Mathematical universe hypothesis: Why shouldn't all imaginable universes exist?

Question

In his paper on mathematical universe hypothesis, Max Tegmark only responses with a single paragraph to this assumption:

The MUH and the Level IV multiverse idea does certainly not imply that all imaginable universes exist. We humans can imagine many things that are mathematically undefined and hence do not correspond to mathematical structures. Mathematicians publish papers with existence proofs and demonstrating the mathematical consistency of various mathematical structures precisely because this is difficult and not possible in all cases.

To me, this argumentation sounds to me like the appeal to ridicule. Personally, I think the theory does imply all imaginable universes exist and thus even those where dragons are presidents if such universes can be logically consistent and the fact such imaginations seem ridiculous doesn't make them invalid. My reasoning for the assumption every possible universe exists stems from a notion Tegmark himself acknowledges:

Stephen Hawking famously asked “what is it that breathes fire into the equations and makes a universe for them to describe?” [92]. In the context of the MUH, there is thus no breathing required, since the point is not that a mathematical structure describes a universe, but that it is a universe.

My understanding of this statement is - there's no reason for a mathematical (=abstract) structure not to exist. That's because there's no "higher logic" which could "dispute" any mathematical system, logic itself is a product of these systems. Thus, even when we formulate a question like "Where does it come from?" we're using the logic which is a product of our universe and doesn't make any sense outside of it.

Have I misunderstood Tegmark or is there any background reasoning that supports his claim MUH does not imply all imaginable universes exist?

Answer 1

I think both summaries of yours are wrong, and both quotes are simpler (but even more profound imo) than you think.

In the first, Tegmark is saying the MUH predicts only mathematical structures exist. A dinosaur president is a perfectly fine mathematical structure, so it could exist in the MUH. Only mathematically undefined things can't exist. Such as an imagined mathematical paradox. I can say the words "a ten sided platonic solid exists" because I can imagine mathematically undefined things, but they can't exist.

If you get the first quote is only trying to saying math is all that exists, the second quote is easier to understand. If math really is all that exists, what gives us subjective experience or qualia? What gives life to the equations? How come fire feels warm, time feels like it flows, etc if there is only math? For Tegmark the fire comes purely from the brain and its tools to survive. There is only math, and we are only math. Any layering on top to explain our subjective experience are just biological processes, which are just math. There is no soul, no "fire" that breathed life into the math. Just a subjective illusion that biologists can explain soon enough, purely with math like everything else. Or said a different way, the fire that breathed life is our brain and it's wonderful subjective reality function. But that's all it is. A mathematical function or relationship.

I have read Our Mathematical Universe and am confident this is what he means.

Answer 2

Maybe Tegmark is referring to descriptions like, "Two universes otherwise identical, where in one of them everything is a nanometer to the right of where everything is in the other." Or, "A universe containing a being who has power over all universes" (IDK if Tegmark allows for transworld deity but he seems to need to rule out at least transworld deities that act from within a specific universe as such; and it's not clear to me that he has "room" for a "zone" above all universes, where a transworld deity would live so that it would not with any single world form a whole).

Since he refers to existence proofs and their difficulty, I wonder how to apply the reference... We can give a negative existence proof for "the largest prime," say, so perhaps Tegmark is speaking of things like "a universe with a largest prime"?