Quote Originally Posted by Mhaeric View Post
The entirety of math is arbitrary if you want to go down that philosophical rabbit hole. But yes, technically, if you want to include 1 as a prime then you can do so... as long as you provide an explicit exception for every single case that it affects. It's far, far simpler, more elegant and to be frank smarter to leave it out.
I would argue that the names we give mathematical objects are more arbitrary than the underlying structure of those objects. You wouldn't say that any of mathematics would break if we renamed 2 to "smorf," or if we called addition "kupo-ing." By that metric it seems disingenuous to say that any theorems would break if prime numbers included 1. Personally, I don't think it's "far, far" simpler to leave it out, as leaving it in would amount to adding a single word to the statement of most theorems involving prime numbers. It is definitely cleaner to leave it out, and I doubt anyone would argue with that, but the distinction isn't particularly important.