Every couple of months I dig out “Cosmos” and watch Carl Sagan again. One nice thing about DVDs is that they die a lot slower with this sort of abuse than the VHS tapes of Cosmos did.

Anyways, one interesting thought went through my head. OK, assume we receive the signal from alien intelligences in some distant start, an in proper Sagan style, it’s a sequence of prime numbers: 2, 3, 5, 7, 11, etc. But no richer message has been found yet, just a short sequence of prime numbers. What can the prime numbers tell us about the alien civilization? That it’s an intelligent species trying to be blatant about it’s intelligence, is the standard Sagan reply. No natural phenomenon could produce such a highly artificial sequence. But is that all we can tell about the alien species, given that clue? I’d argue no.

The first, and by far the most interesting thing, is that the aliens have the same sort of math we do. This is by no means a given, there may be ways to understand the universe that do not involve integers. It’s the only way we know how to understand the universe- counting integers seems to be bound up in our perspective of the universe. Even chimpanzees have the concept of “an apple” as a distinct entity, this apple as being distinct from that apple over there, and thus the concept of “two apples” is natural to us- giving rise (eventually) to the abstract concept of “two-ness”. Early human tribes had different number words for two rocks vr.s two apples, it took us a bit to get the idea of the abstract two concept. But it was the inevitable next step for us humans, I can’t think of any other way it could have gone. But that doesn’t mean an alien species, with a radically different form of intelligence, couldn’t understand the universe without integers. We have only the one sample of an intelligent species (humans)- we can’t begin to even say it’s improbable.

But if they’re sending us a list of primes, they have integers. And integers are really the key into our whole system of mathematics. Even if you just start with the counting integers, the positive integers, you very rapidly start running into problems like what is the number X such that X + 1 = 1? Given the question of what is X given X + 1 = 2, I can solve easily- X = 1. But what is X given X + 1 = 1? Solving this requires me to invent a zero. Then I ask, what is X given that X + 2 = 1? This forces me to invent negative numbers. Note that the questions I am asking all only contain positive integers! So then I ask, what is X given X*3 = 2? This forces me to invent the rationals, so I can say that the answer is X = 2/3. But then I ask what is X given that X*X = 2? This forces me to invent the irrationals. Then I ask what is X given that X*X + 2 = 1? This forces me to invent the imaginary numbes. The imaginary numbers gives me geometery (if I don’t already have it), systems of linear equations gets me linear algebra and geometery in multiple dimensions, and so on. They almost certainly have more math than we do, and may have different math than we do. Integer-based mathematics may be this weird sub-field only taught to a handfull of graduate students, while the undergrads and grade school children are taught some wildly different sort of mathematics, but they have our sort of mathematics.

The other interesting thing is where the sequence stops. Why did it stop there, and not somewhere else? Say the sequence is:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Why stop at 97? Why not go on an include the next prime in the sequence, 101? The reason has to be that there is some “natural stopping point” for the aliens between 97 and 101- there is something different about 97 vr.s 101 which is relevent, “natural” in some sense, to the aliens. We humans are immediately drawn to the answer of 100 being the natural stopping point. For people using base-10 arithemetic, the above list of primes are all the one and two digit primes. What’s different about 97 vr.s 101 is that 97 is two digits, while 101 is 3 digits. If that is the sequence of primes being sent, it’s reasonable to assume that the aliens do their arithmetic in base 10, just like we do. But if the sequence is instead:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167

we might note that 169=13*13 is between 167, the largest prime in the sequence, and 173, the smallest prime not in the sequence. At that point, by exactly the same logic we used above, we can assume that the aliens do their arithmetic in base-13.

One trick to this is that the aliens may be using weird arithmetic forms. We naturally think of representing numbers as a series of digits in base B, where each digit can assume any value in the range 0 to B-1, and a number is represented by the sequence d₀ + d₁B + d₂B² etc., where d_i is the i^th digit. But there are other forms of arithmetic. For example, there is a form where the digits have values in the range of 1 to B, instead of 0 to B-1. Keep B=10 in this system, and using A as the 10^th1 + 10), 21 would be 21 still (2*10¹ + 1), 30 would be 2A, 100 would be 9A, 101 would be A1, and 110 would be AA. If the aliens used this sort of number system, they’d probably send us the prime sequence:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109

This prime sequence would initially deeply confuse us, because if they used base-10, they wouldn’t have included the primes 101, 103, 107, and 109, but if they used base-11, they wouldn’t have stopped at 109, but included the prime 113 (A3 in base-11) as well. It’d be like us stopping at 89. It wouldn’t be until we thought of the zeroless base representation that we’d figure it out.

Unfortunately, in many cases there are multiple different possbilities. We could rule many arithmetics out, but still be left with more than one possibility. For example, say the sequence goes to 29. We might be tempted to conclude that the aliens use a zero-less base-5 arithmetic, as 30 (55 in zero-less base-5) is more than 29 but less than 31- until some joker points out that 29 is also the 10^th prime, and is another sensible stopping point if you use base-10 (zero-less or not). We may not be able to tell if they use a zero-less base 5 or a base 10 (zero-less or not) number system, but we can pretty sure they don’t use a base-7 system.

And some bases tell us less than others. If the aliens come out using base-11, that tells me something about them. But what if they use base-16? There are good reasons that arise out of physics why our computers use base-2 (and powers of two, and powers of powers of two, etc)- reasons that would apply to the aliens as well as us. We’ve only been using computers for a few decades. They have not yet had time to have major impacts on our society. But what of a society that that has been using computers for millions of years? I could see them having migrated over onto binary, or some easy to convert to/from binary system, like base-16. If you hand the problem over to a long time programmer, they are already likely to have the series end at 251, i.e. generate all primes less than 2⁸, or in programmer terms, all primes that fit into a single byte. Finding out the aliens use base-16 tells me that they have more or less the same computer technology we do, but isn’t nearly as interesting as if we figured out they use a zero-less base-11 system.

And some sequences don’t mean anything. In the movie “Contact”, for instance, the aliens terminate their sequence at 101. Other than being the 26th prime, there is no reason I can think of to stop there. It’s not a natural stopping point for any base I can think of, other than base-26.

Popularity: 3% [?]

Trackback URI | Comments RSS