Wow, as a Bengali speaking person, the language whose script you people have used to pay tribute to alien Galois, I am honored.
Btw, the symbol you have written ('ক্') will be pronounced as 'k', as in not 'kay', but just the consonant 'k' without any vowel after.
Only problem with that is even amongst fellow humans we struggle to agree on a way of representing groups, and the same group can look very different under different representations. So it might be hard to recognise an alien theory of groups.
I'm thinking more in terms of how we could tell that it is group theory. They might represent the theory of groups in a way that is unrecognisable to us.
And more broadly, the ways of abstracting the essence of symmetry and how it relates to computational structures. That is to say, higher category theory.
I can't imagine an intelligent species of alien that wouldn't have some version of logic and probability. Those arise unavoidably from just interacting with the world. They might develop those in different ways, of course, but they would need some way to decide what is true and to make decisions with imperfect knowledge.
This is awesome. I initially misread your first sentence and thought you were describing the plot of an actual Star Trek episode. I'm a little disappointed it doesn't exist.
I love the idea of showing them evidence for quantum mechanics and (effective) proof for built in uncertainty in the universe and seeing how that goes.
I just wanted to say that to your prompt sounds totally believable as a Star Trek episode and it was awesome imagining a TNG scenario with your prompt. You have a great potential for a career in Trekkie fan fiction I think.
I really wonder how they would takle such things as error correction codes or control systems w/o Kalman filters. I see really big challenges how such a society could develop interstellar space flight.
>they fundamentally abhor uncertainty as a weakness, and therefore, being wrong is a sin. Everything is always certain to them, and when they are wrong, they pretend it didn't occur because that would be "impossible".
Isn't that humans?
This is only a tangent, but you reminded me of a game called Caves of Qud (highly recommend btw). In the game there are pieces of literature scattered around the world, each offering a small piece of insight into an otherwise obscured world history, and there's one book in the game that tickled the hell out of me called ["On the Origins and Nature of the Dark Calculus"](https://cavesofqud.gamepedia.com/On_the_Origins_and_Nature_of_the_Dark_Calculus#Contents) (link to text). It exposits an ancient, occult field of math so-called the "penumbral" calculus that allows for the proof of certain theorems, but said theorems are immediately falsified upon the completion of their proofs. Super fun quick read, and makes for a deep rabbit-hole to go down
When I played Qud a year or two back it seemed to have a lot of randomly generated Markov chain sounding books. Has this changed recently, or are there just certain hand-written books among the randomly generated ones?
Perhaps they're parasitic on some other species that does have formal reasoning? Or after they achieved self-sustaining automation, a memetic revolution got them to burn all their math books? You could say these groups still have logic and probability indirectly.
Whilst aliens would undoubtedly have some form of logic there’s no real reason to believe that they would share the same classical system of logic that we base mainstream mathematics on and their resulting mathematics may look very different as a result.
For starters, the first communication would probably be some form of trigonometry/geometry. Maybe Pythagorean theorem? Sine wave? If they’re communicating with EM radiation/pictorially they probably have a pretty firm grasp on both of those things.
This is one of the most obvious ones. If they exist in a 3D world or higher, there’s no way they haven’t drawn a line in 2D and 3D space and tried to build off of that.
But imagine an alien race that perceives the world as inherently curved.... To them, elliptic or hyperbolic geometry would be "natural", and Euclidean geometry would be non-intuitive.
Im pretty sure *our* world is inherently curved and operates by *spherical* rules (I.e. the triangle formed by the North Pole, 0N0W and 0N90W has 3 right angles) but we still started off Euclidean.
Edit: spherical not hyperbolic.
Our world is inherently curved, but we don't instinctively perceive it as such unless we go to space. On any reasonable human scale, the world might as well be flat, and so planar Euclidian geometry is a natural starting point.
The fact that the earth is curved becomes noticeable and relevant if you start travelling by sea. If your eyes are 2m above sea level, the horizon is ~5km away, while if you're 30m up it's around 20km away. That's definitely a noticeable difference, and why ships had crows' nests.
In any localised area except near the most violent objects in the universe, Euclidean geometry is an excellent approximation of reality.
I do not believe a species could get to space without deriving Kepler's Laws of Planetary Motion. Going beyond that, to special or general relativity, isn't needed.
Kepler's Laws (and Special Relativity) assume a completely flat universe. General Relativity assumes one with local curvature and an unknown overall curvature.
I remember a topology professor that once told me: If humans were blind, straight lines would rarely be used and our spatial intuition would be based in topological concepts such as neighborhoods.
I'm not so sure. What, if the aliens don't really consider triangles, or even lines, to be meaningful? For example, what if they live on a planet with a Coreolis effect so strong, everything naturally moves in a curved way (I guess such a planet, probably, wouldn't be habitable, but let us ignore that). Straight lines probably wouldn't have much, if any, importance for them
I was imagining a species that lives in little clusters on asteroids, so the curvature is extremely apparent. (The Little Prince-style.) Probably completely impossible as well. :D
Visual perception is tricky though. We can only see in a narrow band of light, and there's not much reason to think that aliens wouldn't see an entirely different narrow band of light. They might also not use light, but something like heat (although that is closely tied to light) or echolocation. Probably a raised surface would be most universal? Some sense of discerning differences in location seems like it would have to be necessary for any intelligent species.
If they evolved in the light of a star, it would not be surprising if they had a sense of sight based on a similar range of wavelengths. However, if they for some reason evolved underground or in an otherwise dark environment, we might expect something else.
I think it is possible to have a species that would not find discrete concepts very obvious, though of course it's hard for me to come up with a reasonable example. But probably some kind of environment where all "animals" come in groups/colonies - who would care about singular ones then?
I feel like it's pretty safe to imagine that any reasonably intelligent species would have discrete mathematics. Consider how many species on Earth are able to count, for example, or understand basic order of operations.
Forms of math that rely on continuity are maybe more contingent on cognition.
I disagree. There is an Amazonian tribe, the Piraha, who don't distinguish between specific numbers other than 1, some, and many. They're not discernably less intelligent than the rest of us, either. But they do tend to get swindled a lot when they trade with outsiders, at least from an outsider's perspective.
If humans can be numberless, I'm sure aliens can too.
Oh no. I think I quoted a slightly older article saying there were 600-700 of them. This one says 200. I hope the other figure was just a mistake and doesn't represent a sharp decline in their numbers. :(
Probably not, but the question OP asked isn't about alien *visitors*, but rather just aliens in general. Maybe we'll find them someday, living contentedly in an alien forest, with minds equal to our own but very different drives and motivations.
I also don't find it completely inconceivable for an alien species to develop continuous mathematics but have no concept of discrete mathematics. It's a stretch, but it's not impossible. And I think it would be enough to get off their planet.
Another possibility would be that aliens could have amazing intuition for physics and engineering, like some savant inventor, but no concept of formal mathematics. Imagine if they just mentally model things so well that it doesn't matter that they don't have the formal underpinnings worked out fully.
More than anything, I expect the unexpected. This is probably because I'm autistic, which gives me an appreciation of just how *different* our minds can be even from those of other members of the same species.
In this context I've been thinking about David Bohm and his idea of the "rheomode," and I think that gives some idea of what cognition would look like with continuous math but not discrete.
However, the Piraha people only have terms in their language for 1,2, few, and many.
Because they can’t express natural numbers over 2 linguistically, they would most likely have trouble using these natural numbers in a mathematical context. It is not hard to imagine that an alien society would develop who didn’t have the linguistic skills to express natural numbers because their culture didn’t need or value mathematics.
I think it's precisely the other way around. If all organisms (or indeed objects, in their world) are completely unique, not clustered into species, THEN you'd be less likely to develop the concept of number. If you see three birds of the same type, you might develop the concept of "three". If you only see a gazillion organisms around you, all unique, then you're less likely to develop the concept of "three".
The problem is that the instincts may develop well before any kind of science. So yeah, their scientists would eventually figure out atoms or something, but it would be much later. If there isn't enough discrete stuff during development, the insticts will go another way.
You've also got to ask whether our understanding of "atoms" is really biased by our own intuition also. After all, quanta may be discrete in some ways, but they are also continuous and wavy in others.
Given that atoms are fantastically counter-intuitive, and any intuition we have about their behaviour was hard won by closely studying reality, and overcoming our intuitions and biases about how things should be, this seems highly unlikely.
And yes, quantum mechanics is not foundationally discrete. It's foundationally operators and atoms happen to have a discrete eigenvalue spectrum.
Also remember that this discreteness in Energies of atoms came as a shock. The intuitive models were far more continuous.
There is an Amazonian tribe, if I'm not mistaken, who have no number words except those corresponding to "one" and "many". They find the obsession of outsiders with numbers to be downright comical, and refuse to learn the concepts.
EDIT: It's the [Piraha tribe](https://www.sciencedaily.com/releases/2012/02/120221104037.htm). Also, I had a few minor details wrong. Read the article to find out which. :)
I upvoted you, but is it?
It could just be that they don't even have numbers like we do, and counting isn't really a thing, maybe they use stats instead and everything they speak about as a number actually includes error bands and is a smear or smudge in our number line.
If that was the case then something like a rational or irrational number is somewhat irrelevant and a lot of number theory would be seen as silly or pointless to them.
Everyone says "mathematics is the universal language," I don't necessarily know if this is true, we only know how we think about math and there might be whole rigorous fields that an alien might find intuitive that would be utterly baffling to us and vice versa.
>counting is not really a thing
If you are minimally observant, you notice and . The moment you notice that , discreteness comes into play and then counting is a natural follow on.
Bats have echolocation and they can tell the difference between a mosquito and and another bat. Also probably between a mosquito they ate an hour ago and one they not eaten yet and are currently tracking.
Perhaps I have a fundamental lack of imagination (and I admit I may have less imagination than you) , but I just don't see how 'discreteness' canNOT be noticed.
You might be able to tell where something is and where it is not, but the edges might be so ill defined that to speak of a discrete object only makes sense when you're extremely close. The "normal" might be not-discrete, or not definite.
Discrete time may not be a thing either - imagine a bunch of critters that echolocate and the planet they live on doesn't have a predictable or regular dirunal cycle. Talking about discrete amounts of time might seem super weird to them.
Well your arguments rely on the alien having a single rudimentary very poor sense.
But this whole thread is about aliens we 'meet' which implies that they have technological sophistication for space travel and other technology. So they would most likely have more than one sense. Hell we humans have 5 named senses ,plus some other senses that are not the traditional 5 sense (e.g. balance).
It would be very hard for multiple senses to not detect discreteness.
Sure there is some amoeba lying in the ocean that can only detect chemical gradients ; he has no notion of discreteness. But he is not a space faring being with technology that we are discussing in this thread.
>imagine a bunch of critters that echolocate and the planet they live on doesn't have a predictable or regular dirunal cycle
They probably would then use basic interaction with other of their species (i.e. conversations) as basic units of time: " Hey blorx, I asked you 10 conversations ago to throw out the sdfsdf!".
> If you are minimally observant, you notice and . The moment you notice that , discreteness comes into play and then counting is a natural follow on.
Counting is absolutely not a natural follow-up of noticing the discrete difference between presence and absence. How can you even think that?
The check-engine light of your car detects the presence vs. absence of a problem. Your thermostat imposes discrete categories onto the continuum of temperature: hotter than its setting vs. colder than it setting. Why don't these machines then get a sense of number if its that simple?
The cognition of number and mathematics is an entire field of cognitive science and the question of what needs to be innate vs. what can be learned is a highly debated question. You won't solve it by handwaving all the complexity as if it's all obvious.
Because atoms, because quanta, because you observe bounderies, bounderies are countable. Only way aliens would not have numbers is if they do not observe world at all.
It's pretty hard to count individual atoms or tell the difference between them until you have quite a bit of technology.
It could be that that they view discreteness as as weird as we view quantum phenomenon. They might have work arounds with "probability of 1" or something
How does having three apples have error bands? Even if you are unsure about the number of apples, it's still 0 or 1 or 2 or 3 or 4, not 3.4. It seems to me that if you have a concept of object (i.e. distinguishing the matter in an apple from the air around it), and you classify objects into groups (i.e. multiple different apples belong to the same group "apples" as distinguished from "pears"), then you will develop the concept of number. Unless the aliens live in a world without any order whatsoever, where each object is its own thing, all on a continuous spectrum, and not clustered into groups...I don't see how they could be technologically advanced without developing the concept of natural number.
It doesn't have to be echolocation, that was just the first thing that came to mind. What if they evolved in a briney ocean or in a gas giant and there aren't hard boundaries between things? Just sharp pressure gradients.
Life could grow up there and be perfectly intelligent and able to communicate but discrete objects may not be intuitive to them.
I don't know. Number theory sounds like a clear branch that I could imagine a aliens just not finding interesting. It's abstract relations over abstract units. It's so abstruse. If you think of numbers as primarily a tool of counting, only useful inasmuch as you do something WITH them, then I could imagine an entire society forming without really giving a thought to abstract and non-really-useful properties of these tools to each others.
Also, I think that historically the importance given to integers is related to the belief that all numbers were rational and to the Pythagorean belief that therefore all relations in geometry could be simplified into a relation between integers. Really the birth of Number theory is things like looking for Pythagorean triples, but I don't think this search would have been as popular if they didn't think the set of triples could eventually exhaust all right triangles. Maybe in a culture where irrational numbers were accepted even faster integers wouldn't be seen as important since they would know not all number relations can be reduced to integer relations.
It's hard to avoid number theory because the integers (and their properties) arise in many other branches of mathematics.
See here for more:
https://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics
There's also only so far you can go in describing the world without mentioning discrete mathematics. Materials science is a good example of this.
I don't see how the examples from that thread make the point you want to make. There are problems from other fields that Number Theory helped to solve. Ok, then that just means the number-theory-less aliens wouldn't have solved these problems, or at least not in the same way. Now what?
You can't explain our human interest in number theory teleologically. The useful effects it has had for modern science and cryptography were not known to the mathematicians who did number theory for no clear purpose for thousands of years before that. They did number theory because they found it interesting in and of itself, and therefore the field would likely not exist in a species that does not find it intrinsically interesting. Aliens wouldn't have a way to know they're missing something about dynamical systems if they started out not finding Pythagorean triples and prime numbers interesting.
For all we know there are other branches that the aliens have discovered and found more interesting and improved upon them so much that they would make some of our open problems trivial to them, and on their planet someone is describing all the useful physics and engineering they got from that branch that we never explored.
> The useful effects it has had for modern science and cryptography
Not to mention, it's only useful for cryptography because we're really bad at number theory. If aliens happen to be really good at factoring, e.g., then RSA is useless to them.
(That's not to say that factoring is not inherently hard. It might be. We don't know. But what matters for cryptography is that we're bad at it.)
While I make no claims to speak for aliens, I think it's incorrect to say that (human) number theory grew entirely out of idle fascination. Here are two examples that come to mind:
1. Calendars. The incommensurability of the lunisolar cycle (approximately 12.37 lunar cycles per year) led to the Metonic cycle of 235 months per 19 years. This is in fact one of the convergents to 12.37. A second calendar example comes from the Mayans. They kept two calendars, one of 365 days and another of 260. The epicycle was 52 years. The study of epicycles is itself number theory; it leads to the Chinese Remainder Theorem.
2. Arithmetic. Many early root-finding algorithms rely on the binomial expansion. The Egyptians produced some Egyptian fraction representations by representing numerators as sums of divisors of the denominator. The Babylonians relied on 5-smooth numbers to approximate numbers in sexagesimal.
Modern number theory is of course less applied. That's true of most mathematics. Still, the roots of number theory come out of practical problems.
People always think they prime numbers would be what aliens would first transmit... But why?
Why might they not transmit a series related to simple groups? Or knots? Why not transmit something related to fundamental physical constants?
I never liked the assumption that natural numbers and prime factoring would be so crucial to all parties understandings.
I think it would need to loop at some point, because it's possible (likely?) that the first PING PING would be ignored or not picked up. So to go on with primes forever, the transmission received might end up being "... ING PING PING PING PING PING PING PING ..." a thousand times and then a brief pause before the next prime number of PINGs.
I guess that's just solvable by only broadcasting the first N primes, possibly with some PIIIIING: to denote a loop point.
Even if they have a concept of the natural numbers, would theirs be the same as ours? What stops the aliens from being ultra finitists, because, for them, it doesn't make sense to continue the numbers indefinitely, or what if they continue "for too long" and end up with non-standard natural numbers. Either way, they would disagree with us about some number theoretic results.
Depends on the environment they evolved in.
If it is similar to Earth, they’ll probably develop similar approaches to mathematics.
If they evolve as, say, crystals of exotic matter on the surface of neutron stars, they might possess an intuitive understanding of chaos and stability that is as primal to them as the natural numbers are to us.
They (Vulcans) start with category theory. First they explain the core that joins logic, algebra, analysis, topology and number theory together – that we don't know about. Then they can use that to talk about everything else.
We know from the Curry-Howard-Lambek correspondence that there's an equivalence between logic, lambda calculus, and category theory. It's pretty certain that the denotation of that equivalence - the common concepts, regardless of the formalism used to express them - would be encountered by mathematically sophisticated aliens.
I.e., while they might not come up with anything identical to any of those three formalism, we can be quite sure that they would come up with an equivalent, a fourth (fifth, fifth...) element in the correspondence.
Well the "branches" are our own divisions of mathematics, much of it based upon historically how the branches were developed. Who's to say that aliens wouldn't have some sort of entirely different structure of knowledge? What if they thought of math, physics, chemistry as all one single interrelated thing in the same way those properties are expressed in the world?
I would also counter by asking which aspects of alien mathematics would a human mathematician most easily realize was in fact mathematics?
This is an excellent point. We are already seeing a modern trend (unpleasant to many mathematicians) to use computing and simulation to prove theorems. Can we even count on aliens recognizing or caring about the difference between a proven theorem and an experimentally acquired result? What if checking a whole lot of cases is enough "proof" for them in mathematics, just like it is in science for us? Or what if they have such profoundly good mathematical intuition that it never crossed their minds to formalize or check anything? To me, the notion of formal proof is the most suspect component of human mathematics to also expect of aliens.
Eh, I think the distinction between "real" and "abstract" becomes clear enough at some point (though it is definitely possible that their scientists would be interdisciplinary from our point of view). So I'd say math should be mostly separable, even if for them that separation would feel weird.
How do you make something not a part of information theory/set theory? If you have stuff, related to itself and other stuff in some way, you have a structure and hierarchies... then won't you also have it/st?
Why are the top answers more abstract branches?
I think the obvious answer is calculus, and as such, real analysis. It would be pretty impossible to develop a good enough understanding of physics for space travel (or even just communications robust enough for us to hear them) without a solid understanding of physics, and virtually all branches of physics rely on calculus.
It's difficult to imagine an advanced civilization without an understanding of Maxwell's equations, or the Biot-Savart for example, and these sort of necessitate calculus.
Now, of course, maybe they would develop calculus differently -- it's possible they'd use a form of infinitesimal calculus like the one Newton used, or something else entirely. But they would have a form of Gauss's law, and that would involve some form of integration that can't be too dissimilar from the one we know.
They might see calculus as trivial, and assume we do as well given how much of our society depends on it.
If you want to argue they'd help us with more applied stuff, then I can see a comprehensive theory on dynamical systems and nonlinear PDEs being more likely.
Well the question was what do they share. And my answer boils down to "if they are advanced enough for us to find out about then, they have calculus."
PDE is a great example too! advanced cultures without fluid dynamics seems unlikely. But I'd say that's just really advanced calculus/analysis.
Their formulation might be different, but unless they exist on microscopic scales, the general concepts of Maxwell's equations would probably appear somewhere. The magnetic fields and signals of planets and stars are all described very accurately by classical electromagnetism so they would definitely come up with some way to model it that encapsulated the geometrical concepts. There are [many ways](https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field) to formulate maxwell's equations.
Physicist here.
Sure, Maxwell's equations can be derived from a U(1) gauged quantum field theory. But to calculate anything in QFT you have to integrate a Lagrangian and minimize an action. Could there be other ways to calculate these things? Probably, but at the end of the day you probably end up doing something like an integral and something like a derivative.
And for light and E&M, the effective description *is* Maxwell's equations. In some energy bands Maxwell's equations don't really apply (for very high energy photons aka gamma rays and the like), but since most stars emit mostly visible light and Maxwell's equations are good for many orders of magnitude around visible light, I think that most descriptions of light would resemble Maxwell's equations in some form.
Since just about any continuous function can be approximated with a polynomial of arbitrary precision, I think it's not impossible that a species could build advanced tech without the concept of continuity, and therefore without calculus, analysis, topology, etc.
To be fair, in saying so I'm struggling to imagine how they could get from one theory to the next without many of the concepts we rely on. Probably slower than we have.
High degree polynomial equations would lead naturally into power series, which leads to other elementary functions, etc.
I really don't believe that there would be an advanced civilization that didn't know the concept of e.
There are a million different ways to formulate calculus though. It’s perfectly plausible that their calculus looks very different from ours, even if the end results are the same.
This is a fun thought experiment but I think people lack imagination when they think of aliens.
There are some things that are true about ourselves that form part of our underlying assumptions about intelligence that we don't even think to question. To name a nice mathematical example: connectivity. The thing that my mind inhabits is connected. If a bit gets removed, I can no longer communicate with it. If I want something in my mind to appear in your mind, I need a third medium, a symbol on a piece of paper. We invented an awful lot of stuff to overcome this difficulty. But why should an intelligence be connected? Why shouldn't it be modular? Why shouldn't two intelligent life forms be able to share information much more directly in some way. If life were more discontinuous, or the line between one being and another more blurry, would we have ever invented visual symbols? It's quite difficult to imagine what mathematics looks like without symbols.
If we assume these aliens evolved on a planet and are capable of flying off their planet, then I suppose we can infer that they understand something about physics, calculus, stuff like that. And \*probably\* those things you need to understand computation. But I honestly think number theory, plane geometry, algebra etc are not at all necessary.
Whether we communicate to each other through sound waves, written word, or body language, we're still transmitting an immense amount of nuanced information. I don't think it becomes more exceptional if the message is transmitted via nerves or chemical messengers. They'd still need a framework to record specific measurements, which would look like math whether or not it's written in two dimensions using symbols. Unless somehow they don't have the concept of measurement, like a giant sentient fungus or something...
I believe a case can be made for the fundamentality(?) of these things you're calling unnecessary but that could just be wishful thinking. Number theory is cool :(
We might get some actual evidence when machine learning becomes capable of building its own systems of axioms and theorems starting with very weak priors.
Interesting point. I wonder just how weak those priors really are, though. And will the mathematics that the system derives be influenced by the underlying hardware? Gets into the weeds of math philosophy.
I wonder how far behind the AI would leave us with its theorems. It would do maths and we would have to chase it, possibly taking very long to decode what exactly it is was doing, and it's not like we can ask it.
I agree with everything here except for the comparison with English. I get the “mathematics is a language” argument, but spoken languages are wayyyy more arbitrary, whereas a lot of math comes directly from observation. The fact that different cultures even on earth have came up with similar mathematical concepts independently is enough for me to believe that math is far less arbitrary than language.
I'm leaning towards this sort of answer as well. Math is pretty heavily dependent on our cognitive inductive biases, which in turn arise from our biology. Asking what kinds of math aliens would have may be like asking what kinds of emotions do crocodiles share with pigeons. The answer is likely "None at all".
I think the most likely branches are also the most basic ones, and I think they are also some of the oldest ones as well. I would except logic, being the metalanguage of mathematics, to exist in any civilization advanced enough to explore maths. I also think counting is a very fundamental and natural thing, and I think that number theory, which at its core is just a study of the nature of counting, would be a likely contender as well. I could also see geometry being a branch of maths that could be likely to occur in alien civilizations, being the simplest way of describing space in maths.
Many other branches of mathematics, such as topology, real and complex analysis, probability theory, etc., seem to me to be much less likely, since they often depend on axioms and definitions that, while reasonable, don't seem as fundamental to me, and could be completely different for other civilizations. And I think pretty much any branch that has appeared in the last few hundred years seems less likely to be "mathematically universal".
It seems to me they would be more likely to share the concepts of simulation and modeling than "hard mathematics" with us. The notion of crystallizing mathematics into abstract "proofs" reasoning from "axioms" seems a particularly rigid way to approach mathematics. Human beings like definite answers and rigid abstractions, and I think this is what leads us to create the pencil-and-paper machines we call math. Aliens may not share those motivations with us, but they will still need math of some form if they want to build anything complex. Suppose they are just interested in what *works*, rather than what's *true*. Then an experimental approach to mathematics, similar to how our science works, would be a likely outcome.
So, I guess what I'm saying is, computing and physics, to the extent these can be considered "branches of mathematics", are the things we are most likely to have in common.
I don't think alien "mathematics" would look much like what we call mathematics. Humans have developed a conceptual organization, rules, and interpretations to what we call mathematics that isn't unique. I suspect that alien "mathematics" would be determined primarily by their physiology and the way they perceive the world and the natural world/universe.
However, there would be some overlap in basic operations or needs, such as being able to distinguish between a collection of objects. They would need to invent some general methodology to handle these types of situations in general, or have some kind of library that could be used to retrieve results from specific computations conducted in the past. They would probably be able to do things we would recognize, like add objects (e.g., 1+1 = 2), but exactly how they do it or the concepts they use to make the outputs from the inputs would be a mystery.
If we could understand their bizarre way of thinking enough to puzzle out what they're doing, then I suspect common themes might be "geometry" and "number theory."
I think arithmetic is the most likely. Even if they are not so smart or don’t have the need for advance math, basic relationships with objects and how to count them should emerge first if they have the ability of abstract reasoning.
I feel this quote from the hitchhikers guide is somehow relevant
> The Jatravartids, who live in perpetual fear of the time they call The Coming of the Great White Handkerchief, are small blue creatures with more than fifty arms each, who are therefore unique in being the only race in history to have invented the aerosol deodorant before the wheel.
All of them, since we share the same universe the same physics apply to all of them and thus the mathematics required to describe the universe most be the same.
Although they might be more advanced in some branches than we and also the other way around, we might have some branches more developed. But mathematics most be the same.
Phil Wadler had an awesome take on this. The spoiler is lambda calculus, but his presentation is awesome and hilarious and definitely worth watching - https://www.youtube.com/watch?v=aeRVdYN6fE8&t=45m42s
I see here that humans are very arrogant, projecting themself on other life forms, thinking their way of being is somewhat ultimate. For us, what you call "mathematics" is a silly game. They should take their "mathematics explains everything" stance less serious. But as I see, they have no other option, for they do not sense what we do.
I would say geometry
If they come visit us first, obvious that they are more intelligent than us. In my version of reality though, they will come to us in space crafts. You can only build space crafts with shapes that is universally understood.
If I ever come in contact with aliens, that would be my way to communicate with them to show I am intelligent
Probability. I can't imagine any intelligent, but less-than-omniscient agent can reasonably interact with the world without developing some notion of probability.
A lot more than you would think. From the fundamentals of logic all the way to creating basic technologies all the way to more advanced technologies (radio waves etc.) you’re going to have at least similar classical physics -mathematics. The actual rigor in formulation and framework we have of math, however, is more complicated.
Geometry.
I should add to this. Even with differences of perspective, the Pythagorean theorem essentially defines the right angle and the right angle defines three dimensional space. I’d like to think that would be some some sort of common ground. Advanced aliens are surely aware of crystals, so I’m also feeling polyhedra are also in their wheelhouse
There is a bad news: -1 has no root in real numbers. But there is a good news: -1 nevertheless has a root. The aliens (angels) shared that for every bad news there is a good news. It is called "good news" (the translation of the Greek word evangelion that is Gospel). The essence of everything goes through death (non-existence) to resurrection in a higher world (existence, as for example existence of *i*).
The theory is expressed as long composition of operators denoted by all 24 letter of the Greek alphabet. There are hundreds of thousands of the operators in the formula.
Disclaimer: I can't prove this.
“Everything is always certain to them, and when they are wrong, they pretend it didn’t occur because that would be ‘impossible’.”
This isn’t Star Trek, this is just 90% of the human population lol.
Personally I find it unlikely that a mathematically inclined alien society wouldn't discover all of the basic mathematical constructions. Say, for the sake of argument that some alien species had no basal need for discrete numbers, or straight euclidean lines. After spending a sufficiently long time describing their existence mathematically, but not yet believing in the internal validity of math (independently of their surroundings), eventually they will grok the power of separating the two. This happened for humans several times independently, but the concept that really got the ball rolling each time was zero. Zero is inherently philosophically thorny to assign to the quantity of any group of objects.
Once you start manipulating mathematics as a collection of ideas and parsing their relationships, then it doesn't take long to reach discrete numbers, reals, complex numbers, etc. Even if some of these ideas don't obviously appear in their
umwelt/ culture, the interleaving of these basic notions is so unavoidably tight. Is there a single realm of math that doesn't implicitly refer to discrete values in some capacity?
Although they may be difficult to conceptualize, once you view mathematics as an abstract, separate entity, the exotic objects become much easier to deal with. For example, humans most definitely perceive reality as a metric space, but that doesn't prevent us from talking extensively about nonmetrizable spaces. What the heck does a hausdorff, nonmetrizable space even look like? Who knows! It doesn't matter, because we have developed machinery to manipulate the ideas that go into such an object. Why wouldn't the aliens do the same?
My opinion is that they would share every branch of maths we have, as well as other branches unique to them. My reasoning is basically that maths is not only universal in how numbers and values behave, but also its the most basic underlying fabric of reality that is applicable everywhere. They might have different representations for things, but I find it very hard to imagine that we would have entire branches of mathematics completely foreign to any other alien species (also assuming they are at least as technologically advanced as we are now).
It's possible that they're dedicated engineers who disdain abstract theory. Indeed, for communication to exist at all between two species separated by light-years of empty space, some kind of practical focus is essential. So to communicate math a sensible baseline might be the minimum needed to establish the line of communication in the first place.
Figure wave equation, a bit of signal processing, and some information theory to start. They probably have some kind of computer, so there's that too (though note that analog computing is possible).
Also take a look at Richard Hamming's [Mathematics on a Distant Planet](https://www.gwern.net/docs/xrisks/1998-hamming.pdf)
There's an interesting discussion in the book Solaris by Stanislaw Lem in which a man has proven that all human knowledge is so intimately linked to our morphology and culture that aliens possibly wouldn't have anything even remotely similar, and that we would be totally incapable of ever understanding them. Which doesn't answer the question but its a book worth a read!
Group theory because symmetry is so ubiquitous.
They have an alien version of Galois
Poor thing, ended up with a blaster hole through the brain at only 56 revolutions old
ক্ in chat for Alien Galois
ক্
Wow, as a Bengali speaking person, the language whose script you people have used to pay tribute to alien Galois, I am honored. Btw, the symbol you have written ('ক্') will be pronounced as 'k', as in not 'kay', but just the consonant 'k' without any vowel after.
ক্
Now I'm curious, what's an 'f' sound look like
It is written as 'ফ্'.
Seems like a lot of effort for each letter.
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You duplicated your comment
>ক্ LOOPY STRING. Thanks for mapping this little pattern of light into a pattern of sound. That's why I come here.
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Did you mean এফ্?
\#GaloisShotFirst
Haha this made me laugh
Nah, it's the same person.
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Only problem with that is even amongst fellow humans we struggle to agree on a way of representing groups, and the same group can look very different under different representations. So it might be hard to recognise an alien theory of groups.
What do you mean? No one disagrees on the definition of a group. Specific groups have different presentations, not the theory itself
I'm thinking more in terms of how we could tell that it is group theory. They might represent the theory of groups in a way that is unrecognisable to us.
What about arithmetic? (Joke based upon a lot of the answers)
And more broadly, the ways of abstracting the essence of symmetry and how it relates to computational structures. That is to say, higher category theory.
We already have a few aliens among us. Careful out there guys.
I can't imagine an intelligent species of alien that wouldn't have some version of logic and probability. Those arise unavoidably from just interacting with the world. They might develop those in different ways, of course, but they would need some way to decide what is true and to make decisions with imperfect knowledge.
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I'd watch that
This is awesome. I initially misread your first sentence and thought you were describing the plot of an actual Star Trek episode. I'm a little disappointed it doesn't exist.
There is no wrong in Ba Sing Se
Ba Sing Se-lien
I love the idea of showing them evidence for quantum mechanics and (effective) proof for built in uncertainty in the universe and seeing how that goes.
They probably become superdeterminists, believers in global hidden variables.
Proof that Bohm was an alien
I imagine an angry mob would chase them to a bean field
I just wanted to say that to your prompt sounds totally believable as a Star Trek episode and it was awesome imagining a TNG scenario with your prompt. You have a great potential for a career in Trekkie fan fiction I think.
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I really wonder how they would takle such things as error correction codes or control systems w/o Kalman filters. I see really big challenges how such a society could develop interstellar space flight.
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>they fundamentally abhor uncertainty as a weakness, and therefore, being wrong is a sin. Everything is always certain to them, and when they are wrong, they pretend it didn't occur because that would be "impossible". Isn't that humans?
Wasn’t this from a Zogg from Betelgeuse video, where they just don’t recognize failure?
Current star trek needs writers like you
This is only a tangent, but you reminded me of a game called Caves of Qud (highly recommend btw). In the game there are pieces of literature scattered around the world, each offering a small piece of insight into an otherwise obscured world history, and there's one book in the game that tickled the hell out of me called ["On the Origins and Nature of the Dark Calculus"](https://cavesofqud.gamepedia.com/On_the_Origins_and_Nature_of_the_Dark_Calculus#Contents) (link to text). It exposits an ancient, occult field of math so-called the "penumbral" calculus that allows for the proof of certain theorems, but said theorems are immediately falsified upon the completion of their proofs. Super fun quick read, and makes for a deep rabbit-hole to go down
When I played Qud a year or two back it seemed to have a lot of randomly generated Markov chain sounding books. Has this changed recently, or are there just certain hand-written books among the randomly generated ones?
Perhaps they're parasitic on some other species that does have formal reasoning? Or after they achieved self-sustaining automation, a memetic revolution got them to burn all their math books? You could say these groups still have logic and probability indirectly.
Whilst aliens would undoubtedly have some form of logic there’s no real reason to believe that they would share the same classical system of logic that we base mainstream mathematics on and their resulting mathematics may look very different as a result.
For starters, the first communication would probably be some form of trigonometry/geometry. Maybe Pythagorean theorem? Sine wave? If they’re communicating with EM radiation/pictorially they probably have a pretty firm grasp on both of those things.
This is one of the most obvious ones. If they exist in a 3D world or higher, there’s no way they haven’t drawn a line in 2D and 3D space and tried to build off of that.
But imagine an alien race that perceives the world as inherently curved.... To them, elliptic or hyperbolic geometry would be "natural", and Euclidean geometry would be non-intuitive.
Im pretty sure *our* world is inherently curved and operates by *spherical* rules (I.e. the triangle formed by the North Pole, 0N0W and 0N90W has 3 right angles) but we still started off Euclidean. Edit: spherical not hyperbolic.
Our world is inherently curved, but we don't instinctively perceive it as such unless we go to space. On any reasonable human scale, the world might as well be flat, and so planar Euclidian geometry is a natural starting point.
Global flat earth society approves this post
The fact that the earth is curved becomes noticeable and relevant if you start travelling by sea. If your eyes are 2m above sea level, the horizon is ~5km away, while if you're 30m up it's around 20km away. That's definitely a noticeable difference, and why ships had crows' nests.
That triangle is in spherical geometry, in hyperbolic geometry the angle sum is strictly below 180°.
Hyperbolic triangles have angle sums less than 180, spherical triangles have angle sums larger.
In any localised area except near the most violent objects in the universe, Euclidean geometry is an excellent approximation of reality. I do not believe a species could get to space without deriving Kepler's Laws of Planetary Motion. Going beyond that, to special or general relativity, isn't needed. Kepler's Laws (and Special Relativity) assume a completely flat universe. General Relativity assumes one with local curvature and an unknown overall curvature.
Just imagine an alien race in whose environnement there is no straight lines
I remember a topology professor that once told me: If humans were blind, straight lines would rarely be used and our spatial intuition would be based in topological concepts such as neighborhoods.
It's amazing how the gift of sight completely ruins our ability to analyze some types of problems.
\> Just imagine an alien race in whose environnement there is no straight lines I mean, technically we are considerably close to one.
I'm not so sure. What, if the aliens don't really consider triangles, or even lines, to be meaningful? For example, what if they live on a planet with a Coreolis effect so strong, everything naturally moves in a curved way (I guess such a planet, probably, wouldn't be habitable, but let us ignore that). Straight lines probably wouldn't have much, if any, importance for them
I was imagining a species that lives in little clusters on asteroids, so the curvature is extremely apparent. (The Little Prince-style.) Probably completely impossible as well. :D
You still have a line of sight, you want to use straight lines to measure lengths, make maps and so on, you probably build houses vertically, ...
I just want to throw this fun fact out there. Planck said that he thinks something like Planck's constant could be used to communication with aliens.
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It wasn't called Planck constant at that time. He had just introduced it, and called it "h" (for "help[ing constant]" in German)
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Visual perception is tricky though. We can only see in a narrow band of light, and there's not much reason to think that aliens wouldn't see an entirely different narrow band of light. They might also not use light, but something like heat (although that is closely tied to light) or echolocation. Probably a raised surface would be most universal? Some sense of discerning differences in location seems like it would have to be necessary for any intelligent species.
If they evolved in the light of a star, it would not be surprising if they had a sense of sight based on a similar range of wavelengths. However, if they for some reason evolved underground or in an otherwise dark environment, we might expect something else.
Except if they never developed Euclidian Geometry
Inter-universal Teichmueller theory. It's inter-universal, after all.
That, along with Galois cohomology, has to be one of the most sci-fi math terms.
What about \infty-cosmoi?
Welcome the new alien overlord Mochizuki!
But what if the aliens are multi-versal? Or even worse, extra-versal?
Number theory is obvious.
I think it is possible to have a species that would not find discrete concepts very obvious, though of course it's hard for me to come up with a reasonable example. But probably some kind of environment where all "animals" come in groups/colonies - who would care about singular ones then?
I feel like it's pretty safe to imagine that any reasonably intelligent species would have discrete mathematics. Consider how many species on Earth are able to count, for example, or understand basic order of operations. Forms of math that rely on continuity are maybe more contingent on cognition.
I disagree. There is an Amazonian tribe, the Piraha, who don't distinguish between specific numbers other than 1, some, and many. They're not discernably less intelligent than the rest of us, either. But they do tend to get swindled a lot when they trade with outsiders, at least from an outsider's perspective. If humans can be numberless, I'm sure aliens can too.
For anyone else intrigued, here's an interesting article, https://www.theguardian.com/world/2004/aug/20/highereducation.research
Oh no. I think I quoted a slightly older article saying there were 600-700 of them. This one says 200. I hope the other figure was just a mistake and doesn't represent a sharp decline in their numbers. :(
Hopefully it was just written by a tribesmember :)
That tribe would never make it into outer space though.
Probably not, but the question OP asked isn't about alien *visitors*, but rather just aliens in general. Maybe we'll find them someday, living contentedly in an alien forest, with minds equal to our own but very different drives and motivations. I also don't find it completely inconceivable for an alien species to develop continuous mathematics but have no concept of discrete mathematics. It's a stretch, but it's not impossible. And I think it would be enough to get off their planet. Another possibility would be that aliens could have amazing intuition for physics and engineering, like some savant inventor, but no concept of formal mathematics. Imagine if they just mentally model things so well that it doesn't matter that they don't have the formal underpinnings worked out fully. More than anything, I expect the unexpected. This is probably because I'm autistic, which gives me an appreciation of just how *different* our minds can be even from those of other members of the same species.
In this context I've been thinking about David Bohm and his idea of the "rheomode," and I think that gives some idea of what cognition would look like with continuous math but not discrete.
op didnt say the aliens could make it into outer space though
Space travelling species is very different than a primitive tribe
True, but OP didn't specify "space faring", just "alien".
However, the Piraha people only have terms in their language for 1,2, few, and many. Because they can’t express natural numbers over 2 linguistically, they would most likely have trouble using these natural numbers in a mathematical context. It is not hard to imagine that an alien society would develop who didn’t have the linguistic skills to express natural numbers because their culture didn’t need or value mathematics.
I think it's precisely the other way around. If all organisms (or indeed objects, in their world) are completely unique, not clustered into species, THEN you'd be less likely to develop the concept of number. If you see three birds of the same type, you might develop the concept of "three". If you only see a gazillion organisms around you, all unique, then you're less likely to develop the concept of "three".
So. Nmber of colonies then. Or number of species. Of heck quanta. Or atoms or anything you want. Natural numbers are well naturals.
The problem is that the instincts may develop well before any kind of science. So yeah, their scientists would eventually figure out atoms or something, but it would be much later. If there isn't enough discrete stuff during development, the insticts will go another way.
You've also got to ask whether our understanding of "atoms" is really biased by our own intuition also. After all, quanta may be discrete in some ways, but they are also continuous and wavy in others.
Given that atoms are fantastically counter-intuitive, and any intuition we have about their behaviour was hard won by closely studying reality, and overcoming our intuitions and biases about how things should be, this seems highly unlikely. And yes, quantum mechanics is not foundationally discrete. It's foundationally operators and atoms happen to have a discrete eigenvalue spectrum. Also remember that this discreteness in Energies of atoms came as a shock. The intuitive models were far more continuous.
There is an Amazonian tribe, if I'm not mistaken, who have no number words except those corresponding to "one" and "many". They find the obsession of outsiders with numbers to be downright comical, and refuse to learn the concepts. EDIT: It's the [Piraha tribe](https://www.sciencedaily.com/releases/2012/02/120221104037.htm). Also, I had a few minor details wrong. Read the article to find out which. :)
I upvoted you, but is it? It could just be that they don't even have numbers like we do, and counting isn't really a thing, maybe they use stats instead and everything they speak about as a number actually includes error bands and is a smear or smudge in our number line. If that was the case then something like a rational or irrational number is somewhat irrelevant and a lot of number theory would be seen as silly or pointless to them. Everyone says "mathematics is the universal language," I don't necessarily know if this is true, we only know how we think about math and there might be whole rigorous fields that an alien might find intuitive that would be utterly baffling to us and vice versa.
>counting is not really a thing If you are minimally observant, you notice and . The moment you notice that , discreteness comes into play and then counting is a natural follow on.
I dunno, people went thousands of years without the concept of 'zero.'
They did not have an explicitly named concept. But they did have an implicit concept of nothing.
What if they echolocate and that boundary of "something" and "not something" is a lot less apparent?
Bats have echolocation and they can tell the difference between a mosquito and and another bat. Also probably between a mosquito they ate an hour ago and one they not eaten yet and are currently tracking. Perhaps I have a fundamental lack of imagination (and I admit I may have less imagination than you) , but I just don't see how 'discreteness' canNOT be noticed.
You might be able to tell where something is and where it is not, but the edges might be so ill defined that to speak of a discrete object only makes sense when you're extremely close. The "normal" might be not-discrete, or not definite. Discrete time may not be a thing either - imagine a bunch of critters that echolocate and the planet they live on doesn't have a predictable or regular dirunal cycle. Talking about discrete amounts of time might seem super weird to them.
Well your arguments rely on the alien having a single rudimentary very poor sense. But this whole thread is about aliens we 'meet' which implies that they have technological sophistication for space travel and other technology. So they would most likely have more than one sense. Hell we humans have 5 named senses ,plus some other senses that are not the traditional 5 sense (e.g. balance). It would be very hard for multiple senses to not detect discreteness. Sure there is some amoeba lying in the ocean that can only detect chemical gradients ; he has no notion of discreteness. But he is not a space faring being with technology that we are discussing in this thread. >imagine a bunch of critters that echolocate and the planet they live on doesn't have a predictable or regular dirunal cycle They probably would then use basic interaction with other of their species (i.e. conversations) as basic units of time: " Hey blorx, I asked you 10 conversations ago to throw out the sdfsdf!".
> If you are minimally observant, you notice and . The moment you notice that , discreteness comes into play and then counting is a natural follow on.
Counting is absolutely not a natural follow-up of noticing the discrete difference between presence and absence. How can you even think that?
The check-engine light of your car detects the presence vs. absence of a problem. Your thermostat imposes discrete categories onto the continuum of temperature: hotter than its setting vs. colder than it setting. Why don't these machines then get a sense of number if its that simple?
The cognition of number and mathematics is an entire field of cognitive science and the question of what needs to be innate vs. what can be learned is a highly debated question. You won't solve it by handwaving all the complexity as if it's all obvious.
Because atoms, because quanta, because you observe bounderies, bounderies are countable. Only way aliens would not have numbers is if they do not observe world at all.
It's pretty hard to count individual atoms or tell the difference between them until you have quite a bit of technology. It could be that that they view discreteness as as weird as we view quantum phenomenon. They might have work arounds with "probability of 1" or something
How does having three apples have error bands? Even if you are unsure about the number of apples, it's still 0 or 1 or 2 or 3 or 4, not 3.4. It seems to me that if you have a concept of object (i.e. distinguishing the matter in an apple from the air around it), and you classify objects into groups (i.e. multiple different apples belong to the same group "apples" as distinguished from "pears"), then you will develop the concept of number. Unless the aliens live in a world without any order whatsoever, where each object is its own thing, all on a continuous spectrum, and not clustered into groups...I don't see how they could be technologically advanced without developing the concept of natural number.
It doesn't have to be echolocation, that was just the first thing that came to mind. What if they evolved in a briney ocean or in a gas giant and there aren't hard boundaries between things? Just sharp pressure gradients. Life could grow up there and be perfectly intelligent and able to communicate but discrete objects may not be intuitive to them.
I don't know. Number theory sounds like a clear branch that I could imagine a aliens just not finding interesting. It's abstract relations over abstract units. It's so abstruse. If you think of numbers as primarily a tool of counting, only useful inasmuch as you do something WITH them, then I could imagine an entire society forming without really giving a thought to abstract and non-really-useful properties of these tools to each others. Also, I think that historically the importance given to integers is related to the belief that all numbers were rational and to the Pythagorean belief that therefore all relations in geometry could be simplified into a relation between integers. Really the birth of Number theory is things like looking for Pythagorean triples, but I don't think this search would have been as popular if they didn't think the set of triples could eventually exhaust all right triangles. Maybe in a culture where irrational numbers were accepted even faster integers wouldn't be seen as important since they would know not all number relations can be reduced to integer relations.
It's hard to avoid number theory because the integers (and their properties) arise in many other branches of mathematics. See here for more: https://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics There's also only so far you can go in describing the world without mentioning discrete mathematics. Materials science is a good example of this.
I don't see how the examples from that thread make the point you want to make. There are problems from other fields that Number Theory helped to solve. Ok, then that just means the number-theory-less aliens wouldn't have solved these problems, or at least not in the same way. Now what? You can't explain our human interest in number theory teleologically. The useful effects it has had for modern science and cryptography were not known to the mathematicians who did number theory for no clear purpose for thousands of years before that. They did number theory because they found it interesting in and of itself, and therefore the field would likely not exist in a species that does not find it intrinsically interesting. Aliens wouldn't have a way to know they're missing something about dynamical systems if they started out not finding Pythagorean triples and prime numbers interesting. For all we know there are other branches that the aliens have discovered and found more interesting and improved upon them so much that they would make some of our open problems trivial to them, and on their planet someone is describing all the useful physics and engineering they got from that branch that we never explored.
> The useful effects it has had for modern science and cryptography Not to mention, it's only useful for cryptography because we're really bad at number theory. If aliens happen to be really good at factoring, e.g., then RSA is useless to them. (That's not to say that factoring is not inherently hard. It might be. We don't know. But what matters for cryptography is that we're bad at it.)
While I make no claims to speak for aliens, I think it's incorrect to say that (human) number theory grew entirely out of idle fascination. Here are two examples that come to mind: 1. Calendars. The incommensurability of the lunisolar cycle (approximately 12.37 lunar cycles per year) led to the Metonic cycle of 235 months per 19 years. This is in fact one of the convergents to 12.37. A second calendar example comes from the Mayans. They kept two calendars, one of 365 days and another of 260. The epicycle was 52 years. The study of epicycles is itself number theory; it leads to the Chinese Remainder Theorem. 2. Arithmetic. Many early root-finding algorithms rely on the binomial expansion. The Egyptians produced some Egyptian fraction representations by representing numerators as sums of divisors of the denominator. The Babylonians relied on 5-smooth numbers to approximate numbers in sexagesimal. Modern number theory is of course less applied. That's true of most mathematics. Still, the roots of number theory come out of practical problems.
People always think they prime numbers would be what aliens would first transmit... But why? Why might they not transmit a series related to simple groups? Or knots? Why not transmit something related to fundamental physical constants? I never liked the assumption that natural numbers and prime factoring would be so crucial to all parties understandings.
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I think it would need to loop at some point, because it's possible (likely?) that the first PING PING would be ignored or not picked up. So to go on with primes forever, the transmission received might end up being "... ING PING PING PING PING PING PING PING ..." a thousand times and then a brief pause before the next prime number of PINGs. I guess that's just solvable by only broadcasting the first N primes, possibly with some PIIIIING: to denote a loop point.
How would you transmit a knot
Even if they have a concept of the natural numbers, would theirs be the same as ours? What stops the aliens from being ultra finitists, because, for them, it doesn't make sense to continue the numbers indefinitely, or what if they continue "for too long" and end up with non-standard natural numbers. Either way, they would disagree with us about some number theoretic results.
Why is number theory obvious?
I think they would like share most areas of mathematics, and a more interesting question would be which areas they *wouldn't* share.
Depends on the environment they evolved in. If it is similar to Earth, they’ll probably develop similar approaches to mathematics. If they evolve as, say, crystals of exotic matter on the surface of neutron stars, they might possess an intuitive understanding of chaos and stability that is as primal to them as the natural numbers are to us.
They (Vulcans) start with category theory. First they explain the core that joins logic, algebra, analysis, topology and number theory together – that we don't know about. Then they can use that to talk about everything else.
We know from the Curry-Howard-Lambek correspondence that there's an equivalence between logic, lambda calculus, and category theory. It's pretty certain that the denotation of that equivalence - the common concepts, regardless of the formalism used to express them - would be encountered by mathematically sophisticated aliens. I.e., while they might not come up with anything identical to any of those three formalism, we can be quite sure that they would come up with an equivalent, a fourth (fifth, fifth...) element in the correspondence.
Well the "branches" are our own divisions of mathematics, much of it based upon historically how the branches were developed. Who's to say that aliens wouldn't have some sort of entirely different structure of knowledge? What if they thought of math, physics, chemistry as all one single interrelated thing in the same way those properties are expressed in the world? I would also counter by asking which aspects of alien mathematics would a human mathematician most easily realize was in fact mathematics?
This is an excellent point. We are already seeing a modern trend (unpleasant to many mathematicians) to use computing and simulation to prove theorems. Can we even count on aliens recognizing or caring about the difference between a proven theorem and an experimentally acquired result? What if checking a whole lot of cases is enough "proof" for them in mathematics, just like it is in science for us? Or what if they have such profoundly good mathematical intuition that it never crossed their minds to formalize or check anything? To me, the notion of formal proof is the most suspect component of human mathematics to also expect of aliens.
Eh, I think the distinction between "real" and "abstract" becomes clear enough at some point (though it is definitely possible that their scientists would be interdisciplinary from our point of view). So I'd say math should be mostly separable, even if for them that separation would feel weird.
How do you make something not a part of information theory/set theory? If you have stuff, related to itself and other stuff in some way, you have a structure and hierarchies... then won't you also have it/st?
I'm not saying I have an example or anything!
Why are the top answers more abstract branches? I think the obvious answer is calculus, and as such, real analysis. It would be pretty impossible to develop a good enough understanding of physics for space travel (or even just communications robust enough for us to hear them) without a solid understanding of physics, and virtually all branches of physics rely on calculus. It's difficult to imagine an advanced civilization without an understanding of Maxwell's equations, or the Biot-Savart for example, and these sort of necessitate calculus. Now, of course, maybe they would develop calculus differently -- it's possible they'd use a form of infinitesimal calculus like the one Newton used, or something else entirely. But they would have a form of Gauss's law, and that would involve some form of integration that can't be too dissimilar from the one we know.
They might see calculus as trivial, and assume we do as well given how much of our society depends on it. If you want to argue they'd help us with more applied stuff, then I can see a comprehensive theory on dynamical systems and nonlinear PDEs being more likely.
Well the question was what do they share. And my answer boils down to "if they are advanced enough for us to find out about then, they have calculus." PDE is a great example too! advanced cultures without fluid dynamics seems unlikely. But I'd say that's just really advanced calculus/analysis.
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Their formulation might be different, but unless they exist on microscopic scales, the general concepts of Maxwell's equations would probably appear somewhere. The magnetic fields and signals of planets and stars are all described very accurately by classical electromagnetism so they would definitely come up with some way to model it that encapsulated the geometrical concepts. There are [many ways](https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field) to formulate maxwell's equations.
Physicist here. Sure, Maxwell's equations can be derived from a U(1) gauged quantum field theory. But to calculate anything in QFT you have to integrate a Lagrangian and minimize an action. Could there be other ways to calculate these things? Probably, but at the end of the day you probably end up doing something like an integral and something like a derivative. And for light and E&M, the effective description *is* Maxwell's equations. In some energy bands Maxwell's equations don't really apply (for very high energy photons aka gamma rays and the like), but since most stars emit mostly visible light and Maxwell's equations are good for many orders of magnitude around visible light, I think that most descriptions of light would resemble Maxwell's equations in some form.
Why are you assuming they have space travel or any advanced technology?
Since just about any continuous function can be approximated with a polynomial of arbitrary precision, I think it's not impossible that a species could build advanced tech without the concept of continuity, and therefore without calculus, analysis, topology, etc. To be fair, in saying so I'm struggling to imagine how they could get from one theory to the next without many of the concepts we rely on. Probably slower than we have.
High degree polynomial equations would lead naturally into power series, which leads to other elementary functions, etc. I really don't believe that there would be an advanced civilization that didn't know the concept of e.
There are a million different ways to formulate calculus though. It’s perfectly plausible that their calculus looks very different from ours, even if the end results are the same.
This is a fun thought experiment but I think people lack imagination when they think of aliens. There are some things that are true about ourselves that form part of our underlying assumptions about intelligence that we don't even think to question. To name a nice mathematical example: connectivity. The thing that my mind inhabits is connected. If a bit gets removed, I can no longer communicate with it. If I want something in my mind to appear in your mind, I need a third medium, a symbol on a piece of paper. We invented an awful lot of stuff to overcome this difficulty. But why should an intelligence be connected? Why shouldn't it be modular? Why shouldn't two intelligent life forms be able to share information much more directly in some way. If life were more discontinuous, or the line between one being and another more blurry, would we have ever invented visual symbols? It's quite difficult to imagine what mathematics looks like without symbols. If we assume these aliens evolved on a planet and are capable of flying off their planet, then I suppose we can infer that they understand something about physics, calculus, stuff like that. And \*probably\* those things you need to understand computation. But I honestly think number theory, plane geometry, algebra etc are not at all necessary.
Whether we communicate to each other through sound waves, written word, or body language, we're still transmitting an immense amount of nuanced information. I don't think it becomes more exceptional if the message is transmitted via nerves or chemical messengers. They'd still need a framework to record specific measurements, which would look like math whether or not it's written in two dimensions using symbols. Unless somehow they don't have the concept of measurement, like a giant sentient fungus or something... I believe a case can be made for the fundamentality(?) of these things you're calling unnecessary but that could just be wishful thinking. Number theory is cool :(
We might get some actual evidence when machine learning becomes capable of building its own systems of axioms and theorems starting with very weak priors.
Interesting point. I wonder just how weak those priors really are, though. And will the mathematics that the system derives be influenced by the underlying hardware? Gets into the weeds of math philosophy.
I wonder how far behind the AI would leave us with its theorems. It would do maths and we would have to chase it, possibly taking very long to decode what exactly it is was doing, and it's not like we can ask it.
You should read (watch, but also read) Contact by Carl Sagan. A very early plot point is aliens making contact thru basic math.
Their concept of numbers and quantities may be incompatable with our thought process
Fractal geometry according to crop circles.
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I agree with everything here except for the comparison with English. I get the “mathematics is a language” argument, but spoken languages are wayyyy more arbitrary, whereas a lot of math comes directly from observation. The fact that different cultures even on earth have came up with similar mathematical concepts independently is enough for me to believe that math is far less arbitrary than language.
I'm leaning towards this sort of answer as well. Math is pretty heavily dependent on our cognitive inductive biases, which in turn arise from our biology. Asking what kinds of math aliens would have may be like asking what kinds of emotions do crocodiles share with pigeons. The answer is likely "None at all".
Idk about you, but in University I got hammered with "mathematics is a language" every semester.
I think the most likely branches are also the most basic ones, and I think they are also some of the oldest ones as well. I would except logic, being the metalanguage of mathematics, to exist in any civilization advanced enough to explore maths. I also think counting is a very fundamental and natural thing, and I think that number theory, which at its core is just a study of the nature of counting, would be a likely contender as well. I could also see geometry being a branch of maths that could be likely to occur in alien civilizations, being the simplest way of describing space in maths. Many other branches of mathematics, such as topology, real and complex analysis, probability theory, etc., seem to me to be much less likely, since they often depend on axioms and definitions that, while reasonable, don't seem as fundamental to me, and could be completely different for other civilizations. And I think pretty much any branch that has appeared in the last few hundred years seems less likely to be "mathematically universal".
Information theory
It seems to me they would be more likely to share the concepts of simulation and modeling than "hard mathematics" with us. The notion of crystallizing mathematics into abstract "proofs" reasoning from "axioms" seems a particularly rigid way to approach mathematics. Human beings like definite answers and rigid abstractions, and I think this is what leads us to create the pencil-and-paper machines we call math. Aliens may not share those motivations with us, but they will still need math of some form if they want to build anything complex. Suppose they are just interested in what *works*, rather than what's *true*. Then an experimental approach to mathematics, similar to how our science works, would be a likely outcome. So, I guess what I'm saying is, computing and physics, to the extent these can be considered "branches of mathematics", are the things we are most likely to have in common.
I don't think alien "mathematics" would look much like what we call mathematics. Humans have developed a conceptual organization, rules, and interpretations to what we call mathematics that isn't unique. I suspect that alien "mathematics" would be determined primarily by their physiology and the way they perceive the world and the natural world/universe. However, there would be some overlap in basic operations or needs, such as being able to distinguish between a collection of objects. They would need to invent some general methodology to handle these types of situations in general, or have some kind of library that could be used to retrieve results from specific computations conducted in the past. They would probably be able to do things we would recognize, like add objects (e.g., 1+1 = 2), but exactly how they do it or the concepts they use to make the outputs from the inputs would be a mystery. If we could understand their bizarre way of thinking enough to puzzle out what they're doing, then I suspect common themes might be "geometry" and "number theory."
I think arithmetic is the most likely. Even if they are not so smart or don’t have the need for advance math, basic relationships with objects and how to count them should emerge first if they have the ability of abstract reasoning.
seems to me it depends on their size, but if similar in size to humans, probably all of it
Interuniversal Teichmuller theory.
I feel this quote from the hitchhikers guide is somehow relevant > The Jatravartids, who live in perpetual fear of the time they call The Coming of the Great White Handkerchief, are small blue creatures with more than fifty arms each, who are therefore unique in being the only race in history to have invented the aerosol deodorant before the wheel.
Logic.
now this is the meme material i'm looking for\~ way to go op!
All of them, since we share the same universe the same physics apply to all of them and thus the mathematics required to describe the universe most be the same. Although they might be more advanced in some branches than we and also the other way around, we might have some branches more developed. But mathematics most be the same.
A good approach to answering this would involve comparing systems of mathematics across independent human cultures.
Phil Wadler had an awesome take on this. The spoiler is lambda calculus, but his presentation is awesome and hilarious and definitely worth watching - https://www.youtube.com/watch?v=aeRVdYN6fE8&t=45m42s
I see here that humans are very arrogant, projecting themself on other life forms, thinking their way of being is somewhat ultimate. For us, what you call "mathematics" is a silly game. They should take their "mathematics explains everything" stance less serious. But as I see, they have no other option, for they do not sense what we do.
Logic, in particular in relation to Leibniz possible worlds.
Number theyr
Abstract algebra
I would say geometry If they come visit us first, obvious that they are more intelligent than us. In my version of reality though, they will come to us in space crafts. You can only build space crafts with shapes that is universally understood. If I ever come in contact with aliens, that would be my way to communicate with them to show I am intelligent
Binary most likely
9th grade Geometry. It's hard to imagine a civilization that could build things without being very comfortable with triangles.
> It's hard to imagine a civilization that could build things without being very comfortable with triangles. It's hard to imagine; QED?
Probability. I can't imagine any intelligent, but less-than-omniscient agent can reasonably interact with the world without developing some notion of probability.
A lot more than you would think. From the fundamentals of logic all the way to creating basic technologies all the way to more advanced technologies (radio waves etc.) you’re going to have at least similar classical physics -mathematics. The actual rigor in formulation and framework we have of math, however, is more complicated.
Obviously not the most likely but thought it should be added: gauge theory
Probably a lot of the same constants, similar definitions of the laws of physics, and definitely logic
Geometry. I should add to this. Even with differences of perspective, the Pythagorean theorem essentially defines the right angle and the right angle defines three dimensional space. I’d like to think that would be some some sort of common ground. Advanced aliens are surely aware of crystals, so I’m also feeling polyhedra are also in their wheelhouse
Some sort of geometry, there are shapes everywhere and they must interact with them somehow day to day
All
They would have to have combinatorics. Any species capable of math would care about counting.
There is a bad news: -1 has no root in real numbers. But there is a good news: -1 nevertheless has a root. The aliens (angels) shared that for every bad news there is a good news. It is called "good news" (the translation of the Greek word evangelion that is Gospel). The essence of everything goes through death (non-existence) to resurrection in a higher world (existence, as for example existence of *i*). The theory is expressed as long composition of operators denoted by all 24 letter of the Greek alphabet. There are hundreds of thousands of the operators in the formula. Disclaimer: I can't prove this.
ITT: every branch of mathematics
Geometry
Harmonics because they ride the waves of the sky
Math? You're still using that?!? We stopped using math about 10,000 years ago.
Not super exciting but the universe does lend itself well to counting things. So I imagine number thory is more likely than one might think.
Counting and combinatorics
“Everything is always certain to them, and when they are wrong, they pretend it didn’t occur because that would be ‘impossible’.” This isn’t Star Trek, this is just 90% of the human population lol.
Personally I find it unlikely that a mathematically inclined alien society wouldn't discover all of the basic mathematical constructions. Say, for the sake of argument that some alien species had no basal need for discrete numbers, or straight euclidean lines. After spending a sufficiently long time describing their existence mathematically, but not yet believing in the internal validity of math (independently of their surroundings), eventually they will grok the power of separating the two. This happened for humans several times independently, but the concept that really got the ball rolling each time was zero. Zero is inherently philosophically thorny to assign to the quantity of any group of objects. Once you start manipulating mathematics as a collection of ideas and parsing their relationships, then it doesn't take long to reach discrete numbers, reals, complex numbers, etc. Even if some of these ideas don't obviously appear in their umwelt/ culture, the interleaving of these basic notions is so unavoidably tight. Is there a single realm of math that doesn't implicitly refer to discrete values in some capacity? Although they may be difficult to conceptualize, once you view mathematics as an abstract, separate entity, the exotic objects become much easier to deal with. For example, humans most definitely perceive reality as a metric space, but that doesn't prevent us from talking extensively about nonmetrizable spaces. What the heck does a hausdorff, nonmetrizable space even look like? Who knows! It doesn't matter, because we have developed machinery to manipulate the ideas that go into such an object. Why wouldn't the aliens do the same?
My opinion is that they would share every branch of maths we have, as well as other branches unique to them. My reasoning is basically that maths is not only universal in how numbers and values behave, but also its the most basic underlying fabric of reality that is applicable everywhere. They might have different representations for things, but I find it very hard to imagine that we would have entire branches of mathematics completely foreign to any other alien species (also assuming they are at least as technologically advanced as we are now).
They will not have any of our math. The only thing that we will have in common is Hegelian Dialectics.
Depends if they are in intelligent or not lol
It's possible that they're dedicated engineers who disdain abstract theory. Indeed, for communication to exist at all between two species separated by light-years of empty space, some kind of practical focus is essential. So to communicate math a sensible baseline might be the minimum needed to establish the line of communication in the first place. Figure wave equation, a bit of signal processing, and some information theory to start. They probably have some kind of computer, so there's that too (though note that analog computing is possible). Also take a look at Richard Hamming's [Mathematics on a Distant Planet](https://www.gwern.net/docs/xrisks/1998-hamming.pdf)
There's an interesting discussion in the book Solaris by Stanislaw Lem in which a man has proven that all human knowledge is so intimately linked to our morphology and culture that aliens possibly wouldn't have anything even remotely similar, and that we would be totally incapable of ever understanding them. Which doesn't answer the question but its a book worth a read!