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Tegmark's mathematical universe hypothesis, posits that reality is a mathematical structure. This mathematical nature of the universe, Tegmark argues, has important consequences for the way researchers should approach many questions of physics. Tegmark's MUH is: Our external physical reality is a mathematical structure. That is, the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure).

If our universe is mathematical، Does not this increase the probability of being complex as well?

Complex Universe Simulation hypothesis

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It's a strange question, if you really look at it. "The Universe" is not a random variable. Probability means nothing. Either the universe is or is not complex. There is no probability to be had unless you define the problem with a random variable.

We could treat this as a Bayesian inference question. You are asking if P(C | M) > P(C) where C is a complex world and M is a mathematical world. By Bayes theorem, P(C|M) = P(M|C)*P(M)/P(C). Since all complex worlds are mathematical, P(M|C) is 1, so this inequality becomes P(M)/P(C) > P(C). Thus if P(M) > P(C)^2, an event which shows the universe is mathematical would increase the likelihood that it is complex. Of course, finding those two probabilities is going to be quite the challenge.

Perhaps diving into a more philosophical approach, we may ask what does it mean for a world to be complex? One approach to this question would be to recognize that real numbers and complex numbers are two of the four real division algebras which have some semblance of associativity (the other two being quaternions and octonions). These are algebras where every multiplication other than by 0 has a corresponding division which undoes it. What does that mean to you philosophically? If you're considering mathematical universes other than division algebras, or other than real algebras, it vastly increases the number of options you have to consider.

Of the four real division algebras with some sense of associativity, each time you go up in dimensionality (from reals to complex to quaternions to octonions) you can describe a wider range of behaviors, but you pay by losing some convenient functionality. One of the more interesting ones we lose going from reals to complex is that there is no longer a total ordering of numbers which behaves nicely with our intuition of addition and multiplication. What does it mean for things in the universe to no longer be totally ordered? (if we move to quaternions, we lose the commutative property, and octonions lose associativity itself, if you're curious. Octonions are merely alternative, which is a much weaker requirement. Beyond the octonions are the sedions, which can describe even more, but don't even have the alternative property, and we are basically yet to find a practical application for them!)

On an aside, one of the interesting underdog theories competing to be a Theory of Everything is using octonions. Should they be "right," that would indicate that the universe isn't complex... it's far more obtuse than that!

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