As expressed in "The Mathematical Philosophy of Giuseppe Peano" by Hubert C. Kennedy, in Philosophy of Science Vol. 30, No. 3 (July 1963)
Peano axioms
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"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)
Kontext: 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.
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
Peano axioms
Kontext: 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.
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)
Kontext: 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
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)
Geometric Calculus (1895) as translated by Lloyd C. Kannenberg (2000) "The Operations of Deductive Logic'" Ch. 1 "Geometric Formations"
“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.”
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.
Arithmetices principia, nova methodo exposita [The Principles of Arithmetic, presented by a new method] (1889)
