# PhilMath Intersem 7. 2016

The PhilMath Intersem (PhilMath Intersem 7) will convene for its seventh annual meetings during June 2016.

The theme of this year's seminar will be the **axiomatic method** and its uses in the history of mathematics. Historical, philosophical, logical and mathematical issues will be considered.

The seminar will meet on the following dates: June 2nd, June 7th, June 9th, June 14th, June 16th, June 21st, June 23rd and June 28th. See the schedule below for times and other information.

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.

The PhilMath Intersem is jointly sponsored by the history and philosophy of science department of the University of Paris 7-Diderot, the University of Notre Dame and the SPHERE group of the University of Paris 7-Diderot.

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

**Notice:** Owing to increased security concerns, entry to the building in which the PhilMath Intersem meetings are held is now more controlled. If you plan to attend a meeting of the seminar, please email Dr. Emmylou Haffner and let her know. She can then add your name to the authorized list of attendees. Security personnel may check your name against this list. Please also come with a photo id. You may be asked to present this as a condition of being admitted to the building.

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**Schedule**

Meeting 1: Thursday, June 2nd

####

• 14h: Pascal Crozet (SPHERE, Université Paris 7-Diderot)

#####

"Thabit ibn Qurra and the fifth postulate"

**Summary:**

Thabit ibn Qurra (826-901) was not the first to propose a demonstration of the parallels postulate. However, his use of movement as a primitive notion of geometry, combined with his use of Archimedes' and Pasch's axioms, provided the starting point for a research tradition that lasted several centuries.

#####

**Reading:**

Roshdi Rashed and Christian Houzel, "Thabit ibn Qurra et la théorie des parallèles", *Arabic Sciences and Philosophy* 15(1) (2005): 9--55.

English translation (but without Thabit's text) in : Roshdi Rashed, *Classical Mathematics from al-Khwarizmi to Descartes* (chapter entitled "Thabit ibn Qurra and Euclid's Fifth Postulate"), Routledge, 2015.

#### • 16h: Vincenzo De Risi (Max Planck Institute for the History of Science, Berlin)

#####

"The axiomatization of space in the modern editions of Euclid’s *Elements"*

**Summary:**

The talk deals with the development of the system of axioms grounding elementary geometry in the editions of Euclid’s *Elements* in the Early Modern Age. Particular importance was given to the axiomatization of purely spatial relations: something that was absent in ancient axiomatics and which reflects a new conception of the object and methods of geometry.

**Reading:**

V. De Risi, "The Development of Euclidean Axiomatics" (euclidean_axiomatics.pdf)

### Meeting 2: Tuesday, June 7th

####

• 14h: Victor Pambuccian (Mathematics, Arizona State University)

#####

"Why should one follow the axiomatic method in geometry? An apology"

**Summary:**

In AD 2016 there are fewer than ten people in the world actively working with the axiomatic method in geometry. On a Hegelian reading, the historical demise of the axiomatic method in geometry, where analytical methods have carried the day for more than a century now, is a sure sign of its intrinsic inferiority, and proof that better approaches have replaced it. In this talk, we will try to make the case for the advantages *sub specie aeternitatis* of the axiomatic method, which opens up a whole new view of truth in geometry, and which presents us with results that have no equivalent whatsoever in algebraic, differential, or topological flavors of geometry.

**Reading:**

V. Pambuccian, Review of Hilbert's Lectures on the foundations of geometry (rev._of_hilbert's_lectures_on_fnds_of_geometry.pdf)

V. Pambuccian, Review of Ivananov (ivanov.pdf)

#### • 16h: Volker Halbach (Philosophy, New College, University of Oxford)

#####

"The axiomatic approach to semantics"

**Summary: **

Etchemendy and others have criticized the model-theoretic account of logical truth and consequence. At the heart of the model-theoretic approach is the reduction of all semantic notions to set theory. I suggest that, at least for certain purposes, we should abstain from a reduction of all semantic notions to set-theory and investigate whether the problems of the model-theoretic account can be overcome by an axiomatic approach to semantic notions. In particular, I will present an account of logical truth and consequence that relies on an axiomatized notion of truth that is supposed to capture the formality constraint of logical truth and consequence.

**Reading:**

Halbach, V. "The substitutional analysis of logical consequence"** halbach.consequence.pdf**

### Meeting 3: Thursday, June 9th

####

• 16h: Volker Peckhaus (Philosophy, Universität Paderborn)

#####

"Zermelo's Set Theory and Hilbert's Philosophy of Axiomatics"

**Summary:**

Ernst Zermelo's early contributions to set theory including his papers on the well-ordering theorem and his axiomatization of set theory are closely related to David Hilbert's early axiomatic program. Originally formulated for deepening the foundations in geometry, arithmetic moved into the focus of the program in order to get rid of relative consistency proofs. Set theory was seen as a competing program for the foundation of mathematics which might be the reason that the Zermelo-Russell paradox was not taken seriously in the beginning, but at least since Hilbert's 1900 Paris lecture on "Mathematical Problems" set theory was included among the research topics in Göttingen. In his early publications Zermelo, who was working in Göttingen at that time, followed closely Hilbert's guidelines. He deviated, however, from Hilbert's constructive approach in his later writings on infinite hierarchies of sets.

**Reading:**

V. Peckhaus, Pro and Contra Hilbert: Zermelo’s Set Theories (pro_contra_hilbert._zermelo_s_set_theories.pdf)

### Meeting 4: Tuesday, June 14th

####

• 14h: David Rabouin (SPHERE, Université Paris 7-Diderot)

#####

"‘En ne laissant passer aucun axiome sans preuve’. On the need to demonstrate axioms in Leibniz"

**Summary:**

It is often believed that axioms were considered as self-evident truths until the XIXth Century, where a radical shift occurred in this regard in mathematical thinking. Leibniz, however, offers a striking counter example. On many occasions, he claims that mathematical axioms have nothing self-evident in them and should always be demonstrated (until one reaches what he calls “identicals” and mere definitions): aucun axiome sans preuve! Moreover, he regularly ascribes this demonstrative strategy to authorities such as Roberval and Pascal, and before them, Proclus and Apollonius. In this talk, I will first present this long tradition on the need to demonstrate axioms. As we will see, there are testimonies indicating that mathematical axioms (and not only postulates) were already the object of heavy debate in antiquity. In the second part of the talk, I will show how Leibniz turned this ancient motto into a general strategy in order to reform mathematics. This proved to be to be a very fruitful strategy and allowed him to initiate entirely new fields of mathematical inquiry. I will give several examples of these various experiments in axiomatization.

**Reading:**

D.Rabouin, "*Analytica Generalissima Humanorum Cognitionum.* Some reflections on the relationship between logical and mathematical analysis in Leibniz", *Studia Leibnitiana* 45(1) (2013): 109--130.

#### • 16h: Alain Prouté (Mathématiques, Université Paris 7-Diderot)

##### "The axiom of choice from an algorithmic viewpoint"

**Summary:**

The analysis we propose for the axiom of choice provides a clear definition of what "choosing" means (in contrast with "computing"), so explaining at the same time why some instances of this schema of axioms are non constructive and why the choice by itself remains nevertheless always effective. This pragmatic approach is likely to largely demystify the axiom of choice and was motivated by the realization of a proof assistant software based on topos theory.

The subject is highly linked to the question of the indiscernibility of proofs, which currently causes heated discussions within the microcosm of logicians and computer scientists. It is likely that our arguments are already known by some proof assistant experts, but the talk will be the occasion to exchange points of view on this question of indiscernibility of proofs, which is to our opinion the most salient feature of mathematics, compared to programming.

**Reading:**

G. Frege, "Über Sinn und Bedeutung" (English translation http://fitelson.org/125/Frege_on_sense_and_reference.pdf )

A. Prouté, "Sur quelques liens entre théorie des topos et théorie de la demonstration (http://www.logique.jussieu.fr/~alp/luminy_05_2007.pdf)

### Meeting 5: Thursday, June 16th

#### • 14h: Akihiro Kanamori (Mathematics, Boston University)

##### "Axioms as procedure and infinity as method, both in ancient Greek geometry and modern set theory"

**Summary:**

Axiomatics have become *de rigueur *in modern mathematics, especially with the development of mathematical logic and formalization. But separate from this, what about axiomatics arising *in* mathematical practice? This was in ancient Greek geometry and modern set theory, which act as parentheses for mathematics in several senses. For both, it is corroborated, by looking at the history and practice, that axioms (and definitions) serve to warrant procedures (e.g. constructions) and to regulate infinity as method. Axioms in practice are thus less involved with mathematical truth or metaphysical existence than with the instrumental properties of concepts. For Greek geometry, this view provides counterweight to the doxography (Plato, Aristotle, Proclus) and historiography (Zeuthen, Unguru controversy), which lean to interpreting constructions as establishing existence. For set theory, this view coheres with extensions of set theory through large cardinal and forcing axioms, and arms the eld of mathematics not as invested in some search for truth but as a study of well-foundedness through the transfinite.

**Reading:**

W. Knorr, "Construction as Existence Proof in Ancient Geometry", in J. Christianidis (ed.), *Classics in the history of Greek Mathematics*, 115--137, Kluwer (2004).

A. Kanamori, "The Mathematical Infinite as a Matter of Method", *Annals of the Japan Association for the Philosophy of Science *20 (2012): 1--13.

A. Kanamori, Notes for Intersem Lecture (ak.intersem.paris16_br.pdf)

#### • 16h: Juliet Floyd (Philosophy, Boston University)

#####

"Gödel on Russell and Axiomatic Method 1942-43"

**Summary:**

Gödel’s “MaxPhil” Gabelsburger notebooks IX-X (1942-3), begun on the day he accepted the invitation to write for Russell’s Schlipp volume, reveal a fascinating attempt on Gödel’s part to come to grips with Russell’s overall philosophy: not only the mathematical and logical aspects of *Principia*, but more purely philosophical ones. The distance between Gödel’s manner of working in these notebooks and his published tribute to Russell (1944) is considerable, evincing a more sophisticated and longstanding grappling with Russell than has been thought. In particular, Gödel did not uncritically hold that we can literally “see” sets, or that axioms “force themselves upon us” as physical objects do. Instead, he aimed in 1942-3 to *rigorize* Russell’s *Principia* idea of truth-as-correspondence, the “multiple relation theory” of judgment, rooted at the atomic level in (what Russell called) “judgments of perception”. Aware of Wittgenstein’s impact on Russell after 1918, and having read Russell’s main philosophical works through *Inquiry into Meaning and Truth*, Gödel holds out for an infinitary version of the multiple relation view, one that takes order, as well as our capacity for other-than-step-by-step, finistic thought, to be basic.

Every axiomatization leaves some interpretive residue behind, the trail where the human serpent brings philosophy and knowledge into the garden of analysis. Gödel’s picture of the rigorizations of truth in axiomatic systems, expressed in the following quote, will be analyzed:

A board game is something purely formal, but in order to play well, one must grasp the corresponding content [the opposite of combinatorically]. On the other hand the formalization is necessary for control [note: for only it is objectively exact], therefore knowledge is an interplay between form and content. [Max Phil IX 16]

**Reading:**

Floyd, J. "The Chains of Life", to appear (jf.chains_of_life.pdf)

Floyd, J. "Turing on "Common Sense": Cambridge Resonances", to appear in J. Floyd and A. Bokulich (eds), *Philosophical Explorations of the Legacy of Alan Turing: Turing 100,* Boston Studies in the Philosophy of Science, Springer, to appear (jf.turing.pdf)

Floyd, J. and A. Kanamori, "Gödel vis-a-vis Russell: Logic and Set Theory to Philosophy", in G. Crocco & E-M Engelen (eds), *Kurt Gödel: Philosopher-Scientist*, Press Universitaires de Provence, 2016

### Meeting 6: Tuesday, June 21st

####

• 14h: Jean Petitot (CAMS, EHESS)

##### 'The intertwining of structures in complex proofs"

**Summary:**

The concept of axiomatized structure, e.g. in Bourbaki's sense, has been deeply investigated philosophically, but its practical usefulness for working mathematicians and its methodological function in proofs have received less attention. The intertwining of very different abstract structures in specific mathematical objects plays a key role in complex proofs. We will give some examples.

[**Note:** Pr. Petitot will give much of his talk in French. The talk will be based on the following parts of the readings, which are available in English:

-- Texts of Connes and a text of Soulé. Will use some of their concepts.

-- Also pieces of the big file (i.e. the file named ``Petitot'') which shares the same underlying philosophy and some of whose technical parts can help:

1. pp.1-12: Unity of mathematics, structural concepts in complex proofs.

2. pp. 42-56. Elliptic curves and L-functions.

3. pp. 57-69; Zeta function (most important).

4. pp. 74-98. Elliptic functions and modular forms.]

##### Reading:

Petitot petitot.redux.pdf

Soulé soule_coursf1.pdf

Connes connes_epicyclic_site_2014.pdf

####

• 16h: Michael Hallett (Philosophy, McGill University)

#####

"Hilbert's Axiomatic Method [AM]: Logic and Set Theory"

**Summary: **

This paper sets out to do two things. First, it will emphasise one of the most important things about Hilbert’s AM, namely that axioms are *schemata*, in which no restriction is placed on the ‘inner nature’ of the primitives, and this is meant to be the case for *all* mature theories. [This is intimately connected to his rejection of what he calls ‘the genetic method’ for the presentation of theories, and also the search for independent axioms.] This makes it clear that axioms can be isolated from a specific context, and above all, it means that much of the weight of the logical investigation will fall on the search for different mathematical *models* of the axioms, which might be from very diverse mathematical fields, with very different interpretations of the primitives. The creative use of disparate models was *crucial* in Hilbert’s initial logical investigation of geometrical theories, often using ‘higher mathematics’, and it was this above all which enabled Hilbert to achieve extraordinary results in geometry. According to Hilbert, the AM was meant to apply to all theories, so in particular to what we might think of as *foundational* theories, including the theory of sets. On the face of it, these theories are different, not least because we presumably expect the interpretation of the primitives to be more circumscribed. I want to argue, however, that a wide conception of *reinterpretation* was still crucial for the investigation of the theory, especially later, and that the use of 'higher mathematics' was also of great importance here.

##### Reading:

Hallett 1 (8.4.1, 8.4.2 and 8.4.3): hallett2008purity_and_hilbert.pdf

Hallett 2 (sections 3 and 4) hallett2010absoluteness_and_the_skolem_paradox.pdf

###

Meeting 7: Thursday, June 23rd

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• 14h: Sebastien Gandon (Philosophie, Université Blaise Pascal)

#####

"Axiomatization as Analysis"

**Summary:**

Russell's conception of axiomatization is puzzling. On the one hand, axiomatization plays a central role in logicism, since the whole of mathematics (and not only arithmetic as in Frege) is said to be deducible from a few logical principles; on the other, Russell's view of axiomatization is markedly different from the contemporary Hilbertian approach. In this talk, my aim is to put Russell's conception in its historical context. Discussing some writings of Moore and Sidgwick, I will suggest that Russell's view of axiomatization takes its roots in the debate opposing utilitarians and intuitionists in XIXth Century moral philosophy.

**Reading:**

Moore, G. E., *Principia Ethica*, §§9-15, §§39-46.

Levine J., "The Place of Vagueness in Russell's Philosophical Development" in, Sorin Costreie (ed), *Early Analytic Philosophy-New Perspectives on the Tradition*, Cham, Switzerland, Springer, 2016, pp. 161 - 212.

### Meeting 8: Tuesday, June 28th

#### • 16h: Gilles Dowek (Deducteam, INRIA, Paris)

##### "Are there good and bad axioms?"

**Summary: **

The axiomatic method suggests that any set of axioms forms a theory,

even those jeopardizing the cut-elimination property, the witness

property of constructive proofs, and even consistency. When trying to

replace axioms with rewrite rules, in order to recover these

properties, we are led to make a distinction between "good" and "bad"

axiomatic theories: those that can and those that cannot be expressed

as rewrite rules. We shall, in particular, discuss the differences

between two formulations of arithmetic: Peano's original one and the

more modern presentation where Peano's axioms 1 and 2 have been

dropped.

##### Reading:

Dowek, G., *Proofs in Theories, *dowek.proofs_in_theories.pdf chapter 4.

♦ **Sponsors: University of Notre Dame, Université de Paris 7-Diderot, PICS 'Proof and Computation' Project**

**♦ Directors: Pr. Michael Detlefsen (University of Notre Dame), Pr. Jean-Jacques Szczeciniarz (Université de Paris 7-Diderot**)