Peano's Axioms

In 1889 Giuseppe Peano published a slim pamphlet in Latin and in it performed the most consequential act of mathematical economy in the history of thought: he took the counting numbers—one, two, three, all the way to infinity—and showed that the whole structure could be generated from five stipulations. There is a starting number; every number has a unique successor; different numbers have different successors; the starting number is the successor of nothing; and any property that holds for the starting number and is inherited by successors holds for every number. From those five, and only those, the entire infinite sequence of integers follows. Peano did not discover new numbers. He revealed that the numbers we have always used rest on a finite, statable skeleton. This is the founding gesture of the automatable view of thought: a being with no insight at all—a clerk, a mechanism, a program—could in principle generate all true statements of arithmetic by grinding those rules. Every claim that intelligence