[Health] [Math] Proofs, forgetting, and an eldritch god

by aaq 4 min read11th Nov 20194 comments

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This post made me realize something very important, and in hindsight very obvious about myself. I'm leaving it up for future reference. I don't think it's actually very well written.

Originally, it was tagged [Math], not [Health] and [Math].

I envy computers sometimes for their memory.

My basic intuition from learning signals and Fourier transforms in my EE major is that any limited-space memory system that has to adapt to new surroundings, also has to have some factor by which old memories decay over time -- otherwise the system would become saturated with information content. (I am not being precise with these terms. Look where I point.)

But computers don't usually have as ruthless a "use-it-or-lose-it" forgetting mechanism as human brains do built into them. Their decay is mostly due to (i) other conscious agents going in and rearranging their memories; (ii) the slow encroaching wave of entropy on their hard disks. (i) can be dealt with by locking the computer in a room away from human hands, and praying the Poincare recurrence theorem combined with thermodynamic noise doesn't imply it will eventually birth an AGI all on its own. But that's a symptom of (ii), which can be mostly dealt with via RAID-5 and an influx of hard drives. With those two in place, on a human timescale, computers essentially never have to forget something once it's encoded in memory.

Human beings, we aren't like that. We forget. We forget so easily. And we warp the memories we do have to fit our twisted little narratives in the moment. I'm not a terribly existential person, but the fact that we ship of Theseus ourselves according to the ridiculous whims of future us, people just like us now but ever-so-slightly slower, crazier, and more tied down to the local flora for their survival, you have to admit -- it's a little unsettling.

I've been having a bit of a crisis over the last few days. It's about math, as all my crises are.

The problem is simple: I prove $X$ once. I walk away for a few days. When I go back to look at $X$ again, I suddenly realize the steps of the proof don't immediately spring to mind any more. I have to prove it again, don't I; or accept it on faith that $X$ is a fact, that former me proved $X$ to be a fact, that no, there's really nothing much to be gained from re-proving something in roughly the same way that I did before.

The cognition is contrarian: Proofs take effort. I don't like wasting effort. And just because I could prove $X$ once is no guarantee at all that I could prove $X$ again right now -- memories decay, I only have limited vision and limited CPU cycles, maybe I actually can't prove $X$ again, and that means I don't really understand $X$ in some sense, and I won't know unless I try, right? And math is supposed to be this tower of logic, where -- in theory -- we could break everything all the way down to set/category theory and build it back up, right? That's what good mathematicians do, right? Aren't we supposed to be the true Scotsmen of deductive reasoning?

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I once ran a campaign where there were 4 deities. One of them was called the White Noise; the Noise derived its power from all friction generated in the environment. When you and a stranger are walking down a street and don't want to bump into each other, but you step in the same direction they step, and then you step the other way at the same time that they do -- the White Noise feeds. When you're short 2 cents at the convenience store and the clerk says "Don't worry about it", but then you leave and worry that maybe the clerk will get fired because of the disparity at the end of the day. When lovers quarrel. When soldiers don't fight as hard as they should because they don't fully believe in their cause -- but also (scholars posit) the whole friction generated by war itself, the ultimate (pacifists say) zero-sum waste of resources. Eventually all becomes friction, and time stops.

If I had to choose a deity out of the 4 to follow, it would be the White Noise in a heartbeat. To me, it represents the ultimate decision that pointless human bickering is far preferable to brutal, paperclip-maximizing, mechanized efficiency.

Which is why it's so fucking stressful to me that the one time I do want mechanized efficiency, I can't seem to acquire it. I just want to be able to prove something once, and then store the memory of that proof in my head so that I don't have to keep doing it over and over again, man. Is that too much to ask?

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Yes, it is. But let's go back to what we said before.

Mathematicians (human ones at least, not Coq or something) don't actually work in a strictly deductive fashion. Even if they wanted to, they realistically couldn't. We all have limits to what we can store in our short term memory, and the limits vary, but they presumably don't vary by orders of magnitude; similarly, we all have limits to how fast we can think through the logical implications of a thing, but again, this doesn't vary by orders of magnitude. And the subset of math we care about here is math-as-group-enterprise; so the fact that mathematicians are all competing for roughly the same rewards suggests that they would find ways to work around those fundamental limits in order to outdo each other. So it's certainly not the case that every other good mathematician on the planet does a proof once, memoizes exactly how to do it, and then just keeps that memory fresh.

What does happen, then? Probably what we would expect from a common sense view of human nature: They forget. They forget almost everything they learn at the undergraduate and graduate level, just like anyone else would. Oh, they probably don't forget as much relevant to their field of interest -- but I highly doubt there's any tenured, non-set theory professor out there who consciously reruns through proving Zorn's lemma from scratch once or twice a year, just to make sure they can still do it. It's a difficult, convoluted proof, and they have more important work to be done. New work.

Now, I don't think any working mathematician will admit this. They'll probably say something along the lines of, "Well of course I can't prove it on sight right now, but give me a couple days and I can probably get back to you with one. Just need to refresh my memory." That might even be true. But they're not just refreshing their memory -- they're turning a highly-tuned neural network, full of a lifetime of proving difficult propositions and engaging in clever logical tricks, back towards a problem they already understood the solution to once when they were less well-attuned.

Maybe, someday, we'll be able to fuck around with human memory enough that nobody will forget how to do a proof after the first time they spend energy on figuring out how to do it. But that day isn't today. Today, the White Noise scores a point for human imperfection. And while that might be locally frustrating -- think how much easier I would have it if that were the case! -- I wouldn't actually want to live in a world where everyone, universally, had those powers of memory. The competition would just adapt, quickly, to the newfound power, and I wouldn't end up with an obvious improvement in the group effort as before. I want it for me, and me alone; and that's a sign that I'm not actually upset for a good reason. I'm just being selfish. Even my OCD channels human bias.


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