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 discuss how this paradigmatic case study supports the recent approaches (...) to problem-solving and philosophy of mathematics, and how it suggests refinements of them. In more detail, I argue that the inferential micro-structures enable us to shed more light on the informal, heuristic side of mathematical practice, and its inferential and rational procedures. I show how they enable the generation of a problem, the construction of its conditions of solvability, the search for a hypothesis to solve it, and how these processes are representation-sensitive. On this base, I argue that: 1.the recent development of the philosophy of mathematics was right in moving from Lakatos’ initial investigation of the formal side of a mathematical proof to the investigation of the semi-formal, heuristic side of the mathematical practice as a way of understanding mathematical knowledge and its advancement. 2.The investigation of mathematical practice and discovery can be improved by a finer-grained study of the inferential micro-structures that are built during mathematical problem-solving. (shrink)

Thought experiments (TEs) are important tools in science, used to both undermine and support theories, and communicate and explain complex phenomena. Their interest within philosophy of science has been dominated by a narrow question: How do TEs increase knowledge? My aim is to push beyond this to consider their broader value in scientific practice. I do this through an investigation into the scientific imagination. Part one explores questions regarding TEs as “experiments in the imagination” via a debate concerning the epistemic (...) status of computer simulations in science. I outline how, against Hacking, TEs also have “a life of their own” and I argue against accounts that privilege experiments over simulations (and by extension TEs) in light of their capacity to surprise in a productive way. Part two develops a pluralist account of the nature of the imagination in science. At its core, my view is that when we attend to a number of examples of TEs and consider the context in which they are used, we see that TEs engage a variety of our imaginative capacities. Existing monistic views fail to recognise the richness of the imagination and its potential in science. Part three looks to the “beauty” of TEs which is currently overlooked in the aesthetics of science literature. I put forward a new account that demonstrates the epistemic value of aesthetic features in science by showing how an appropriate fit between form and content enhances the usability of a TE, and its effectiveness as a prompt for our imagination. This also enables a more nuanced take on the proposed similarities between TEs and literary fictions. In the concluding chapter, I outline ways in which the core features of my account can be extended beyond TEs to illuminate the significance of the imagination and aesthetic values in other areas of science. (shrink)