Peano arithmetic

<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 Peano

The 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)

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<2001-03-16>

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Nearby terms: pattern matching « pattern recognition « PDP « Peano arithmetic » Peano Giuseppe » Peirce Charles Sanders » perceive

Peano Giuseppe

<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]

<2002-03-04>

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Nearby terms: pattern recognition « PDP « Peano arithmetic « Peano Giuseppe » Peirce Charles Sanders » perceive » perception