"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. (Giuseppe Peano, "I fondamenti dell’aritmetica nel Formulario del 1898" ["The Principles of Arithmetic, presented by a new method"], 1889)
"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." (Giuseppe Peano, "I fondamenti dell’aritmetica nel Formulario del 1898" ["The Principles of Arithmetic, presented by a new method"], 1889)
"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." (Giuseppe Peano, "Sui fondamenti della geometria", 1894)
"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. (Giuseppe Peano, "Notations de Logique Mathématique", 1894)
"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." (Giuseppe Peano, "Geometric Calculus", 1895)
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." (Giuseppe Peano)
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