So I’ve been listening to podcasts about the search for extra-terrestrial life. Several mention that if a SETI received transmission contain prime numbers, we could expect that other intelligent life forms broadcast them.
But then I started thinking that our math is probably probably base 10 because we evolved ten fingers. If another intelligent life form evolved, say, 30 digits wouldn’t their math be radically different? Is there prime numbers in base 30 and would we recognize them if they came in on a signal from deep space?
It’s the kind of thing that seems like a deep and difficult question, but has very down to Earth proofs if you look at it from the right angle. Take these 14 instances of a randomly chosen symbol:
XXXXXXXXXXXXXX
The only other way of arranging them in rows, columns and … whatever you want to call higher dimension orderings is:
XXXXXXX
XXXXXXX
What you are talking about is: suppose you want to send a message to aliens. The problem is that you have no idea how their minds work, if they even have such a thing you would recognize, but sweeping all that completely under the rug you may as well hypothesize some interest in mathematics, i.e. counting, in which case a sequence of prime numbers is unmistakable. So you can, and people have, constructed messages starting with that. Here is one dated just a few months ago, for what it is worth:
ETA they would recognize the prime numbers (which have nothing to do with any base notation, which not even all human cultures have used), maybe, let’s say, but would the low-res illustrations of humans and chemical compounds mean anything?
I think there’s a tangentially related strategy for strategically communicating with aliens that involves the product of two primes, both of which are perhaps in the range from a few dozen to a few thousand. You’d transmit two different states, in a pattern that repeats every time that product has cycled. It should be obvious to advanced thinkers to try creating a raster image whose height and width were those two factors, a binary black and white sketch, which could have whatever interesting meaning you like.
A Beacon in the Galaxy: Updated Arecibo Message for Potential FAST and SETI Projects
Now that the famous Arecibo Telescope is retired, is “Arecibo message” a term of art for “message intended for interstellar transmission”? Perhaps a term that has been in use for some time, even before the Arecibo Telescope collapse?
Making naita’s point more explicit: the properties of numbers don’t usually depend on their representations. 997 (base ten) is prime (that is, not divisible into whole numbers other than itself and one) even if we write it as 0x3E5, 0b1111100101, or CMXCVII.
The actual question we should be asking is “would aliens understand whole numbers?”. Every human culture with numbers has counting numbers. It seems simple to us and we use them to construct all our other number systems (fractions, negative numbers, real numbers, complex numbers, finite groups, etc).
But it’s possible an alien species could use something other than counting as the basis of their mathematics. Maybe, geometric lengths, or complex phases. Who knows? It’s hard for a counting species like ours to imagine math without it. But I don’t think we can necessarily exclude.
Still, if I were trying to communicate with aliens, establishing how to express counting numbers would be the first thing I tried. Even before logical expressions.
True, but didn’t math develop independently several times here on earth? Or to ask it a better way, has any earth culture ever developed a math system that involved anything other than counting?
(maybe another math system would depend on a radically different brain function - like octopuses brains for example)
Modern mathematics tends to think of mathematics deriving not from counting but from set theory. A huge amount of mathematics is pretty much oblivious to counting, or integers as anything special.
There is also category theory which looks to provide a unifying view of most of the various areas of mathematics.
I have a really fun unit that I do for fifth grade math students, where we learn about the Arecibo message and why it works for aliens, and then students create their own “Arecibo messages” and then decode one another’s messages.
I let them do multicolored messages, unlike the original Arecibo, but they make an array with prime number dimensions and then stretch it out into a long string. There’s a Google Sheet with all the long strings, and students have to figure out each string’s length (e.g., 247), figure out what prime numbers the length is a product of (e.g., 19 x 13), and then create both arrays using those dimensions, looking for the array that recreates the picture.
It’s super fun, and is based on the idea that primes don’t depend on a specific number system.
As far as I know, every human culture that has numbers has counting numbers. That is, I’ve read that some cultures only have one and many as “numbers” (sorry, no cite), but anything beyond that involves counting.
Our culture does having mathematical systems not necessarily based on counting: geometry (think of strict Euclidean construction) and phases (think of various acoustic, rotational, and electronic devices that depend on phase differences). But we already have counting.
Yes, I think we can’t make assumptions about alien brain functions.
On preview: Francis_Vaughan, good point, but historically all of that was developed after we had counting numbers. And as individuals, we all learn counting numbers before we learn other types of numbers. An alien species could possibly have equivalent mathematics to ours as their state of the art, but the larvae still learn something different. Counting could wait until grad school.
I think it would be fair to expect that any critter with the technology needed to receive a coded signal would have developed enough mathematics to be familiar with any concept we would encode.
The bit of these signals that bugs me is the picture of humans. I doubt they would make any sense. If they did, about the best they might convey is whether we are good eating or not. I always worry that these messages are not so much a greeting as a menu.
Back in the 70’s the long dead British magazine Punch opined that an alien species would regard the inclusion of a picture of a naked blonde as indicating that the whole message was a low grade joke.
ETA - this is of course in reference to the Voyager plate, but the same pic is included in the paper @DPRK lined to above.
Ultimately, we have to make some assumptions. Are our assumptions any good?
Prime numbers are easily discoverable, easily observable, and have lots of uses in mathematics. How universal is mathematics? Hard to say, but it seems like the most universal thing we’ve come up with. Could another entity receive a radio broadcast without developing math enough to recognize what a prime number is? Plausibly, but what do we have that’s better?
Understood. I was just wondering whether today, astronomers can properly refer to, say, “an Arecibo message to be sent from Germany’s Effelsberg Radio Telescope next week”. Or if one can say “Researchers will continue to refine our Arecibo messages in preparation for alien radio contact”.
Just my opinion but I’m sure the Voyager plate was intended for humans alone whatever they may have said. The needle is too small and the haystack is too large for there to be any realistic chance of non-human intelligence, no matter how advanced, ever seeing it.
