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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).

Mikael
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    You might be amused by Greg Egan's short story "Luminous", which treats some of these themes as carefully and rigorously as seems possible. (Summary and reviews) – MJD Aug 28 '13 at 17:55
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    My view is that mathematics is immutable throughout the universe and indeed throughout all possible universes – because mathematics is unphysical. Thus, even if you could alter the laws of mathematics, nothing would happen in the physical world! – Zhen Lin Aug 28 '13 at 17:59
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    http://en.wikipedia.org/wiki/Flatland – Amzoti Aug 28 '13 at 18:00
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    @MJD Nice references! So my idea was quite an old one then... Anyway, I'll check them and see what new themes I could add. – Mikael Aug 28 '13 at 18:10
  • Every idea is an old one. You don't write a good story by having a new idea. – MJD Aug 28 '13 at 18:11
  • @Amzoti thank you for your information. I actually read that before, but it appeared too abstract to me... – Mikael Aug 28 '13 at 18:11
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    The idea of "mathematical laws" isn't too clear to begin with. If I come up with a new type of number system I don't call it a new "mathematical law". It just takes a logical place alongside the old ones, without really being considered as part of "another universe's mathematics". Systems with contradictory axioms live happily together in math as alternatives, which is a bit different from the other sciences that cleanly separate "real" from "hypothetical". You might go for a universe that uses number systems differently? e.g. counting isn't done with naturals, but modular arithmetic. – Robert Mastragostino Aug 28 '13 at 18:15
  • @RobertMastragostino you made a good point. I have edited my questions. – Mikael Aug 28 '13 at 18:38
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    This is why I hated Contact. It's not possible to embed messages in mathematical constants. As Robert Mastragostino suggests, people in another universe (or even another world, or our own future) may very well choose to study different forms of mathematics, and will likely be guided by their own scientific/engineering/religious needs, but wherever their mathematics hits ours, they should agree. – dfeuer Aug 28 '13 at 18:41
  • Aside from the possibilities that this universe is a computer simulation of some higher beings, I've strong doubt about changing mathematical laws within this universe. Unlike physicals laws, there is no middle ground for a change in mathematical laws. If two piece of universe or two universes with incompatible truth comes into contact, everything become inconsistent and the most likely outcome is both realities will collapsed to nothing. It is perfectly possible other universes follow different mathematical laws. As long as we can't contact them, nothing really matters. – achille hui Aug 28 '13 at 18:47
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    @dfeuer I'm not sure I follow. The idea was that mathematics itself was set up from the beginning with coded messages (such as the circle in $\pi$). The message sent by "the others" was not embedded in a constant, was pulses that came with too much regularity to be produced by natural causes. And the way Sagan imagined the message itself was carried out is based on a protocol that has a serious theoretical background (lincos), see here. – Andrés E. Caicedo Aug 28 '13 at 18:48
  • @AndresCaicedo: how can mathematics be set up from the beginning with coded messages? That's the problem... the creators of the universe have no way to influence $\pi$. – dfeuer Aug 28 '13 at 19:39
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    3 is a prime number in any base, so 3 is a prime number no matter what – zerosofthezeta Aug 28 '13 at 19:52
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    @dfeuer You'd have to ask Sagan. It seems you are advancing an argument against omnipotence. – Andrés E. Caicedo Aug 28 '13 at 20:43
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    @AndresCaicedo: sufficiently strong forms of omnipotence have serious philosophical problems. "Can God create a rock so heavy God cannot lift it?" is a classic example. If you counter that the concept of God creating a rock God cannot lift it is not sensible, then I must respond by asking if God can make that concept sensible, and we've left all forms of reasoning behind. – dfeuer Aug 28 '13 at 20:55
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    @dfeuer No need to argue with me on this topic. As I said, I didn't follow your comment, it seemed to me you were either saying that "the others" did something they did not do, or you thought the idea behind their coding machinery was flawed, which does not seem to be. You have clarified that this is not what you were saying. – Andrés E. Caicedo Aug 28 '13 at 21:02
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    But 5 isn't a Gaussian prime... – vonbrand May 25 '14 at 15:12
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    Since mathematics is a social activity you could have alien mathematicians start from a different place. For example they could have thought about weaving mathematically before inventing counting numbers. Earth mathematicians have already investigated the case you mention $1+2 \neq 2+1$. Presumably, though, if they understood our arguments they would reach the same conclusions about the topics we consider. – isomorphismes Feb 23 '15 at 16:25
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    H. P. Lovecraft invoked non-Euclidean geometry in The Call of Cthulhu. I think that's more effective for those who don't understand maths. There’s often a "magical" feeling of unknown possibility to whatever mathematics is currently beyond one’s reach. I think what Lovecraft wanted to evoke was the unfathomable or even the impossible—which seems like what you're going for as well. But while mathematics is baffling to laypeople, it's the opposite to the cognoscenti. – isomorphismes Feb 23 '15 at 23:43
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    (Doug Hofstadter discusses the feeling I refer to in my previous comment. He knew what a superscript was—an exponent—but what is this funny x₃? Not knowing that subscripts are merely indexes, he ran to his father with unbounded ideas (not just a logarithm) for the exotic things x₃ could mean. (And was disappointed when the magic was pulled away: the meaning of x₃ is pretty boring.)) – isomorphismes Feb 24 '15 at 00:18
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    @MJD: Greg Egan wrote a sequel to "Luminous", called "Dark Integers". If you liked the former, you will certainly enjoy the latter. Greg Egan has made many of his stories available online for free, but I don't think these two are officially available. – TonyK Dec 08 '15 at 14:04

9 Answers9

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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.:)

Alecos Papadopoulos
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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.

Ryan
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  • I thought physics must change if mathematics changes, so it is not necessarily true? Interesting... – Mikael Aug 28 '13 at 18:59
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    In your situation, it is best to think of math as a language which is used to describe various physical phenomena. You can't change the language itself. But if the rules of the universe became really whacky, we would have to start using different words in our language to describe it. – Ryan Aug 28 '13 at 19:04
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    So math is like French if I was born in France, but I need to learn German if I want to do business with German? – Mikael Aug 28 '13 at 19:19
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    Yes, I think that is a fair analogy. – Ryan Aug 28 '13 at 19:26
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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.

Eric Stucky
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    Can I interpret your last paragraph that, any computer-like things are bounded to have some computation limit? – Mikael Aug 28 '13 at 19:48
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    Yes, it is well known and formalized in computer science. A simple diagonalization shows that you can't have a computer (universal machine) and an halting solver in the same scope. It is linked to the absurdity of the set of all sets. – Xoff Aug 28 '13 at 19:51
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    OK, every computer has its own limits. But are there any theorems suggesting or preventing infinitely many new-generation computer? Like, we have iphone-5 now, can we have iphone-$\infty$ in the (infinite) future? – Mikael Aug 28 '13 at 19:55
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    @Mikael: With infinite time, sure. But the idea you've been getting at is to compress infinities into finite time (since we only have finite time as far as we know). Then an interesting question becomes if it is theoretically impossible to have computers of a certain power. I would be very surprised if this question is answerable with what we know about the universe right now. – Eric Stucky Aug 28 '13 at 19:58
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    I'm sure that I will say something very strange to you, but computing power is like ordinals : you can always define a theoretical computer that is much more powerful than another theoretical system. But even if you consider an infinite series of such systems $s_0<s_1<s_2<\dots$, there will be another system $s_\omega$ that will be more powerful than all of them. But as all other systems, there will be another system more powerful than $s_\omega$. No system can "do everything" without being inconsistent. – Xoff Aug 28 '13 at 20:01
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    @Xoff: so can I say that "there is always a better computer" and "no computer can do everything"? – Mikael Aug 28 '13 at 20:06
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    @Mikael: See my point above: You can say "There is always a better theoretical computer" and "No computer can do everything imaginable". (But you might now be able to say "there is always a better constructible computer": physics may give a hard upper limit. Similarly, the physics may also allow you to say "there is a [theoretical?] computer that can do everything that is possible.") – Eric Stucky Aug 28 '13 at 20:13
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    yes, both sentences are in fact the same. But you must understand this is purely speculative. Because, what we can do in a mathematical scope is very different of what we can do in reality : to build those super computers, we need to find new physical laws that we don't know yet. In mathematics, we have no such limits. So practically, all computers on earth have the same limits, and we just improve the speed a little bit, but not what they can really do. If quantum computers work as expected, we'll be able to answer much more rapidly to some answers, but we won't be able to solve new problems. – Xoff Aug 28 '13 at 20:13
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    +1 for "quantum computers… won't be able to solve new problems" Now, the speed may decrease so that currently possible problems become practical. But please don't push aside theoretical computability issues with the magic word "quantum", you will make CS people very sad. – Eric Stucky Aug 28 '13 at 20:19
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    To write a last comment, I would say that if an alien race would be able to build such super computers, they will be like gods to us. Because they could analyze so much datas, and solve so many problems, they will be like people that can see the future. They will be just unbeatable. The super computers could be used to design flawless weapons and ships, so such aliens would need to be perfect themselves to avoid self destruction. I'm not sure it's the right angle for a good book. – Xoff Aug 28 '13 at 20:25
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    Because no computer can do everything, so there must be something that cannot be done. And assume that we can find this cannot-be-done thing mathematically, so it cannot be done by any possible computer in any possible space-time. Does this imply that we indeed have some truly-universal mathematical truth? – Mikael Aug 28 '13 at 20:31
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    @Xoff: I have already given up hope on wining such alien war 20 minutes ago:) Now I am thinking whether these god-like aliens can have a rock so heavy that they themselves cannot lift up – Mikael Aug 28 '13 at 20:35
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    Egan also has a story, "The Planck Dive", about some people who build a spaceship and shoot themselves into orbit around a black hole in an attempt to perform non-Turing computations. – MJD Aug 28 '13 at 21:26
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    @Mikael : Does it imply ? no, because now that you can do that things that cannot-be-done, you just realized you have new questions with no answers, and new things that cannot-be-done. This is endless. – Xoff Aug 28 '13 at 21:47
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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.

goblin GONE
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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.)

Chris Culter
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    You or the OP might be interested in the Wikipedia page on Hypercomputation, which states that "According to a 1992 paper,[13] a computer operating in a Malament-Hogarth spacetime or in orbit around a rotating black hole[14] could theoretically perform non-Turing computations.[15][16]" So these hypercomputations might be available to anyone who can generate (and mitigate the effects of) a black hole. – Eric Stucky Aug 28 '13 at 19:25
  • Do you mean that Gödel's incompleteness theorems can be overturned by some aliens, and they can have an algorithm for halting problem? – Mikael Aug 28 '13 at 19:25
  • @EricStucky This is truly cool! Now my fiction is not only a fiction anymore. – Mikael Aug 28 '13 at 19:28
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My answer to the OP question is yes. I believe "alternate" mathematical systems are possible.

But creating such a mathematics would be quite dramatic. It would require a completely different approach to integers (which ultimately derive from a universe that contains discrete objects).

So a universe that did not have completely discrete objects may develop a completely different mathematics. Our mathematics is based on binary logic: example: is an object in a set? the answer is yes or no. There is nothing between those two answers. In other mathematical systems this would not be the case.

One last comment: you may not need another universe for this. It may be possible in parts of our own universe. Indeed in certain instances (e.g. at the quantum level, in black holes) the concept of discrete objects breaks down. Hence there is an opportunity here to formulate an alternate mathematics. Indeed, an alternate mathematics may be REQUIRED for us to try understand these situations.

Ken Abbott
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If you ever get to writing your science fiction novel you might want to use the following variation on the theme of the variability of the laws of mathematics so as to amuse your readers:

A fellow is eating a meal in a restaurant, and confides to a waiter who is holding a bill in his hand: "I have always been fascinated by the idea that the laws of arithmetic may contain a contradiction, which would then invalidate these calculations."

Mikhail Katz
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Mathematics tends to follow from deterministic causality, or at least be a tool used to understand things in a world ruled by deterministic causality:

(1) Deterministic: The same action taken in two equivalent circumstances will have two equivalent results

(2) Causal (sort of...): The universe occurs in a sequential nature, with any point in time being determined only by the state of the universe at a previous point in time

I wouldn't want to live in a Universe where those two things aren't true, and I wouldn't want to read a book that takes place in such a universe, because it would just be a mess.

But if those two things weren't true about the Universe, then mathematics either wouldn't exist or it would be very very different, and our form of mathematics would seem like useless nonsense.

DanielV
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There are different approaches to mathematics, so it is unclear what the original poster understood as “mathematical laws”.

For a Platonist, we are unable to change the world of ideas (although God might be able, if God is envisaged). We can’t change how the ideal objects behave, but we can change denotation of the symbols (make $1$ and $2$ to denote not natural numbers, but something else).

If we speak of formal systems, then we are unable to influence which propositions are theorems in a specified formal system, but we have the choice to use one system or another. We can add or remove axioms, inference rules, and symbols. A discussion about “natural numbers”, from the formalist perspective, is meaningless until we specify the formal system.

If we admit a sociocultural foundation of mathematics, then yes, we can change “mathematical laws”. There were no formal proofs in many ancient mathematical schools, but they became a requisite by the late 19th century. But there were no computer proofs, whereas now they are.

Incnis Mrsi
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