I'd like to pose the community here a challenge.
It's often supposed that if we encounter intelligent alien life, mathematics will probably be how we start communicating.
We have seen Rosetta-stone-type codebreaking, which depends on assuming some common ideas (e.g. proper nouns are phonetically about the same as in another language), and Enigma-type codebreaking, which depends on our ability to analyse a given mathematical system.
But if aliens landed and immediately offered us a proof of the Riemann Hypothesis, would we recognise it?
As a fun exercise, invent a new mathematical notation and see if other people in this community can figure out which axiom/theorem/concept you are demonstrating with it.
Your notation should have an unambiguous (albeit initially secret) specification.
Kudos for
a) expressing a nontrivial axiom/theorem/concept, and
b) inventing a notation which is as dissimilar as possible to anything you'd find in an existing textbook.
I don't think that it is worthless to play with symbols without knowing their meaning. For instance writing programs to prove theorems require that one be familiar with the game of meaningless formalism.
From the point of view of philosophy it is especially interesting to try to imagine how one would build an intuition in (him/her/it)-self given a formal system and no clue of its meaning. Would it be necessary that two creatures like two Earthlings or two Martians or two intelligent Machines would build the same intuition or would there be a way to share this intuition among each other.
So, I've taken the freedom to give here an example:
I will use the following undefined words (concepts) "pötty", "igenyes", "közt" and I will use the language of naive set theory and logic (in English).
Axiom 1. There exist at least two pötty $A$ and $B$ such that $A\not =B$. (no plural after numbers in that martian language : ( )
Axiom 2. If $A\not = B$ are pötty then there exists a pötty $C$ such that $B$ is közt $A$ and $C$.
Axiom 3. If $B$ is közt $A$ and $C$ then $A\not=C$.
Axiom 4. If $B$ is közt $A$ and $C$ then $A$ is not közt $B$ and $C$.
Definition An igenyes of $A$ and $B$ is the set of those pötty for which either $P$ is közt $A$ and $B$ or $A$ is közt $P$ and $B$ or $B$ is közt $A$ and $P$.
Axiom 5. If the pötty $C \not = D$ and both belong to the igenyes of the pötty $A$ and $B$ then the pötty $A$ belongs to the igenyes of the pötty $C$ and $D$.
Axiom 6. There is a pötty $C$ wich does not belong to the igenyes of the pötty $A$ and $B$.
Axiom 7. If $B$ is közt $A$ and $P$ and $C$ does not belong to the igenyes of the pötty $A$ and $B$ and if there exits a pötty $F$ which is közt $B$ and $C$ then there exists a pötty such that it is közt $A$ and $C$.
Theorem For any pair of pötty $A\not =B$ there is a pötty $C$ such that $C$ is közt $A$ and $B$.
Proof ???
Intution ???
If I get enough votes up then I disclose the intuition behind.