Formalize!(?) - 4 

Schedule

15:15 - 15:30 Introduction

- perspectives from the (history of) science
15:30 - 16:00 Talk 1: Sikunder Ali: How to bring social critique to mathematical structures as AI?
16:10 - 16:40 Talk 2: Sepehr Ehsani: Formalisms in scientific laws and principles: some potential implications
16:50 - 17:20 Talk 3: Michele Luchetti: Epistemic circularity and the origins of quantitative psychology

17:20 - 17:50 Break and zoom photo

perspective from the (history of) mathematics
17:50 - 18:20 Talk 4: Arilès Remaki: The Leibnizian notation laboratory: the case of exponents
18:30 - 19:00 Talk 5: Nicolas Michel: Creativity under constraint: On E. Study's line geometry
19:10 - 19:40 Talk 6: Jemma Lorenat: Seeking and hiding knots at the turn of the twentieth-century
19:40 - 19:45 closing

19:45 -   The room will stay open for unofficial discussion

Abstracts and title (in order of the talks): 

How to bring social critique to mathematical structures as AI?
Sikunder Ali,
Generative AI (as mathematical algorithms) as ChatGPT are foregrounding speed and efficiency as part of technological solutionism to various social and educational problems. These technical artifacts are bringing pressing challenges to the work of equity, diversity, and inclusion. Among several challenges, the pressing challenge is manipulation of data (according to predefined criteria) in terms of spread of misinformation and disinformation (through internet). This is bringing variety of ethical issues that actions of learning and teaching specially in mathematics education must consider and tackle. Through my contribution, I will articulate some specific challenges that these mathematical structures as AI are bringing for us as educators and researcher in mathematics education. I will then articulate some ways to bring social critique to these mathematical structures. I will use lens of critical mathematics education and science and technology studies to develop this social critique.

Formalisms in scientific laws and principles: some potential implications

Sepehr Ehsani
Lawlike generalizations, such as the laws of thermodynamics or principles of parsimony, figure in many scientific explanations. Even in fields such as cell biology where explanations are mostly mechanistic, lawlike generalizations can play an important role; namely, they could help answer contrastive questions such as why a certain protein, say, interacts the way it does and not some other way with another protein. This talk is divided into two parts. First, I will describe a framework that can distinguish scientifically-relevant 'laws' from 'principles': even though these two terms have been used loosely and without clear definitions in the historical development of scientific generalizations, the simple framework I propose can help to differentiate them using the criteria of (i) scope and (ii) expectability of predictions. Moreover, the framework helps to bring into dialogue different laws and principles that can explain some aspect(s) of the same target phenomenon. In the second part, using some example cases of generalizations that have stood the test of time, I discuss how expressing laws and principles in formal vs. natural language terms can impact our understanding and application of those generalizations.

Epistemic circularity and the origins of quantitative psychology
Michele Luchetti

Quantitative measurement is generally characterized by resorting to formal models of the measurement process. These characterizations usually cannot address the historical process that made the formal representation of quantitative measurement possible in the first place. In particular, they rarely account for how a certain research program has overcome, if at all, the threat of epistemic circularity in quantitative measurement. This threat is due to the fact that answers to the questions “What counts as a measurement of the quantity X?” and “What is the quantity X?” often seem to presuppose one another when a theoretical understanding of the quantity of interest is weak.

In this paper, I will address how the problem of epistemic circularity in measurement was tackled by Gustav Theodor Fechner’s in his psychophysical research, one of the of pioneering projects of quantitative psychology in the second half of the nineteenth century. To establish psychophysics as a fully quantitative discipline, Fechner set out to construct a mapping between the intensity of sensory stimuli and sensory experience, which he operationalized as intensity of sensation. In Fechner’s view, this mapping was necessary to ground the possibility of quantifying the mental by measuring the physical. However, a circularity issue was threatening the success of this mapping: How, in fact, could Fechner identify the functional relationship between intensity of stimulus and intensity of sensation without already presupposing some quantitative understanding of intensity of sensation? My analysis will situate Fechner’s approach to the problem of epistemic circularity in measurement within the context of his overarching goal of providing a quantification of experience, of his philosophical and methodological commitments, and of the understanding of measurement and quantification dominant in the context of mid-nineteenth century German science.

The Leibnizian notation laboratory: the case of exponents
Arilès Remaki
For certain mathematical notions, we find within Leibniz's work a great diversity of notations and symbols used, which testifies to a research activity focused on the elaboration of the writing of the concepts themselves. This scriptural laboratory has already been well documented in the case of ambiguous signs, for which Leibniz undertook very rich work, over a very short period, during his youth. The case of exponents was treated mainly in the context of the development of the theory of determinants. Concerning the exponential calculus itself, the work on notations remains little known and involves a large number of unpublished documents. The period during which this work took place is much broader, i.e. about ten years, and therefore implies an additional difficulty of an epistemological nature: are the conceptual frameworks described by these notations commensurable, in Kuhn's sense? 

The presentation will focus on Leibniz's attempt to develop a functional notation of exponents, capable of expressing all possible relations. This endeavour highlights the complex relationship between notation and concept in the process of generalisation.

Creativity under constraint: On E. Study's line geometry

Nicolas Michel

The German geometer Eduard Study (1862-1930) constructed a view of mathematical objects and knowledge that sits uncomfortably with usual pictures of early 20th century philosophy of mathematics. Whilst maintaining that mathematical concepts are free creations of the human mind, he rejected axiomatics as a viable form of mathematical practice and espoused a strict form of ontological realism. Whilst combatting the intrusion of geometrical intuition in proofs and reasonings, he nonetheless demanded that algebraic constructions be motivated by critical engagement with spatial forms. The thrust of his arguments is directed against formalism and intuitionism alike, all the while borrowing little from contemporary developments in logic.

Focusing on his Klein-inspired theory of line complexes, as well as some unpublished typescripts, we shall connect Study's philosophy and practice of mathematics. In so doing, we shall discuss his original conception of the role of algebraic formalism in geometrical investigations.

Seeking and hiding knots at the turn of the twentieth-century
Jemma Lorenat

Mary Gertrude Haseman published her dissertation on knot theory in 1918. Almost half of her text is a meticulous reflection on the values of symbolic representation and the potential for multiple modalities to display complementary features and obscure irrelevant particularities. Certainly knot theory presented a novel site on which to explore a variety of ​"aids to the imagination.'' To visually convey relevant features of knots, Haseman employed drawn figures and three types of symbols: alphabetical, intrinsic, and compartment. This talk surveys these complementary ways of seeing (or obscuring) knot features for purposes of representation, calculation, and identification.