Colloque final Les mathématiques en action

Jeremy Avigad ((Carnegie Mellon), The Mechanization of Mathematics

The phrase "formal methods" is used to describe a body of methods in computer science for specifying, developing, and verifying complex hardware and software systems. Such methods hold great promise for mathematical discovery and verification of mathematics as well. In this talk, I will survey some applications, including verifying mathematical proofs, verifying the correctness of mathematical computation, searching for mathematical objects, and storing and communicating mathematical results.

Even though such methods have had little effect on mathematical activity to date, they are likely to have a profound impact on the subject in the years to come. I will discuss some of the philosophical questions they raise regarding the nature of mathematical knowledge and understanding, as well as interesting methodological questions as to the ways formal methods can be used most effectively.

 

Jessica Carter (University of Southern Denmark), The role of diagrams in contemporary mathematics: Tools for discovery?

I will present two examples where diagrams function as tools for discovery in contemporary analysis. One purpose of the talk is to analyse why these diagrams are fruitful and whether they are exclusively so. I will show that in both cases properties as well as relations between properties can be shown in the diagrams – in contrast to being described by symbolic expressions. Furthermore it is possible to experiment on, or manipulate, diagrams, revealing new insights. These features, that is, diagrams as representations, that can be manipulated, so that new properties can be seen, resemble C.S. Peirce’s characterisation of ‘diagrammatic reasoning’. With Peirce I will note that diagrams are not exclusively fruitful in this sense. (Here taking ‘diagram’ as a visual representation composed of points, lines, etc.) Having said this, however, I will argue that diagrams offer a cognitive advantage, or an advantage for human understanding.

 

Silvia De Toffoli (Stanford University), Mathematical Justification in Practice

In my talk, I will focus on the norms for doxastic justification at play in actual mathematical practice. Although such norms are about individual agents, they present an important social component. In fact, in my view the bar on justification changes according to the social role the subject 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 agent. I will characterize SPs as sharable. Having a SP implies grasping how it supports its thesis and also being able to share it in the appropriate context. That is, 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 (doxastic) justification differs from more traditional ones is that it has a fallibilist flavor: justification comes apart from truth since a subject may be justified in believing a false proposition or in believing a true proposition by improperly grasping a fallacious argument.

 

Yacin Hamami (Vrije Universiteit Brussel), Cognitive Bases of Euclidean Diagrammatic Reasoning: Some Empirical Results

In this talk, I will present a series of psychological experiments on the cognitive bases of Euclidean diagrammatic reasoning. These experiments are organized around two lines of research concerning (1) the cognitive processing of spatial relations in Euclidean diagrams (joint work with Ineke van der Ham (Leiden), Milan van der Kuil (Leiden), and John Mumma (San Bernardino)), and (2) the psychology of Euclidean diagrammatic reasoning per se (joint work with Marie Amalric (CMU), and John Mumma). In both cases, I will discuss the philosophical implications of the empirical results obtained for the epistemological analysis of Euclid's diagram-based geometric practice.

 

Matthew Inglis (Loughborough University), Understanding expert and novice mathematical reading

Reading printed mathematics is an important activity for mathematics students at all levels, but particularly undergraduates. Here I report a series of studies designed to understand how undergraduate students and expert mathematicians engage with written mathematical proofs. First, I describe an expert/novice eye-movement study designed to probe the differences in reading behaviour between undergraduates and research mathematicians. Second, I report a study which investigated whether reading behaviours vary depending on the goals of the readers. Finally, I report a series of experiments which investigated how a tailored program of self-explanation training can help novices read in a manner more consistent with the reading behaviours exhibited by experts.

 

Amirouche Moktefi (Tallin University of Technology), From logic diagrams to diagrammatic logics

The role of diagrams in mathematical proofs is disputed. Yet, diagrams have long been used in various mathematical disciplines. In logic, they knew a golden age after their popularisation by Leonhard Euler in his Letters to a German Princess (1768). By the end of the nineteenth century, several schemes were in existence, and to some extent, in competition. In particular, diagrams were invented to handle logic problems known as elimination problems which consist in finding what information regarding any combination of terms follows from a set of premises. For the purpose, John Venn published in 1880 a scheme offered as an improvement over Euler’s well-known circles. The method consisted in representing the complete information contained in the premises on a single diagram, then to see ‘at a glance’ the conclusion regarding specific terms. An inconvenience of this scheme, as pointed out by Louis Couturat (1914), is that it does not really tell how the conclusion is to be ‘extracted’ from the diagram. A rival scheme, published in 1886 by Lewis Carroll, demands that information is transferred from the premises-diagram to another diagram that would depict the conclusion. This transfer is achieved by following rules which are explicitly defined and strictly applied. Although both Venn and Carroll introduced diagrammatic methods for the problem of elimination, they differ in their practices and demands on how a diagram ought to be manipulated. Venn appealed to imagination to work out the conclusion with a single diagram while Carroll applied rules on a diagram to derive other diagrams. The former method was said to lack rigor, but the latter may be accused of lacking naturalness and economy. This difference of practices, and the philosophical views that they embody, will be shown to resurface in the recent debates on the role of diagrams in mathematical practice.

 

David Rabouin (CNRS,SPHERE), On the opacity of mathematical representation

What I call the “opacity” of representation is a phenomenon similar to what Leibniz had in mind when talking about a cognitio caeca. According to him, this phenomenon was ubiquitous in human knowledge, but it was held in particular in mathematics (“we make use of it in algebra and arithmetic, indeed virtually everywhere”). However, the recent literature on mathematical representations barely mentions this phenomenon (or transforms it into the sole question of “de-semantification”). In this talk, I would like to recall Leibniz’s views and, in particular, how cognitio caeca was related to the use of external aids, which he called “characters”. I will indicate that a similar idea was already present in Proclus’ commentary on Euclid and that he gives us some important information about the Ancient Geometrical practice.

 

Roi Wagner (ETH Zurich), The concrete practice of mathematical abstraction

The notion of abstraction is most often explained in terms of subtraction: we select only some features of available objects, and construct new, abstract mind objects, giving rise to a hierarchical structure. Against this common narrative, I would like to suggest a different approach, according to which abstraction is a cumulative practice of incomplete, intermittent and open-ended translations between various concrete presentations and practices. My argument will rely on research from math education, cognitive science and history of mathematics.

Evidence from math education shows that abstraction does not simply arise from subtracting something from concrete representations, but rather from correlating diagrams, words, symbols and gestures. I interpret this work as showing that abstract reasoning depends not on extracting a “common core” of the above, but on the ability to translate between the different presentations in controlled ways without committing to any of the specific presentation.

In order to provide this story with a cognitive-theoretical underpinning, we must move away from representational approaches, which assume that the brain transforms concrete stimuli into abstract computational symbols, to embodied and enactivist approaches. Here I will follow the approach of Vincent Walsh, who argues for the existence of a combined space-time-quantity unit in the brain (instead of distinct modules for each), and of Walter Freeman III, who argues that cognition is reflected in a large scale dynamic, evolving and somewhat chaotic coordination of various brain parts related to different kinds of sensory modules.

Given this theoretical grounding, I will move on to historical examples. I will show how my approach plays out in the understanding of infinitesimals in the 18^th century and the evolution of variables from the Abaco school to 18^th century algebraic analysis.