Peano and the Chimps: Act I, Where we meet our hero Peano.

Peano makes his entrance during a class at a rather small (enrollment approx. 5,000) institution of higher learning in an equally small (pop. approx. 24,000), but with a large attitude, town. It is a grand entrance with all due pomp and circumstance. Then with a nod to the professorial staff as they immortalize his impeccable logic and their love of dominoes, Peano leaves. Years later he will enjoy an unacknowledged cameo appearance (along with dominoes) on the television show NUMB3RS.

Peano will return to this same school several years later when older, and more subdued, students are learning how to create number sets using his axioms.

But what is Peano’s motive? The pamphlet describing these axioms (Arithmetices principia, nova methodo exposita, 1889) is in Latin! For the benefit of these untutored rubes taking up valuable educational space, the axioms are proffered in English with explanations by the harried professor.

P(1) There is a natural (counting) number * (the first number which
the students may think of as 1).
P(2) For each natural number n, there is a natural number n’ (called the successor of n and which the students may think of as n + 1).
P(3) There is no natural number n that has a successor equal to *.
P(4) If m and n are both natural numbers and m is not equal to n, then the successor of m does not equal the successor of n..

P(1) and P(2) starts the sequence of counting numbers. P(3) gives the it a beginning and P(4) makes sure that each number in the sequence has a distinct successor (otherwise 4 could be the successor to 3 and another number – maybe 2?)

To complete the axioms, there is the most perplexing of all, but oh so important if the student is going to count properly.

P(5) If A is a subset of the set of all the natural numbers and if * is in subset A and n’ is in subset A for every n in subset A, then A is the set of all the natural
numbers.(1)

Ah such confusion reigns. Is our hero really a hero? And who (or what) are the chimps? Perhaps Act II will provide the insight we desperately search for.

(1)This is a rather lose translation of Peano’s Axioms from class notes for Intro to Abstract Math (2002).