Manufacturing a Mathematical Group: A Study in Heuristics
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Manufacturing a Mathematical Group: A Study in Heuristics Emiliano Ippoliti1
© Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forward. Keywords Heuristics · Hypotheses · Inference · Inferential micro-structures · Logic · Discovery
1 Introduction In this paper I examine the construction of the concept of group in mathematics in order to discuss fundamental heuristic procedures. This concept and its history have been studied extensively (see e.g. Barnett 2010, 2017; Birkhoff 1937; Chakraborty and Chowdhury 2005; Chowdhury 1995; Kleiner 1986, 2007; Ronan 2006; Wussing 1984), so I will approach it from a heuristic viewpoint, that is, by focussing on the heuristics that gradually have led generations of mathematicians to its formation and refinement, and I will consider what we can learn from it. To this end, I will examine specific aspects of the works of Lagrange, and then I will look at how these seminal works have been developed by Cauchy, Galois and Cayley in order to produce a mature group theory. In more detail I will examine the heuristic reasoning of Lagrange’s work (Sect. 2.1), its development in the works on permutations by Cauchy, the introduction of the notion, and term, of groups provided by Galois, and then the first abstract treatment of group produced by Cayley (Sect. 2.2). This analysis enables us to shed light on mathematical practice and to show how the manufacturing of the group concept displays paradigmatic features of the core of * Emiliano Ippoliti [email protected] 1
Sapienza University of Rome, Rome, Italy
problem-solving, that is, the generation of a new hypothesis. To provide a bit more detail, I will examine how a new mathematical term, and concept, are introduced (i.e. the term group, groupe in French) and I will discuss if this case study agrees with major accounts of this issue, in particular I will look at Lakatos (1976); Grosholz (2007); Grosholz and Breger (2000); Cellucci (2013, 2017). I will also consider the notion of solution to a problem, or better what counts as a solution, and the way it changes. Tellingly, this examination enables us also to shed light on the very first steps that make possible the formation of a h
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