Remember that mathematics is something we make up; mathematics isn’t fundamental to, prior to, or indeed related to existence itself at all; mathematics is the process of formalizing rules and seeing what happens. You can invent whatever rules you want, although the interesting stuff generally doesn’t really happen unless the rules are consistent / satisfiable with respect to one another.
The fact that mathematics happen to be useful in describing reality doesn’t imply that reality is fundamentally mathematical, except in the sense that reality does something like follow a consistent set of rules, and there may be a deep isomorphy between all sets of self-consistent rules—but it’s also entirely possible that the mathematics we invent, we invent for a reason, and those reasons have an inherent relationship with the rules the universe itself follows. Personally I lean towards “deep isomorphy”, which would imply that there’s only one “mathematical universe”.
Let’s consider inconsistent mathematics for a moment, however, because I will observe that your paradox does not depend upon the rules of the “mathematical universe” actually being consistent / satisfiable—the paradox doesn’t depend on the idea that the universe being described can exist, only on the idea that an entity described by the mathematical framework can be aware of its own existence. Suppose for a moment that there exists a mathematical system with one or more contradictions, which is still capable of “running” a dynamical system for a given set of limited parameters such as to give rise to an entity in that system which is “aware of its existence” (suppose for the sake of argument that the contradictions do not present a problem for that limited set of parameters, which includes a finite extent of time). Does that entity “exist” for your purposes?
Note, as you consider this question, that one of the central claims of a religion which focuses largely on examining oneself is that you do not, in fact, exist. Granted I think this claim is misleading, and I’d say the proper claim is something more like “The you that you think of as yourself is more like a mental image of yourself and is wholly imaginary”, except that is also misleading, and “You don’t exist” is actually somewhat closer to the true claim being made. However, I think it is particularly applicable here, because the entity that makes the claim “Cogito ergo sum” is, in a particular sense, not actually real; or at least is real in the same sense that a mathematical entity which is examined to see whether or not it thinks it exists, which itself is real in the same sense as these words are real.
Are these words real? They’re embedded in physical hardware somewhere. The act of running the dynamical system to see whether or not an entity thinks it exists, is also the act of embedding the entity being examined in physical hardware somewhere. Unrolling the function to see whether or not an entity thinks it exists is equivalent to making that entity exist.
What if we could prove such an entity exists in a mathematical framework, without instantiating the specific entity by actually running the function? Well, I suspect such a proof is impossible, but supposing it isn’t; does that entity actually think that it exists? It would have to exist in order to do so, no? This no longer seems particularly paradoxical; I would analogize to a human being whose existence is contingent on my having sex with a particular person at a particular time. They would think they exist, if they come to exist; this doesn’t imply existence.
But supposing the paradox is still unresolved, I’d add the following considerations: Is the proof that such an entity would exist, if the function were run, itself a proof that the entity exists regardless of whether or not the function is run? Does the proof cause the entity to exist, or does it exist regardless of whether or not any such proof is attempted? (Do entities exist in the infinite possible encodings of the digits of irrational numbers?)
Remember that mathematics is something we make up; mathematics isn’t fundamental to, prior to, or indeed related to existence itself at all; mathematics is the process of formalizing rules and seeing what happens. You can invent whatever rules you want, although the interesting stuff generally doesn’t really happen unless the rules are consistent / satisfiable with respect to one another.
The fact that mathematics happen to be useful in describing reality doesn’t imply that reality is fundamentally mathematical, except in the sense that reality does something like follow a consistent set of rules, and there may be a deep isomorphy between all sets of self-consistent rules—but it’s also entirely possible that the mathematics we invent, we invent for a reason, and those reasons have an inherent relationship with the rules the universe itself follows. Personally I lean towards “deep isomorphy”, which would imply that there’s only one “mathematical universe”.
Let’s consider inconsistent mathematics for a moment, however, because I will observe that your paradox does not depend upon the rules of the “mathematical universe” actually being consistent / satisfiable—the paradox doesn’t depend on the idea that the universe being described can exist, only on the idea that an entity described by the mathematical framework can be aware of its own existence. Suppose for a moment that there exists a mathematical system with one or more contradictions, which is still capable of “running” a dynamical system for a given set of limited parameters such as to give rise to an entity in that system which is “aware of its existence” (suppose for the sake of argument that the contradictions do not present a problem for that limited set of parameters, which includes a finite extent of time). Does that entity “exist” for your purposes?
Note, as you consider this question, that one of the central claims of a religion which focuses largely on examining oneself is that you do not, in fact, exist. Granted I think this claim is misleading, and I’d say the proper claim is something more like “The you that you think of as yourself is more like a mental image of yourself and is wholly imaginary”, except that is also misleading, and “You don’t exist” is actually somewhat closer to the true claim being made. However, I think it is particularly applicable here, because the entity that makes the claim “Cogito ergo sum” is, in a particular sense, not actually real; or at least is real in the same sense that a mathematical entity which is examined to see whether or not it thinks it exists, which itself is real in the same sense as these words are real.
Are these words real? They’re embedded in physical hardware somewhere. The act of running the dynamical system to see whether or not an entity thinks it exists, is also the act of embedding the entity being examined in physical hardware somewhere. Unrolling the function to see whether or not an entity thinks it exists is equivalent to making that entity exist.
What if we could prove such an entity exists in a mathematical framework, without instantiating the specific entity by actually running the function? Well, I suspect such a proof is impossible, but supposing it isn’t; does that entity actually think that it exists? It would have to exist in order to do so, no? This no longer seems particularly paradoxical; I would analogize to a human being whose existence is contingent on my having sex with a particular person at a particular time. They would think they exist, if they come to exist; this doesn’t imply existence.
But supposing the paradox is still unresolved, I’d add the following considerations: Is the proof that such an entity would exist, if the function were run, itself a proof that the entity exists regardless of whether or not the function is run? Does the proof cause the entity to exist, or does it exist regardless of whether or not any such proof is attempted? (Do entities exist in the infinite possible encodings of the digits of irrational numbers?)