Mathematics is considered to be the language of the universe since it is the most fundamental type of logic (in physics anyway) and therefore the best way of expressing the universe (which is a physical environment). From this it is safe to assume that any civilization advanced enough to reach the stars has a good grasp of math, in whatever form it make take for them. Therein lies the context of my question.

If two civilizations meet, let say humans and martians for ease, could they communicate with math? And if so, how would this look?

Basically, I'm looking to avoid an Arrival type meeting, and use a non-verbal based form of communication (at least initially). But I can't figure out how this would work. I chose math because it's "the language of the universe" and as postulated above, fair to assume everyone involved would know.

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    $\begingroup$ Although every intelligent species would likely come up with the same math independently, math isn't used for communication of everyday ideas so it would be quite useless to use it to do communication. What would likely happen is we'd be able to establish a basis for intelligence with exchange of some math and from there developed a intermediate language based on very simple logic (which can be thought as Math I guess) and then we'd be able to translate that into the respective languages. It will be very much like using binary to communicate with computers. $\endgroup$ Dec 22, 2017 at 17:55
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    $\begingroup$ Have you read or watched Sagan's "Contact"? Learning how aliens do math is a major plot point. $\endgroup$
    – user25818
    Dec 22, 2017 at 18:13
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    $\begingroup$ Possible duplicate of worldbuilding.stackexchange.com/questions/22882/… $\endgroup$
    – Olga
    Dec 22, 2017 at 18:29
  • $\begingroup$ @Olga I disagree that it is a duplicate. I think that question deals with the universality of math, while this the specific application of math to establishing communication. It doesn't help, though, that the accepted answer to that question would be a more appropriate answer to this one... $\endgroup$
    – kingledion
    Dec 22, 2017 at 18:47
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    $\begingroup$ @kingledion, the wording is different, but the essence is quite similar. I am not suggesting to close this question, but it would be nice if the OP could edit the question to make it a bit more specific. For example, it would be nice to clarify and specify what 'communication' means. Is simple exchange of maths facts will be enough? Or do we need to talk about everyday things? Or maybe the goal is to share philosophies? These are very different types and levels of communication. I am not sure if all of them can be accomplished with maths. $\endgroup$
    – Olga
    Dec 22, 2017 at 19:05

4 Answers 4


"Omnilingual", written by H. Beam Piper back in 1954 or so, explains how it can work. There are universals in science and math describing the real world, which two technologically advanced cultures are going to have to share.

So, math leads to communication: basic math leads to understanding each others symbols for numbering and mathematical operations, as well as the base system used for counting. Once you have that, you have the capability of translating any number in one language to a number in another.

Now, take those numbers. The aliens send you a list that looks like this:

  • 1 1.0 blarg
  • 2 4.0 gurgle
  • 3 6.9 blech
  • 4 9.0 yech
  • 5 10.0 burble
  • 6 12.0 rez
  • 7 14.0 skurp
  • 8 16.0 xowill, and so on all the way up to
  • 92 238.0 fraznip

It's not going to take someone long to recognize the periodic table and the names of elements. Now, ignoring the linguistics you could derive from the names themselves, consider now what's possible: blarg is hydrogen, xowill is oxygen, so if you see (blarg)(blarg)(xowill) = kerblud, that gives you a reasonable assumption that kerblud means "water". And so on and so forth. That allows you to create a vocabulary of translated words, and once you have that, the process of mutually understanding languages becomes faster and faster.


You can build a language from anything

Language is just a collection of building blocks. Human language is, of course, very complicated, but it must have been formed from the sorts of basic communication events that mammals the world over still use.

So 'math' is as good a way as any to communicate with someone else.

What will aliens and us mutually understand?

This is harder to grasp conceptually. If we just send the aliens some numbers, it may not be clear to them how to associate the symbols with any numerical meaning, without some sort of logical framework. How can you know that the symbol "2" represents the counting concept of two?

However, there should be some things that should be universally understood. A signal that oscillates between two values should be easily interpreted as binary. From there, we can send messages that are consistently framed that would be interpreted as integer numbers. By consistent framing, I mean that you would send your numbers in sets of 8 or 16 or 64 bits, just like computers do when they store integers.

Once you can send numbers, you can send patterns that will be obvious indications of intelligence: you could send a series of prime numbers or a Fibonacci sequence, the sides of all integral right triangles (i.e. 3-4-5, 5-12-13, 8-15-17, etc). Doing this should at the very least establish that you are talking with someone who can talk back.

Build from mathematics upwards

Obviously, we aren't the first ones to tackle this problem. There are already proposals for various way to build from mathematics into full fledged communication. Lincos is one, designed in the 60s, while a more recent one is Lone Signal. Lone Signal is designed around sending two things together: a very simple set of mathematical statements about he laws of physics coupled with messages in fully complex language. The idea is that the simple physical statements would act as a 'Rosetta stone' allowing the aliens to translate the full language.

There are doubts as to whether or not aliens would be able to translate human language at all. Noam Chomsky, using his theory of genetically determined grammar, suggests that without a built in faculty to understand human grammar, aliens might not be able to make sense of our language. In my opinion, while that would stop, say, Han Solo from talking to Chewbacca, it would not stop a computer from being able to decode an alien language.

How does the message work?

Technical details of the transmission setup are contained here.

Here is report on the test message. It was devised by Michael Busch from CalTech and broken by a colleague, Rachel Reddick from Stanford. Obviously, it would be easier for a someone with a human understanding of math and science to decipher a human-built code, but the recipient was able to quickly decipher the message with just pencil and paper.

In the test message link, the decoded message itself starts on page 9. The message starts with several tautologies to establish a vocabulary. I've tried to reproduce the decoded message and the quad side by side. The quad is simply two binary digits.

( ___0 = ___0 )  00000000 10000000 01000001 10000000 00010000
( ___1 = ___1 )  00000000 10000001 01000001 10000001 00010000
( ___2 = ___2 )  00000000 10000002 01000001 10000002 00010000
( ___3 = ___3 )  00000000 10000003 01000001 10000003 00010000
( _671 = _671 )  00000000 10022133 01000001 10022133 00010000
( ___0 = -__0 )  00000000 10000000 01000001 20000000 00010000
( ___1 ≠ -__1 )  00000000 10000001 01000100 20000001 00010000
( ___1 ≠ ___0 )  00000000 10000001 01000100 10000000 00010000
( _870 = _870 )  00000000 10031212 01000001 10031212 00010000
( _870 ≠ _871 )  00000000 10031212 01000100 10031213 00010000

From here, you see that we have established some symbolic 'language' already. What is encoded as parenthesis are frame divisions (00000000 and 00010000). Equality (a concept!) is expressed as 01000001 while inequality is 01000100.

From there, the message goes to describe the characteristics of our sun and of the elements, among other things. If the listener can identify the element, then they can associate the 'symbol' for the element. For example, oxygen is identified by 31020020; building off this water is identified by its chemical composition as 31021001.

Read the paper for more; there are 70+ pages of encoded and decoded message, along with an encryption key. Both concrete (water, electric charge, solar years) and abstract (equality, time) concepts are expressed within a mathematical framework with only a simple message.

Will it work?

Of course, this is the most important question, and one that we don't have the answer for.

But in summary, mathematics are important for two things: for establishing that we are communicating with an intelligent lifeform; and for building a simple 'Rosetta stone' of concepts that can be use to translate more complex language.

  • $\begingroup$ To be a pedant, those examples seem like constructed languages not math. It may attempt to use mathematics to bootstrap an understanding of the language but it isn't math that is communicating the meaning. $\endgroup$
    – sphennings
    Dec 22, 2017 at 19:43
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    $\begingroup$ This allows building a common vocabulary to describe the world. But how should we go about expressing intentions, emotions, cultural and philosophical concepts? $\endgroup$
    – Olga
    Dec 23, 2017 at 17:13
  • $\begingroup$ @Olga Once you start with a basic language, you can add more expansive concepts with each successive message. The listener will have to extrapolate the meaning with each new term heard; this is the same way a child learns new vocabulary. $\endgroup$
    – kingledion
    Dec 23, 2017 at 22:06

Another book which explains part of this process is Contact, by Carl Sagan. The movie that was made based on his novel glossed over the maths part, but it's a very important part of the process of understanding an alien message.

Becoming Aware of Each Other
The first step in receiving a message is becoming aware of its existence. Space is a lot noisier than a lot of people think and the reality is that we've come across many different signals that were first thought to be messages that we now know were natural phenomena, like Pulsars. So, how does one send a signal that can be recognised as a signal from an intelligent source?

Prime Numbers.

Send two pulses, then rest. Send three pulses, then rest. Send Five. Seven. Eleven etc, up to an arbitrary count. Rinse and repeat. This CANNOT be a natural signal (alright, perhaps it can, but the odds are as close to zero as anything else I can think of) because although it's regular, it's regular based on a mathematical pattern that is too complex for normal phenomena but which can be easily understood by any intelligent life with even a rudimentary grasp of mathematics. They're the ones you really want to communicate with anyway.

Once You Have Their Attention
The next step is to build some basic math rules. One pulse (symbol W) One pulse (symbol E) two pulses. Symbol W is plus, Symbol E is Equals. Do this for a whole range of mathematical equations, until they understand. Then do it for Symbol X, or multiply. Again, teach the symbol through the math. Keep going. Minus. Divide, etc.

Then, when you're done with that; one pulse (symbol W) one pulse (symbol N) three pulses. You're building the symbol for not equals. Keep showing this with random answers to which no logical connection to the real answer is possible (real answer + number of wrong questions, for example) and you've built the concepts for Yes and No, true and false out of math.

Bridging from Math to Broader Concepts
You can now start introducing items like the Periodic table (as has already been suggested) with new symbols reflecting the elements. Then, you can start adding in things like decay rates, meaning that you can introduce the concept of time. Keep going. Eventually, you've built concepts that can in turn be built upon to get to what you really want to talk about.

What we're talking about here is really just using math as the 'crack in the ice'; the starting point that builds on the few things we're likely to see conceptually similarly so that we can form an understanding of the things close by, then build out in a radiating field of understanding.

Don't Anthropomorphise
There are some things we can't make assumptions about during this process. Don't assume that another alien civilisation will work the same as ours. They may not have eyes (and therefore not 'see' the same way we do), they may not have emotions. The human brain is a very complex piece of equipment designed over 3.5 billion years of evolution; to assume that aliens feel 'happy' or 'scared' etc. in the same way we do would be a gross over-estimation of the commonality of our genes. Remember that as we build outwards from maths, the less likely our concepts can converge in any way. Be prepared to hit a barrier against which further conceptual development isn't possible. But still, you've got something to work with and until you learn more about them (and they learn more about us) we have to accept some limits as inevitable in the early to mid stages of the communication process.

The short story on which the movie Arrival is based extends beyond real science the concept that our psychology is largely shaped by our language. This is true but not to the extent that our psychology can transcend physics and we can remember in both temporal directions as a result of learning a new language.

What's important to remember out of that concept though is that we do use our language as a form of operational base for our conceptual frameworks. Every language we currently know however is human based which is why we all think reasonably similarly, although if you're in any doubt about the power of language to shape thought, sit back and enjoy a coffee while watching a .NET programmer discuss process with a sculptor. :)

Why is this important? It's because while there is much that we think we know that can't be described by maths, an alien species with a completely different development path to humans is going to struggle even more at understanding us than we do understanding each other when we don't know each others' language.

In essence, this means that we have no choice but to revert to the one form of language that cannot change between us, and that's mathematics. If we can communicate via that, then if we're lucky the broader concepts will follow.


You can't communicate with maths.

Maths are a collection of related formal systems that are somewhat an extension of formal logic. Languages are systems for communication. These are two very different types of systems that aren't necessarily compatible.

While some mathematicians will discuss the universality of math they are talking about a very different thing than communication . Try translating "pass me that" into maths and see how it works out. What maths is great at doing is reasoning about things. Assuming there was a method of mathing back and forth at each other you'd find out that you both agreed about whatever mathematical proof you happened to exchange.

If you remember from The Arrival there was another team that was trying to communicate with the aliens via math. The result was that they got the aliens to solve simple math problems and produce sequences of prime numbers, not an answer to the question of the movie "What is your purpose here?".

To quote from Artelius' answer to a similar question:


  • Mathematics can't be used to say "we come in peace"
  • Mathematics can be used to establish common ground with most conceivable intellectually mature beings
  • Although there is no guarantee they will "get" it, it will be obvious if they do
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    $\begingroup$ I wonder why people are downvoting this answer. As you say, maths and formal logic cannot be used to convey all possible types of information. $\endgroup$
    – Olga
    Dec 22, 2017 at 19:08
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    $\begingroup$ Math cannot be used to communicate "vague" concepts. Math language is either absolutely true or not defined. $\endgroup$
    – L.Dutch
    Dec 22, 2017 at 20:20
  • $\begingroup$ The concept or trope of math as a universal "language" does not posit that counting or doing math is the goal, but that it is a common reference from which non-mathematical communication can be built. $\endgroup$
    – rek
    Dec 23, 2017 at 7:06
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    $\begingroup$ The value of math is the ability to bridge differences in semiotics. Once that minor Rosetta Stone has been achieved, the subtle nuances of Nietzsche aren't too far behind. (OK, so "Urggle-koo-ARAN-broack" means "not equal to..." Now we're getting somewhere!) $\endgroup$
    – JBH
    May 28, 2019 at 20:40

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