"Remember how we managed to build 'or' out of 'implies' and 'not'?  And that wasn't even set up on purpose by anyone or anything, it's just the human mind being thrown together by a design process that included more structure than the strict minimum?  Each time you say something like '\k. k + 0 = k', you constrain the meaning that + and 0 can have.  Imagine looking at these blue circles, each a possible world; imagine that instead of colored shapes inside them, there are objects that might be numbers, a function that might be plus.  Every time you make another statement like '\k. k + 0 = k', you kick out some of the worlds and mappings where the function you mapped onto '+' and the object you mapped onto '0' didn't always eat an object and 0 and spit that same object back out again.  Make enough statements like that, and maybe you can narrow down the possible worlds to ones that only contain objects that look like the numbers you know?  That, from a certain perspective, is what it means to define numbers and addition - to make statements such that anything they are true about must be numbers and addition.  Got any more statements like it?  Somebody wipe this wall, please, we'll want to start writing down the statements like forall k, k plus 0 equals k."

"Remember how we managed to build 'or' out of 'implies' and 'not'?  And that wasn't even set up on purpose by anyone or anything, it's just the human mind being thrown together by a design process that included more structure than the strict minimum?  Each time you say something like '\k. k + 0 = k', you constrain the meaning that + and 0 can have.  Imagine looking at these blue circles, each a possible world; imagine that instead of colored shapes inside them, there are objects that might be numbers, a function that might be plus.  Every time you make another statement like '\k. k + 0 = k', you kick out some of the worlds and mappings where the function you mapped onto '+' and the object you mapped onto '0' didn't always eat an object and 0 and spit that same object back out again.  Make enough statements like that, and maybe you can narrow down the possible worlds to ones that only contain objects that look like the numbers you know?  That, from a certain perspective, is what it means to define numbers and addition - to find statements such that anything they are true about must be numbers and addition.  Got any more statements like it?  Somebody wipe this wall, please, we'll want to start writing down the statements like forall k, k plus 0 equals k."

"Remember how we managed to build 'or' out of 'implies' and 'not'?  And that wasn't even set up on purpose by anyone or anything, it's just the human mind being thrown together by a design process that included more structure than the strict minimum?  Each time you say something like '\k. k + 0 = k', you constrain the meaning that + and 0 can have.  Imagine looking at these blue circles, each a possible world; imagine that instead of colored shapes inside them, there are objects that might be numbers, a function that might be plus.  Every time you make another statement like '\k. k + 0 = k', you kick out some of the worlds and mappings where the function you mapped onto '+' and the object you mapped onto '0' didn't always eat an object and 0 and spit that same object back out again.  Make enough statements like that, and maybe you can narrow down the possible worlds to ones that only contain objects that look like the numbers you know?  That, from a certain perspective, is what it means to define numbers and arithmetic - to find statements such that anything they are true about must be numbers and arithmetic.  Got any more statements like it?  Somebody wipe this wall, please, we'll want to start writing down the statements like forall k, k plus 0 equals k."