# PhilMath Intersem 10.2019

The PhilMath Intersem will convene for its tenth annual meetings during June 2019.

The theme of this year's seminar is the nature and place of rigor in mathematics. The seminar is interdisciplinary, and we will consider a variety of philosophical, historical, logical and mathematical issues.

The seminar will meet on the following dates: June 4th, June 6th, June 11th, June 13th, June 18th, June 20th and June 25th.

Some sessions will feature talks and discussions with one speaker, some with two. See the schedule below for further information, including titles, summaries and suggested readings.

All but one meeting of the seminar will take place on the Rive Gauche campus of the University of Paris 7-Diderot, in the Condorcet Building. That exception is the meeting of June 13th. It will take place at the IHPST.

The meeting of June 20th will be held in the Condorcet Building on the Rive Gauche campus of Paris 7. It will meet in the salle Mondrian (room 646A), however, rather than in the usual room (i.e., the salle Klimt, room 366A).

All the rest of the meetings will be in the salle Klimt.

Here's a meeting-by-meeting summary with times and places:

• Meeting 1: June 4th, Marcus Giaquinto, salle Klimt (366A), 14h-16h

• Meeting 2: June 6th, Part A: Hourya Sinaceur, salle Klimt (366A), 14h-16h; Part B: Silvia de Toffoli, salle Klimt (366A), 16h15-18h15

• Meeting 3: June 11th, Part A: Erika Luciano, salle Klimt (366A), 14h-16h; Part B: Yacin Hamami, salle Klimt (366A), 16h15-18h15

• Meeting 4: June 13th. Joint meeting with *Themes from the work of Göran Sundholm*. IHPST. Schedule.

• Meeting 5: June 18th, Antoni Malet, salle Klimt (366A), 16h-18h

• Meeting 6: June 20th, Michael Harris, Kevin Buzzard and Patrick Massot, salle Mondrian (646A), 15h-18h

• Meeting 7: June 25th, Part A: Renaud Chorlay, salle Klimt (366A), 14h-16h; Part B: Jean-Jacques Szczeciniarz, salle Klimt (366A), 16h15-18h15

Please continue to check here for updates.

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**Schedule**

**Meeting 1:** Tuesday, June 4th, 14h-16h

#### Pr. Emer. Marcus Giaquinto, Philosophy, University College London

####
**"**Rigorous and non-rigorous thinking using diagrams: both can be epistemically valuable"

#### Summary:

Consider the following claims: "In mathematics, a diagrammatic argument which is non-rigorous (even by standards of non-formal rigour) cannot warrant high confidence in its conclusion. A (non-formal) diagrammatic argument in which a diagram is used as visual evidence for a proposition of the argument is not mathematically rigorous, and therefore not a proof." By focussing on two or three examples, I will cast doubt on these claims.

#### Background Reading:

Jaffe, A. and F. Quinn, "Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics", *Bulletin of the American Mathematics Society* Vol. 29 (1), 1-13, 1993.

Thurston, W., "On Proof and Progress in Mathematics", *Bulletin of the American Mathematical Society, *vol. 30 (2), 1994.

**• Related event of interest:** Paolo Mancosu, "The company you keep: Some recent work on neo-logicism and abstraction principles", Tuesday, June 4th, 17h, salle E314, 24 rue Lhomond, École normale supérieure, Paris.

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**Meeting 2:** Thursday, June 6th, 14h-18:15h

####
**Part A:** 14h-16h

Pr. Hourya Sinaceur, History and Philosophy of Science, Paris-Sorbonne, CNRS

#### ``“Proving what everyone already knows”: Bolzano’s conceptual analysis''

Summary:

I will sketch the main traits of Bolzano’s conceptual analysis: determining precisely what the concepts and propositions mean, stating definitions for fundamental and non primitive concepts such as the concept of continuity, seeking proofs even for obvious truths, revealing the ``objective connection'' of truths, distinguishing between principles [Grundsätze] and theorems [Folgewahrheiten], ``unfolding all truths of mathematics down to their ultimate grounds'', making clear the intrinsic nature of concepts so as to describe the architecture of mathematics (``classification of pure mathematics'' Bolzano 1810, §7) and promoting a neo-Aristotelian demand for purity of methods.

#### Background Reading:

Benis-Sinaceur H., ``Bolzano et les mathématiques'', in *Les Philosophes et les mathématiques* (Barbin É. & Caveing M. eds.), Paris, Ellipses, 1996, p. 150-173.

Bolzano B., ``Betrachtungen über einige Gegenstände der Elementargeometrie'', Prag, 1804, Schriften 5: Geom. Arbeiten, Prag 1948. English transl. in Russ.

Bolzano B., ``Beyträge zu einer begründeteren Darstellung der Mathematik'', Prag, 1810, English transl. by Russ S. in Russ and in Ewald.

Bolzano B., ``Der binomische Lehrsatz und als Folgerung aus ihm der polynomische, und die Reihen, die zur Berechnung der Logarithmen und Exponentialgrössen dienen, genauer als bisher erwiesen'', Prag, 1816, English transl. in Russ.

Bolzano B., ``Rein analytischer Beweis, dass zwischen je zwey Werthen, die ein entgegensetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege'', Prag, 1817. English Transl. in Russ and in Ewald.

Ewald W., *From Kant to Hilbert*, Volume 1, n° 6, p. 168-248.

Russ S., *The mathematical works of Bernard Bolzano*, Oxford University Press, 2004.

####
**Part B:** 16h15-18h15

#### Dr. Silvia de Toffoli, Philosophy, Princeton University

#### "Rigorous Diagrammatic Proofs"

#### Summary:** **

The aim of my talk is to investigate the role of diagrams, and in particular of diagrammatic notations, in the context of proving in mathematics. First of all, I will address the preliminary issue of characterizing mathematical diagrams and I will propose a technical definition to delineate the phenomenon under investigation. I will then focus on the features of diagrams that underwrite the possibility for them to enter in the inferential structure of proofs. My main philosophical point will be to show 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 of diagrams used in different areas of contemporary mathematics.

#### Background Reading:

De Toﬀoli, S. “‘Chasing’ the diagram--the use of visualizations in algebraic reasoning.” *The Review of Symbolic Logic*, 10(1):158–186, 2017.

De Toﬀoli, S. and V. Giardino. “Roles and forms of diagrams in knot theory”, *Erkenntnis*, 79(3):829–842, 2014.

Giaquinto, M. “Visualizing in mathematics”, in P. Mancosu (ed.) *The Philosophy of Mathematical Practice*, 22–42. Oxford University Press, 2008.

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**Meeting 3:** Tuesday, June 11, 14h-18h

####
**Part A:** 14h-16h

#### Dr. Erika Luciano, Mathematics, University of Turin

#### "A matter of style and praxis: On Peano’s concept of rigour in mathematics education"

#### Summary:

Between the late 19^{th} century and the 1920s, many of the greatest Italian mathematicians showed a particular sensitivity regarding educational issues. These were sometimes spurred by their own scientific activity, and sometimes by interests of a pedagogical and epistemological or social type. It is in this context that we consider the reflections of the members of the two Italian research ‘schools’ that flourished in Turin (those of G. Peano and C. Segre) on the possibility of introducing the results of studies on the foundations of mathematics into teaching. These reflections renewed at various levels the traditional treatment of rational arithmetic and geometry.

The range of topics addressed by Peano, Segre and their protégés, in keeping with international influences (of Klein, Poincaré, Borel, Young, Halsted, …), included the dialectic between rigour and intuition; the cognitive aspects of the axiomatic-deductive approach in mathematics education and the reflexions on mathematical language, in a constant search for the right balance between the opposing poles of natural narrative and logical symbolism (ideography).

My talk is to be taken in this context. It is intended

- to pinpoint the different ways in which rigour and intuition were characterized by the ‘Peanians’, in relation to both teaching practice and pedagogic theory, and
- to examine the debates that hypothetico-deductive teaching ignited within the context of Italian mathematical instruction, particularly those concerning the features and the role of rigour in textbooks by Peano and his followers

#### Background Reading:

Segre C., Su alcuni indirizzi nelle investigazioni geometriche, *Rivista di Matematica*, 1, 42-65, translated by J.W. Young in 1904, and published with the title “On some tendencies in geometric investigations” in *Bulletin of the American Mathematical Society*, 2, 10, 442-468, 1891.

Peano G., Osservazioni del Direttore sull’articolo precedente* *[Segre, “Su alcuni indirizzi nelle investigazioni geometriche”], *Rivista di Matematica*, 1, 66-69, 1891.

Peano G., Sui fondamenti dell’Analisi, *Bollettino della Mathesis*, 31-37, translated by H. Kennedy in *Selected works of Giuseppe Peano*, Toronto: Univ. Press, 219-226, 1891.

####
**Part B:** 16h15-18h15

#### Dr. Yacin Hamami, Center for Logic & Philosophy of Science, Free University, Brussells

#### ``Judgments of Rigor in Mathematical Practice''

#### Summary:

How are mathematical proofs judged to be rigorous in mathematical practice? Traditional answers to this question have usually considered that judging the rigor of a mathematical proof proceeds through some sort of comparison with the standards of formal proof. Several authors have argued, however, that this kind of view is implausible (see, e.g., Robinson, 1997; Detlefsen, 2009; Antonutti Marfori, 2010), and have thus called for the development of a more realistic account of rigor judgments in mathematical practice. In this talk, I will sketch a framework aiming to move forward in this direction. My starting point is the observation that judging a mathematical proof to be correct or rigorous amounts to judging the validity of each of the inferences that comprise it. Accordingly, the framework focuses on the processes by which mathematical agents identify and judge the validity of inferences when processing the text of an ordinary mathematical proof. From the perspective of the resulting framework, I will then discuss what is sometimes called the standard view of mathematical rigor, by examining whether there is any ground supporting the thesis that whenever a proof has been judged to be rigorous in mathematical practice it can be routinely translated into a formal proof.

#### Background Reading:

Antonutti Marfori, M. ``Informal proofs and mathematical rigour'', *Studia Logica* 96(2): 261–272, 2010.

Detlefsen, M. ``Proof: Its nature and significance'', in Bonnie Gold and Roger A. Simons (eds), *Proof and Other Dilemmas: Mathematics and Philosophy*. The Mathematical Association of America, Washington, DC, 2009.

Robinson, J. A. ``Informal rigor and mathematical understanding'', In G. Gottlob, A. Leitsch, and D. Mundici (eds.), *Computational Logic and Proof Theory: Proceedings of the 5th Annual Kurt Gödel Colloquium*, August 25-29, 1997, volume 1289 of *Lecture Notes in Computer Science*, pages 54–64, Heidelberg & New York, Springer, 1997.

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**Meeting 4:** Thursday, June 13

####
**Note:** This will be a joint meeting of the PhilMath Intersem with the conference:

*Formalization vs. Meaning in Mathematics: Formal theories as tools for understanding.*

####
*Themes from the work of Göran Sundholm.*

The meeting will take place in the Grande salle of the IHPST.

Here's the Schedule.

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**Meeting 5:** Tuesday, June 18, 16h-18h

#### Pr. Antoni Malet, History of Science, Pompeu Fabra University

#### "Circular definitions, and circular reasoning in 17th-century mathematics"

#### Summary:

Questions about rigor in early modern mathematical arguments are more often than not cast in foundational terms, particularly but not exclusively about the well-trod issue of the foundations of the calculus. Furthermore, questions about early modern mathematical rigor have been widely and generally raised, with a large number of historians pointing to it as a major feature of modern mathematics—remember Morris Kline's celebrated chapter title, “The Instillation of Rigor in Analysis”, in his influential account of mathematics’ history. I aim at problematizing the traditional historiographical approach by pointing to a different but also prominent case in point of early modern mathematics, the handling of ratios and proportionality. As explicitly shown in John Wallis's well-known *Mathesis universalis* (1657), the so-called ‘arithmetical’ understanding of ratio and proportionality involves an obvious, explicit logical loop, or *petitio principii*. Interestingly, the fact that most mathematicians were embracing a central notion that they defined fallaciously was publicly exposed—to no avail. I will argue that historical accounts of past understandings of mathematical rigor (when historical artefacts are disregarded) were in part responding to presentist concerns, concerns that were relevant to mathematics in historians's own days. Moreover, reconstructions of past discussions on rigor have crucially disregarded the role of social convention in determining what can or cannot pass by a rigorous argument.

#### Background Reading:

Hacking, I. "What makes mathematics mathematics?", J. Lear, A. Oliver (eds.) *The Force of Argument, Oxford University Press,* 82-106, 2000.

Jesseph, D. "The ‘Merely Mechanical’ vs. the ‘Scab of Symbols’: Seventeenth Century Debates over the Criteria for Mathematical Rigor,” *Philosophical Aspects of Symbolic Reasoning in Early Modern Mathematics*, ed. Albrecht Heeffer and Maarten Van Dyck. Special Issue of Studies in Logic, 273-288, 2010.

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**Meeting 6:** Thursday, June 20, 15h-18h

####
**Reminder:** This meeting of the intersem will be held in room 646A (the salle Mondrian) of the Condorcet building.

#### Pr. Michael Harris, Mathematics, Columbia University

Pr. Patrick Massot, Mathematics, University of Paris-Sud

Pr. Kevin Buzzard, Mathematics, Imperial College London

#### "Mechanization of Rigor"

#### Summary:

Most mathematicians have been generally satisfied in recent decades with the state of rigor, as enforced by the peer review process as well as by informal means. Increasingly, however, some proofs have become so long or complex, or both, that they cannot be checked for errors by human experts. They may also rely on results which were announced but whose proofs never appeared in the literature. In response, a small but growing community of mathematicians, collaborating with computer scientists, have begun to work with systems designed by the latter that allow proofs to be verified by machine. Do the development of automated proof checkers and the promise of effective automatic theorem provers, represent a turning point in mathematical research? If so, how will future human mathematicians adapt — or will there still be a place for human mathematicians?

#### Background Reading:

Avigad, J., ``The mechanization of mathematics'', *Notices of the AMS *65(6), (June-July, 2018), 681-690

Baker, A., Non-deductive methods in mathematics, Stanford Encyclopedia of Philosophy.

Barany, M. and D. MacKenzie, ``Chalk: Materials and Concepts in Mathematics Research'', in C. Coopman, J. Vertesi et al (eds.), *Representation in Scientific Practice Revisited, *MIT Press, Cambridge, MA, 2014.

Dick, S. After Math: Reconfiguring Minds, Proof, and Computing in the Postwar US

Harris, M. ``Do Androids Prove Theorems in their Sleep'', in *Circles Disturbed: The interplay between mathematics and narrative*, A. Doxiadis and B. Mazur (eds.), Princeton University Press, Princeton, 2012.

Wiedijk, F. The seventeen provers of the world

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**Meeting 7:** Tuesday, June 25, 14h-18h

####
**Part A:** 14h-16h

#### Dr. Renaud Chorlay, Didactique et Histoire des Mathématiques, Sorbonne Université

#### "Did Bourkabi care about rigor?"

#### Summary:

We will study the epistemological values put forward by the founders of the Bourbaki group, and their connections with mathematical practice. We will endeavor to challenge the received view that “rigor” was a key value for the Bourbaki group by comparing three sets of texts; texts of different genres, and associated with three facets of mathematical practice: (1) the manuscripts and discussions which document the collective writing of the first volumes of the "Elements of Mathematics", (2) the manifesto entitled "L’architecture des mathématiques", (3) and the research papers which lead to the emergence of the sheaf concept.

#### Background Reading:

Chorlay, R., *From Problems to Structures: The Cousin Problems and the Emergence of the Sheaf Concept. *Archive for History of Exact Sciences 64(1), 1-73, 2010. click here

####
**Part B:** 16h15-18h15

#### Pr. Jean-Jacques Szczeciniarz, History and Philosophy of Science, University of Paris 7-Diderot

#### "Towards a history of the concept of rigor"

#### Summary:

I propose steps in the history of the concept of rigor. I then propose a philosophical exercise---namely, to answer the following question: What are the differences between rigor and accuracy and rigor and precision?

My examples will be taken from category theory but not only from it. The goal will be to refine hypothetical historical reasoning.

#### Background Reading:

Cavaillès, J. ``Sur la logique et la théorie de la science'', *Oeuvres complètes*, Philosophie des Sciences, Hermann (ed.), Paris, 1994.

Detlefsen, M., ``Brouwerian Intuitionism'', *Mind* (99) 501-534, 1999.

Desanti, J. ``Idéalités Mathématiques'', Paris, 1968.

Descartes, ``Discours de la méthode'', in Oeuvres de Descartes*,* ed. C. Adam and P. Tannery, Paris, 1897.

Merker, J. and Jean-Jacques Szczeciniarz, ``Théorème de Gauss-Bonnet pour les surfaces : vers une philosophie intrinsèque de la géométrie différentielle (Partie I)'', HAL archives, 2019.