Aliens doing algebra

Suppose we have an intelligent alien who has landed on Earth and has somehow found his way into a human high school math class (for the sake of scenic stability, the alien has the ability to cloak itself as a human form).

The typical trope is the alien finds the human math elementary, and quietly chuckles to himself about how primitive humans are. Sometimes the alien, when interacting with his human friends, mutters advanced mathematical equations to himself, which goes right over the humans’ heads. In our high school math class, the alien gets called up to the board and sometimes inadvertently fills up the entire blackboard to the shock of the human teacher.

For example, in the Animorphs series, there’s a line where Ax (an alien posing as a human teen) says in a train of thought:

I think I remember the equations ... in an equation where t is time, z is Zero-space, w inversely cubed represents the nexus of…

But how realistic is it that an alien would have the same concept of algebra as we do? Would our equations even make sense to them? Would theirs make sense to us? Would they even have “equations”? Would they have the concept of things like a variable or a square root? Would things like “subtract x from both sides” make sense to them?

Note that I’m not asking about symbols—we assume that, just as our friendly space alien has learned the English alphabet, he also knows about human symbols such as + – = × / √, etc. He also speaks the English language well (or has a translator chip in his brain). I’m also not asking if the alien would have problems with arbitrary conventions in our mathematical system (i.e. base 10, 360° in a circle, etc), rather the underlying system of logic. I’m asking that, if our mathematical glyphs were intelligible to our alien, would he feel right at home performing algebraic operations on Earth? If our alien came from a planet with a completely foreign algebraic system, how quickly could he pick up “human” algebra? How much of his native algebra knowledge could be transferred to his new context?

• not very likely, even our use of 10 as a base is arbitrary, 12 would have been a much more useful base – Chris J Aug 10 '16 at 15:17
• i'm not really asking about numeric bases (which are arbitrary), more the underlying system of logic. clarified in the question – taylor swift Aug 10 '16 at 15:18
• Algebra IS elementary, that's why it's taught in grade school. – Seeds Aug 10 '16 at 16:02
• @Seeds the vast majority of the population (in the US at least) would beg to differ – taylor swift Aug 10 '16 at 16:06
• It doesn't matter if they differ, the simple fact is that it's taught in grade school. That makes it elementary, not as elementary as counting, addition, and subtraction. I am not saying the concepts couldn't be taught better, but it's amazing how much people use the stuff they learned in algebra without ever realizing it as adults. fwiw easy is not the same as elementary. – Seeds Aug 10 '16 at 16:19

While the symbols we use to describe mathematics are a human creation, the underlying truths of math are not. The relationship between π and the radius/circunference of a circle, the square-cube law and the relationship among speed, time and distance were part of the universe before humans existed, and will continue to be after humans are gone.

Any aliens who have enough technological advancement to come to Earth and disguise themselves as an act of engineering should have an understanding of Math in the very least equal to ours, even if the way they abstract math in their heads and recordings is different from ours.

For such aliens, learning our algebra could be like what learning chinese or japanese is to a regular western person. Not only the symbols are different, the way you have to organize your thoughts so that you can give meaning to an idea and express it is different too. But the things you can talk about in those languages are the same things we can talk about in English, even if some things area easier to express in one language or another.

As an exercise on reorganizing your thoughs to abstract and express mathematical ideas differently, and with different symbols (or different meanings for the same symbols we use), you can also learn a low-level programming language (languages that force you to learn how a computer works in very minute detail) and play with it. You'll never see an equation the same way again. In the very least you will start using less and less base 10 and more and more bases 2 and 16 (and sometimes 8) in your head. I have been doing it for years, to the point that reading time from a watch like the one below is easier for me than reading time on an analog watch.

(It says 12:15 by the way)

Another example of how some programmers are practically aliens when it comes to Math. The "equation" below (actually a simple program written in Brainfuck) is a simple addition. It works if both values are zeroes or positive integers and the sum of their values is no greater than 255:

[->+<]


The best part of it is that the variables to be summed are not represented in that equation. And the "+" symbol there does not mean "add", it means "increase by one". I could not explain that in an answer here, and if I tried, the more I talked about it, the more you would think I am an actual alien.

Once you've put yourself through the effort of learning a language that uses different symbols and requires you to reorder your thougths, you can use your experience to describe how learning our algebra should feel to an alien. How easy or hard that would be for said alien is up to your tayloring as the write of your story.

There is alternative here, though. If the aliens come to Earth as an act of magic or psionics, done intuitively or instinctively, then they may not need mathematical concepts. But this does not mean they could not have them.

• That watch is wrong. A real nerd would count the number of seconds since Thursday, 1 January 1970. – Aron Aug 22 '16 at 5:38
• @Aron In UTC, at that. – user Aug 22 '16 at 9:51
• Interesting that the watch is that way around. It took me a few minutes of puzzling before I realized it should be read right to left, not left to right. As an english speaker, I normally read the other direction... – Benubird Jun 26 '18 at 13:12
• @Benubird What do you mean? Bit 0 is at the beginning…? – wizzwizz4 Sep 14 '20 at 22:08
• @wizzwizz4 the high value bits are to the left. – The Square-Cube Law Sep 14 '20 at 22:48

There was a SETI Weekly Seminar a while back on just how alien maths might be.

We like to think that intelligent aliens would have the same basic ideas about numbers and geometry as us, but, even if they do, they might express those ideas very differently. To illustrate what different forms a concept can take, I will show how differently the law ab=ba has been interpreted at different times in human mathematical culture. This seemingly basic law has several different origins -- in geometry, number theory, and set theory -- some of which seem alien even to experienced mathematicians.

Interesting question. I tend to say it's "just" a matter of learning the mathematical symbols and the rules for manipulating them. Mathematic is a language and this one needs to learn it. Now one is able to discover patterns and constants. For example, the alien might learn that pi is the ratio of the circumference and the diameter of a circle and that it is always the same, say a constant. If the aliens have a concept of a circle then it is likely that they have also a symbol for pi. In that sense it may look familiar to the alien once it understands our mathematical language.

• An alien might have a well-known value for 2π or 1/π and tnink we’re weird. – JDługosz Aug 10 '16 at 19:15
• @JDługosz Who uses π? Real mathematicians use τ. – Aron Aug 22 '16 at 3:02
• I agree, that it is "just" a case of learning the mathematical language of the primitive humans. However it is just that, a language. I had real trouble transitioning to multi-dimensional non-euclidean calculus required for general relativity because it require me to re-learn all my calculus notation to accommodate for the 3,1 space-time metrics. Even simple equations were confusing for me. – Aron Aug 22 '16 at 3:14
• @Aron that’s my point. Alien mathematicians might more readily have a name for 2π. Or, turn the ratio around: would you immediatly recognise .3183… as being a special iconic number that symbolizes circles? – JDługosz Aug 22 '16 at 4:58
• @Aron: Are real mathematicians something like true Scotsmen? – celtschk Aug 22 '16 at 7:54

Any species would have a notion of counting - at the very least, a farmer needs to know whether he has the same number of sheep at the end of the day as he did at the beginning. Once you have counting, you have addition of whole numbers.

If the concept of a rectangle occurs to you, then multiplication is the natural next step. Now, you might not consider a rectangle to be a sensible choice of shape - an alien species might perhaps think that triangles are the right way to arrange groups of objects, and then instead of $x \cdot y$ they might use an operation $x \oplus y = (x \cdot y) / 2$, or something weirder.

Here's where I think the first major divergence opportunity kicks in. Humans, at this point, abstracted a little and moved from counting objects to measuring lengths and areas; an alien species might not make that leap at all, and might decide that the only number that exist are the whole numbers. Then division becomes complicated - if there are no non-integers, then $3/2$ makes just as little sense as $1/0$. Unfortunately, there wouldn't be much else to work with unless they eventually did decide that non-integers exist - math would stall here.

If they accept rational numbers, then division is the natural next step. Here's the next divergence - if they made the leap of connecting numbers to distances, then square roots will happen immediately, because there's no other way to deal with triangles. If they didn't, and they just decided division was okay anyway, then they might stall again.

Algebra would happen as usual, at least to begin with - if you care about patterns, which you have to if you're going to do math at all, then replacing numbers with symbols is the objectively correct next step.

BUT - exponents could easily go a different way. $x$ is a length, $x^2$ is an area, $x^3$ is a volume - you could insist that $x^4$ doesn't exist, because it doesn't have a physical analogue. So an alien civilization might have functions $S(x)$ and $C(x)$ meaning $x^2$ and $x^3$ respectively, and they might know that $x \cdot S(x) = C(x)$, but they might think $x \cdot C(x)$ is gibberish in the same way that $1/0$ is.

All our mathematics is built on top of our logic, with concepts of "true" and "false". However what if the alien logic doesn't have those concepts? What if they have a sort of fuzzy logic where things can be "more true" and "less true", but not "absolutely true" or "absolutely false"?

Such an "inconclusive logic" would certainly affect all of their mathematics. For a start, many of out paradoxes would be completely incomprehensible to them. First, they'd have a problem to even understand the concept of a contradiction, as in their logic, a sentence could be neither true nor false, as they don't have those concepts. Also, they couldn't even formulate a sentence like "This sentence is false", the best they could do is "This sentence is less true than its negation".

Now, one important proof method in mathematics is proof by contradiction. They wouldn't have that in their toolbox. On the other hand, they would likely have developed other tools that work better in their logic (but may be problematic to describe in ours).

Not to mention that a "proof" in their mathematics would be something very different from a proof in our mathematics, as their proofs wouldn't establish truth (remember, that's not a concept in their thinking), they would just increase the trueness. In turn, several different proofs of the same fact might not be seen as redundant by them, but each independent proof increases the trueness of the claim.

• Fuzzy logic is a real mathematical concept. en.wikipedia.org/wiki/Fuzzy_logic – Donald Hobson Aug 22 '16 at 10:03
• @DonaldHobson: Yes, but our fuzzy logic has values for "absolutely true" (1) and "absolutely false" (0), and is built on top of the real numbers and boolean logic. The alien "fuzzy logic" I described has no concept of "absolutely true" or "absolutely false", and is their fundamental way of thinking. – celtschk Aug 22 '16 at 10:56
• While it's certainly feasible to have a society that thinks in degrees of truth, not unlike many people, I found it extremely unlikely that a mathematician would be able to grasp the difference between true and false. Simply put: 1+1 = 2, 1+1 != 3 – ventsyv Aug 22 '16 at 15:08
• @ventsyv: Your brain is wired like a human brain, not like an alien brain. No wonder that you have trouble with the idea. And who tells you that there is a "true" or "false" outside our heads? – celtschk Aug 23 '16 at 7:29

Learning the symbols and getting used to base 10 will be the two major problems, then there are certain conventions such as Cartesian coordinates, the "width" of a degree etc, but everything else is transferable.

I'm reading a book on the history of the computer and it said that a lot of mathematicians in the early 20th century had problems working with bases other than 10 which prevented many of them from coming up with the idea of electronic computer. Given that your alien is fairly used to different bases, it should be fairly simple for him to "learn" high school algebra.

First thing would be to get familiar with the integer set and the basic operators. Properties of sets and set operations are universal, same for boolean logic, therefore it should be fairly straightforward to learn the basic operations (+, -, *, /, power, =, parenthesis) by looking at equations such as:

... Identity for +,-
0 + 0 = 0

... Transitive Property, similarly for the other equivalence properties
0 + 1 = 1
1 + 0 = 1

1 + 1 = 2
... So forth demonstrating 0 - 9

9 + 1 = 10


If you take a leap of fiction and stipulate that the whole world uses SI units, you'll make your alien's life much simpler.

• Why would the system of units in use, or even a single system of units in use worldwide, have any significant effect on an alien's ability to learn algebra? – user Aug 22 '16 at 15:03

Mathematics is completely made up by humans, so the answer is very probably no. Counting is present in a lot of species; the ability to track the number of objects is important. This later evolved from a simple tally system to a numeral system like we have today. This sort of system is a reasonable assumption, it facilitates all sorts of things like including trade and agriculture.

As a presumably space-faring civilisation, they would have to have some understanding of physics, so they'd need a way of dealing with that too. It's possible this would involve mathematics, but not necessarily certain. Variables play an important roll here, so I'd guess so.

They would have equations, you can have an equation without mathematics after all. Indices aren't actually necessary for maths, they're technically just shorthand for lots of multiplication.

As for algebra, that's the border of the realm of pure mathematics, which has little to do with the real world. Algebra as a concept would likely make little sense to an alien, why are the earthlings adding letters together? Since algebra has a simple, consistent logic I don't think it would take long for them to figure out how it works.

• There is nothing made up about the mathematical concepts you noted. They are in fact all based on simple axioms, which are very evident in SIMPLE Physics (one apple + one apple stuff). Even algebra is actually just application of those same axioms + some logic. – Aron Aug 22 '16 at 3:10
• Arithmetic is a natural outgrowth of counting. Basic algebra is a natural outgrowth of arithmetic (if arithmetic is "find the answer to this question", algebra is "find the question to this answer"). Beyond that, I could easily see things diverging (eg. developing iterated algebra in place of calculus). – Mark Aug 22 '16 at 5:08

Mathematics is based on logical Euclidian type proof. That type of "straightforward" rational logic Is not necessarily shared by aliens. Even for human populations, non-westerners often have problems with I.Q. tests, logic, or problem solving designed for westerners.

Aliens could use a different type, like irrational, associative, emotional or non-linear logic. Even something as simple as 1+1=2 could be radically different.

The logic could be:

• 1+1=1

Like in 1 water + 1 water = 1 water, 1 white light + 1 white light = 1 white light, 1 red + 1 yellow = 1 orange, 1 hunger + 1 hunger = 1 hunger, 1 eater and 1 sandwich the sandwich is gone, eaten but the eater remains, 1 male and 1 female become 1 couple,...the goal of yoga is union, so this is fusion so, 1 + 1 = 1

• 1+1=3

1 billing hour + 1 billing hour = 3 billing hours, ask lawyers or financiers about this one, 1 husband and 1 wife make 1 baby, so 2 becomes 3, the red-yellow previous addition could also be counted as orange which is made of red and yellow, so 3 colors are there, so it really depends on perception. All procreation relies on this so, 1 + 1 = 3

• 1+1=0

1 alcohol + 1 alcohol = 0 alcohol, they both evaporate, 1 stain + 1 cleaning product cancel each other so = 0, 1 traveler and 1 plane, there are no longer here, so they are both gone, so = 0, 1 army and another conflicting army kill each other so no one left at the end, 1 fat man + 1 diet = no more fat man...This could be seen as mutual destruction, in a way it can be seen as 1+ (-1) = 0, but in other cases it is dissolution, were 1 + 1 = 0

• 1+1=x

All the previous examples are still based on human logic and perceptions, maybe for the aliens 1 cat and 1 trombone = 42 or µ²

Actually, maybe the aliens wouldn’t even understand the concept, or the need for addition.

Edit

I admit that my examples are not very good and some have no have no logical consistency, but I am not a logician and I just came up with some various illustrations, and interpretations, of how 1+1 may not be 2.

I have rarely such number of DV in so little time, plus a request to close and being put on hold, particularly given the research and time it took me to seriously answer. DVs are supposed to be for bad answers with no research, not because you don’t agree.

I wanted to give an alternative answer, I could easily have just unthinkingly brayed: “Math is universal! End of the question!", like some fanatical priest.

To me this shows that without really questioningt it, they just accept this axiom as true, and automatically rejecting any notion that math may not be a universal language.

You may not agree, that math is subjectively based on our perception of the world and there are valid arguments why you may not, but the universality of mathematics has been a valid debate for over 2000 years.

Some constants like Pi, plank units, the speed of light are fixed, they are inherent and not mathematically constructed, or dependend, on our mathematical understanding.

Yes, mathematics has self logic and consistency but it is like a closed circuit philosophical system. I could also argue that if it was even close to an “universal” system, or even a good system, we would know all about prime numbers. They seem to be the fundamental building blocks of the universe and our human math only know they are there and where their shadows are.

What you are describing is nothing to do with mathematics. – Aron

In a way that's my point, math is a human logic language based on arguments. It is a way for our primitive brains to grasp the universe. It is based on our perceptions of counting units like pebles and 2D surfaces like circles.

They are other types of logic, which math, in its limited scope, declares false or not relevant, like the 9 tailed cats of equivocation.

A basic search reveals a plethora of them like alternative logics, Logic and Mathematics not Universal nor Absolute , Math: the Not-So-Universal Language . Plus there is lots of evidence of math being culturally subjective, Math a NOT so Universal Language , or a universal language… or is it?

• Mathematics is universal under the same axioms of choice. What you are describing is nothing to do with mathematics. – Aron Aug 22 '16 at 3:05