Example: space-time exists in n(>=4) dimensions, there are several quantum fields that exist in that space-time and evolve/interact according to the equations of qft, gravity has some mathematical description (better than the ones we have now). And that's it, there's nothing else for us to know. Of course, we can never know for certain whether we've bottomed-out our description of reality, so physicists will always have some hope.
Now, if the universe is described entirely by mathematics, bootstrapped by a few base cases (the existence of fields/particles/space-time/strings/etc), I'd be comfortable with that. I'm not sure what a truly satisfying understanding of reality would look like beyond that.
But we have that in math and logic: axioms!
> Of course, we can never know for certain whether we've bottomed-out our description of reality
That we do not know is a fact; but is it a fact that we can never know?
What if one fine day we do know?
Yes and no. Axioms are chosen such that the math that is based on them makes sense to us (because the math reflects what we'd expect in the real world) and isn't self-contradictory.
So they do exist for a reason.
> That we do not know is a fact; but is it a fact that we can never know?
What if our universe actually has randomness that appears to come from "outside the system", i.e. there is really no way to predict the outcome of certain things from inside the system.
Picture you're sitting in a box trying to play Mikado. In theory you should be able to perfectly predict the outcome of every move you make - except there's a stranger shaking the box you're sitting in at truly random intervals.
For our reality this would mean that there's no way for us to develop an understanding of certain things and predict their behavior.
So essentially the end of the line for all sciences except theology and philosophy.
This possibility makes me somewhat uncomfortable.
On the other hand maybe everything is perfectly deterministic within our universe.
> Won't that description consist of nothing but math?
Math is just words. Any description of anything can be reduced to math if you assume the proper math will be developed to describe it. But the same could be said of any subject. You could say the universe is Mandarin Chinese if you find a perfect description in that language. You could say the universe is binary code if you found a perfect description of it in binary. You could say the universe is tree bark if you found a perfect description of it in tree bark.
You're saying nothing useful. You're just confusing the map for the territory. (And arguably, you will never find a complete description of the entire universe that fits in a network of human brains, so the universe can't be math for that reason alone.)
I am not at all! We do math with symbols and diagrams, but those symbols and diagrams are about something: they are maps, and there is a territory that they are about. When I say "math", I include the concepts, not just the symbolic, graphical and other representations.
So what I'm saying is that if/when mathematics provides a complete map to that territory which is "the world", that territory must then be made of exactly the same stuff as other described-by-math territory-entities like "ellipse", "pi", "finite field", "Hilbert space", ... the map of mathematics doesn't describe any other kind of territory.
I'm not saying that the symbols and pictures used to carry out math (the "maps") are the world.
> you will never find a complete description of the entire universe that fits in a network of human brains
The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be. E.g. the Mandelbrot set is just iterating on z^2 + c for various c. To describe isn't necessarily to evaluate.
>if/when mathematics provides a complete map to that territory which is "the world",
Then it will encode all the information that exists in the universe. Which will not fit in your head, and thus will not be mathematics. Just because a territory is trivially a map of itself doesn't mean it is useful to call it a map.
>The description of the state of it as it is before us is monstrously large, but the initial conditions and rules might not be.
Reality isn't just the initial rules. It has state. That state must be described for you to have a complete description of reality. Yes, we could mathematically describe the complete laws of nature. But reality is necessarily more than just that, because the state is really damn relevant. You can't just ignore it because it doesn't fit your argument.
If reality is deterministic, then all the state is derivable from the rules and the initial state, and it's possible we could figure out the rules and initial state and become relatively sure of them. (We probably wouldn't be able to calculate the future, since evaluating the rules within reality probably necessarily takes more steps than reality takes during that time. But we might be able to calculate enough to see that it matches up with what we think the early universe must have looked like. We might notice that even the smallest of tweaks to the rules or initial state creates an avalanche of changes that doesn't line up with what we believe the early universe looked like, so we can be sure there are no free parameters.) It's fully possible that the rules and initial state could fit in someone's head. They might be ridiculously simple, like Conway's game of life in a few more dimensions.
(If you think "reality can't be deterministic because true randomness or quantum events exist", well you can fit true randomness into a deterministic theory by having the world branch of every random event outcome. The Many-Worlds Interpretation could be viewed as doing something like this for quantum events; it's a deterministic theory.)
No it isn't! Either it is math all the way down, or else it isn't.
If it isn't, then what is at the bottom?
So this is actually an existential question.
Suppose we reach a final description of reality that is entirely accurate (no longer an approximation that fails to account for some structures and processes).
Won't that description consist of nothing but math?
Then, if it is complete, how does that description account for the fact that reality isn't math? If there is a difference between reality and math, can't that be described with math, so then it goes away?
If you want math to just be a description of a structure that itself isn't math, then the structure has to have attributes that are not covered by the math. The math is then using a substitute structure with fewer properties as a proxy for that structure. Once all the attributes are rolled into the math, there ceases to be a difference between that target structure and the math; it is that mathematical object. To prove otherwise requires one to name the difference. If two entities are completely described and found to be the same, they are just one entity.
> You may as well be declaring the world to be flat.
It is testable that the world isn't flat.