Finite Factored Sets
Nitpick; I thought “finite factored sets” were being rebranded as factored space models, because the finiteness was inessential. I think this name is better and would like to see the switch-over happen before the framework gets more popularity. But maybe I’m wrong about this intention?
As I learn mathematics I try to deeply question everything, and pay attention to which assumptions are really necessary for the results that we care about. Over time I have accumulated a bunch of “hot takes” or opinions about how conventional math should be done differently. I essentially never have time to fully work out whether these takes end up with consistent alternative theories, but I keep them around.
In this quick-takes post, I’m just going to really quickly write out my thoughts about one of these hot takes. That’s because I’m doing Inkhaven and am very tired and wish to go to sleep. Please point out all of my mistakes politely.
The classic methods of defining numbers (naturals, integers, rationals, algebraic, reals, complex) are “wrong” in the sense that it doesn’t match how people actually think about numbers (correctly) in their heads. That is to say, it doesn’t match the epistemically most natural conceptualization of them: the one that carves nature at its joints.
For example, addition and multiplication are not two equally basic operations that just so happen to be related through the distributivity property, forming a ring. Instead, multiplication is repeated addition. It’s a theorem that repeated addition is commutative. Similarly, exponentiation is repeated multiplication. You can keep defining repeated operations, resulting in the hyperoperator. I think this is natural, but I’ve never taken a math class or read a textbook that talked about the hyperoperators. (If they do, it will be via the much less natural version that is the Ackermann function.)
This actually goes backwards one more step; addition is repeated “add 1”. Associativity is an axiom, and commutativity of addition is a theorem. You start with 1 as the only number. Zero is not a natural number, and comes from the next step.
The negative numbers are not the “additive inverse”. You get the negatives (epistemically) by deciding you want to work with solutions to all equations of the form for naturals and . The fact that declaring these objects to exist is consistent should be a theorem, as should the fact that some of the solutions to different equations are equal (e.g. that the solution to is the same as the solution to ).
This idea is also iterated up through the hyperoperations. The rational numbers are (again, epistemically) the set of all solutions to the equations where and are integers. The fact that this set is not consistent when is should also be a theorem.
Since the third-degree hyperoperation is not commutative, you can demand two new types of solutions, those of and those of . This gives you roots and logs. The fact that roots require you to define the imaginary numbers, and therefore lose the total ordering over the numbers, should be a theorem.
Some of the solutions to these equations are numbers we already had, and some of them are new numbers that we didn’t have This setup leads to the natural question of whether we will keep on producing new types of numbers or not. The complex numbers are special in part because they are closed under all these inverse operations. But do we need new types of numbers if we demand solutions to the fourth-order hyperoperator? What happens if we go all the way up? I have no idea.
Yeah, I agree. An earlier deadline would cut out procrastination time, or more accurately, it would just make people write more efficiently per unit time. I’d honestly be pretty excited to try having a deadline even as early as 6pm.
Some subtle signals perhaps?
(I have no idea what y’all are using KL-divergence for, so I have no opinion about whether you should have been using it in this theorem.)
a metric, a feature sorely lacking from
I have a pet peeve around this, which is hopefully a useful comment for someone to read; KL-divergence should not be symmetric, because of the whole thing that it is. If you’re using KL-divergence and thinking to yourself “I wish this was symmetric”, then that should be a red flag that you’re using the wrong tool!
I think it’s easy for people to think, “hm, I’d like a way to quantify the how different two probability distributions are from each other” and then they grab the nearest hammer, which happens to be KL-divergence. But mathematical definitions are not for things, instead they mean things.
You should use KL-divergence when you want to measure the cost of modelling a true distribution using a false distribution. The asymmetry comes from the fact that one of them is the true one (and therefore the one that you take the expected value with respect to).
I enjoyed reading this post quite a bit, but the exact reason why eludes me. I think I’ve had or almost had a lot of these thoughts before, and this post clarified them for me.
I have a tendency to be constantly asking something like “but what is the real thing going on?”. This is obviously very useful overall, but as in the post, it is sometimes also to useful to say “within this context, let’s only reason as if this is all that’s going on” and then try to improve my understanding of something that way. I think this would have significantly improved my experience of learning thermo and stat mech.
For anyone who’s interested in going deeper on a formalism of thermo with no stat mech, I once found this niche book that seems to do exactly that: A First Course in the Mathematical Foundations of Thermodynamics. (It’s published by Springer so it’s probably legit, but I didn’t actually get that far into it, so this isn’t a recommendation per se.)
Oops, I said two things.
Er, bug report, the right-click menu is where my spell-check’s suggestions were. So now I get a red underline telling me a word is not in the dictionary, but to correct it I have to just, try to type the correct spelling. Or google it I guess.
Turing completeness, let alone error correction, has not been shown for leptonic matter on its own.
Curious if you have a citation for this!
There are, as far as I can tell, no book-length biographies of Kolmogorov’s life. The world is worse for it! His life was long, and his accomplishments innumerable and broad-reaching. An aspiring biographer could establish a career with such a book.
(There are a few different compilations of essays about Kolmogorov, but they’re by fellow academics, and largely focus on his work rather than him.)
“Inkhaven” is already modifying the name of the campus “Lighthaven” (which is already a modification of “Lightcone”). Since you’re still doing the “ink” part by writing, but not doing the haven part (by not being at Lighthaven) I’d recommend a different name.
May I suggest “Inkhalven”/”Inkhalfen”? Maybe the verb “Inkhalfing”?
Now that I think about it, it could be useful to have a timed sequence of posts that are just for discussing the book chapters, sort of like a read-along.
I think something like “accountability” is a much more accurate term for this generalization than “dominance”. Parents, teachers, bosses etc. are roughly as motivated by their accountability to their children, students and employees.