# Workshop: Proofs and Representations (6 - 8 July 2018)

## Idea and Motivation

The notion of representation plays a central role in mathematics: without particular representations the abstract objects of mathematics would be unthinkable. Reflections on the means of expression of abstract thoughts have a long tradition in philosophy, for example in the work of Leibniz and Frege. Notation systems, spatial and symbolic representations, and representation theorems are among key notions across all mathematical subfields. In this vein, the aim of this workshop is to explore various ways in which proofs and representations advance mathematical knowledge and mathematical understanding. This workshop will address questions such as: Are representations mere instruments for conveying and illustrating mathematics, or do they play a more substantial role in the generation of mathematical knowledge and understanding, and if so, how? What is the relation between conceptual shifts in the history of mathematics and logic, and changes of representations? How do representations inform us about theoretical virtues of proofs and arguments, such as explanatoriness and purity? These are just some of the questions that we intend to get a better grasp of during this workshop, by looking at a wide range of representations––models, diagrams, notations, and more––as they have been used in historical cases, as well as through empirically based reflection and systematic analysis.

## Invited Speakers

- Jeremy Avigad (Carnegie Mellon University)
- Karine Chemla (Université Paris 7 – CNRS)
- Silvia De Toffoli (Stanford University)
- Walter Dean (University of Warwick)
- Valeria Giardino (Université de Lorraine – CNRS)
- Yacin Hamami (Vrije Universiteit Brussel)
- Emmylou Haffner (Bergische Universität Wuppertal)
- Brendan Larvor (University of Hertfordshire)
- Sarah Ottinger (LMU Munich)

## Organisers

- Marianna Antonutti Marfori (MCMP/LMU Munich)
- Dirk Schlimm (McGill University/MCMP)

## Attendance

If you want to attend this workshop, please send notice to Marianna.AntonuttiMarfori@lrz.uni-muenchen.de. Attendance is free.

## Program

### Room Arrangement

Date | Address, Room |
---|---|

06.07.2018 | Ludwigstraße 28, room 026 |

07.07.2018 | Ludwigstraße 28, room 026 |

08.07.2018 | Geschwister-Scholl-Platz 1, room M203 |

### Day 1 (6 July 2018)

Time | Event |
---|---|

15:00 - 16:30 | Silvia De Toffoli: Heterogenous Notations for Mathematical Proofs |

16:30 - 17:00 | Coffee Break |

17:00 - 18:30 | Yacin Hamami: The Rationality of Mathematical Proofs |

20:00 | Informal Dinner |

### Day 2 (7 July 2018)

Time | Event |
---|---|

09:00 - 10:30 | Sarah Ottinger: Content and Formal Representations of Proofs – How Do Both Quality Aspects Affect Each Other? |

10:30 - 10:50 | Coffee Break |

10:50 - 12:20 | Brendan Larvor: Conditions on the Possibility of Mathematical Proving Practices |

12:20 - 14:00 | Lunch Break |

14:00 - 15:30 | Jeremy Avigad: The History of Dirichlet's Theorem on Primes in an Arithmetic Progression |

15:30 - 17:00 | Emmylou Haffner: Arithmetic, computations and devices in the development of Dedekind's concept of Dualgruppe |

17:00 - 17:30 | Coffee Break |

17:30 - 19:00 | Walter Dean: Two Routes Between Number Theory and Set Theory |

20:00 | Workshop Dinner |

### Day 3 (8 July 2018)

Time | Event |
---|---|

10:00 - 11:30 | Karine Chemla: Equations, Representations and Proofs in 13th Century China |

11:30 - 12:00 | Coffee Break |

12:00 - 13:30 | Valeria Giardino: Representations and Their Cognitive Significance in Mathematics |

## Abstracts

### Jeremy Avigad (Carnegie Mellon University): The History of Dirichlet's Theorem on Primes in an Arithmetic Progression

In 1837, Dirichlet proved that there are infinitely many prime numbers in any arithmetic progression in which the first two terms have no common factor. Modern presentations of the proof are explicitly higher-order, allowing quantification and summation over certain types of functions known as "Dirichlet characters." I will discuss the history of the theorem, and explain how it illustrates profound ontological and methodological shifts in nineteenth century language and method.top

### Karine Chemla (SPHERE, CNRS-University Paris Diderot): Equations, Representations and Proofs in 13th Century China

In 13th century China, different sources attest to the fact that establishing an algebraic equation solving a problem can be done using two completely different ways of inscribing the equation, and accordingly two different ways of working with the notation. This talk relies on a corpus of documents in which these two ways of working with algebraic equations can be compared. In some cases, different authors address the same problem, using different ways of working and different methods of establishing the equations. In other cases, they combine parts of text in which they deal with the two inscriptions one after the other. How do authors understand the difference between the two inscriptions and their related potentialities? How do they understand the possible relationships between the two types of inscription, and which kind of cooperation do they envisage between them? These are some of the questions about the relationship between proof and representation that my talk intends to address.top

### Silvia De Toffoli (Stanford University): Heterogenous Notations for Mathematical Proofs

The aim of my talk is to investigate some aspects of the nature and justificatory role of mathematical proofs by focusing on mathematical notations – broadly understood, to include diagrammatic notations. I will focus on the features of mathematical notations that underwrite the possibility for them to enter in the inferential structure of proofs. Moreover, I will present an account of doxastic justification in mathematics as often subject to social norms deriving from the fact that proofs in practice, as I will characterize them, must be sharable. In the case of mathematics, epistemic justification is then shown to be an inter-subjective affair with an important social role. My main philosophical point is that in some cases not only proof presentations, but proofs as well are dependent on certain features of the notations they deploy, the ones which I will label constitutive features. I will argue for this point through examples from elementary arithmetic and knot theory.top

### Walter Dean (University of Warwick): Two Routes Between Number Theory and Set Theory

This talk will explore two means of interpreting fragments of set theory in arithmetic: via the Arithmetized Completeness Theorem and via the Ackermann interpretation. I will discuss both of these methods in their original contexts, making particular contact with Hilbert & Bernays's *method of arithmetization*. I will then discuss subsequent results which I will suggest highlight some exigencies concerning the role of representation in mathematics.top

### Valeria Giardino (Université de Lorraine – CNRS): Representations and Their Cognitive Significance in Mathematics

In the talk, I will briefly present some examples of mathematical representations that I have treated elsewhere (De Toffoli & Giardino 2014, 2015 and 2016, Eckes and Giardino 2018), and I will pinpoint the cognitive roles that they have in guiding the mathematical reasoning and (in some cases) in proving a result. In particular, I will focus on two crucial issues: the relationship between imagined procedures and external representations, and the distinction between diagrammatic and symbolic reasoning.top

### Emmylou Haffner (Bergische Universität Wuppertal): Arithmetic, computations and devices in the development of Dedekind's concept of Dualgruppe

When Richard Dedekind introduced the notions of module and ideal in his famous 1871 *Supplement* X to Lejeune-Dirichlet's *Vorlesungen über Zahlentheorie*, he also defined notions of divisibility (e.g. a module a is divisible by a module b if a is included in b) and related arithmetical notions for modules and ideals (e.g. LCM and GCD of modules or of ideals). In 1877, in *Über die Anzahl der Ideal-Klassen in den verschiedenen Ordnungen eines endlichen Körpers*, the introduction of notations for divisibility, LCMs and GCDs of modules allowed Dedekind to state new theorems, which are now recognized as the modular laws in lattice theory. Observing the “peculiar dualism” displayed by the operations, Dedekind pursued his investigations on the matter. This led him to the introduction of the notion of *Dualgruppe* (equivalent to our modern-day lattice), presented in two papers: *Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler* (1897) and *Über die von drei Moduln erzeugte Dualgruppe* (1900).

It was, in his words, obtained “not without great effort” (Dedekind 1897, 113) and indeed after two decades of work. Dedekind's *Nachlass* contains several hundred of pages laying the ground for his research in module theory and towards the concept of *Dualgruppe*. In this talk, I propose to explore some parts of this extremely rich archival material, so as to understand the steps that led from the introduction of a notation for modules to the invention of a new concept. In a first time, I will show how the notation for the LCM and GCD of modules turned these arithmetical notions into actual operations. This allowed Dedekind to go from simply using a terminology mimicking arithmetic to actually computing with modules. In a second time, I will look closer at Dedekind's *Nachlass* and show how the computations and the devices introduced throughout the research process were a way to explore module theory and its properties, looking for remarkable properties, generally valid properties, and “fundamental” properties,in a way that is not reflected in Dedekind's published works. Finally, I will turn to the slow, stepwise process of generalization that paves the way from modules to *Dualgruppen*.top

### Yacin Hamami (Vrije Universiteit Brussel): The Rationality of Mathematical Proofs

On the traditional view, a mathematical proof is nothing more than a sequence of deductive steps, the only requirement being that each deductive step be valid. However, it has been noted by several leading mathematicians that this view offers a too impoverished conception of the *nature* of mathematical proofs. A case in point is Henri Poincaré who writes in *Science et Méthode* that: “A mathematical demonstration is not a simple juxtaposition of syllogisms; it consists of syllogisms *placed in a certain order*, and the order in which these elements are placed is much more important than the elements themselves”. What Poincaré is contesting here is the idea that a mathematical proof is a sequence of *arbitrary* deductive steps. In this talk, I will explore a potentially more satisfying view according to which a mathematical proof is conceived as a sequence of *rational* deductive steps. To this end, I will propose an account of what it means for a deductive step in a mathematical proof to qualify as rational by exploiting various resources from the philosophy of action, most notably Michael Bratman’s theory of *planning agency*. [This is joint work with Rebecca Morris (Stanford University, USA).]

top

### Brendan Larvor (University of Hertfordshire): Conditions on the Possibility of Mathematical Proving Practices

Under what conditions does a representational practice offer the possibility of rigorous mathematical proof? Reflection on recent work on mathematical practices suggests three:

- the representation should be open to manipulation by adepts in the practice. Manipulation can include selective attention, imaginary manipulation, the haptic imagination.
- the information thus displayed is not metrical and
- it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices.

This talk will expound these proposed conditions with reference to empirical work by Keith Weber and others on the distinction between metrical and non-metrical graphical inferences.top

### Sarah Ottinger, Stefan Ufer (LMU Munich): Content and Formal Representations of Proofs – How Do Both Quality Aspects Affect Each Other?

The construction of proofs poses problems to students at all educational levels. In the transition to tertiary education, one major challenge is the focus on formal-deductive proofs in university mathematics. Creating acceptable mathematical proofs requires not only to find deductive lines of reasoning, but also to communicate these proofs with adequate formal precision. In a study with N=159 students at the transition to a university mathematics programme, we examined how the content quality of mathematical proofs and of several formal aspects of their representations are interrelated. The results show formal aspects vary in the degree to which they are connected with the quality of the content of the proofs. We discuss implications for research as well as for support of students at the beginning of their mathematics studies.top

## Venue

### Day 1 & 2

Ludwigstraße 28

80539 München

### Day 3

Main University Building

Geschwister-Scholl-Platz 1

80539 München

## Acknowledgement

This workshop is generously funded by the Alexander von Humboldt Foundation and the European Union's Horizon 2020 research and innovation programme under the Marie SkÅ‚odowska-Curie grant agreement No 709265.