‘woman’ to define what a woman is. However I am yet to find a clear definition that isn’t self referential from those who take this line, so this is the best I can do), then clearly trans-women are women. That is true by definition. Likewise if you define a woman as someone born female, or someone with XX chromosomes, or someone born with female reproductive parts or something similar, then clearly trans-women are not women. Again, this is simply by definition.

So why point this out? Because I think it typifies a wider problem, which is where the title of this blog comes in. You could theoretically have two people in total agreement about a particular person (i.e. both agree that said person was not born female and yet identifies as a woman), and yet this agreement is totally clouded by the fact that one calls her a woman and the other doesn’t, because they’re defining the word differently from each other. This could be extended to other issues- theological arguments going round in circles because the two arguing both mean completely different things by the words ‘faith’ or ‘justification’. Which is where I hope that thinking mathematically might help us get along more.

In mathematics, it is absolutely crucial to define things, and to do so carefully. Thus, when mathematicians talk about ‘continuous functions’ or some other idea, they’re all in agreement as to what is meant. As you start learning a new branch of mathematics, everything is defined carefully and clearly. So there’s relatively little controversy in maths. People argue about the truth of statements that have not yet been proved. People argue about the philosophy of maths (questions like ‘is maths created or discovered’). People argue about the ‘right’ way of approaching something. But roughly speaking people agree about definitions (and if they don’t, because things are defined clearly it is easy to tell when something is defined differently), about what has been proven and what hasn’t. This is in stark contrast with the humanities, where just about every essay title you can come up with has the potential to cause controversy, and even the most basic definitions are slippery and contentious.

Now I’m not suggesting that if we were simply more upfront about definitions then all our disagreements would go away- after all, people may well take offense at our definitions, and indeed the way we choose to define words says something much deeper about the way we think about the topic in question, which may still leave plenty to disagree about.

But I do think it could help. Surely such fighting can only be exacerbated if it’s happening in a context where the two sides don’t even really know what each other mean, as if they’re speaking different languages. In some sense this is worse than speaking different languages- someone who only speaks English may well struggle to understand someone who only speaks French, but they’re also unlikely to misinterpret what the other is saying, as they just won’t interpret it at all. But when you have two groups that seem to be using all the same words, only some have different meanings, confusion is bound to ensue. Imagine an Englishman in the USA who orders chips and is dismayed when he is brought a packet of Doritos, only what’s at stake is often much deeper issues of how people perceive themselves and the world around them.

Of course perhaps it might make things worse, when two people who once thought they agreed suddenly discover they mean totally different things by what they’re saying. But even this is helpful- surely it’s better to realise that you disagree than to go years thinking you’re in agreement, simply biding time for the ever-growing awkwardness of the moment you realise you aren’t.

Even if being clear on your definitions doesn’t avoid offense, it should avoid confusion. It should mean that at the very least you and your debate-partner are speaking the same language. It should show up when supposedly contentious claims are really just trivial (in a mathematical sense, which is to say nothing of their significance or profundity), dependent on nothing more than how you’ve defined the words you’re using to make them. And it should help expose precisely where it is that you are disagreeing, if in fact there is any disagreement at all.

So next time someone says ‘trans-women are/aren’t women’, perhaps instead of launching off into why their claim is bigoted/unbiblical/unscientific/irrational/harmful, try asking them ‘Ok, but what exactly do you mean by ‘women’? How would you define that word? What makes a woman a woman?’- it will almost certainly either help you understand their position better or expose the fact that they don’t really understand their own position very well. It’ll probably result in a more fruitful discussion than if you’d gone straight for the jugular. And the same goes for any other discussion which seems rife for similar confusion. So release your inner mathematician, and get clear on your definitions!