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I’m not a big fan of the idea of “court packing”, but if the only thing the Republicans are going to listen to at this point is exercise of raw power, then yes, the Democrats should absolutely promise to do that.It’s not an unreasonable thing to do because there are 12 federal circuits and going to 13 justices would get us back to having one justice available to work with each circuit and the Chief available to fill in if one is absent or otherwise unable. I’ve thought this for a while but figured that it was impossible in the current climate to get a bipartisan agreement on how to expand the court. But if the GOP is simply going to ignore what they’ve said in the past in order to put another conservative justice on the court, then it becomes an entirely appropriate response to do it unilaterally.

Here’s the deal: it is true that if you write out a proof of a theorem in full, including all logical steps, then there is an algorithmic process for determining whether the proof is valid or not. There is, indeed, no leeway here—either it is correct, or it isn’t. But, in practice, no one ever writes out all logical steps. To do so would be monstrously time consuming, and would usually render the final document unreadable to anything other than a computer specifically designed for such a task. Anyone who has ever had to prove anything remotely non-trivial from first principles knows what I am talking about.So it is completely normal for mathematicians to leave small gaps in their proofs. Every human written proof is really a proof sketch, written in such a way that if you had to fill in all of the gaps, it would be immediately evident how to do so. This is assuming, however, that the paper is written well, such that it is easy to follow and understand, at least for experts in the field. Sadly, from experience I know that all too often, this is not the case. (Incidentally, this is reason why I recommend to anyone interested in doing research: learn to write. There is too much awful mathematical writing already. Don’t contribute more.) This means that you will have mathematicians arguing over whether something really follows or not—one will argue that it is obvious, the other will point out that it is not, and then they will talk it out until they come to an agreement about what needs to be changed, if anything.There are plenty of other things to argue about, too, such as definitions. Definitions are arbitrary, so you should choose them in such a way as to maximize utility—they should be as simple as possible, such that you can still prove as much as possible. But, what the right definitions are is usually only obvious in hindsight, when you have a good idea of what is and isn’t true. When you are still at the stage of feeling around in the dark, trying to get some intuition, and trying to figure out how to phrase the question in a productive way, it is often far from obvious what the right definitions should be. And so mathematicians will argue about this as well.In short, bear in mind that mathematics is a fundamentally social activity, and I don’t know if humans know how to be social without arguing at least sometimes.

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