Is it possible to alternate the law of mathematics?

A "horrendously-advanced alien civilization maybe able to" -carry their physical laws with them while being in our universe -in a local periphery of their spaceships. What better "shield" could there be? Since what you intent to write seems like a space-opera with evil aliens, this plot device will help you make their spaceships almost invincible - at least for long enough in your book to build up drama, as well as to narrate the heroic mission to uncover either their laws, or a way to map their laws to ours, or a way to just break this shield (and then imagine what would happen to them when exposed to the laws of our universe). Since this forum seems to have given you enough good ideas, I believe that it is appropriate that you commit publicly that, the final version of your book will contain at least some pages of mathematical and physics philosophy, preferably together with some mathematical symbols - and that you will guard these pages with your life against any publisher that will try to delete them arguing that they will alienate your potential readers.:)

As other commenters have said, you are probably going to get more milage by looking for alternative (local?) spacetimes that would allow the aliens to use their own laws of physics. The "laws" of mathematics can be thought of as a language, in the sense that if you change them* then you wouldn't necessarily have any effect on the universe.

(*And it's not clear what exactly this would mean: mathematics as usually studied is a form of an axiom system called ZFC, and people experiment often with axiom systems that are weaker than, stronger than, or inconsistent with ZFC.)

However, here is an excerpt from the Wikipedia page on hpyercomputation which seems like it would be of interest to you. [Citations have been removed]

According to a 1992 paper, a computer operating in a Malament-Hogarth spacetime or in orbit around a rotating black hole could theoretically perform non-Turing computations.

So these hypercomputations might be available to anyone who can generate (and mitigate the effects of) a black hole.

A super-Turing machine would definitely be able to solve the classical halting problem, that is, the halting problem for Turing machines. The explanation is given very briefly in another Wikpedia page. However, my intuition is that it would be unlikely to be able to solve the halting problem that they would be concerned with, the halting problem for their name brand of super-Turing machines.

I would be careful about throwing Gödel around, however. It might be that when you get down to the very low-level nuts and bolts that there is some finiteness-of-proofs assumption. If that were the case then you might be able to get around those pesky incompleteness theorems (well, the classical incompleteness theorems, at least…). However the way that it's been described to me has made it seem like you could very easily run into a classical-Gödel problem regardless of what sort of infinities you have access to.

As others have already mentioned, it doesn't make much sense to "alter" mathematics. You can however invent "new" mathematics, by starting from a different set of axioms.

If what you are looking for is a mathematical system which seems totally counterintuitive, I would recommend you look into the p-adic numbers. In basic arithmetic we can make statements like "$9$ is closer to $10$ than it is to $27$." When dealing with p-adic numbers, however, $9$ would be closer to $27$ since they both contain a factor of $3^2$. Such a strange notion of distance would throw euclidean distance straight out the window.

As for your second question, I don't think there is a definitive answer. It doesn't make much sense to say that the universe is governed by mathematical laws in the first place. Rather, we have found mathematical laws which model it very well. For an alien species to change the "math" of the universe is equivalent to them changing the very laws of the universe itself. But there is no reason for this to limit you - the genre of sci-fi isn't usually constrained to reality, so I don't see a reason why a fictional alien species couldn't alter mathematical laws while keeping the laws of physics intact.

Cool question.

I agree with the other answers that changing the mathematics doesn't really make sense. But you can also change something other than the physics. For example: perhaps make the differences "anthropological." (ahh, alienalogical?)

So maybe if I give you $1$ gift, followed by $2$ gifts the next day, it means you're winning me over; therefore you only need to reciprocate with $2$ gifts to keep us on good terms. However, if you reciprocate with strictly fewer than $2$ gifts, our friendship will suffer. On the other hand, if I give you $2$ gifts, followed by $1$ gift the next day, it means you're losing me, and you need to give me $4$ gifts else our friendship will suffer.

Perhaps aliens are very cunning, and they're constantly trying to use the non-commutativity of their gift-giving system to their own advantage.

There's alien courts, of course. Legal theorists have long worked on a "true model of gift-giving," an exact number system that describes exactly what you owe and what to expect. However, there are controversies. Sure, everyone agrees that $7+3+4$ gifts equals $14$ gifts, but does $7+4+3$ really equal $16,$ as the (by now, canon) Rara Blockfeel equations predict? Some lawyers have argued no, that under such extreme conditions, $17$ are in order.

Indeed, alien anthropologists, having visited (alien) hunter-gatherers in very secluded locations, report a consensus feeling that $17$ gifts are in order. "$17$ gifts!!" splutters a head scientist. It means completely reworking the Blockfeel formula.

Meanwhile, the lawyers are having a field day. With all this scientific controversy, there's cash to be made in the courts. Fortunes change hands in the blink of an eye, and entire empires crumble on the non-commutativity of alien addition.