# Giuseppe Peano In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that can be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction.

Giuseppe Peano (27 August 185820 April 1932) was an Italian mathematician, logician, and one of the founders of modern mathematical logic and set theory. His work, summarized in Formulario mathematico (1895) was highly influential and the standard Peano axioms of the natural numbers are named in his honor.

## Quotes Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use. Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define.
• Quaestiones, quae ad mathematicae fundamenta pertinent, etsi hisce temporibus a multis tractatae, satisfacienti solutione et adhuc carent. Hic difficultas maxime en sermonis ambiguitate oritur. Quare summi interest verba ipsa, quibus utimur attente perpendere.
• Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.
• Arithmetices principia, nova methodo exposita [The Principles of Arithmetic, presented by a new method] (1889)
• In every science, after having analysed the ideas, expressing the more complicated by means of the more simple, one finds a certain number that cannot be reduced among them, and that one can define no further. These are the primitive ideas of the science; it is necessary to acquire them through experience, or through induction; it is impossible to explain them by deduction.
• Notations de Logique Mathématique (1894), p. 173, as quoted in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
• Certainly it is permitted to anyone to put forward whatever hypotheses he wishes, and to develop the logical consequences contained in those hypotheses. But in order that this work merit the name of Geometry, it is necessary that these hypotheses or postulates express the result of the more simple and elementary observations of physical figures.
• "Sui fondamenti della geometria" (1894), p. 141, as quoted in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
• Geometric calculus consists in a system of operations analogous to those of algebraic calculus, but in which the entities on which the calculations are carried out, instead of being numbers, are geometric entities which we shall define.
• Geometric Calculus (1895) as translated by Lloyd C. Kannenberg (2000) "The Operations of Deductive Logic'" Ch. 1 "Geometric Formations"
• These primitive propositions … suffice to deduce all the properties of the numbers that we shall meet in the sequel. There is, however, an infinity of systems which satisfy the five primitive propositions. … All systems which satisfy the five primitive propositions are in one-to-one correspondence with the natural numbers. The natural numbers are what one obtains by abstraction from all these systems; in other words, the natural numbers are the system which has all the properties and only those properties listed in the five primitive propositions
• On what became knows as the Peano axioms, in "I fondamenti dell’aritmetica nel Formulario del 1898", in Opere Scelte Vol. III (1959), edited by Ugo Cassina, as quoted in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)

### Peano axioms

Translations of the Peano axioms from various sources.
• 1. 0 is a number.
2. The immediate successor of a number is also a number.
3. 0 is not the immediate successor of any number.
4. No two numbers have the same immediate successor.
5. Any property belonging to 0 and to the immediate successor of any number that also has that property belongs to all numbers.
• As expressed in Galileo's Finger: The Ten Great Ideas of Science (2003) by Peter Atkins, Ch. 10 "Arithmetic : The Limits of Reason", p. 333
• 1. Zero is a number.
2. The successor of any number is another number.
3. There are no two numbers with the same successor.
4. Zero is not the successor of a number.
5. Every property of zero, which belongs to the successor of every number with this property, belongs to all numbers.
• As expressed in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963) 