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Here is an excerpt from ESSAI D’UNE THÉORIE ALGÉBRIQUE DES NOMBRES ENTIERS, PRÉCÉDÉ D’UNE INTRODUCTION LOGIQUE A UNE THÉORIE DÉDUCTIVE QUELCONQUE from Alessandro Padoa (as can be found here):

nous choisissons les idées que nous représentons par les symboles non-définis et les faits que nous énonçons par les P non-démontrées; (...) Alors, le système des idées que nous avons choisi d’abord n’est qu'une interprétation du système des symboles non- définis; mais, au point de vue déductif, cette interprétation peut être ignorée par le lecteur, qui peut librement la remplacer, dans sa pensée, par une autre interprétation qui vérifie les conditions énoncées par les P non-démontrées. (...) Ainsi les questions logiques acquièrent une complète indépendance à l’égard des questions empiriques ou psychologiques (et, eu particulier, du problème de la connaissance) ;

On the Wikipedia article on Euclidean geometry, Padoa is also quoted (the quote, translated in English, is very similar to some parts of the previous text) to illustrate the modern standards of rigor in mathematics.

Since this is the first time I hear about Padoa, I am wondering:

  • What was the context of his intervention?
  • What role did Padoa play in the development of modern mathematics and logic?
  • Was the view he expressed here new to mathematicians of the time?
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There are many sources in Italian about Alessandro Padoa, two in-depht sources, with a wide bibliography, are:

Alessandro Padoa, in Enciclopedia Treccani.

Cantù, P. (2007). Il carteggio Padoa-Vailati. Un’introduzione alle lettere inviate da Chioggia [The Correspondence between Padoa and Vailati. An Introduction to some letters sent from Chioggia] (Italian). Chioggia. Rivista di Studi e ricerche, 30, 45-70. $$$$ Alessandro Padoa took part, with Burali-Forti, Peri, Vailati, and others, in the Peano's school of logic. His main contributions were on occasion of the International Conference of Mathematics and that of Philosophy, held in Paris in 1900.

In English you can refer, for information about his biography and his contribution to logic and mathematics, to

https://mathshistory.st-andrews.ac.uk/Biographies/Padoa/

He belonged to Peano's school of mathematical logic, popularising this type of work. He gave the important lecture Essay of an algebraic theory of whole numbers, preceded by a logical introduction to any deductive theory at the International Congress of Philosophy in Paris in 1900. He had discovered an important method in the theory of definition. He had drawn a parallel between the axiomatic method, where theorems are deduced from axioms, and deducibility from primitive terms. Padoa was the first to present a method to prove that a primitive term of a theory cannot be defined within the system using the remaining primitive terms. This result was first made public in his lecture at the Paris Congress referred to above. […]This work became even more important when model theory was developed and Tarski proved Padoa's method in 1924. Immediately following the Congress of Philosophy in Paris, the Second International Congress of Mathematicians took place in 1900. Padoa spoke on A new system of definitions for Euclidean geometry but began with a summary of his lecture at the Philosophy Congress.

Jeremy Gray, in his book Plato’s Ghost. The modernist Transformation of Mathematics, Princeton University Press, 2008, writes about Peano ‘s school, Padoa and his role in what he calls modernism in mathematics, and the cultural milieu, in particular around 1900, and the role of Alessandro Padoa are well outlined:

The year 1900 was a time for a great number of celebrations of the new century. It was an opportunity to appraise the past and to sing the praises of the immediate future, and many people seized it. Paris played host to no less than six months of international congresses, of which those of the philosophers and mathematicians stayed back to back, and several stayed for both. Famously, it was at the Congress of Philosophers that Russel and Peano met and realized that there was a deep, subtle subject called mathematical logic that had a lot to offer him, in which the Italian grouped around Peano were particularly strong.

[…]Peano read the paper on mathematical definitions that excited Russel’s admiration. It was discussed by quite a number of people: Jules Tannery, Schöreder, and Padoa. Burali-Forty’s paper was on similar theme […]. Padoa’s paper continued the impressive Italian presence. It was entitled in part “A logical Introduction to Any Deductive Theory.” He described a theory as a set of undefined symbols and unproved propositions, from which other propositions are derived logically and other symbols introduced by definition. He then proposed a test to see if a symbol cannot be deduced from the propositions of the theory: it is necessary and sufficient to find an interpretation of the system that “verifies the system of unproved symbols and continues to do so if we vary the meaning only of the symbols considered”.

[…] Russell did not stay for the International Congress of mathematicians, but the Italians did. Some had papers to present, some stayed out of interest. The most long-lasting paper presented there did not please them at all. This was Hilbert’s plenary address “Mathematical Problems” […] After the talk, Peano pointed out that more was known about the problem of axiomatizing arithmetic than Hilbert seemed to recognize, and went on to advertise the fact that Padoa would be reporting on the topic at the Congress. Indeed, said Peano, Padoa and others had in fact solved the problem, insofar as that was possible. Two years later Padoa published a paper in which he claimed that the only way to show that a set of axioms for the natural numbers is consistent is to exhibit a set of objects that exist and satisfy the axioms. But what could they be other than the natural numbers? There is a patent risk of vicious circle, but to call for a proof of consistency without exhibiting objects struck Padoa as absurd. There is another way, but it took logicians many years to develop; Hilbert, it must be said, seems not to have engaged with Padoa‘s criticism at all. (J. Gray, cit. pp. 171-172.)

$$$$ A part of Padoa’s Essai d’une théorie algébrique des nombres entieres, précedé d’une introduction logique à une théorie déductive quelconque (1900) has been translated in English, with an introduction of the editor to Padoa’s work, in Van Hejernoort, Jean, (ed.), From Frege to Gödel. A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, 1967.

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