Usually, if a die lands on edge we say it was a spoiled throw and do it over. Similarly if a Dark Lord writes 37 on the face that lands on top, we complain that the Dark Lord is spoiling our game and we don't count it.

We count 6 possibilities for a 6-sided die, 5 possibilities for a 5-sided die, 2 possibilities for a 2-sided die, and if you have a die with just one face -- a spherical die -- what's the chance that face will come up?

I think it would be interesting to develop probability theory with no boundaries, with no 0 and 1. It works fine to do it the way it's done now, and the alternative might turn up something interesting too.

Chris and Ben, we create axiom systems and we discover parts of "mathematics". There are probably only a finite number of theorems that can be stated with only 10 characters, or 20 characters, or 30 characters, provided we don't add new definitions. But the number of possible theorems quickly gets very very large. Will each independent group of mathematicians come up with the same theorems? Probably not. So we get different mathematics.

How different could alien mathematics be? I don't know. We could look at a variety of alien mathematics and see. Except, we don't have much of that. We presumably had different mathematical traditions in china, india, and europe, and we got some minor differences. But they were solving similar real-life problems and they could have been in communication. If you want trade in silk then you need a lot of it for it to make much difference. A very few mathematicians traveling could spread ideas easily.

It's easy to see alternate physics, and alternate technology is kind of arbitrary. I think alien math might be pretty different depending on which theorems they proved first. But I don't have good examples to demonstrate it.

The Pennyologist who notices the O's is not that different from the Pennyologist who notices there's one L on each side. A particular Pennyology might notice one of those, or the other one, or both. Out of the many relationships you could pick out from the penny, which ones will people pay attention to?