Prayer is also a framework for solving problems, albeit less useful in the general case. The description is not sufficient.
Personally, I'm quite influenced by Pi's "Mathematics is the language of nature", though the sentiment dates at least to Galileo. For those of us who believe that alien intelligence must BY DEFINITION have recognizable mathematics (because we have no other way to recognize intelligence), this points to that universal quality, and also hints at the improbable success of mathematics in describing and predicting the physical universe.
Agreed. They stay on top of different assumptions: Prayer supposes that there is "god" that can overrule everything, mathematics stands on top of axioms.
Mathematics, even though can be amazingly beautiful in certain cases, is nothing more than a collection of maps (theorems, problems) from assumptions (axioms,definitions) to statements(solutions, proofs).
One of the awesome things about it is that the axioms and definitions are extremely portable to many different domains (in nature). The notion of "number" can be applied to almost everything around us, euclidean geometry laws fit pretty well (yet not perfect) the surrounding physics reality.
Yet, aliens may have completely different mathematics, built on completely different set of axioms, applicable to their needs and easy to process for their "brains". IMHO, it is a little arrogant of us humans to assume that alien intelligence would have the same type of mathematics as us.
I also think that aliens might well have completely different mathematics. To me, the most compelling reason to think that is that the development of mathematics is rooted in our conscious perception of the world around us. Our conscious perception is rooted in how our brains work, how our world looks, and how the stimuli from the world are connected to the brain.
Consider for example the theory of differential geometry. Would we care about objects such as curves, surfaces, and manifolds if they (well, their low dimensional special cases) weren't ubiquitous in our world?
Would we care about permutations and symmetry if some people hadn't played around with arranging sets of objects?
Would we care about waves (sonic, fluid, E&M) if we didn't observe water flowing?
Even problems that seem to be mere puzzles with no basis in the ``real world'' are inextricably linked to the conscious perceptions that we learned to form by living in the world.
There was a great push to axiomatize all of mathematics beginning in the early 20th century and now it can be said that a lot of mathematics really is a collection of maps from assumptions to statements.
It could be that alien mathematicians have reached a collection of maps isomorphic to the one we have, but they ascribe completely different meanings to them.
Another possible scenario is that within the universe of all mathematics, our conscious perceptions and cognitive biases have carved out some subset, and alien mathematicians have carved out another. These two subsets could be so vastly different that they are mutually unintelligible, and very difficult (maybe impossible) to translate between the two mathematical languages. Even if we could translate the alien mathematics to a symbolic language familiar to us, and check that their theorems are correct, because of the vast difference in meanings ascribed to these two sets of mathematics, we just might not be able to understand what the alien mathematics means!
(I want to apologize. I am reading HN on an iPad. I decided to upvote this comment and instead fat-fingered it down, apparently to zero. Why do the voting arrows disappear after a vote is cast?)
Personally, I'm quite influenced by Pi's "Mathematics is the language of nature", though the sentiment dates at least to Galileo. For those of us who believe that alien intelligence must BY DEFINITION have recognizable mathematics (because we have no other way to recognize intelligence), this points to that universal quality, and also hints at the improbable success of mathematics in describing and predicting the physical universe.