# The Universe of Discourse

Tue, 10 Apr 2018

Philosophy makes a distinguish between necessary and contingent facts, but I'm not exactly sure what it is. I think they would say that the election of Al Gore in 2000 is contingent because it's easy to imagine a universe in which it went the other way and the other guy won. But that seems to depend on our powers of imagination, which doesn't seem very rigorous. Is the mass of the electron necessary or contingent? What about the fine-structure constant?

What facts are necessary? Often in this context people fall back on mathematical truths, for example !!1+1=2!!, which does seem hard to assail. But I recently thought of something even farther down the scale, which seems to me even harder to argue.

Mathematics deals with many sorts of objects which are more or less like the ordinary numbers. Some are more complicated, and ordinary numbers are special cases, for example functions and matrices. Some are simpler, and are special cases themselves. Mathematicians can and do define !!2!! in many different ways. There are mathematical systems with !!1!! and !!+!! in which there is no !!2!!, and instead of !!1+1=2!! we have !!1+1=0!!. Well, not quite; there is !!2!!, but !!2=0!!. So one can say that !!1+1=2!! still, but the !!2!! is not very much like the !!2!! that we usually mean when we say !!1+1=2!!. Anyway certainly there is such a system, and I can certainly conceive of it, so there might be a philosophical argument that could be made that !!1+1=2!! is a contingent fact about how numbers happen to work in the universe in which we happen to find ourselves: we are not living in a universe where numbers form a field of characteristic 2.

But here's a fact that I think is unassailably necessary: rubies are red. Why? By definition! A ruby is a kind of gemstone, a type of aluminum oxide called a corundum, that has a deep red color. There are non-red corundums, but they are sapphires, not rubies, because a ruby is a red corundum. There is no such thing as a blue or a green ruby; a blue or green ruby is not a ruby at all, but a sapphire.

How about over in Narnia, where rubies are blue? Well, maybe the Narnians people call hats “avocadoes”, but whether those things are hats or avocadoes depends not on what the Narnians call them but on their properties. If those things are made of felt and the Narnians wear them on their heads, they are hats, regardless of what the Narnians call them; they are avocadoes only if they are globular and can be eaten on toast. Narnians might put actual avocadoes on their heads and then there might be an argument that these things were hats, but if the avocado is a hat it is only because it is customarily worn on the head.

And so too the Narnians can call !!2!! an avocado and say that !!1+1=\text{avocado}!! but that doesn't mean that !!1+1!! is an avocado, even in Narnia. Maybe the Narnians call avocadoes “rubies”, but they're still avocadoes, not rubies. And maybe the Narnians call blue corundums “rubies”, but they're still sapphires, not rubies, because rubies are red.

So I think it might be conceivable that !!1+1=2!! is contingent, and it's certainly easy to conceive of a universe with no rubies at all, but I can't conceive of any way that a ruby could be other than red.