Nice to know there's a name for this concept!

You can take the idea further and abandon any notion of time or even discreteness altogether: If the universe is essentially information, this information can be _completely_ described by some (mathemtical) language. Note, this description doesn't have to be finite.

For the finite and discrete case this description may simply be a very long bitstring/vector which contains every detail of our universe. If time is discrete as well we can write a whole timeline as a matrix, a set of bitstrings each representing one slice of 'now'. Notice how we now have one static thing being a perfect representation of the universe. No need for a computer or interpreter. Just a big table of ones and zeroes representing everything from the big bang to your mother.

However, our universe might be continuous. You might still be able to describe it with finite amount of information tough. A function like x -> x*x is continuous and defined for infinitiely many values but can still be written down just fine. Even infinite complex things like fractals can have a very short descriptions. Or take pi. An infinite sequence of numbers describable in a few sentences.

Point is, a universe need not to be discrete in order to be written down on a sheet of paper. It just needs to be finite in an information theoretical sense. And you may need a lot of paper.

I strongly suspect the universe is bounded. But what if it isn't? Can we still se it as a mathematical entitiy? I think so, but I'd like some comments on that. We have the well-defined set of real numbers and it contains _every_ real number, even those who are completely random up until the last of their infinite digit. There are numbers in R which have an infinite Kolmogorov complexity [1]. And yet, we can put them all in one set. We can even draw a picture of all these numbers (it's just a line, but that's more than our analysis professor would draw to illustrate something).

In conclusion: I think (and didn't proof) that even if the universe contains infinite information and cannot be the result of some turing machine it can still be tought of as a mathematical object. A thing like sqrt(2) or a triangle.

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A second and orthogonal tought is about the 'reality' of all this. This is a bit metaphysical and philosophical. If you are still reading please tell me I went crazy.
The idea is, if you have a perfect description of our universe (be it a turing machine, a big matrix or some other complicated mathematical thingumabob) is there a difference between the description and the real thing? Obviously not, one is just words on a paper or bits in a computer and the other a world full of loving and caring people and other stuff. How can I even ask such a question? (Hint: I'm on the Internet and can write what I want.) Well, imagine doing this whole computer simulation of our universe. Would you feel any different when being run on some sort of computer? If you and everything around you would be perfectly simulated would you notice it? Would you even be able to? For the ones running the simulation the difference would be obvious. But being part of the simulation you may have no way of knowing that you are a piece of software. It would all seem and feel real. It would BE real. And yet you are just bits. You could stop the computer, dump it's memory and write it all down. Doesn't seem like such a difference between words on paper and reality after all.

What I'm trying to explain is that a perfect description of something is actually, in some sense, the _same_ thing as the object being described. It's a bit like uploading the whole universe instead of just a brain. In the end all you have is information. Doesn't matter if this information flows trough transistors, is written in ink or just an element of the set of real numbers.

And if you say the number '4' 'exists' and so does 'pi' then why not 'the universe' as well? If you say this then you just answered the question 'why do we exist?' Because _everything_ exists. In the same sense as every possible triangle 'exists' so does every universe. We just happen to observe one specific instance.

The sad thing is: This doesn't really explain anything. If you say absolutely everything exists you haven't made any predictable or verifiable claim. So it may give you a philosophical answer to why are we here but it isn't much more than '42'...

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[1] Some formal proof would be nice here. I didn't think it trough down to the simplest axiom. Maybe it's enough to argue that there are only countable many turing machines but uncountable many real numbers and therefore some real numbers cannot have a finite description.