Publications

Refereed Journal Articles

(Forth.) Longa, G., “The occurrences of analysis in Greek Geometry”.

Abstract: In this paper, I will address a seemingly simple question: How often is the method of analysis and synthesis applied in the ancient Greek mathematical corpus? Despite its central role in understanding Greek geometric analysis, this question has not been addressed extensively in the existing literature. This paper aims to fill this gap by introducing a linguistic criterion for determining whether a given proposition (be it a theorem or a problem) has been proved by the method of analysis and synthesis. The outcome will be somewhat unexpected. Indeed, the method of analysis and synthesis has a much wider application than is commonly acknowledged.

(Forth.) Longa, G., “Pappus between Rhetoric and Mathematics: A New Interpretation of the Preface to Book VII of the Collectio”.

Abstract: The description of the method of analysis and synthesis provided by Pappus in the introduction to Book VII of his Collectio has served as the starting point for nearly all modern interpretations of this method in ancient Greek geometry. We argue that this is a mistake. Indeed, we aim to show that Pappus’s account should not be regarded as “the most elaborate utterance on the subject” (Heath), but rather as a rhetorically and stylistically sophisticated attempt to assemble and interpret available texts with didactic and rhetorical aims that do not necessarily reflect the original intentions behind those texts. Consequently, this article contends that taking Pappus as the principal point of departure for interpreting ancient analysis ultimately leads to a partial—if not misleading—understanding of the practice of analysis and synthesis.

Panza, M., Longa, G., “Diagrams in Intra-Configurational Analysis“, Philosophia Scientiæ 25(3), 2021: 81–102.

Abstract: In this paper we would like to attempt to shed some light on the way in which diagrams enter into the practice of ancient Greek geometrical analysis. To this end, we will first distinguish two main forms of this practice, i.e., trans-configurational and intra-configurational. We will then argue that, while in the former diagrams enter in the proof essentially in the same way (mutatis mutandis) they enter in canonical synthetic demonstrations, in the latter, they take part in the analytic argument in a specific way, which has no correlation in other aspects of classical geometry. In intra-configurational analysis, diagrams represent in fact the result of a purely material gesture, which is not codified by any construction canon, but permitted only by the (theoretical) practice of the method of analysis and synthesis.

Gandon, S., Longa, G., “Identité des mots, identité des diagrammes: une approche kaplanienne“, Cahiers Philosophiques 62(3), 2021: 132–149.

Abstract: Identity conditions play a fundamental role in the debate over the epistemological status of diagrams. In this respect, both critics and proponents of their use as means of proof share a common assumption: the extension of the type/token distinction—originally developed for words—to diagrams. In this article, we aim to challenge that assumption. In the first part, we will show that the type/token distinction, as a criterion for word identity, is far from uncontroversial in the philosophy of language. We will examine, in particular, how David Kaplan rejected this criterion and proposed an alternative approach, inspired by Saul Kripke’s notion of common currency. In the second part, we will explore the possibility of extending Kaplan’s approach to diagrams, drawing on historical considerations concerning the transmission of Greek mathematical texts.

Edited Volumes and Journal Special Issues

Longa, G., Gandon, S., Geometrical Analysis in Ancient Greek Mathematics, Special Issue of Philosophia Scientiæ, 25/3, 2021.

Abstract: This special issue aims to reassess the meaning and significance of the analytical method in ancient Greek geometry. While the importance of this mode of reasoning in premodern mathematics has been acknowledged since the earliest historical inquiries (e.g., Montucla 1758), its precise nature remains a matter of debate. From early modern interpretations by Descartes, Newton, and Leibniz to more recent epistemological and logical analyses (Polya 1945; Hintikka & Remes 1974; Lakatos 1978), the method has attracted sustained interest. Yet no consensus has emerged regarding its definition or scope. Bringing together recent scholarship (Netz 2000; Menn 2002; Fournarakis & Christianidis 2006; Acerbi 2007, 2011; Sidoli 2018), this volume offers a state-of-the-art assessment of the topic and highlights the need for a genuinely interdisciplinary approach to the historical and philosophical dimensions of ancient analysis.

Book Reviews

Review of J. Feke, Ptolemy’s Philosophy: Mathematics as a Way of Life, Princeton University Press, 2018, Historia Mathematica, 54(1), 2021: 117–122.

Review of G. Lolli, Numeri. La creazione continua della matematica, Turin, Bollati Boringhieri, 2015, Lo Sguardo. Rivista di Filosofia, 21, 2018: 377–380.

Review of D. Molinini, Che cos’è una spiegazione matematica ?, Milan, Carocci, 2014, Lo Sguardo. Rivista di Filosofia, 20, 2018: 325–327.

Review of E. Cinnella, L’altro Marx, Pisa-Cagliari, Della Porta Editori, 2014, Quaderni Materialisti, 13, 2017: 230–236.

Review of E. Cinnella, Review of A. Guidi, Un segretario militante. Politica, diplomazia e armi nel cancelliere Machiavelli, Bologna, Il Mulino, 2010, Quaderni Materialisti, 13, 2017: 245–251.

Translations

(Forth.) Longa, G., “Ciò che i numeri non dovrebbero essere“, Italian translation of “Benacerraf, P., ‘What Numbers Could not Be’, The Philosophical Review, 74, 1965: 47–73″.

Longa, G., “Aristotle and Greek Geometrical Analysis“, English translation of “Berti, E., Aristotele e l’analisi matematica greca”, Philosophia Scientiæ 25(3), 2021: 9–21.

Longa, G., Storni, M., “Questo è Marxismo ortodosso : la Weltanschauung materialista condivisa di Marx ed Engels“, Italian translation of “Johnston, A., This is Orthodox Marxism […]”, Quaderni Materialisti 11, 2016: 53–69.

Short Notices and Research Reports

Mathematical Explanation (in ‘Varieties of Explanation’), Philpapers: Online Research in Philosophy.

Ancient Greek and Roman Philosophy of Mathematics, Philpapers: Online Research in Philosophy.