Suppose you, and you alone, were to make first contact with an alien species.
This is kind of an unfair question if you're cut off from the internet. We're a technological civilization, we've realized our wet brains are inadequate for storing all our knowledge so we put most of it on silicon where we can access it if we need it. It isn't really fair to expect us to demonstrate our rationality without giving us access to the primary repository of our knowledge.
BUT:
It should be pretty easy to explain our formalization for the real number system and propositional logic.
|+||=|||
1+2=3
||+||=||||
2+2=4
| : 1
1+1=2
+1=3
+1=4
+1=5... and so on
1+1=3 :: F
1+1=2 :: T
Then give truth tables for elementary logical connectors. This probably only works if they have some kind of analogous formalization for math, but if they're building and flying spaceships instinctively- instead of doing the math ... well, first damn. Second, there isn't any kind of symbolic proof of rationality you could give, but if they want to know if you are they'll probably want to see how you play games and will test you themselves.
You could then give a written description of updating by the conditionality principle. But I'm not sure you could express the significance of that exactly.
Then you could draw a periodic table- though I probably couldn't do it from memory. You could relate the elements to elements in the solar system by labeling the bodies with the proportions of the major elements found in the sun and on each planet.
Could you make a rough sketch of the first few atoms on the periodic table or other such universal phenomena so that it would be generally recognizable?
Do physical observations determine our visual models for atoms with a sufficient degree of specificity for this to work?
Depending on how much of the periodic table you have memorized you could go a little further with decimal notation to give atomic number, atomic weights, and specialized symbols.
H, 1, 1.01
He, 2, 4.00
Li, 3, 6.94
Be, 4, 9.01
B, 5, 10.8
C, 6, 12.01
Past that point you can probably just get away with reference by atomic number and you only have to have the periodic table's order memorized (...boron, carbon, nitrogen, oxygen, fluorine, neon, sodium, magnesium, etc). Then you can diagram basic molecules like water as "(1)-(8)-(1)". Nail down the meaning ...
Allow me to propose a thought experiment. Suppose you, and you alone, were to make first contact with an alien species. Since your survival and the survival of the entire human race may depend on the extraterrestrials recognizing you as a member of a rational species, how would you convey your knowledge of mathematics, logic, and the scientific method to them using only your personal knowledge and whatever tools you might reasonably have on your person on an average day?
When I thought of this question, the two methods that immediately came to mind were the Pythagorean Theorem and prime number sequences. For instance, I could draw a rough right triangle and label one side with three dots, the other with four, and the hypotenuse with five. However, I realized that these are fairly primitive maths. After all, the ancient Greeks knew of them, and yet had no concept of the scientific method. Would these likely be sufficient, and if not what would be? Could you make a rough sketch of the first few atoms on the periodic table or other such universal phenomena so that it would be generally recognizable? Could you convey a proof of rationality in a manner that even aliens who cannot hear human vocalizations, or see in a completely different part of the EM spectrum? Is it even in principle possible to express rationality without a common linguistic grounding?
In other words, what is the most rational thought you could convey without the benefit of common language, culture, psychology, or biology, and how would you do it?
Bonus point: Could you convey Bayes' theorem to said ET?