# PhilMath Intersem 9. 2018

The PhilMath Intersem will convene for its ninth annual meeting during June 2018.

The theme of this year's seminar is the use of non-deductive methods 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 5th, June 7th, June 12th, June 14th, June 19th, June 21st and June 26th. Some sessions are single sessions and some double. See the schedule below for details.

All meetings will take place on the Rive Gauche campus of the University of Paris 7-Diderot, in the salle Klimt (room 366A) of the Condorcet Building.

See the schedule below for further information, including titles, summaries and suggested readings.

Please continue to check here for updates.

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**Schedule**

**Meeting 1: Tuesday, June 5th**

#### • 14h00: Catherine Goldstein (CNRS, Institut de mathématiques de Jussieu-Paris Gauche)

**Title: "**Baconian mathematics"

**Summary:**

It is usual to oppose mathematics to the so-called Baconian, that is experimental, sciences. Thomas Kuhn even suggested that modern science was created by the separation of experimental science from mathematical sciences during the seventeenth century. However, a number of early-modern mathematicians shared Francis Bacon’s criticism of deductive procedures, either syllogistic or Euclidean. Focusing on Marin Mersenne's circle, in particular the work of Bernard Frenicle de Bessy (a founding member of the French Academy of Sciences), I shall discuss how some of the mathematical production in this circle went even further, following closely Baconian precepts. My key questions will be how induction and observation are used to produce mathematics, and in particular which type of induction for which mathematics.

**Reading:**

C. Goldstein, "Ecrire l'expérience des mathématiques au XVIIe siècle", in *Réduire en art*, éd. H. Vérin et P. Dubourg-Glatigny, Paris, MSH, 2008, pp. 213-234.

C. Goldstein, "How to Generate Mathematical Experimentation and Does it Provide Mathematical Knowledge?", in *Generating Experimental Knowledge*, ed. U. Fest, G. Hon, Hans-Joerg Rheinberger, Jutta Schickore, Friedrich Steinle, PMPIWG 340, Berlin, Max Planck Institute for the History of Science, 2008, p. 61-85 (available online here).

*T. S. Kuhn, "Mathematical versus Experimental Traditions in the Development of Physical Science', The Journal of interdisciplinary History* 7 (1976), 1-31 (repr. in *The Essential Tension*, Chicago: University of Chicago Press, 1977, pp. 31-65).

#### • 16h00: Eberhard Knobloch (Institut für Philosophie, Literatur-, Wissenschafts- und Technikgeschichte, Berlin)

**Title: **"Archimedes and his successors Kepler and Leibniz: Their use of analogies and equivalences in mathematics"

**Summary:**

Archimedes used analogies in the context of discovery. Especially interesting examples are his calculation of the surface of the sphere and his way of transferring the structure of quantities to the structure of non-quantities or indivisibles. Kepler amply used analogies in his "New solid geometry of wine barrels“ (1615) also in the context of justification referring to Archimedes without knowing his famous letter to Eratosthenes (so-called Method). Leibniz, too, referred to Archimedes in order to justify his infinitesimal calculus. He considered analogies as special harmonies that are closely connected to his law of continuity. He transferred the structure and laws of finite quantities to that of infinitely small and infinite quantities. Curves have to be considered as equivalent to a polygon of infinitely many sides. An instructive example is his geometrical integration of the logarithmic curve. Mathematical creativity is closely connected to the surpassing of limits.

**Reading:**

J. Kepler, *Nova stereometria doliorum vinariorum / New solid geometry of wine barrels.* Accessit Stereometriae Archimedeae supplementum / A supplement too the Archimedean solid geometry has been added. Edited and translated, with an introduction by E. Knobloch. Paris 2018 (in the press) (Sciences et savoir vol. IV).

E. Knobloch, "Analogy and the growth of mathematical knowledge", in: E Grosholz, H. Breger (eds.), *The growth of mathematical knowledge*. Dordrecht etc. 2000, 295-314.

E. Knobloch, "Analogien und mathematisches Denken", in K. Hentschel (ed.), *Analogien in Naturwissenschaft, Medizin und Technik*, *Acta historica Leopoldina* 56 (2010), 309-327.

G. W. Leibniz, *De quadratura arithmetica circuli ellipseos et hyyperbolae cujus corollarium est trigonometria sine tabulis*, hrsg. und mit eine Nachwort versehen von E. Knobloch, Aus dem Lateinischen von O. Hamborg. Berlin-Heidelberg 2016.

**Meeting 2: Thursday, June 7th**

#### • 16h00: James Franklin (School of mathematics and statistics, University of New South Wales, Sydney)

**Title: **"Evidence for mathematical conjectures and non-deductive logic"

**Summary:**

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as the Riemann Hypothesis, have to be considered in terms of the evidence for and against them in the same way as legal verdicts and scientific hypotheses. After showing some examples, it is argued that it is not adequate to describe the relation of evidence to hypothesis in these cases as "subjective", "heuristic" or "pragmatic", but that it has an element of what it is strictly rational to believe on the evidence, that is, of non-deductive logic (or logical probability or objective Bayesianism with imprecise probabilities).

**Reading:**

George Pólya, *Mathematics and Plausible Reasoning *(Princeton University Press, 1954).

James Franklin, "Logical probability and the strength of mathematical conjectures", *Mathematical Intelligencer* 38 (3) (Sept 2016), 14-19.

**Meeting 3: Tuesday, June 12th**

#### • 14h00: Jean Paul Van Bendegem (Centrum voor Logica en Wetenschapsfilosofie, Center for Logic and Philosophy of Science, Vrije Universiteit, Brussells)

**Title: **"Can there be experiments, real or imaginary, in mathematics?"

**Summary:**

At first sight the idea of a mathematical experiment seems an impossibility. Should mathematics, seen as the deductive discipline par excellence, not be ‘immune’ from empirical and experimental considerations? In Van Bendegem (1998) I firmly believed this to be the case but in Van Bendegem (1996) I equally firmly changed my mind on the issue (incidentally, the order of publication is the inverse of the order of writing) and defended the possibility of (particular types of) mathematical experiments. Faced with Starikova & Giaquinto (2018) on the specific issue of thought experiments and given the rapid development of so-called ‘experimental mathematics’, a re-evaluation of the issue seems appropriate. The conclusion is that mathematical experiments are indeed a genuine possibility but that a single definition of what they are seems unlikely, given the heterogeneity of cases.

**Reading:**

Jean Paul Van Bendegem (1998), “What, if anything, is an experiment in mathematics?”, in Dionysios Anapolitanos, Aristides Baltas & Stavroula Tsinorema (eds.), *Philosophy and the Many Faces of Science*, (CPS Publications in the Philosophy of Science). London: Rowman & Littlefield, pp. 172-182. Downloadable here.

Jean Paul Van Bendegem (1996), “Mathematical Experiments and Mathematical Pictures”, in: Igor Douven & Leon Horsten (eds.), *Realism in the Sciences. Proceedings of the Ernan McMullin Symposium Leuven 1995*. Louvain Philosophical Studies 10. Leuven: Leuven University Press, pp. 203-216. Downloadable here.

Irina Starikova & Marcus Giaquinto (2018), “Thought Experiments in Mathematics”, in Michael T. Stuart, Yiftach Fehige & James Robert Brown (eds.), *The Routledge Companion to Thought Experiments*. London: Routledge, pp. 257-278.

#### • 16h00: Emmylou Haffner (Arbeitsgruppe Didaktik und Geschichte der Mathematik, Bergische Universität Wuppertal & SPHERE, University of Paris Diderot)

**Title: "**On the computations underlying (some of?) Dedekind's “conceptual mathematics”"

**Summary:**

Richard Dedekind is usually presented as one of the founding fathers, with Bernhard Riemann, of so-called conceptual mathematics. Both advocated for the importance of grounding a theory on concepts rather than computations. Computations, they wrote, should follow from rather than lead the definitions of the fundamental notions of a theory. Yet, if one looks more closely into their *Nachlässe*, one notices the remarkable quantity of computations on which their preliminary researches rely. In this talk, I will concentrate on Richard Dedekind's mathematical works, and propose to dive into some of his manuscripts so as to understand how these computations were for him a stepwise way to come to more general (“conceptual”) algebraic laws. I will consider, in particular, the researches that let to his papers on *Dualgruppen* (similar to what is today called lattices) (Dedekind 1897, 1900). An analysis of his drafts suggests that Dedekind's statements concerned the way one should write and present a theory to the scientific community and for its further uses, rather than the ways in which one investigates and elaborates new mathematical ideas properly speaking. I will argue that in the latter, the mathematics “in the making” so to speak, mathematical practices take on different forms and are closer to mathematical exploration and experimentation through computations and gradual generalization.

**Reading:**

Dedekind, R. (1897), "Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler", in *Gesammelte mathematische Werke*, volume II, pages 103–147.

Dedekind, R. (1900), "Über die von drei Moduln erzeugte Dualgruppe," in *Gesammelte mathematische Werke*, volume II, pages 236–271.

Edwards, Harold M. (2010), "The algorithmic side of Riemann’s mathematics," in *A Celebration of the Mathematical Legacy of Raoul Bott*, R. Bott & P. Kotiuga (eds), Providence : American Mathematical Society, CRM proceedings & lecture notes.

Schlimm, D. (2011), "On the creative role of axiomatics.The discovery of lattices by Schröder, Dedekind, Birkhoff, and others", *Synthese* 183(1): 47–68.

**Meeting 4: Thursday, June 14th**

#### • 16h00: James Franklin (School of mathematics and statistics, University of New South Wales, Sydney)

**Title: "**The nature of non-deductive logic in the light of its use in mathematics"

**Summary:**

The fact that non-deductive/probabilistic methods work in pure mathematics shows they are fully logical. For example, since induction works in pure mathematics, it cannot depend on any contingent fact such as the "uniformity of nature". It is argued that non-deductive methods are as strictly logic as deductive logic, as in Keynes' view that logical probability is a degree of partial implication. Philosophies of the nature of logic are canvassed, in the light of which non-deductive inference forms like the confirmation of theories by their consequences and the proportional syllogism are exhibited as purely logical. Non-deductive logic is no less formal than deductive logic.

**Reading:**

George Pólya, *Mathematics and Plausible Reasoning *(Princeton University Press, 1954).

James Franklin, "Logical probability and the strength of mathematical conjectures", *Mathematical Intelligencer* 38 (3) (Sept 2016), 14-19.

**Meeting 5: Tuesday, June 19th**

#### • 14h00: Alex Paseau (Faculty of Philosophy, Wadham College, University of Oxford)

**Title: "**Knowledge of Mathematics without Proof"

**Summary:**

Mathematicians do not claim to know a proposition unless they think they possess a proof of it. For all their confidence in the truth of a proposition with weighty nondeductive support, they maintain that, strictly speaking, the proposition remains unknown until such time as someone has proved it. My talk will present and further develop some arguments in my 2015 BJPS paper of the same title that challenge this conception of knowledge within mathematics.

**Reading:**

Fallis, D. (1997), "The Epistemic Status of Probabilistic Proof", *The Journal of Philosophy* 94, pp. 165–86.

Echeverria, J. (1996), "Empirical Methods in Mathematics. A Case-Study: Goldbach’s Conjecture", in G. Munevar (ed.), *Spanish Studies in the Philosophy of Science*, Dordrecht: Kluwer, pp. 19–55.

Paseau, A. (2015), "Knowledge of Mathematics without Proof", *The British Journal for the Philosophy of Science* 66, pp. 775-99.

#### • 16h00: Karine Chemla (CNRS, SPHERE, University of Paris Diderot)

**Title: **"When a diagram writes a proof"

**Summary:**

The talk will focus on a specific diagram, which, I claim, writes the correctness of an algorithm solving a quadratic equation numerically. After having outlined elements of context that are required to interpret the diagram, I will analyze the techniques used to formulate the proof in this way. We will thus see how the meaning of a single diagram unfolds while reading the text of the algorithm.

**Reading:**

Karine Chemla, "Changing mathematical cultures, conceptual history and the circulation of knowledge" in K. Chemla & E. Fox-Keller (eds.), *Cultures without Culturalism: The Making of Scientific Knowledge*, Duke U Press, 2017.

**Meeting 6: Thursday, June 21st**

#### • 16h00: Christine Proust & Adeline Reynaud (CNRS, SPHERE, University of Paris Diderot)

**Title: "**Reasoning with diagrams and layouts, examples from Mesopotamia"

**Summary:**

Algorithms and procedures are generally explained in cuneiform mathematical texts in a quite terse manner. Most often, the modalities of reasoning underlying them can be detected only through indirect evidence, such as terminology, structures of texts, diagrams, layouts, tabular formats, marginal notations, or epigraphy. A landmark work on evidence provided by terminology has been accomplished by Jens Høyrup (2002).

This presentation proposes a reflexion on the way in which other elements of reasoning can be grasped through the observation of diagrams and layouts. Adeline Reynaud will present some mathematical problems in which key elements of the solving procedure are absent from the discursive text and suggested only by a diagram drawn beneath it. She will analyse how these visual elements are articulated to those provided by means of words, and what this may reveal on the (non-)deductive character of the procedures. Christine Proust will present numerical texts that do not exhibit any discursive element. She will also analyze how the arrangement of the numbers on the surface of a clay tablet reflects the implementation and justification of an algorithm.

**Reading:**

Chemla, Karine (ed) (2012), *The History of Mathematical Proof in Ancient Traditions*. Cambridge: Cambridge University Press. See especially the Prologue and chs. 2, 11 and 12.

**Meeting 7: Tuesday, June 26th**

#### • 16h00: Alan Baker (Swarthmore College, Swarthmore, PA)

**Title: "**Induction, Explanation, and Applied Mathematics"

**Summary:**

Some mathematical proofs are explanatory and others are not. I shall refer to a true, general mathematical proposition that lacks an explanatory proof as a “mathematical accident.” My interest in this talk is how mathematical accidents function in the context of applied mathematics. I shall focus on two main questions. Firstly, can mathematical accidents be used to explain scientific phenomena? Secondly, can the use of mathematical accidents in an applied context provide them with non-deductive support?

**Reading:**

Baker, A. (2009), “Mathematical Accidents and the End of Explanation,” in *New Waves in the Philosophy of Mathematics*, ed. O. Bueno & Ø. Linnebo, Palgrave Macmillan, 137-159.