1st Axdef workshop "Implicit definitions"
1st AXDEF WORKSHOP "Implicit Definitions"
May 27-29, 2024
Centre Gilles Gaston Granger, Aix-Marseille Université, Salle de la Frumam, Campus Saint-Charles - 1, place Victor Hugo, 13001 Marseille
Criticism of the idea that theorems can be derived from definitions was advanced by several authors and came along with new proposals to distinguish between definitions, axioms, and postulates. Yet, a radical change occurred at the end of 19th century, when axioms were considered as implicitly defining the meaning of a term or group of terms. Hilbert’s Grundlagen (1899) are usually cited as a prime example, but the expression ‘implicit definition’ was never used for definitions by axioms neither by Hilbert nor by Frege, but first used by Gergonne and then by Pasch, Poincaré, the members of the Peano School, Enriques, Schlick and Carnap in quite different senses. The exact history of the emergence and development of implicit definitions is still to be written. The workshop aims at elucidating not only the well-known debate between Frege and Hilbert, but also the complex connections between the German, the French, the Italian, and the English tradition.
The workshop will investigate the debates around definition among 19th century French geometers (Gergonne, Chasles, Poncelet, and German mathematicians (Kummer, Grassmann, etc. who read the French closely). Gergonne would undoubtedly deserve special attention regarding the question of definition beyond what one usually reads in Enriques-style histories about the principle of duality and implicit definitions. Henri Poincaré's geometric conventionalism will also be investigated with special attention to the status of definitions, both in geometry and in arithmetic. Pasch, the Peano School and Weyl will also be investigated. In Pasch the expression “implicit definition” corresponds to a creative act determining the meaning of sentences containing a non-referential noun that is treated as if it were the proper name of a thing: a typical example is the extension of the number concept (Pasch 1921). Peano’s use of the term, as well as Schlick or Carnap’s uses (Wagner 2022), are definitions of a higher order concept, whereas Vailati and Burali-Forti’s uses correspond to definitions of the primitive terms. They differ both from Gentzen’s idea of implicitly defining the meaning of logical constants (Gentzen 1935), and from Boghossian’s use of implicit definitions to explain the concept of analyticity (see Giovannini and Schiemer 2019). Implicit definitions, and a typical example, definitions by abstraction, will be investigated not only in the writings by Grassmann, Frege, Russell, Peano and Weyl, or in the writings by the Vienna Circle members, but also in the phenomenological tradition.
Finally, a history of implicit definitions cannot be separated from an analysis of axiomatic programs, in order to better elucidate the role assigned to definitions in the hypothetico-deductive approach: the best known Hilbertian and Russellian tradition, the Italian tradition (the Peano School, Enriques), Frege but also Klein, Cassirer and Helmholtz, and the French tradition, from Poincaré and Couturat up to Vuillemin, will be compared, allowing not only a better understanding of ‘implicit’ definitions but also of the development of axiomatics.
Scientific Committee: Paola Cantù, Gabriella Crocco, Gerhard Heinzmann, Pierre Wagner
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Program
Monday 27 Mai 2024
14h30 Welcome
15h00-16h15 Laura Fontanella, On the computational content of axioms
16h15-17h45 Gabriella Crocco et Fabien Carbo-Gil, Les axiomes comme définitions implicites des concepts : le point de vue de Gödel
Tuesday 28 Mai 2024
9h00-10h15 Gerhard Heinzmann, Poincaré’s “disguised definitions”: are they implicit or explicit?
10h30-11h45 Pierre Wagner, How to define computability?
12h00 Pause
14h00-15h15 Paola Cantù, Axiomatic definitions in the Peano School
15h30-16h45 Wilfried Sieg (via Zoom), Dedekind & Hilbert's mathematical structuralism
16h45 Pause
17h15-18h45 Eduardo Giovannini (via Zoom), Implicit definitions and the development of modern axiomatics
Wednesday 29 Mai 2024
9h00-10h15 Jean-Baptiste Joinet Sur la forme et la conservativité des définitions explicites au delà du premier ordre
10h15-11h30 Pierre Joray, Lesniewski et Pasch: la définition dans les fondements des mathématiques
11h30 Pause
12h00-13h15 Dirk Schlimm Pasch on logic and definitions
The languages of the workshop will be French and English. The workshop will be broadcast via Zoom.
Write paola.cantu@univ-amu.fr for the link, mentioning why you are interested in the event.
Talks by Sieg and Giovannini will be exclusively via Zoom.