Alessandro Padoa

Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano. He is remembered for a method for deciding whether, given some formal theory, a new primitive notion is truly independent of the other primitive notions. There is an analogous problem in axiomatic theories, namely deciding whether a given axiom is independent of the other axioms. The following description of Padoa's career is included in a biography of Peano: He attended secondary school in Venice, engineering school in Padua, and the University of Turin, from which he received a degree in mathematics in 1895. Although he was never a student of Peano, he was an ardent disciple and, from 1896 on, a collaborator and friend. He taught in secondary schools in Pinerolo, Rome, Cagliari, and (from 1909) at the Technical Institute in Genoa. He also held positions at the Normal School in Aquila and the Naval School in Genoa, and, beginning in 1898, he gave a series of lectures at the Universities of Brussels, Pavia, Berne, Padua, Cagliari, and Geneva. He gave papers at congresses of philosophy and mathematics in Paris, Cambridge, Livorno, Parma, Padua, and Bologna. In 1934 he was awarded the ministerial prize in mathematics by the Accademia dei Lincei (Rome). The congresses in Paris in 1900 were particularly notable. Padoa's addresses at these congresses have been well remembered for their clear and unconfused exposition of the modern axiomatic method in mathematics. In fact, he is said to be "the first ... to get all the ideas concerning defined and undefined concepts completely straight". At the International Congress of Philosophy Padoa spoke on "Logical Introduction to Any Deductive Theory". He says during the period of elaboration of any deductive theory we choose the ideas to be represented by the undefined symbols and the facts to be stated by the unproved propositions; but, when we begin to formulate the theory, we can imagine that the undefined symbols are completely devoid of meaning and that the unproved propositions (instead of stating facts, that is, relations between the ideas represented by the undefined symbols) are simply conditions imposed upon undefined symbols.

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Concepts associés (4)

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Foundations of geometry

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view.

Primitive notion

In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem).

Giuseppe Peano

Giuseppe Peano (Spinetta di Cuneo (Coni), - Cavoretto, près de Turin, ) est un mathématicien et linguiste italien. Pionnier de l’approche formaliste des mathématiques, il développa, parallèlement à l’Allemand Richard Dedekind, une axiomatisation de l'arithmétique (1889). Il est par ailleurs l’inventeur d'une langue auxiliaire internationale, le Latino sine flexione (LsF) (le latin sans déclinaisons) en 1903. Il fut membre du comité qui créa la délégation pour l'adoption d'une langue auxiliaire internationale.

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