Suppose that human beings had absolutely no idea how they performed arithmetic. Imagine that human beings had evolved, rather than having learned, the ability to count sheep and add sheep. People using this built-in ability have no idea how it worked, the way Aristotle had no idea how his visual cortex supported his ability to see things. Peano Arithmetic as we know it has not been invented. There are philosophers working to formalize numerical intuitions, but they employ notations such as

Plus-Of(Seven, Six) = Thirteen

to formalize the intuitively obvious fact that when you add "seven" plus "six", of course you get "thirteen".

In this world, pocket calculators work by storing a giant lookup table of arithmetical facts, entered manually by a team of expert Artificial Arithmeticians, for starting values that range between zero and one hundred. While these calculators may be helpful in a pragmatic sense, many philosophers argue that they're only simulating addition, rather than really adding. No machine can really count - that's why humans have to count thirteen sheep before typing "thirteen" into the calculator. Calculators can recite back stored facts, but they can never know what the statements mean - if you type in "two hundred plus two hundred" the calculator says "Error: Outrange", when it's intuitively obvious, if you know what the words mean, that the answer is "four hundred".

Philosophers, of course, are not so naive as to be taken in by these intuitions. Numbers are really a purely formal system - the label "thirty-seven" is meaningful, not because of any inherent property of the words themselves, but because the label refers to thirty-seven sheep in the external world. A number is given this referential property by its semantic network of relations to other numbers. That's why, in computer programs, the LISP token for "thirty-seven" doesn't need any internal structure - it's only meaningful because of reference and relation, not some computational property of "thirty-seven" itself.

No one has ever developed an Artificial General Arithmetician, though of course there are plenty of domain-specific, narrow Artificial Arithmeticians that work on numbers between "twenty" and "thirty"...

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