Ricerc@Sapienza

I investigate the construction of the mathematical concept of quaternion from a methodological and heuristic viewpoint to examine what we can learn from it for the study of the advancement of mathematical knowledge. I will look, in particular, at the inferential microstructures that shape this construction, that is, the study of both the very first, ampliative inferential steps, and their tentative outcomes—i.e. small ‘structures’ such as provisional entities and relations.

I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail, I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into it that it did not contain at the beginning of the process.

In this paper I examine the heuristic view of mathematics focussing on an updated version of it. I discuss how the seminal work of Lakatos has been improved under several respects and I point at four issues that are particularly relevant for the philosophy of mathematics and mathematical practice from a heuristic point of view: 1. the quest for a method of discovery (§ 2); 2. the construction of heuristic procedure, that is rational and inferential ways of producing a hypothesis to solve a problem (§ 3); 3. the nature of mathematical objects (§ 4); 4.

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.

In this paper I lay out a non-formal kernel for a heuristic logic—a set of rational procedures for scientific discovery and ampliative reasoning—specifically, the rules that govern how we generate hypotheses to solve problems. To this end, first I outline the reasons for a heuristic logic (Sect. 1) and then I discuss the theoretical framework needed to back it (Sect. 2). I examine the methodological machinery of a heuristic logic (Sect. 3), and the meaning of notions like ‘logic’, ‘rule’, and ‘method’. Then I offer a characterization of a heuristic logic (Sect.

Theory-building is the engine of the scientific enterprise and it entails (1) the generation of new hypotheses, (2) their justification, and (3) their selection, as well as collecting data. The orthodox views maintain that there is a clear logical and temporal order, and distinction, between these three stages. As a matter of fact, not only is this tenet defective, but also there is no way to solve these three issues in the way advocated by traditional philosophy of science.

The way scientific discovery has been conceptualized has changed drastically in the last few decades: its relation to logic, inference, methods, and evolution has been deeply reloaded. The ‘philosophical matrix’ moulded by logical empiricism and analytical tradition has been challenged by the ‘friends of discovery’, who opened up the way to a rational investigation of discovery. This has produced not only new theories of discovery (like the deductive, cognitive, and evolutionary), but also new ways of practicing it in a rational and more systematic way.

This book is a basic English grammar, based on decades of teaching. It is organized around two core points: 1) the discovery of grammatical regularities, and 2) language variation. The first core point uses both a top-down approach, using a set of simple discovery procedures (heuristics), and a bottom-up approach, in which the learner is invited to notice similarities and differences and to build their own rules.