Taking a shot at this, I think your mistake is that you think in terms of mathematical properties of a universe, rather than mathematical properties period. For example, I think the existence of something like a prime number is not a property of a universe but a logical truth that couldn’t be false in any universe. More generally, I would say the same about any sentence in a formal language.
There may also be mathematical properties that are universe-specific (the best candidates here are natural constants), but the extent to which these exist is questionable—perhaps understanding general math more would reveal that these properties are logically necessary. Even if not, you may still get far by only pondering non-universe-specific math.
To turn this into a suggestion, I think you should view mathematical insights as being independent of the universe they are made in, and independent of the agent who makes them. Then, if it is possible to argue from pure mathematics that only one universe makes sense, that has to be the one. Thus, I accept (1) and (2) but reject (3), and in particular, the sentence
we cannot see, purely from a mathematical description of a theory, whether that theory describes reality or not
I have high error bars around how credible LW will think this is, but personally, I think this video presents the most valuable ideas on the subject that I’ve ever heard, and I’d probably describe it as figuring out whether a theory is true purely from the description of the theory. (I’ve sadly seen similar ideas presented elsewhere in combination with extremely sloppy epistemology.) The idea is roughly that “nothing exists” is, in fact, as true as it is logically possible to be; our intuitions about what “nothing” means are just wrong.
“There may also be mathematical properties that are universe-specific (the best candidates here are natural constants), but the extent to which these exist is questionable”
The exact position of every atom in the universe at time t=10^10 years is a “mathematical property of our universe” in my terminology. The fact that some human somewhere uttered the words “good morning” at some point today, is a complicated mathematical property of our universe, in principle derivable from the fundamental theory of physics.
Taking a shot at this, I think your mistake is that you think in terms of mathematical properties of a universe, rather than mathematical properties period. For example, I think the existence of something like a prime number is not a property of a universe but a logical truth that couldn’t be false in any universe. More generally, I would say the same about any sentence in a formal language.
There may also be mathematical properties that are universe-specific (the best candidates here are natural constants), but the extent to which these exist is questionable—perhaps understanding general math more would reveal that these properties are logically necessary. Even if not, you may still get far by only pondering non-universe-specific math.
To turn this into a suggestion, I think you should view mathematical insights as being independent of the universe they are made in, and independent of the agent who makes them. Then, if it is possible to argue from pure mathematics that only one universe makes sense, that has to be the one. Thus, I accept (1) and (2) but reject (3), and in particular, the sentence
I have high error bars around how credible LW will think this is, but personally, I think this video presents the most valuable ideas on the subject that I’ve ever heard, and I’d probably describe it as figuring out whether a theory is true purely from the description of the theory. (I’ve sadly seen similar ideas presented elsewhere in combination with extremely sloppy epistemology.) The idea is roughly that “nothing exists” is, in fact, as true as it is logically possible to be; our intuitions about what “nothing” means are just wrong.
“There may also be mathematical properties that are universe-specific (the best candidates here are natural constants), but the extent to which these exist is questionable”
The exact position of every atom in the universe at time t=10^10 years is a “mathematical property of our universe” in my terminology. The fact that some human somewhere uttered the words “good morning” at some point today, is a complicated mathematical property of our universe, in principle derivable from the fundamental theory of physics.
Sure, but those properties are upstream of the laws of physics, so you don’t need to figure them out to answer your main question.