# Talk (Work in Progress): Matteo De Benedetto (MCMP/LMU)

Location: Ludwigstr. 31, ground floor, Room 021.

16.01.2020 12:00 – 14:00

## Title:

Lakatosian Populations and Mathematical Selection

## Abstract:

According to Lakatos, conceptual change is one of the main engines of mathematical progress. Specifically, Lakatos' heuristics of proofs and refutations (Lakatos 1976) provided the first explicit theory of conceptual change in mathematics, centered around the notion of 'concept-stretching'. Despite its historical significance, Lakatos' proposal is nowadays considered too narrow as a general model of conceptual change in mathematics.

In this talk I will propose a way of generalizing and improving Lakatos' model of conceptual change, accounting for the variety of conceptual dynamics exhibited by mathematical practices. Building upon Mormann's evolutionary reading of Lakatos (Mormann 2002) and Godfrey-Smith's population-based Darwinism (Godfrey-Smith 2009), I will present a conceptual framework centered around the family of notions of a Lakatosian population. I will augment my framework with four parameters: conceptual variation, reproductive competition, environmental stability, and continuity. These parameters track how much the evolution of a given mathematical population exhibits Lakatosian features. These four parameters form what I call Lakatosian space. Based on these parameters, conceptual populations in mathematics can be more or less paradigmatically Lakatosian, occupying different regions of the Lakatosian space.

I will show how different examples from the history of mathematics exhibit different tunings of the four parameters of a Lakatosian space and thus different evolutionary dynamics. As case studies, I will use Lakatos’ own example of Euler’s conjecture and the polyhedron concept (Lakatos, 1976), Hamilton’s discovery of the quaternions (Hamilton, 1843a,b, 1853), and the pre-abstract concepts of group (Wussing, 1984). I will also show how my framework, augmented with a time-dimension, is able to model the evolution of a given mathematical field from one Lakatosian population to another one. Different changes in the parameters of the augmented Lakatosian space correspond to different rational intra-practice and inter-practice transitions between mathematical practices (Kitcher, 1983). Thanks to these applications, I will argue that my framework is able to defend a Lakatosian view on conceptual evolution in mathematics against many of the well-known critiques of Lakatos’ philosophy (Werndl, 2009; Feferman, 1978; Fine, 1978; Corfield, 2003).