# Giuseppe Peano and his School: logic, epistemology and didactics

# Thematic issue of Philosophia Scientiæ

## Guest editors: Paola Cantù and Erika Luciano

### Submission Deadline: 1 Mars 2020

### Acceptance Notification: 1 July 2020

### Final version due: 1 September 2020

### Please send submissions to: paola.cantu@univ-amu.fr, erika.luciano@unito.it

Peano’s axioms for arithmetic, published in 1889, are ubiquitously cited in the writings on modern axiomatics. And his *Formulary* is often quoted as the precursor of Russell’s *Principia Mathematica*, but a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics is still to be achieved.

Even if the contributions by Padoa, Peano, Burali-Forti, Pieri, and Vailati impressed Russell as well as most participants in the 1900 Paris Conferences in Mathematics, Philosophy and Psychology, several reasons explain the loss of philosophical interest for the member of the school: the non-academic nature of the group, the multiform topics of interest (going from mathematical analysis to geometry, from linguistics to universal languages, from philosophical pragmatism to logicism), the scarce attention given to the transformation of mathematics and to the development of set theory after 1910, and the non monolithic philosophical perspective developed in the school.

Yet, notwithstanding a general lack of interest in the school (among the few notable exceptions I will cite here Kennedy, Grattan-Guinness, Ferreiros, and several Italian scholars, including Ugo Cassina, Paolo Mancosu, Gabriele Lolli, Clara Silvia Roero, Livia Giacardi, Erika Luciano, Enrico Pasini and Paola Cantù), the views held by Peano and other members of the school not only had a strong impact on the writings by Frege, Russell, Carnap and Gödel, but can also be fruitfully explored in order to understand the development of certain philosophical isms, such as logicism and structuralism.

The thematic issue will publish two kinds of contributions: historical analyses of the logical, mathematical, foundational and didactical writings by Peano and the members of the school; philosophical investigations on the relation between Peano’s axiomatics and the approaches by Dedekind, Frege, Hilbert, Russell, Carnap, and Gödel.

One could deal with the following issues:

- What are the relations between Peano’s interests for mathematical analysis, linguistics, universal language, foundations of arithmetic and logic?
- Is there any relation between the interdisciplinary and collaborative approach of the group and its mathematical, logical and foundational results?
- What was the impact of the school on contemporaries? Was Peano’s figure dominant or were the figures of Pieri, Padoa, Burali-Forti and Vailati also influential?
- Was there any significant difference between Peano’s and Dedekind’s approach? If the latter is considered as a form of structuralism, should the former also be considered as such?
- What is the relation between Frege’s logicism and Peano’s approach?
- Was there any specific influence of the Peano School on the Vienna Circle?
- What are the main differences between the project of the Principia Mathematica and the project of the Formulary?
- Was there any relevant influence of the Peano School on Hilbert’s geometry, or vice-versa?
- What was the impact of Peano’s approach to didactics on the school’s epistemology?

Manuscripts should be submitted in French, English, or German, and prepared for anonymous peer review.

Abstracts in French and English of 200-300 words in length should be included.

Articles should not exceed 50 000 characters (spaces, list of references and footnotes included).

Please send submissions to: paola.cantu@univ-amu.fr, erika.luciano@unito.it

Guidelines for authors are to be found on the journal’s website: http://philosophiascientiae.revues.org/633.

## Bibliography

Cantù, P. (2014). The right order of concepts: Grassmann, Peano, Gödel and the inheritance of Leibniz’s universal characteristic. *Philosophia Scientiae – Studies in History and Philosophy of Science*, 18(1):157–182.

Cantù, P. (2016). Peano and Gödel. In Crocco, G. and Engelen, E.-M., editors, *Kurt Gödel: Philosopher-Scientist*, pages 107–126. Presses Universitaires de Provence.

Cantù, P. (2016). Louis Rougier’s reception of the Peano School. In Fréderic Brechenmacher, Guillaume Jouve, L. M. and Tazzioli, R., editors, *Images of Italian Mathematics in France. The Latin Sisters, from Risorgimento to Fascism*, Series New trends in history of science, pages 213–254. Springer.

Ferreirós, J. (2005). Peano and Dedekind. In Grattan-Guinness, I., editor, *Landmark Writings in Western Mathematics 1640–1940*, pages 613–626. Elsevier, Amsterdam, San Diego, Oxford and London.

Grattan-Guinness, I. (2000). *The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor Through Russell to Gödel*. Princeton University Press, Princeton (NJ).

Lolli, G. (1985). Le forme della logica: G. Vailati. In Lolli, G., editor, *Le ragioni fisiche e le dimostrazioni matematiche*, pages 107–132. Il Mulino, Bologna.

Lolli, G. (2012). *Cesare Burali-Forti (1861-1931) e la logica matematica*. Edizioni della Normale, Pisa.

Luciano, E. (2012). Peano and His School Between Leibniz and Couturat: The Influence in Mathematics and in International Language. In Krömer, R. and Chin-Drian, Y., editors, *New Essays on Leibniz Reception*, Publications des Archives Henri Poincaré, pages 41–64. Springer Basel.

Mancosu, P., Zach, R., and Badesa, C. (2009). The development of mathematical logic from Russell to Tarski, 1900–1935. In L. Haaparanta (ed.), *The development of modern logic*, Oxford University Press, pages 318–470.

Mancosu, P. (2018). Definitions by abstraction in the Peano school. In Giordani, A. and de Florio, C., editors, From Arithmetic to Metaphysics: A Path Through Philosophical Logic, pages 261– 288. De Gruyter.

Pasini, E. (2004). Peano e la filosofia della matematica. In Roero, S., Giacardi L., Luciano E., editor, *Conferenze E Seminari 2003-2004*, pages 203–220. Associazione Subalpina Mathesis.

Peano, G. (1958*). Opere scelte*, Cremonese, Roma, a cura di U. Cassina, voll. 1-3.

Roero, C. S. (2010), editors. Peano e la sua scuola fra matematica, logica e interlingua. Atti del Congresso Internazionale di Studi (Torino, 6-7 Ottobre 2008), pages xi–xviii, Torino. Università degli Studi di Torino.

Skof, F., editor (2011). *Giuseppe Peano between Mathematics and Logic. Proceedings of the International Conference in honour of Giuseppe Peano on the 150th anniversary of his birth and the centennial of the Formulario Mathematico* Torino (Italy) October 2-3, 2008. Springer-Verlag, Milan.