Is it possible to alternate the law of mathematics?

Question

I am freelance writer. Recently I have been planning a science fiction - just planning, nothing solid yet - and I was wondering would it be possible for some other universes that have different set of mathematical laws? By alternative mathematical laws, I mean $1+2\ne 2+1$ or $1\times2\ne2\times1$, or that the prime number 3 is no longer a prime in other universe, that kind of stuff. To push things further, I am imaging that a horrendously-advanced alien civilization maybe able to, by some mysterious method, change the mathematical laws here on earth, so as to disarm our spaceships and conquer our world.

I have read some popular science books, and was informed that we could have different physical laws in other universe, if the other universe exists. This is the motivation behind my (naive) question. I think, from a writer's prospective, I may have more freedom in creating a plot, well, at least I myself is the rule-maker.

My major was biology, so I haven't really learned any maths at all, plus the fact that I left college some twenty years ago. I can merely make sure I wasn't in any debt using my math skills, so I hope my questions aren't bad or stupid.

Sorry for these irrelevant details, I repeat my two questions below:

• Is it possible to have different mathematical laws in other universe?
• Is it possible for some aliens to alter the mathematical laws in our universe?

Thank you for your consideration!

EDIT I'd like to mention that I have read the novel "Contact" by Carl Sagan. I was truly amazed that those aliens sent prime numbers to Ellie, who then be able to make the contact happens. But then I wonder, what if those aliens somehow do not have the same prime number we have on earth? This is just my random thought.

EDIT 2 @Robert Mastragostino made a good point in the comment. I think what I am asking is that, could the deductive reasoning, which I suppose mathematics is mostly based on, be violated? That is, if those aliens are powerful enough, we no longer have $1+1=2$ on earth but $1+1=3$?

EDIT 3 After reading all your comments so far, I now start to think that I may have asked the wrong question. I used to think that math is "superior" than physics, as being convinced by a comic. The analogy by Ryan showed me that math is somewhat like a language, and I think that, well, what's the big deal about changing a language? It appears that physics is still all that matters if you want to win an alien war. Let me know if I am wrong (again).

Answer 1

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.

Answer 2

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.

Answer 3

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.:)

Answer 4

As far as we know, physical objects in our universe are finite. If an alien universe has laws of physics that somehow grant them access to infinite objects, then questions of "set theory" that are formally undecidable for us would be everyday matters of fact for them. Maybe aliens in one universe have a magical balance scale that can weigh every set of real numbers. Maybe aliens in a larger universe have a well-ordering of their real numbers inscribed on an obelisk, and they view the former aliens as we view bacteria.

As far as we know, our physical phenomena are computable by Turing machines. If the alien universe has laws of physics that somehow grant them access to oracles or hypercomputers, what wonders might they be capable of? If their conquering force on Earth can communicate with the computers on their native plane, how quickly might their technology overwhelm ours?

(Feel free to let me know why none of this makes sense.)