Epistemology
epistemology
Analogical inference in mathematics: from epistemology to the classroom (and back)
In this presentation, we will discuss adaptations of historical examples of mathematical research to bring out some of the intuitive judgments that accompany the working practice of mathematicians when reasoning by analogy. The main epistemological claim that we will aim to illustrate is that a central part of mathematical training consists in developing a quasi-perceptual capacity to distinguish superficial from deep analogies. We think of this capacity as an instance of Hadamard’s (1954) discriminating faculty of the mathematical mind, whereby one is led to distinguish between mere “hookings” (77) and “relay-results” (80): on the one hand, suggestions or ‘hints’, useful to raise questions but not to back up conjectures; on the other, more significant discoveries, which can be used as an evidentiary source in further mathematical inquiry. In the second part of the presentation, we will present some recent applications of this epistemological framework to mathematics education projects for middle and high schools in Italy.
A Functional Approach to Analogical Reasoning in Scientific Practice
The talk argues for a new approach to analysing analogical reasoning in scientific practice. Traditionally, philosophers of science tend to analyse analogical reasoning in either a top-down way or a bottom-up way. Examples of top-down approaches include Mary Hesse’s seminal work (1963) and Paul Bartha’s articulation model (2010), while most popular bottom-up approach is John Norton’s material approach (2018). I will address the problems of these traditional approaches and introduce an alternative approach, which is motivated by my exemplar-based approach to the history of science, defended in my recent book (2020).