# Is mathematics a truly universal language?

The quote has many forms, but is basically "mathematics is a universal language." My question is if this is true? If we met aliens could we use mathematics to talk to them?

The fact that we could show them we know what 10 is doesn't seem like much of help in saying "do you come in peace?"

• Sure. Prime numbers have been proposed (and used in spacecraft, I believe). – HDE 226868 Aug 18 '15 at 16:58
• The maths yes, but I somehow doubt that Vegans use Greek symbols or Fraktur letters. – his Aug 18 '15 at 18:33
• mathematic is a great tool to describe the universe around us, we use them to develop physics and chemistry etc. unless your computer would like to initate a conversation with an intelligent extraterrestial being using assembly language (0s and 1s), don't assume that their turing machine runs on earthly OS. – user6760 Aug 19 '15 at 3:53
• definitely YES, at least in our universe, that has been proved that the law of phisycs described by math are valid. Just think about math constants PI, kelper number, speed light... – albanx Aug 19 '15 at 18:20
• As a maths teacher with a few students that does'nt speak a word of french (i'm french) i can confirm they understand maths much better than any other field. I know, aliens could have a whole different logic, but maths are very adaptable (we can deal with the fact they do not use the decimal basis, it's very easy). Maths can't bring a clear and immediate message such as "you're welcome" but it's a very good start to start a comunication. If aliens communicate, our first try to understand will use maths (stats, lenght of sounds, visual check of sound waves for example) – Tyrabel Sep 28 '15 at 17:47

The use of prime numbers in communication is talked about in Prime Numbers and the Search for Extraterrestrial Intelligence. Here's one method:

1. Create a rectangle.
2. Divide it into units, such that each side has the length of a certain prime number.
3. Encode images into the rectangular grid by making each square black or white (or a dot or a dash).
4. Take apart the image row-by-row or column-by-column and make it into one long pattern.
5. Transmit the message.

The Arecibo message used this technique (in a 23 by 73 grid). This ensures that a civilization must be relatively advanced to factor the multiple of the two numbers (which is 1,679).

You can transmit plenty of information like this, using only basic algebra!

Here's the message, gridded and colored:

Also, the consensus in A World Without Mathematics is that advanced technology is near impossible without mathematics. If they contacted us - or we simultaneously met - chances are good that they would have access to mathematics.

A more detailed way to communicate (used once) is to use a language like Lincos. Lincos is not entirely based in mathematics - in fact, much of it is not - but it uses mathematics in the "dictionary" that is supposed to be broadcast that explains the use of the language. Using Lincos, complicated conversations are possible.

• Note that a civilization capable of receiving radio messages from the stars is probably capable of factoring 1679. :) The same technique is (fictionally) used by aliens in Carl Sagan's 1985 novel Contact; he was surely aware of the 1974 Arecibo message. – Quuxplusone Aug 19 '15 at 2:59
• @Quuxplusone Ah, true, I didn't think of that. I've been meaning to read Contact for a long time; I've only seen the movie. :-) – HDE 226868 Aug 19 '15 at 16:22
• @HDE226868 The movie is quite different from the book in many ways. Probably some of those for dramatic effect, but not only. Not necessarily better or worse (that depends on your personal preferences), just different. – a CVn Aug 20 '15 at 12:21
• @MichaelKjörling My teacher did warn us of that. – HDE 226868 Aug 20 '15 at 15:27
• All but your last paragraph are basically equivalent to what the OP meant by saying "show them we know what 10 is". It is just about demonstrating intelligence, opening the door to the possibility of communicating. But actually communicating, the realm of language - there math is little more than a limited use tool. Your last paragraph is closer to the mark, especially in mentioning that much of Lincos is not based on math. In fact, most of meaning is established in it not by the math, but by repetition, contrast and establishing context. – user11864 Dec 15 '15 at 17:59

No, mathematics is not a universal language. It is, however, the study of universal truths.

As long as both parties have studied the same truths, differences in language can quickly be figured out by both sides. For instance if we met a race of aliens that use the octal number system we could very quickly figure it out and share discoveries with each other.

But mathematics is infinitely big; we've only explored some universal truths. How do we know aliens would be attracted to the same areas of mathematics? Well, we use mathematics for reasoning about the world around us, so beings from other planets will almost certainly have discovered a lot of what we have, by studying objects, movement, heat, and time. But beings that inhabit quite different environments may have studied mathematics quite different to ours. For instance, energy-beings that exist outside time (or in 2-dimensional time?) or intelligent microbes that can teleport using quantum tunnelling. It is hard to imagine technologically advanced beings that don't know how to count, but beings that exist outside of space and time, where it doesn't make sense to ask "how many things are within <blah>", may have had no reason to study counting!

As to whether we could communicate about practical matters ("we come in peace"), the language(s) of mathematics are not designed for this sort of thing. However, the language of mathematical logic would be a good starting point: thing A has property B; provided C and D are true then E is true; for every F there must be at least one G. We could ensure that the aliens have the same understanding of these terms to us by applying them to mathematical examples. Then we could apply the same terms to real-world entities, and the properties they may have (e.g. "We" have the property "wanting-spaceships").

If the aliens are reasonably similar to us (social species with physical form) then it will probably be easier to just learn each other's language through immersion, like we do when we meet an unfamiliar human culture. But if the aliens do not really have a language and thus are poorly adapted to learning languages (e.g. they communicate using hive mind or telepathy, or we meet one individual from an asocial species) then mathematics could be a good way to establish a frame of reference. If they are not good at language but they are good at reasoning about the world around them, they will realise how our mathematical terms correspond to elements of their reasoning system. Adding in a few important concepts ("I", "you", "good", "bad", "hour", ...) allows some basic practical statements to be made about the real world. This is essentially the approach taken by Lincos, however Lincos is designed for broadcasting messages that may take millennia to reach their destination. If we actually met aliens it would be much easier to "explain by showing", even if their senses are considerably different to ours.

Do we have any guarantee it will work? No, but if it does work, it will be clear, because they will be able to respond to us with valid mathematical statements. Their understanding of our terms will either be correct and very precise, or obviously incorrect. And once they grasp the basics, expanding with more concepts will become much easier. So it is definitely a worthwhile approach to try.

tl;dr

• 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
• Actually, there should be no need for "hour", at least initially, even if you want to convey information relating to time. Just go back to the definition of the second, and borrow a page from Carl Sagan's book (that's Contact) in discussing "elements that have X protons and Y electrons" and similar. – a CVn Aug 20 '15 at 12:30
• But then we'd be talking physics, not just mathematics (which is a valid approach, of course, especially if you expect the message to be decoded by top scientists, but otherwise consider that far more human adults could tell you that 3x4=12 than the atomic number of Caesium). Lincos defines a second by simply placing n-second delays in the transmission. – Artelius Aug 20 '15 at 22:07

Maybe.

Our sample size of aliens to make such a claim is 0, so all we can do is give it our best guess.

The advantage mathematics has is that most of its content is fully defined within the system of mathematics. The number of axioms at the bottom of the chain is very small, and we have trouble comprehending an alien culture without them. Because most of the content is defined within the system, we can easily send a great number of statements to the aliens which can be cross-checked to develop a mutual understanding. It is harder to do that with natural language because the words often have more subtle meanings.

For example, arithmetic is defined within what are known as the Peano axioms. The first defines a constant to start from. The next four define properties of 'equality,' and the final one defines a 'successor' function used to define the rest of numbers:

1. 0 is a natural number
2. (reflexive) x = x
3. (symmetric) if x = y then y = x
4. (transitive) if x = y and y = z then x = z
5. (closure) if b is a natural number and a = b then a is a natural number
6. There exists a "successor function" S(n), such that if n is a natural number, S(n) is a natural number

The successor function is pretty simple, when viewed as a table:

S(0) = 1
S(1) = 2
S(2) = 3
...


That's it. Everything else in arithmetic (addition, subtraction, multiplication, etc.) is defined from those axioms. We consider those to be pretty darn hard to avoid, so it seems reasonable that any space-faring race will have a mathematics that is at least congruent to ours.

There could be issues if they approach math from a different direction, especially in the higher levels of mathematics (a sound wave graphed against time looks really different from its Fourier transform), but in theory we could use agreement in the lower levels to develop agreement in the upper levels.

Now we do know that mathematics is not universal. The Pirahã people have a language which lacks recursion, as best as we understand it. Those Peano axioms depend heavily on recursion to do their work. It is yet unknown whether the Pirahã actually have a system of numbers or not. The language barrier is great enough that it's a real challenge to make heads or tails of it. However, the assumption is that it will be impossible to become a space faring civilization without developing something sufficiently similar to math that we can adapt our notation to meet theirs.

• Surely our sample size is $1$? – gerrit Aug 18 '15 at 17:21
• @gerrit I'll reword: sample size of aliens – Cort Ammon Aug 18 '15 at 17:31
• I suspect that you are mixing two definitions of recursion here. The fact that the Pirahã lacks linguistic recursion does not mean that we are unable to define mathematical recursion using this language. – Taemyr Aug 19 '15 at 7:44
• @Taemyr It has been shown that the Pirahã have no concept of recursion at all (or at least none we relate to). For a while, we thought they had words for 1 and 2, but interactions with them have suggested they actually have no concept of numbers at all (the number of items which they described using the word we thought was 2 varied from 2 to 5, depending on how you get there). – Cort Ammon Aug 19 '15 at 15:01

The problem isn't the math itself, it's the context.

We can throw all the math we know at the aliens, but it's going to be hard to find a version of it that sticks. For instance, the aliens might be blind; there goes any equations on paper, computer screens, or calculators. They'll most certainly have a different set of symbols used to describe their mathematical concepts, so we'll have to translate those somehow. I personally cannot think of a way to define any symbol in mathematics without defining another symbol first. It may help if both sides are actively working on the problem, but if we're just transmitting some theorems to the nearest flying saucer, they may think we're sending them a greeting, or a declaration of war, or the Hitchhiker's Guide to the Galaxy entry for Earth. Keep in mind that even though the idea has been presented that mathematics is a universal constant, that idea may not have gained any traction among aliens; similarly, they might not have brought any mathematicians along, or let them anywhere near the signal receivers. Their culture is alien to ours, so for all we know they drive their ships via religious zeal, with no knowledge of the divine mysteries of numbers.

The second problem with using mathematics to communicate is that it only goes so far. The only use for math as a form of communication that I've heard is as a confirmation of intelligence; if we met a vaguely humanoid alien at a party, we could show them some proofs to let them know we're smarter than the average primate, but we couldn't tell them our names, or define the abstract concept of a name. These things take context, and 'pure' mathematics is generally free of context.

Thus, I'd say mathematics can be used at best as a tool to begin communication, rather than the language to communicate with. You might be able to throw the Fibonacci sequence at them and hope someone recognizes it, but after that I think the first steps towards communication would be trying to get our natural languages to work rather than our mathematical ones.

The concepts of mathematics may be universal; I think that as far as we known, they are.

However, we do not speak mathematics. We use certain encodings: geometric forms, formulae using roman, arabic and greek symbols, and some other funky inventions.

While aliens with advanced technology probably have discovered the same concepts we do, they most certainly represent them very differently. So we have to think very carefully on how to communicate something that illustrates we understand a concept without defaulting to using an encoding only we can comprehend.

Some ideas have been put into objects we actually sent into space, e.g. the Voyager Golden Record. See also How Aliens Do Math by J.B. Nation for some thoughts.

Demonstrating technology my be suitable to demonstrate mastery of underlying physics, and via that some mathematics. For instance, if we meet in space, both parties probably understand mechanics and relativity -- along with the corresponding mathematics.

But then, mathematics ultimately only serves to explain things in a model world. It is not inconceivable that other models exist which can explain the same phyiscal phenomena, but are not immediately compatible with ours.

That said, if you sit a human and alien mathematician plus some linguists of both sides together, I'm optimistic that they can figure out a way to understand each other. Assuming we both talk about the same thing when pointing at the moon, we can learn each other's encoding of mathematics.

We just have to not shoot each other for long enough.

As a reference in fiction, I loved Story of Your Life by Ted Chiang for a believable tale of understanding completely different minds.

Possibly.

I would argue that math is a universal language as long as your perceptions are equivalent. So we could use it to develop communication with similar species, like star trek/star wars aliens.

But what about really, really different aliens? What if they only communicate and perceive scents, but at a level thousands of times more sensitive than any lifeform on earth? In that case we might be able to use math to decode and start understanding their language structure. But we wouldn't necessarily be able to use it to actually translate or communicate - their concepts could be so alien that starting with math isn't sufficient to jumpstart to other understandings.

• You might use math as a tool to help you analyze the patterns in a scent language, just as we use math as a tool to help us study the patterns in sound. But language is about connecting those patterns to meaning, and math would be of little use there. Math is universal, and it is a science, but it is not a language in the sense of communicating the wide range of meaning (most of it abstract) that constitutes a language. – user11864 Dec 15 '15 at 17:15

The short answer is that we don't know.

There is an active field of academic philosophy that deals with the underpinnings of mathematics trying to answer this question, and one of the more alarming notions to emerge from it is Fictionalism.

The gist of it that numbers (and thus math) are convenient conceptual tools that make it easier for us to describe categories of ideas, but aren't essential to do so. One of its proponents, Hartry Fields, went so far as to accurately and in detail restate Newton's laws of gravitation without using a single number or formula - science without the math. (Every highschooler's dream made nightmare by academic philosophy's rigor!)

Fictionalism is not widely accepted, but so far has proven very resilient to most of the arguments challenging it, so it's not crackpottery, either.

So mathematics may not be an absolute, because mathematics may not exist in and of itself outside our human minds.

I work as a mathematician. In many instances, the same mathematical idea seem to come up in different places, and independently. The most famous example is perhaps Newton vs. Leibniz with calculus.

This somehow shows that mathematics is universal in some sense.

• Newton and Leibniz were both human. All those "different places" you mention are human cultures. The "some sense" you suggest is at best human. This question asks about aliens. – AakashM Aug 19 '15 at 7:32
• Of course, but these examples shows that among humans, in different places, math still tend to be developed in a similar fashion. It is not so strange then to assume that even aliens have math similar to ours. – Per Alexandersson Aug 19 '15 at 12:11
• Universal,yes, a language, no. Language is about communicating meaning, thought. While math requires thought, it does not concern itself with describing thought. Most of the ideas we use in communicating have nothing to do with concrete, math accessible concepts. The very words "idea" or "thought" have no equivalent differential equation. We may use math as a tool to study sound patterns or brain scans, but to go into meaning (the core of language) we step out of math into a realm beyond it. – user11864 Dec 15 '15 at 17:11
• @AgapwIesu: I think it is implicit in the question that maths is considered a language only able to express mathematical concepts. However, advances in maths is probably a much better way to "measure" a civilization, compared to many other things, such as number of books printed, or number of episodes of Friends produced... – Per Alexandersson Dec 15 '15 at 17:17
• @PerAlexandersson - the question is very explicit about the core of what the OP is asking - can math be a language that goes beyond mathematical concepts. The OP admits we can demonstrate we know what 10 is (a mathematical concept) but asks if we can use math, as a language can be used, to communicate meaning ("we come in peace"). Yes, math can demonstrate intelligence, but once you know the other party is intelligent enough to communicate, then what? How will you convey meaning? How will you say "we come in peace"? There, you have gone well beyond math. – user11864 Dec 15 '15 at 17:37

Mathematics is universal. It was developed by observing the natural world (including things outside of our planet). One of the fundamental aspects of our understanding of the universe is that the physical laws (conservation of energy, momentum, charge, etc.) are equally applicable any where you are in the universe. Assuming that this is the case (we have never observed anything supporting the contrary) then an intelligent alien race existing in the same universe will necessarily have to develop the same physical laws that we have.

Whether or not conversation could occur, like "We come in peace", is somewhat of an ambiguos question. To illustrate my point consider a dolphin. It is entirely possible that a dolphin is intelligent and has the ability to, at some point in time, comprehend and develop the laws of physics (maybe in 10,000 years the dolphin will have developed the ability for space travel? It took us about this long to get to the moon). However, at this point in the history of the dolphin we have never really tried to start a conversation with it and likewise, as far as we know, the same is true for the dolphin. Indeed, it has only been recently that we have even had the ability to "hear" a dolphin (SONAR). It may be that in order to fully communicate with a dolphin it would take 20 or even a hundred years of intense study by many people in order to develop the skills needed. The fact is that a human discussing the complexities of calculus with a dolphin will most likely never occur since we have considered ourselves to be the superior intelligent race and have never really needed to interact with a dolphin on an "intelligent" basis.

Now consider an alien race that comes to Earth. We could try and make contact but they must be willing and able to listen. By ability I mean that the alien race must have some way of actually detecting our message, be it sent via sound waves, light waves, particles waves, or any other type of wave (sound for a dolphin). By willing I mean that they must not view themselves as the superior intelligent race since we can only assume that they would take the position we have with the dolphin. Provided that these two things are present then we would need to begin communication from a perspective that is familiar to both species. Since this species has travelled the cosmos to get to us it is not a bad assumption that they would understand physics and mathematics.

Note that we would also have to be willing and able to listen to the alien species as well (if they, for example, communicated with a beam of neutrinos, then we don't have the technology to listen right now)...

No matter how you look at it, it will take time to develop the skills necessary to communicate if communication is possible. Mathematics and physics, being universal, would be a good place to start.

You don't have to go very far afield to encounter "aliens" that use very different math. The typical counting system for tribes in the Amazon is 1, 2, 3, 4, "many". That's it. If you tried to exchange differential equations with them, they simply would not care and find you incomprehensible. And when trying to communicate with these or other cultures with far more math in their knowledge base, linguists never resort to math as a "universal". The thing is, math has in it some universal things, but it does nothing for the concepts that language most often deals with.

Relationships, words like "peace", concepts like greetings and honorifics, expressions of concern, empathy, disgust, all of these are abstract concepts that have no correspondent in mathematics. A linguist trying to get the word for "friend" in another language will never use mathematics as a help.

At some point, in the collection of language data, a linguist will elicit numbers and ordinals, but that is a small, very tiny subset of language. Nowhere in the vicinity of being a language in itself.

• Linguists study human language. Aliens may be different. – 458 Dec 14 '15 at 8:37
• @fredsbend - But math is math, and not able to communicate meaning beyond its very limited realm. As for linguistics, what you say is not exactly true. I am a linguist (speak three languages and have studied others, and have MA in Linguistics), and linguistics is the study of languages, period. Until now, that is only human ones, but if we were to encounter aliens, it would probably be the linguists studying their languages, and using quite similar principles as we use for human languages. Linguistics concerns itself with the communication of thought and meaning, well beyond math's realm. – user11864 Dec 15 '15 at 16:46
• @fredsbend - by the way, in thinking about "alien" type languages, one of my professors once told us about coming up on two speakers of a language he was studying, and seeing one of them speaking like the other was answering, but the other was not "saying" anything. It took a few moments for him to realize that the other was not using what laymen normally think of as words, but was simply humming his answers. The language is so tonal, that whistling or humming is a perfectly fine form of "speech". – user11864 Dec 15 '15 at 16:52

How would we choose to convey the known relationship between the idea of numbers. Surely not computers or waves of electronic messages. The best solution would be to leave large celestial objects coded in relationship to other natural objects within the vicinity where life MAY come to gaining the intelligence to decipher the improbable relationship, which would be designed to be unnaturally occurring.

The chances of two different beings from different planets, surviving and being alive in the same point of space time seems unlikely. But to leave a relic for the future to understand seems like the best mode of communication among the expanse of time.

There was a talk, perhaps in SETI Seminars, but it might have been elsewhere. I saw it on youtube.

The presenter showed particular esoteric and mind-twisting mathematical objects that had obvious 2D sketches representing them.

He points out that an alien might communicate a concept by saying "what this has"..."and this does not have". With different basis for approaching mathematics, or knowledge of those things through culture, it makes sense. But to us it would be baffling.

There are many things not included, or not yet included, in mathematics. We have calculus now, but before a few centuries there was no calculus. Mathematics does not deal with biological decisions — it's just too complicated. Maths is only useful where it has applications. So, no, it isn't.

# Kind of

## Math itself is omniversal

While the exact representations, rules, and applications may vary, the observational strategy of logic within mathematics apply not only to this universe but can apply to everything in all universes.

Geometry in other universes might not be Euclidean, but that doesn't rule it out as being non-mathematical. It would simply be based on different assumptions. Technically spacetime is non-Euclidean so this is actually the case for our own universe, however, Euclidian geometry is still extremely useful at the scales we typically operate.

While unlikely or perhaps impossible, you might have some universe where 1 + 1 = 2 and 2 + 2 = 4 but 2 + 1 + 1 = 5, but that just means certain numeric axioms no longer hold. Beings in such a universe could conceivably work with a mathematical system that assumes associativity of addition of integers even if their world doesn't actually work like that. After all, we use imaginary numbers even though there isn't really a physical analog for their meaning. (Though they are useful for certain physical properties like electromagnetism) (As a tangent, there actually are mathematical number spaces where addition is non-associative, but I don't know of any that have a practical application.)

The only way math can be useless in some universe is if it is so chaotic that there is no consistency of behavior and there are no universal laws. Such a universe could not support structure, let alone life.

But bringing it back to just our universe, it's extremely unlikely that, say, prime numbers will suddenly stop working in some remote corner of the universe. Geometry and algebra will still work elsewhere. The rules are going to be the same no matter where you are in the universe or even what universe you are in.

## Notation is not universal

As far as communicating with aliens, the only barrier is representation. Humans have more or less agreed on the use of base 10 using arabic big-endian place-value numerals, symbols used for operators and common functions, using pi rather than tau, degrees, radians, and a couple of different notations for calculus (due to the Leibniz/Newton split). We've even decided on a binary encoding for all the symbols we ever would want to use. But all this was after thousands of years of development of humanity and most of the standardization has come in the last 500 years or so (with Unicode occupying the last 30 years or so).

Aliens almost certainly developed differently than humans, resulting in different representations of mathematical concepts. This wouldn't be a simple matter of different symbols to learn. For integers alone, there are hundreds of plausible notations from base, mixed-radixes, big and little endian representations, or even more complex representations of numbers like graphs. Real and complex numbers then multiply that further. Adding in operators, there are at least three plausible notations (prefix, postfix, and infix) if expressions are represented in a linear fashion. Expressions could also be represented in a tree structure. As you go through every mathematical concept, it's not unreasonable that you'll find some similar notation somewhere, but to have it match everywhere is extraordinarily unlikely.

Mathematics isn't truly universal - if you're not using the same factor of scaling.

Base 10 is the most common version I can think of, where every number we use scales on a factor of 10. (10 = 10)(11 = 11)(12 = 12)

However, on a factor 5 scale: (10 = 5)(11 = 6)(12 = 7)

Factor 2 (binary): (10 = 2)(11 = 3)(100 = 4, because '12' doesnt exist in binary)

Factor 3: (10 = 3)(11 = 4)(12 = 5)(20 = 6, because '13' doesnt exist in factor 3)

That being said, if you know the scale, you can directly communicate.

• Mathematics is independent of representation. 7 (in base 10) is a prime regardless of wheter it is written as 7, or 11(base 6) or ####### (tally) etc. – Taemyr Aug 19 '15 at 8:00
• We are generally referring to mathematics in the abstract, not to a particular way of spelling. Integers are universal. Ways of encoding them in some medium will vary. – JDługosz Aug 19 '15 at 8:31
• There's actually an universal way to encode natural numbers (which however gets impractical for large number): Just repeat the same thing $n$ times. – celtschk Aug 21 '15 at 18:46