The universe is not a mathematical object
TL;DR: Mathematical objects are idealizations we use to describe the world, not the world itself
This week I gave a colloquium and I was asked the following. Since I put emphasis on developing precise mapping between physical objects and their mathematical representation, do I believe the universe consists of mathematical objects? I do not. In fact, I don’t even think the proposition makes any sense. It’s a bit like saying that, since we use words to identify objects, the objects are actually words. Now, the question itself is legitimate because there are some people that indeed hold that position. So, let’s see how we can quickly poke holes in that idea.
All of math is based on some type of logic: we need to have well-formed propositions and be able to assign them a truth value. In the simplest case, the truth value is either TRUE or FALSE. In a slightly more complicated case, one may have TRUE, FALSE and some type of UNDEFINED. In an even more complicated case, one can actually define the truth value to be a real number from 0 to 1. Now, the meaning of what these truth values represent is actually up to debate, and it may depend on what one is doing.
In a strict formal sense, TRUE and FALSE in math may simply mean that the proposition can be proven from a set of given axioms. For example, the assertion “4 is not a prime number” is TRUE because four can be shown to factorize. On the other hand, the statement “there is no set whose cardinality is strictly between that of the integers and the real numbers,” the continuum hypothesis, cannot be proven from the standard axioms of set theory. In fact, it can be proven that it could be either TRUE or FALSE without contradicting any of the standard axioms. The proposition is independent, so one may say that it is neither TRUE nor FALSE but UNDEFINED. However, we can add the continuum hypothesis to the set of axioms, and now it would be trivially TRUE. What is TRUE or FALSE, what is provable, depends on the choice of axioms. So, not only can TRUE mean something different (i.e. ontologically real vs provable from a set of axioms), but what is TRUE may depend on context.
Even if two mathematicians start from the same axioms, they may disagree on what constitutes a proof. For example, most mathematicians accept proofs by contradiction: if I can show that the negation of a well-formed statement is FALSE, then the original statement must be TRUE. A constructivist would disagree: you still have to prove that the original statement is TRUE. One may also disagree on what a well-formed proposition is. For example, one may say that infinite objects are nonsense, and therefore any proposition about infinite objects is not well-formed. So, if one wants to argue that the universe consists of mathematical objects… the first problem is: which ones? What do you count as well-formed formulas? What axioms are you willing to accept? What rules of deduction do you allow?
But all of this is not even the biggest problem. If we want to map mathematical objects to the real world and do science, TRUE and FALSE cannot simply mean provable. They must correspond to something along the lines of experimentally confirmed and experimentally disconfirmed. There is an issue already that some statements can be shown to be experimentally true, but not false. For example, if we didn’t find life in the observable universe, it would not mean that there was no life. But the bigger issue is: what does it mean “to find life?” What is the definition of life? What evidence is considered conclusive? Again, this may depend on what definitions one may accept, and we bump into the same problem: all definitions are context dependent.
The subtle thing here is that even if we agreed with some experimental notion of truth, we still need to find a good set of statements with agreed upon definitions that allow us to do physics. The main goal of physics, in fact, is finding those systems, those carved out parts of the universe, that can be sufficiently isolated and manipulated to be studied independently from the rest. For example, we can study thermodynamics only after we realize that, at equilibrium, statements about pressure, volume, temperature, energy and so on have relationships that can be studied independently from all other variables. If we are not in equilibrium, notions of temperature and pressure, for example, lose meaning. In the same vein, we can study the motion of a cannonball here on earth. On the surface of the sun, the notion of a cannonball would not last long.
Only after we have found a set of circumstances and a “nice” set of statements and tests, after we have agreed on what matters and what can be disregarded in those conditions, can we proceed and formalize the physical description in mathematical language. The mathematical objects we use in physics, then, are contingent upon the idealization we made while defining the context. The mathematical objects are crisp precisely because we hid complexity through the modeling. Understanding physics is not simply understanding the model: it is also understanding the idealizations and the contexts in which those idealizations are valid so that the model is applied appropriately. None of this is within the mathematical model. Therefore the universe cannot consist of mathematical objects because they do not capture all the physics.
I strongly believe that this is inescapable: we cannot construct a formal system without limiting ourselves to a well-defined but idealized context. It’s a hunch that comes from having done mathematical modeling at the lowest levels. At this point, I am not able to construct a tight argument that shows how this is an inevitable part of the process of formalization, but I do think it should be possible. If only I were philosophically more sophisticated…