Abstract
Mathematicians frequently attend their peers’ lectures to learn new mathematical content. The goal of this paper is to investigate what mathematicians learned from the lectures. Our research took place at a 2-week workshop on inner model theory, a topic of set theory, which was largely comprised of a series of lectures. We asked the six workshop organizers and seven conference attendees what could be learned from the lectures in the workshop, and from mathematics lectures in general. A key finding was that participants felt the motivation and road maps that were provided by the lecturers could facilitate the attendees’ future individual studying of the material. We conclude by discussing how our findings inform the development of theory on how individuals can learn from lectures and suggest interesting directions for future research.
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The data set generated for and analyzed in this paper is available from the corresponding author upon reasonable request.
Notes
There are two exceptions. Rood (2003) presented a qualitative analysis of student comments saying that lectures could provoke awe and wonder, and can provide students with a sense of community. Bergsten (2007) analyzed one professor’s teaching of a proof-based calculus course. Although Bergsten thought inspiration should be a goal of lecturing, he found little activity in the professor’s teaching that could elicit students’ inspiration.
We are eliding some subtle metamathematical issues here. By Gödel’s second incompleteness theorem, ZFC cannot prove its own consistency and hence we cannot prove that any model of ZFC exists. It is widely accepted that ZFC is consistent, and it can be shown that if ZFC has one model, then it has infinitely many. Kunen (1980) is a standard text for showing independence results. The status of different ZFC models is discussed in Hamkins (2012).
However, this viewpoint is not universal. Hamkins (2012), for instance, celebrates the plethora of ZFC models.
Social goals that the participants stated included finding potential collaborators and talking to organizers who could subsequently provide guidance and support.
Quotes in this paper were lightly edited to increase their readability. In particular, stutters, repeated words, and words and phrases that did not appear to carry meaning were removed from the transcript we reported. We do not believe that we have altered the intended meaning that the interviewed participant intended to convey.
Technical details: Prikry forcing is a well-known forcing technique that starts with a set theory universe with a measurable cardinal \(\mathrm\kappa\) and expands the universe so that \(\mathrm\kappa\) is still a cardinal but is no longer regular (and hence not measurable). The standard treatment uses a normal measure of \(\mathrm\kappa\) to construct the forcing partially ordered set. The variant presented by O4 was more flexible and could use any measure of \(\mathrm\kappa\), even one that was not a normal measure.
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Weber, K., Fukawa-Connelly, T. What mathematicians learn from attending other mathematicians’ lectures. Educ Stud Math 112, 123–139 (2023). https://doi.org/10.1007/s10649-022-10177-x
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DOI: https://doi.org/10.1007/s10649-022-10177-x