GIUSEPPE PEANO
(1858 - 1932)

Giuseppe Peano was an Italian mathematician and logician. He was the great master of the art of formal reasoning and was Professor of the University of Turin. His work was motivated by a desire to express all mathematics in terms of a logical calculus. He is noted particularly for his researches in vector algebra and formal logic. He reduced the greater part of mathematics to strict symbolic form, in which there are no words at all. The symbolism of his ideographic language was widely adopted by mathematical logicians because of its simplicity.

Around 1900, Peano invented a new system of symbols for use in symbolic logic. This enormously increased the range of symbolic logic by introducing symbols to represent other logical notions, such as: "is contained in," "the aggregate of all x's such that," "there exists," "is a," "the only," etc.

Peano is well-known for his work Formulario Mathematico in which he formulated the set of nonnegative integers on the basis of three undefined terms: 0 (zero), number and successor. This is known as Peano's Axiom System As A Basis For Mathematics. Peano devised a postulate system from which the entire arithmetic of the natural numbers can be derived. The primitives of this system are the terms mentioned above (0, number and successor). No definition of these terms is given within the theory, but the symbol "0" is intended to designate the number zero in its usual meaning and the term "number" is meant to refer to the natural numbers 0, 1, 2, 3 ..., exclusively. By the successor of a natural number n, which will be called n', is meant the natural number immediately following n in the natural order. Peano's system contains the following five postulates:

The last postulate embodies the principle of mathematical induction and illustrates the enforcement of a mathematical "truth" by stipulation. The construction of elementary arithmetic on this basis begins with the definition of the various natural numbers. 1 is defined as the successor of 0, or as O'; 2 as 1', 3 as 2', and so on. Based on P2, this process can be continued indefinitely, because of P3 (in combination with P5) it never leads back to one of the numbers previously defined, and in view of P4, it does not lead back to zero either.

As the next step, we can set up a definition of addition which expresses in a precise form the idea that the addition of any natural number to some given number may be considered as a repeated addition of 1; the latter operation is readily expressible by means of the successor relation. This definition of addition is as follows:

a) n + o = n
b) n + k'= (n + k)' The two stipulations of this recursive definition completely determine the sum of any two integers.

For example, consider the sum 3 + 2:

  1. According to the definitions of the numbers 2 and 1, we have 3 + 2 = 3 + 1' = 3 + (0')'.
  2. According to b above, 3 + (0')' = (3 + 0')' = ((3 + 0)')'.
  3. From a above and by the definitions of the numbers 4 and 5,
    we have ((3 + 0)')' = (3')' = 4' = 5.
The multiplication of natural numbers may be defined by means of the following recursive definition, which expresses the idea that a product of nk of two integers may be considered as the sum of k terms each of which equals n:

a) n x o = o
b) n x k' = (n x k) + n For example, if n=5 and k'=3 then:
  1. n x k' = 5 x 3 = 15
  2. (n x k) + n = 5 x 2 + 5 = 15
In terms of addition and multiplication, the inverse operations of subtraction and division can be defined. But it turns out that the difference and quotient are not defined for every couple of numbers; for example, 7 - 10 and 7 divided by 10 are undefined in the natural number system. This situation suggests an enlargement of the number system by the introduction of negative and rational numbers.

The negative and rational numbers can be obtained from Peano's primitives by constructing explicit definitions for them, without the introduction of any new postulates or assumptions. Every positive and negative integer is definable as a certain set of ordered couples of natural numbers, thus, the integer +2 is definable as the set of all ordered couples (m,n) of natural numbers where m = n + 2. The integer -2 is the set of all ordered couples (m,n) of natural numbers with n = m + 2. Similarly, rational numbers are defined as classes of ordered couples of integers. The various arithmetic operations can then be defined with reference to these new type of numbers, and the validity of all arithmetical laws governing these operations can be proved by using Peano's postulates and the definitions of the various arithmetical concepts involved.

The much broader system thus obtained is still incomplete in the sense that not every number in it has a square root, and more generally, not every algebraic equation whose coefficients are all numbers of the system has a solution in the system. This suggests further expansion of the number system by the introduction of real and complex numbers. This enormous extension can be effected by mere definition, without the introduction of a single new postulate. This means that every concept of mathematics can be defined by means of Peano's three primitives, and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms. These deductions can be carried out, in most cases, by means of nothing more than the principles of formal logic.

Cheryl Haefner

[Ed. note] Although today the postulates described above are universally referred to as the Peano Postulates, they were derived directly from earlier work by Richard Dedekind. Note, also, the use of recursion to define addition and multiplication. Such definition technique is a characteristic of both PASCAL and the C/C++ programming languages. Finally, the definitions of subtraction and division, while making use of the Peano Postulates, also rely on the notion of an ordered pair which, while not explicitly violating the concept of successor, does broaden its definition. Ordered pairs are discussed as part of the topic Functions and Relations.

Robert H. Orr