# CSHPM/SCHPM Online Colloquia

## CSHPM/SCHPM Online Colloquia

The Canadian Society for the History and Philosophy of Science and Technology is hosting occasional online colloquia. For regular updates on the programme, and joining instructions, see the News section of their website http://www.cshpm.org/ or follow them on Facebook.

Talks typically take place on Fridays, at times to be confirmed, and last 30 mins followed by Q&A.

Past talks include:

November 27th

SILVIA DE TOFFOLI, Postdoc Philosophy at Princeton University

TITLE: A Fallibilist Account of Mathematical Justification

ABSTRACT: In this talk, I will put forward an account of mathematical justification that is faithful to actual mathematical practice. I will focus on mathematical doxastic justification, that is, justification for an agent's belief in a mathematical claim for mathematical reasons. In contrast to traditional views, I will argue that even in the case of mathematics justification and knowledge can come apart. I will argue that the norms for doxastic justiﬁcation at play in actual mathematical practice apply to individual agents but present an important social component. Moreover, in my view the bar on justification changes according to the social role the agent is playing. Whereas for the laywoman pure testimony is enough and for the clairvoyant the reliability of her super-power would suﬃces, for the expert mathematician a mathematical argument is needed. Such argument is what I label a simil-proof (SP), that is, an argument that looks like a proof to the relevant agents. I will characterize SPs as sharable: having a SP implies grasping how it supports its conclusion and also being able to share it in the appropriate context. This implies that being justiﬁed is connected to the ability not only of responding to criticism adequately, but also of justifying. One striking respect in which my account of mathematical justification differs from more traditional ones is that it has a fallibilist flavor: justification comes apart from truth since an agent may be justiﬁed in believing a false proposition or in believing a true proposition by improperly grasping a fallacious argument.

October 23rd

DAVID ORENSTEIN, retired from the Toronto District School Board, will debate MICHAEL BARANY, Lecturer in history of science at the University of Edinburgh.

ABSTRACT: Following a controversy-laden 1920 revival of the International Mathematical Congresses in Strasbourg in the wake of the Great War and an abortive American bid to host the subsequent Congress in the United States, University of Toronto mathematics professor John Charles Fields proposed to hold the 1924 International Mathematical Congress in conjunction with the forthcoming Toronto meeting of the British Association for the Advancement of Science. The Toronto IMC, held in August 1924, was a milestone for international mathematics, but what kind of milestone? The speakers will examine the background, events, and legacies of the 1924 Toronto meeting with the hope of illuminating the question of whether it should be considered a success, and if so in what senses. (The 1924 Proceedings are available at https://www.mathunion.org/icm/proceedings/1924)

September 25th

VALÉRIE LYNN THERRIEN, PhD Candidate in Philosophy at McGill University

TITLE: On Counting as Mathematical Progress: Kuratowski-Zorn's Lemma and the Path Not Taken

ABSTRACT: In her "Naturalism in Mathematics," Maddy claims that historical case studies give us sufficient reason to exclude extra-mathematical considerations from our account of mathematical progress. Indeed, she vouches that historical case studies can be tested against the predictions of a reconstructed means-end analysis. In this paper, we will take up this formidable challenge. We aim to do so via a carefully chosen case study designed to test the limits of a rational reconstruction's ability to predict not only the path taken by mathematics, but also the path not taken by mathematics: the case of the Kuratowski-Zorn Lemma. Can Maddy's framework account for why Zorn’s Lemma counts as mathematical progress, but Kuratowski's prior equivalent maximal and minimal principle does not? While Maddy has done ground-breaking work in rationally reconstructing the path taken by set theory, it is not clear that her account can provide a convincing rationale for the path not taken. Our conclusion is that, while Maddy's account provides a razor-thin margin of success, it also does not take into account salient extra-mathematical considerations. Ultimately, it is unlikely to be convincing to anyone not epistemologically committed to mathematical naturalism.

September 4th

JEAN-PIERRE MARQUIS, Professor of Philosophy at the Université de Montréal

TITLE: On Mathematical Style

ABSTRACT: In this short talk, I will propose a notion of mathematical style, based on a specific case, namely Bourbaki's mathematics. The main goal is to capture the "structuralist style" of mathematics, but also to provide a general and supple framework to capture other types of style. In the spirit of Paolo Mancosu's challenge presented in his article in the Stanford Encyclopedia of Philosophy on the same topic, I want to show that a style in the sense that I propose has an inherent epistemic component and is not merely an aesthetic addition to a discourse.

21 August

JAMIE TAPPENDEN, Professor of Philosophy at the University of Michigan, Ann Arbour, will speak on:

TITLE: Frege on Computation and Deduction: Herbart, Fischer and

"Aggregative, Mechanical Thinking"

ABSTRACT: This paper reconstructs some details of Frege's early

intellectual environment and reads "Grundlagen" in light of them. The contextual information is of considerable interest in its own right, but here I'll concentrate on using the information to interpret some passages and features of "Grundlagen". The reading identifies unnoticed dialectical structure and thematic cohesion linking the introduction and conclusion of "Grundlagen" pertaining to, among other things, the deductive character of mathematics versus the "aggregative mechanical thinking" proposed by Kuno Fischer. Specific points include:

a) The opening pages of "Grundlagen" present interrelated goals in a way that has not so far been noticed; The successful achievement of these goals is implicitly announced in sections 87-8, the beginning of "Grundlagen"'s conclusion.

b) The goals include establishing the value of mathematical reasoning and the "fine" structure of mathematical concepts as well as establishing the nature of arithmetic (and mathematics more generally) as deductive rather than computational. The solution (among other things) binds together the deductiveness of mathematical reasoning, the fine structure of mathematical concepts, explanation and the possibility (due to the

fruitfulness of mathematical concepts) of extending knowledge via deduction alone.

c) The goals are framed by a contrast between Kuno Fischer and Johann

Herbart on the nature and value of arithmetic, a contrast whose significance and ramifications would have been obvious to those in Frege's environment but which slips past us today.

d) Frege's rejection of Fischer's picture --- on the surface just a rejection of the phrase "aggregative mechanical thinking" in connection with arithmetic --- is motivated by a broader opposition to Fischer's dismissive stance on the value of thinking in arithmetic. Fischer's evaluation had significant consequences for education and academic

politics as well as philosophy, points to which Frege clearly alludes.

e) Further complexity that would have been clear to Frege's intended

readers is implicit in the reference to Herbart. This is true in particular of Frege's use of a Herbartian technical expression "working out" (Bearbeitung). Frege's effort to define number in "Grundlagen" would have been recognized by his readers as a clear example of "working

out" in Herbart's sense.

7 August

BRENDA DAVISON, Senior Lecturer in Mathematics at Simon Fraser University

TITLE: Divergent series and Numeric Computation

ABSTRACT: In a paper published in 1856, G.G. Stokes (1819-1903) used a divergent series to compute many values of the Airy integral. Some of these values had been previously computed via a convergent series but this method was too laborious to make all of the desired calculations. This talk will examine how Stokes numerically computed a

class of definite integrals, including the Airy integral, using divergent infinite series. Emphasis will be placed on what lead Stokes to use this method, what types of physical problems required these solutions, how Stokes justified using his method, and how the results obtained were verified. How, when and for what purpose did other mathematicians and physicists use this method during the mid-19th century, before divergent series were given a rigorous treatment, will also be discussed.