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 research unit SPHERE (UMR7219) of the University of Paris Diderot

The seventh annual PhilMath Intersem (PhilMath Intersem 7) will take place this coming June 2016. The theme will be the axiomatic method and its uses in the history of mathematics. Historical, philosophical and logical issues will be considered.

Abstracts, advices of reading are available on this website: http://mdetlefsen.nd.edu/philmath-intersem/philmath-intersem-7/

Contact: **Emmylou Haffner** (University of Lorraine & SPHERE),

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

Thema 2016: **Axiomatic method and its uses in history of mathematics **

To sessions: June 2, 7, 9, 14, 16, 21, 23, 28

- 14:00
Pascal Crozet (CNRS, SPHERE)
*Thābit ibn Qurra and the fifth postulate*

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.

- 16:00 Vincenzo De Risi (Max Planck Institute for the History of Science, Berlin)
*The axiomatization of space in the modern editions of Euclid’s*Elements.

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.

Tues. June 7

- 14:00
Victor Pambuccian (Mathematics, Arizona State University)
*Why should one follow the axiomatic method in geometry? An apology.*

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 historic 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.

- 16:00 Volker Halbach (Philosophy, New College, University of Oxford)
*The axiomatic approach to semantics.*

- 14:00
Volker Peckhaus (Wissenschaftstheorie und Philosophie der Technik, Universität Paderborn)
*Zermelo’s Set Theory and Hilbert’s Philosophy of Axiomatics.*

Thu. June 6

- 14:00
Akihiro Kanamori (Mathematics, Boston University)
*Axioms as procedure and infinity as method, both in ancient Greek geometry and modern set theory.*

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.

- 16:00 Juliet Floyd (Philosophy, Boston University)
*Gödel on Russell and Axiomatic Method 1942-43*

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].

- 14:00
Jean Petitot (CAMS, EHESS, École Polytechnique)
*The intertwining of structures in complex proofs.*

- 16:00 Michael Hallett (Philosophy, McGill University)
*Hilbert’s Axiomatic Method: Logic and Set Theory.*

- 16:00 Sébastien Gandon (Philosophie, Université Blaise Pascal)
*Axiomatization as Analysis*

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.

- 16:00 Gilles Dowek (INRIA)

tba

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

Also in this section :

- Seminar SAW 2015-2016: Exploring 19th and 20th centuries historiographies of mathematics in the ancient world
- Séminar "(Id)entity :: (Id)entification" 2015-2016
- Seminar of Philosophy and mathematical physics 2015–2016
- Science and Philosophy from Ancient to Modern Age 2015–2016
- Powers of Imagination. Historical Approach.2015–2016
- Meetings HPS of Paris Diderot 2015–2016
- AXIS HISTORY AND PHILOSOPHY OF MATHEMATICS
- History and Philosophy of Mathematics 2015–2016
- Reading Mathematical Texts 2015–2016
- "Arabic" Mathematics 2015–2016
- Mathematics in the Renaissance 2015–2016
- Mathematics at Modern Age 2015–2016
- Mathematics 19th and 20eth, History and Philosophy 2015–2016
- AXIS HISTORY AND PHILOSOPHY OF SCIENCE OF NATURE<BR>
- History and Philosophy of Physics 2015–2016
- Averroes’s Cosmology: The Middle Commentary on the De Caelo of Aristotle 2015–2016
- History of Light 2015–2016
- Striving for Coherence : Readings in Averroes’ Incoherence of the Incoherence 2015–2016
- Power and Acting in Averroes: the Great Commentary to Metaphysics Theta 2015–2016
- AXIS HISTORY AND PHILOSOPHY OF MEDECINE