What is reality?
The most complete answer that physics allows us to date is: a vector in Hilbert space.
For those who are not familiar with the mathematics of quantum mechanics, this statement may seem completely meaningless; all the others, however, will agree that this is the most succinct and yet the most complete description that we can provide of the universe, given our current knowledge.
It is somewhat ironic that the best elucidation (with respect to the MDL principle) that we have for the physical world consist of a 100% mathematical narrative. You already seem to hear the Platonists shouting “we told you so!”
Can this be enough to prove that our universe is mathematical in nature? If not, how can we explain that the best theories we have for the functioning of everything are expressed in a language made up of numbers and operations on them?
As with many facts about reality, the short answer is that we don’t know.
However, this does not mean that such interrogations are completely beyond our investigate capacity, after all we have come a long way since we thought that light was the fart of angels.
I personally do not believe that nature IS mathematical, but I think it is the case that mathematics was (among all) the best possible tool for coding natural dynamics. If I have to be honest I have difficulty understanding the very concept of nature = mathematics. We must always remember that arithmetic is a language that possesses certain properties. It is possible to express Newton’s law of universal gravitation through English which, together with other natural languages, is powerful enough to express mathematics. Does this mean that reality is English?
I hope I have made my point clear. Nature is composed of relationships between objects, which system, among all those we own, better explains those relationships? Nature is made up of lengths, masses, temperatures, rotations, waves, fields.
What language best encompasses the functioning of these entities? The answer, of course, is math. But this doesn’t mean anything, the best we can say is that mathematics is the least imperfect tool when it comes to doing science. This probably comes from two major properties:
1) Universality: 2 + 2 means the same thing in Iceland, Costa Rica and Japan, while this does not happen with natural languages.
2) Simplicity: as already stated it is possible to write equations in English but this has the disadvantage of making statements more expensive (from the point of view of the number of symbols / Kolmogorov complexity) and strongly subject to the interpretation of the reader.
The proposition nature = mathematics is true if and only if EVERY aspect of nature can be explained and “simulated” only through mathematics and nothing else. I seem to see at least one major problems with this … how do you produce natural language through math? This is a fact that all linguists know: the grammars that produce natural languages are higher (or even absent) in the Chomsky hierarchy than the grammars that generate formal languages, of which arithmetic is one. This is because natural languages are the most powerful, from an expressive point of view, that men know, if you don’t believe it try to encode a poem through a formal system.
A slightly different point of view that has recently been spreading among philosophers and physicists is that of reality as information.
Sadly, I’m having trouble accepting this too.
By information, at the end of the day, we mean a resolution of uncertainty: before rolling a dice, the maximum information I have regarding the result of that system is of probabilistic type, there is an epistemological uncertainty that underlies the whole system. When I look at the result, the uncertainty resolves and of the 6 possibilities, only one remains.
Based on this definition it seems rather obvious that, in order to acquire information about anything, there must be an a priori uncertainty, therefore to affirm that at the basis of reality there is information seems to logically imply that at the basis of reality there is unpredictability.
But what does this even mean?
Uncertainty must be something that exists in a cognitive dimension, not in reality. Basically the same problem which is raised with all this information thing is similar to the perplexities Einstein had when he said “God doesn’t play dice with the universe.” Reality cannot be uncertain, it must know, in a crystal clear way, everything it does, otherwise we will live in a completely inconsistent world. Before saying “Reality is information” I believe we should deepen our theoretical knowledge on the subject, clarifying the definition and eliminating uncertainty from the pictures … but I don’t think it’s possible. After all, what information can you acquire if you don’t have uncertainty first? Without uncertainty everything is known and information does not exist by definition.
If reality at its foundations is not mathematics and it is not even information, then what is it? Energy? Mmm the same information problem can also be raised for energy. Energy is a quantity of work, but how can you have work if you have no other entity? What do you apply that work to? I believe that the answer to this question is still quite far from our current standards but I think it can be found on much lower scales than we are currently able to inspect. At these scales, perhaps close to the Planck length, fundamental entities are lurking. From the set of properties that compose them, everything else is derived: space-time, energy, the standard model, QFT, general relativity. From this perspective, the programs aimed at identifying a so-called Theory of Everything (e.g. M-theory) are extremely exciting, even if at present the theoretical research in these fields is centuries ahead of experimental capacities.
However things are resolved in the future, it is incredible what we have come to know through the lenses of science.