Mathematics is not only concerned with what logical statements is true but also how those statements can be related to intuition. Along with mathematics comes communication. Mathematics is the true language of the Universe.
I would like to think that an alien civilization would bring about mathematics that would be at least recognizable as a pattern, show a glimmer of some strange internal logic, other than gathering no information at all. It would amazingly surprise me if it wasn’t at least recognizable in some form or function. There’s an infinite amount of mathematics that you could come up with, but the bits we discovered first and the bits we teach are the bits that are useful for representing the world.
So here’s one immediate difference: a singular being has no need for communication (with whom would it communicate?). So no language, no pictures. What is mathematics without a language, symbols, and diagrams? Also, there is nothing to count: there’s me and that’s it (stars, because of the dense atmosphere, were not visible from Solaris). That’s some food for thought…
We have to think of beings whose makeup and history is drastically different from ours. Stanisław Lem was such a thinker and he came up with Solaris, a planet with a thinking ocean. How could we ever expect to talk to a sentient ocean?
There has to be some kind universal constants that might align so some kind of cognitive meaning can be formed. A dialog could then possibly occur. No matter how very infinitesimally, incredibly unlikely, that is to ever happen.
Let’s ponder …
Provided aliens come in one alien, two aliens, three aliens, etc, and live on a planet where there’s one rock, two rocks, three rocks etc, and one tasty creature, two tasty creatures, three tasty creatures etc, they’re going to have a concept of integers. If they fight over the tasty creatures, they’re going to have a concept of fractions (three tasty creatures between two aliens gives one and a half each). If they live in the same space we do, they’re going to have a concept of real numbers and three-dimensional vector spaces. And so on and so on.
Of course, there will probably be lots of minor differences. The notation will be very different. Important results will be named after famous little green men (or hopefully, women). Instead of a symbol equivalent to our , they may have a symbol equivalent to the common combination (like the that some people are trying to popularize), and write our as squiggle/2. Maybe they’ll teach a whole lot more geometry via Euclid-style ruler-and-compass proofs, and think of s as impossibly advanced. Conversely, maybe they’ll define sin and cosine primarily in terms of calculus (as the functions whose second derivatives are the same as the function but negative) rather than trigonometry.
Suppose that aliens based their mathematics on a foundation equivalent to ZFC set theory, a set of axioms accepted by most mathematicians. They could send us the formal definitions for their mathematical concepts in terms of set-theoretic definitions, but we might have no intuitive basis for understanding those definitions. Conversely, those aliens might not have a concept of “real number.” We could send them the set-theoretic definition of “real numbers,” which are equivalence classes of certain infinite nested sets. However, they might wonder, “Why do these humans care so much about these infinite nested sets?”
It would not be that difficult to write a program which could systematically enumerate all true theorems in mathematics given an infinite running time. In some finite time, it will have enumerated all true theorems published so far by both humans and any alien species which also use ZFC. However, the bulk of these automatically generated “theorems,” despite being non-trivial and true, would be of little use to a human mathematician, alien mathematician, or anyone else.
They would be statements like “a ruzzly flabber with pointiferous bargles is either guzly or mimsy” where “ruzzly”, “flabber”, “pointiferous”, etc. could only be interpreted by us in terms of set-theoretic definitions. Yet, the definition for “guzly” might be no more complicated when written in set theory than our definition for “real number.” It just happens that the definition of “guzly” has no relevance to anything we can find in our universe or even imagine. An infinite amount of ways for it to be uncomprehending and only one way for it to make sense. The odds suck on this one.
Then, in that case, our primitive primate brains are the limiting factor to this potential communication. It was an incredibly long shot to start with to hope for meaning and an exchange of ideas with E.T. Maybe the next alien bunch will be understandable. Keep positive.
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