Mathematics can work in an arbitrary number of dimensions imagined or not, and does so regularly. The fact that an alternate universe may not have any dimensions that we recognize certainly doesn't invalidate the fact that the mathematics is perfectly capable of operating in the dimensions that it does have. Space and time are arbitrary labels that we give dimensions in our universe, but mathematics only makes the distinction when applied to our universe and universes like it, mostly for our own convenience. Note that theoretical physicists regularly work on models of the universe with vastly more dimensions than four even though that is the only dimensions we can perceive, and often work in spaces that are entirely constructs of the imagination. Everything is still correctly derivable in those spaces if you take those spaces as axioms for deriving applied mathematics.
Logic in mathematics has no concept of time or any other property of our universe, hence why it is easily applied to all. You have confused applied logic, which takes our universe as an axiomatic environment and derives the consequences of mathematical logic in that environment, with pure logic from which the applied logic you are referring to was actually derived. If you look up first-order logic, it is essentially set theory type mathematics and spaceless. Rhetorical or applied logic is first-order logic applied to our universe (and sometimes not even that). You could just as easily re-derive "applied logic" for any other universe as well. If you stick with strict mathematical logic in arguments and avoid derivative applied logic, your logic is portable to other universes.
Numbers definitely do exist in mathematics independent of a physical existence. We take them to mean something slightly different in practice (i.e. there are some subtle differences based on the axiomatic existence of our universe), but a material existence is immaterial (no pun intended). You can count things that don't physically exist.
A lot of people understand first-order logic as it is applied to our universe, but few understand the sense in which it exists and works when we are talking about universes that aren't strictly like ours.
Well now ... I'm no expert here, and you do seem quite certain of what you're saying. Perhaps what I'm about to say is wrong; it's certainly simplistic. But to me, if there's no universe there are no numbers. I think of numbers as nothing but abstractions. They are imposed upon us by the nature of the universe, but that imperative, if gone, would obviate the numbers too. Or so it seems to me. (I'm pinging some folks who know more than I do about this. I'm always willing to learn.)
IMHO, first order logic is quite handy for computer science but fails in natural science and thus would not be of a necessity portable to another universe or domain which would be subject to other physical laws.
For lurkers: there are various schools of mathematics and tortoise and I are at odds because I fall in the Platonist school.
Formalism is the view that mathematical statements are not about anything, but are rather to be regarded as meaningless marks. The formalists are interested in the rules that govern how these marks are manipulated. Mathematics, in other words is the manipulation of symbols.
Unfortunately, this philosophy was proven unfit by Gödel's incompleteness theorem [see below] ...
A break off of Formalists school is the Logicists school, championed by Bertrand Russell and Gottlob Frege, who sought to show that are knowledge of mathematical truth was as certain as our knowledge of logical truth. They attempted to define mathematics in the language of logic. Their efforts resulted in some important ideas, such as the relationship between number theory and set theory, but ultimately this enterprise was found faulty as well due to paradoxes such as Russell's Paradox, an important principle of set theory which could not be based on logic.
Inventionism
Inventionism, also sometimes called Constructivism, holds that true mathematical statements are true because we say they are. Mathematicians do not discover mathematics, as the Platonists claim, they invent new mathematics.
Intuitionism
The easiest way to define intuitionism is that it is the corollary of logicism. The Logicists want to define mathematics in the language of logic. The intuitionists want to define logic in the language of mathematics.
Platonism
The view as pointed out earlier is this: Mathematics exists. It transcends the human creative process, and is out there to be discovered. Pi as the ratio of the circumference of a circle to its diameter is just as true and real here on Earth as it is on the other side of the galaxy.
The above definitions are from What is Mathematics?
Incompleteness Theorum (Gödel)