Attack of the Killer Differential Equations

"Alignment is like rocket science" is, like all analogies including great ones[1], misleading[2] beyond a certain point[3]: in particular, it makes the obstinacy of some regarding Agent Foundations incomprehensible. Indeed, if one tries to imagine a situation in which rocket science is the unknown paradigm that has to be found, it's easy to converge around a story like this:The year is 1666, and an asteroid headed toward Earth appears over the skies of England. Different scholars attempt interpretability research on cannonballs, on stones thrown from towers, on balls rolled down planes, but a certain Isaac proposes that what is needed is a more general understanding of the concept of Throwability, so we can backchain which thing we should be throwing at the asteroid to stop it.Is this an appropriate analogy to understand why someone would insist in doing AF and not prosaic alignment? No[4], because there is no real conflict between cannonball interpretability and Throwability Foundations. At some point one should of course abstract away from cannonballs entirely, but gathering more cannonball data is never going to hurt. The relevance of Throwability for our universe routes through the single fact that matter obeys the laws of Throwability; otherwise it would be just a mathematical curiosity. More important: The laws of Throwability that our universe follows can only be found by looking how matter obeys them. They are, a possible mathematical object among infinitely many others, but we can't find that object mathematically because there is nothing mathematically special about it.Here is a better analogy:[5]The year is 1666. Isaac has an intuition that few people share or understand: he has been thinking about something he calls "fluxions", which he can't really describe beyond vaguely gesturing at coffee cooling down faster when very hot or pendulums moving faster at the edges of their arc. He thinks some of those fluxions will be very dangerous for England, so he s