<*mathematics*> A system for representing natural numbers
inductively using only two symbols, "0" (zero) and "S"
(successor).

This could be expressed as a recursive data type with the following Haskell definition:

data Peano = Zero | Succ PeanoThe number three, usually written "SSS0", would be Succ (Succ (Succ Zero)). Addition of Peano numbers can be expressed as a simple syntactic transformation:

plus Zero n = n plus (Succ m) n = Succ (plus m n)[FOLDOC]

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<*history of philosophy, biography*> Italian mathematician and
logician (1858-1932) who formalized Dedekind's insight that the
arithmetic of natural numbers could be constructed as an
axiomatic system. In Arithmetices principia nova methodo
exposita (The principles of arithmetic, presented by a new
method) (1889) Peano showed how to derive all of arithmetic
from the principles of logic, together with a set of nine
postulates about numbers: 1 is a number. Every number is
equal to itself. Numerical equality is commutative. Numbers
both equal to a third are equal to each other. Anything equal
to a number is a number. The successor of any number is a
number. No two distinct numbers have the same successor. 1 is not
the successor of any number. Any property that is: (a) true
of 0, and (b) if true of any number is true of its
successor, must be true of all numbers. This foundation for
mathematical induction was an important step toward the
twentieth-century logicization of arithmetic.
Recommended Reading: Selected works of Giuseppe Peano (Toronto,
1973); Hubert Kennedy, Peano: Life and Work of Guiseppe Peano
(Kluwer, 1980); and D. A. Gillies, Frege, Dedekind, and Peano
on the Foundation of Arithmetic (Van Gorcum, 1988).

[A Dictionary of Philosophical Terms and Names]

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