Some people ask, “Why does mathematics so eerily and accurately predict the universe? Is there not something deeply mathematical about the universe?”
This seems a great mystery. There is a greater mystery, however.
Why does mathematics so eerily and accurately predict mathematics?
This may seem silly, but it’s quite serious. You can observe analogous mathematical structures between mathematics in Euclid's original five axioms, and ZFC. Or between different coordinate systems, which can be expressed as axiomatic frameworks. Or between set theory and category theory.
But these are all mathematics, you might argue, of course they predict each other.
But, here’s the thing. Pick any set of entirely arbitrary (in the mathematical sense of arbitrary, not in the colloquial sense) set of rules, and insofar as they are self-consistent, you can then prove that the structures in these rules are in some sense the same as the structures in some other, entirely different, arbitrary set of rules. Set theory and category theory are, when you get down to it, different, arbitrary, sets of rules, chosen on the basis of certain properties like self-consistency.
The mystery is - at some deep level all self-consistent systems are, apparently, expressions of the same thing. The universe doesn't run on mathematics, the universe is just self-consistent, and so the structures in the universe are in some sense the same as the structures in mathematics.
The challenge here is that self-consistency cannot be proven for nontrivial systems. (Nontrivial also used in a technical sense, rather than the colloquial sense, although the colloquial sense is pretty close).
There have been some mathematical attempts to get at the root thing underlying all things - but the nasty little fact of it is, mathematics itself isn’t general enough. You can express the physical universe, or at least a simplification of it, in mathematics - but you can also express mathematics, or at least an approximation of mathematics, in physical media. Hence, computing. But as our understanding of computing has deepened, at a conceptual level, we have slowly come around to an idea that should really be quite startling, if it weren’t expressed in the way that it generally is.
The general way it is expressed is that a rock falling is a perfect computation of a rock falling. People sometimes call this idea “analog computing”; it is one of the core criticisms of quantum computing, that it is, in a significant way, sort of like computing how sand will fall in an hourglass by putting sand in an hourglass and watching it fall.
People who hold this view tend to be unimpressed by things like “accoustic black holes”, experiments where incomprehensible phenomena are expressed in another medium and observed there.
I’ll admit I have some skepticism about the specific experiment in question - the behavior doesn’t necessarily tell you anything about black holes, and that entire framing, in my opinion, makes the experiment seem kind of dumb. No, the exciting thing there is -
Analog computing can be used to compute things other than the thing they are actually doing.
Obviously this is true; computers work, and when you drill down deep enough, all they’re doing is analog computing.
If you can figure out how to encode addition in the falling of sand in an hourglass, an hourglass and a pile of sand becomes a powerful tool.
Most of mathematics is, when you get down to it, “encoding”, or, alternatively, mapping. Mapping in this case is used in a technical way, a set of pointers from one thing to another thing. The things in question can each be a set of things. A function is a map, from one set of things to another set of things. Isomorphies between mathematical frameworks, or between mathematics and physical reality, are identified by identifying a mapping between them.
Many NP problems are mapping problems. They are all at least isomorphic to mapping problems, since at least one NP problem is determining whether or not two graphs are isomorphic to one another, meaning each node in one graph can be mapped to a node in the other graph with all relationships remaining intact (given a potential mapping between relationships for a full generalization of the problem).
I’d suggest that NP problems, then, may be isomorphic to the fundamental mystery of why mathematics predicts physical reality - which in the end is a mapping problem. Or at least isomorphic to one.
And just like the graph isomorphy problem can, in specific cases, be made trivial by assigning indices - it’s easy to check if two graphs are isomorphic if the relevant equivalent nodes share an index - while not solving the general problem, we can find deep symmetries between mathematics and physical reality by indexing reality. We define a measurement for distance - we define an index - and suddenly we can calculate distances we haven’t even measured, by measuring other things. Defining measurements for angles may or may not also be necessary.
An index is a form of mapping, you see. Or a mapping is a form of index. Measurements are a form of mapping. Indices are a form of measurement.
There is a deep symmetry here, perhaps made challenging by the fact that we can’t prove that a system is self-consistent, among other qualities we kind of need. And made even more challenging by the incompleteness theorem - which suggests that there is no single mathematical framework with a complete set of mappings to every other mathematical framework; there’s always some (true) theorem that cannot be proven in a given framework.
But perhaps we can work around that, because it kind of seems like reality has to. Unless, perhaps, reality itself is mathematically incomplete, which seems … interesting.