Q&A: Category Theory and the Status of Mathematics in Describing Reality
Category Theory and the Status of Mathematics in Describing Reality
Question
Hello, Rabbi.
I would like to ask about a question that has been bothering me for a long time.
As is well known, mathematics is founded on a finite number of basic assumptions (called "axioms"). According to Gödel's incompleteness theorems, there can never be an axiomatic system that is both consistent and complete. That is, there can always be further mathematical propositions that cannot be proved within the system.
My question concerns the existence of additional mathematical systems and their place in describing reality.
First, if we assume that a finite number of mathematical statements is required in order to describe our world adequately (see the idea of a "final theory"), then it follows that only a finite number of mathematical systems is needed in order to describe reality. Let us note that the fact that more than one mathematical system is needed to describe reality is not surprising, since already today we know of such a system (see non-Euclidean geometry and its place in Einstein's general theory of relativity).
But even in such a case, one must examine the issue of general mathematical propositions whose intention is not to describe reality. (As is well known, mathematics is not limited only to describing our world…)
In this way we have covered the case in which only one system with a finite set of axioms is required (after unifying the different mathematical systems) in order to describe reality. But what about the case in which the number of mathematical systems needed to describe the world is infinite? Is such a situation even coherent?
In addition, I was wondering whether you have any knowledge of category theory that might clarify for me whether there is indeed a connection between this theory (which is sometimes called "the mathematics of mathematics") and the issue I raised. I have a strong feeling that there is, but unfortunately I do not have the mathematical background required to study the subject in depth, and the Wikipedia page is not helpful (though it does reinforce my intuition about the relevance of this topic).
To sum up,
I would be glad if you would share your opinion, and I would be even more glad for any corrections, comments, and so on. Thank you.
Answer
Hello Yossi.
First, I am not a mathematician and not an expert in this field. It would be better to ask mathematicians. But one should also remember that the question is not about mathematics but about its application in our world (and that is a question for a physicist). I will try to address it anyway.
1. I do not know where you get the idea that mathematics is based on a finite number of basic assumptions. Gödel's theorem deals with systems whose number of assumptions is countable (and not necessarily finite), but I am not familiar with the claim that every mathematical axiomatic system is of that kind. By the way, this also depends on what you call an assumption. I recall reading a few weeks ago about ChatGPT: they asked it to translate an argument stated in ordinary language into a logical argument with premises and a conclusion, and it asked how many premises they wanted. The same argument can be translated into any number of premises, depending on the level of resolution and on what you regard as a premise. I no longer remember the source.
2. You assume (at least for the sake of the discussion) that a description of the world is based on a finite number of assumptions. Here too I do not see the basis for this, but I will adopt it for the sake of the discussion.
3. You also assume that a description of the world (that is, a physical theory) is supposed to look like an axiomatic system in mathematics. That itself is a problematic assumption. Tarski, in his book on logic, presented such formalizations for biology and for various areas of physics, but that is far from making a claim about every scientific description of reality. Physics operates at a different level of rigor from mathematics. Can one create an axiomatic system in the strict mathematical sense in order to describe the world? I do not know.
4. A finite number of assumptions does not imply a finite number of systems. That also depends on the rules of inference.
5. The use of non-Euclidean systems does not reflect a need for several systems to describe the world. On the contrary, the claim of relativity theory is that the description of the world is non-Euclidean, not that one also needs non-Euclidean geometry in addition to Euclidean geometry.
6. You assume that the systems can be unified into one system. But there may be a contradiction between them, in which case unification is impossible.
7. After all this, I still have no idea what your question is. In the second-to-last paragraph you say that you have covered the case in which one system is required. What does "covered" mean? What are you claiming? I really do not understand what you want.
8. Then in that same paragraph you move to the possibility that infinitely many systems are required and ask whether that makes sense. I did not understand the question. First of all, why not? Second, even if it does not make sense (why not?), then maybe it will not happen. What is the question?
In short, think for a moment about what exactly you want to ask. Formulate the question clearly, and then explain it by means of axiomatic systems. I did not understand anything here.
As for the last paragraph, I have no knowledge of category theory, but I did not understand the issue you raised, so I certainly cannot answer.
Discussion on Answer
I think you need to sort out the questions before you start looking for answers.
Contradictory mathematical theories are theories that describe different worlds. The fact that there is a round world does not contradict the fact that there is another square world. So a contradiction between descriptions, and between worlds, should not trouble you.
The threshold condition is consistency and content. I'm not sure that everything meeting that threshold will be included in the Platonic world, but without it, apparently not.
Right, I wasn't clear.
According to Platonism, every mathematical theory is an objective truth. But we agreed that there can be contradictory axiomatic systems (= "mathematical theories"), so this opens the door to the question: can there be inconsistency in the "Platonic world"?
In addition, another relevant question: what is the "threshold condition" for a mathematical theory to "enter" the Platonic world of ideas? Must there be some domain in the world in which its axioms are instantiated? If not, then there certainly are contradictory axiomatic systems, and therefore there certainly is inconsistency in the "Platonic world." If yes, then it still remains to be checked whether a situation is possible in which there are two contradictory axiomatic systems that faithfully describe the world, or at least part of the world.
I got tangled up. I'd appreciate it if you could help me make some order out of this. Thanks.