> The question is whether we can communicate our mathematics to non-humans, whose senses may be radically different from our own.
I'm not sure why people find this so relevant. Regardless of the sense, there must be enough discernible structure to detect whether 0 of something is there, whether 1 of something is there, or whether there are many more than 1.
Human sense of smell might top out at differentiating maybe 4 or 5 different things, dogs can probably sense a lot more, but either way it lets us set up basic counting, and that's generally all you need for most of our math.
Well, first we could send a series of integers, each being a sequence of pulses, to convey the fact that we are talking about integers. Then express each axiom by a series of concrete examples. And repeat. Intelligent species should be good at recognizing patterns.
The ancient Egyptians expressed axioms by a series of concrete examples. Stating them as actual axioms or theorems with proofs (by Greeks, Indians and Chinese, among others) was an improvement.
How can we communicate actual axioms to extra-terrestrials?
Presumably we can create a kind of Rosetta Stone with different representations of logical and algebraic expressions, and maybe they can decode that if they can figure out how to decode whatever "broadcast" representation of this we come up with. And that assumes that they recognize such a broadcast as something from intelligent life.
Why would we identify a sequence of pulses as integers if, for example, the passage of time is perceived differently by aliens? Or if mathematics within alien life is based on atomic units of group like set logic, and so the sequence of pulses itself is seen as "1".
I'm not sure why people find this so relevant. Regardless of the sense, there must be enough discernible structure to detect whether 0 of something is there, whether 1 of something is there, or whether there are many more than 1.
Human sense of smell might top out at differentiating maybe 4 or 5 different things, dogs can probably sense a lot more, but either way it lets us set up basic counting, and that's generally all you need for most of our math.