Objectivity and Rigor in Classical Italian Algebraic Geometry

TL;DR

This paper examines the historical development of rigor and objectivity in Italian algebraic geometry, highlighting the diverse conceptions of rigor among key mathematicians and their implications for mathematical objectivity.

Contribution

It provides a nuanced analysis of how Italian algebraic geometers' notions of rigor and objectivity differ from modern standards, emphasizing their contextual and multi-dimensional nature.

Findings

Italian geometers distinguished between small-scale and large-scale rigor.

Severi differentiated between formal and substantial rigor.

Multiple conceptions of objectivity are linked to different types of rigor.

Abstract

The classification of algebraic surfaces by the Italian School of algebraic geometry is universally recognized as a breakthrough in 20th-century mathematics. The methods by which it was achieved do not, however, meet the modern standard of rigor and therefore appear dubious from a contemporary viewpoint. In this article, we offer a glimpse into the mathematical practice of the three leading exponents of the Italian School of algebraic geometry: Castelnuovo, Enriques, and Severi. We then bring into focus their distinctive conception of rigor and intuition. Unlike what is often assumed today, from their perspective, rigor is neither opposed to intuition nor understood as a unitary phenomenon - Enriques distinguishes between small-scale rigor and large-scale rigor and Severi between formal rigor and substantial rigor. Finally, we turn to the notion of mathematical objectivity. We draw from our case study in order to advance a multi-dimensional analysis of objectivity. Specifically, we suggest that various types of rigor may be associated with different conceptions of objectivity: namely objectivity as faithfulness to facts and objectivity as intersubjectivity.