Hilbert’s program then and now

Zach, Richard. 2007. “Hilbert’s Program Then and Now.” In Philosophy of Logic, edited by Dale Jacquette, 5:411–47. Handbook of the Philosophy of Science. Amsterdam: North-Holland. DOI: 10.1016/B978-044451541-4/50014-2

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s.

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Teaching Logic from Historical Sources

This is an interesting project: teach discrete mathematics not from a textbook, but using the historical papers that first dealt with the topics taught. A bunch of mathematicians and computer scientists at New Mexico State are doing that, and they’re asking for your help: try it out in your courses, write them letters of support for NSF funding. They have two modules on logic: one on set theory (Cantor), and one on computability (Turing’s 1936 paper). Here’s their email that came over FOM yesterday.

A team of mathematicians and computer scientists at New Mexico State University and Colorado State University at Pueblo has developed an innovative pedagogical technique for teaching material in discrete mathematics, combinatorics, logic, and computer science, with National Science Foundation support for a pilot project. Topics are introduced and studied via primary historical sources, allowing students to participate in the sense of discovery, and to appreciate and gain motivation from the context in which concepts were developed.

For example, we have authored classroom modules in which students learn mathematical induction from Pascal’s “Treatise on the Arithmetical Triangle,” written in the 1660’s. Another module develops the short recursion relation for the Catalan numbers from a seminal paper of G. Lame in 1838 (based on a start by Euler!!) We also have authored modules on binary arithmetic, based on the original historical sources by Leibniz and von Neumann; on infinite sets, based on original historical sources by Cantor; and on Turing machines, and Church’s Thesis, based on original historical sources by Goedel, Church, Turing, and Kleene.

We have authored 18 modules so far; all these modules and more information can be found at www.math.nmsu.edu/hist_projects/. The modules will appear in a chapter of a forthcoming MAA resource book for teaching discrete mathematics. We found that 65% of the students who completed a course with these historical projects performed equally well or better than the mean GPA in subsequent mathematics and computer science courses.

We are seeking to expand our pilot program with further major support from the National Science Foundation to create a full book with a comprehensive collection of classroom projects based on historical sources. We would like to invite any instructors of mathematics or computer science courses to agree to site test future projects in related courses in discrete mathematics, combinatorics, logic, or computer science, or perhaps even to design your own projects. We hope to be able to provide a little NSF support as travel and/or consulting for site testers.

If you think that you (or a colleague) would be interested in teaching with a project during 2008-2011, we would like to hear from you. We plan to finalize our new NSF proposal by mid-December, and would like to attach a brief letter of support from you if you are interested. It would be nice if it indicated the institution, the course, nature of students, rough timeframe, why you think it would be good for your students, and possible choice of projects for your testing.

Contact persons:
Guram Bezhanishvili (gbezhani@nmsu.edu)
Jerry Lodder (jlodder@nmsu.edu)
David Pengelley (davidp@nmsu.edu)

Universal Logic in China

2nd World Congress and School on Universal Logic
Call for papers
Xi’an, China, August 16-22, 2007

This event is the second in a series of events whose objective is to gather logicians from all orientations (philosophy, mathematics, computer science, linguistics, artificial intelligence etc) – people not focusing only on some specific systems of logic or some particular problems, but inquiring the fundamental concepts of logic.

There will be a four days school with about 20 tutorials followed by a 3 days congress. Among the participants there will be Walter Carnielli, Hartry Field, Valentin Goranko, Vincent Hendricks, Wilfrid Hodges, Istvan Nmeti, Gabriel Sandu, Stan Surma, Heinrich Wansing and many others.

The deadline for submission of contributed papers is March 15, 2007. There will also be a contest with subject: How to translate a logic into another one?

This event will take place in Xi’an, the ancient capital of China, just after the 13th LMPS to happen in Beijing. For further information, please visit the website below:


Applied Logic Job in Darmstadt

The Department of Mathematics of the Darmstadt University of Technology (TUD) invites applications for a

Juniorprofessorship in Mathematics – Applied Logic (W1 B BesG)

to be filled 1.4.2007.

Applicants must be qualified in Applied Logic. Applications from candidates working in “Mathematical Proof Theory” (e.g. proof mining, proofs as programs, reverse mathematics, constructive formal systems) and/or “Computability in Mathematics” (e.g. effective algebra and analysis, symbolic computation) are particularly welcome.

Besides pedagogical skills the ability to conduct high quality scientific research must be proven by a PhD of exceptionally high quality and additional research papers. The duration of the PhD studies plus subsequent academic employment should in total not have exceeded 6 years.

Candidates should be willing to cooperate with colleagues from neighboring areas in the department as well as be open towards problems in other scientific areas such as Computer Science.

Duties of the successful candidate are to take part in the teaching provided by the department, to conduct original research in the area mentioned above and to further develop pedagogical skills. Willingness to take part in academic administrative issues is also expected.

The employment is initially for 3 years. After a successful evaluation as professor it will be extended for another 3 years.

Darmstadt University of Technology aims at increasing the number of women among the faculty and particularly encourages applications from women. Disabled candidates are considered — given equal qualification — with preference.

Applicants are asked to have 3 letters of recommendation sent to the Dekan of the Department of Mathematics of TUD.

Applications with the usual documents (CV, list of publications, teaching record) should be sent (referring to the Kenn-Nr. 337 of the position) by 31.12.2006 to: Dekan, Department of Mathematics, Darmstadt University of Technology, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany.

The official German version of this advertisement can be found at http://www.tu-darmstadt.de/pvw/dez_iii/stellen/337.tud

Canadian PhD programs in the 2006-08 PGR

table.lines td { vertical-align: top; border: 1px dashed gray; empty-cells: show; padding: 2pxWith the kind permission of Brian Leiter, here’s a breakout of the Canadian philosophy departments by specialty according to the Philosophical Gourmet Report 2006-08. The same programs are ranked in 2006-08 as in the 2004-06 edition. This year, only the rank ordering of the top four departments was given in the PGR. As two years ago, I’m providing the the rank ordering based on both the entire survey responses and the responses from Canadian evaluators (with mean scores in parentheses). The numbers following the specialties are: the peer group the program falls in and the rounded mean/median score. The “Notable” category (median of 3.0) is no longer included in the PGR (according to Brian Leiter, merely for reasons of time). See the overall rankings and the specialty rankings from the PGR for explanations.

PS: Because Canadian students wanting to study at a Canadian school don’t exactly have many options for any given specialty, you might consider consulting last year’s rankings as well. That still gives the “notable” category for the various specialties (ie, median scores of 3.0 that just barely didn’t make the official rankings).

Program Ranked Specialties
University of Toronto
Overall rank: 1 (3.7)
Canada rank: 1 (4.1)
Philosophy of Language (4/24-36, 3/3)
Philosophy of Mind (3/13-25, 3.5/3.5)
Metaphysics (5/25-36, 3.0/3.0)
Philosophical Logic (3/13-21, 3.5/4)
Normative Ethics and Moral Psychology (3/6-12, 4.0/4)
Political Philosophy (2/4-13, 4.0/4)
Philosophy of Law (2/3-11, 4.0/4)
Applied Ethics (2/7-18, 3.5/3.5)
Philosophy of Science (4/13-29, 3.5/3.5)
Philosophy of Biology (3/8-16, 3.5/3)
Philosophy of Cognitive Science (4/12-27, 3.0/3.5)
Philosophy of Social Science (4/16-34, 3.0/3.25)
Philosophy of Mathematics (4/13-22, 3.5/3.5)
Mathematical Logic (3/10-19, 3.5/3.5)
Ancient Philosophy (2/2-4, 4.5/4.5)
Medieval Philosophy (1/1-3, 5.0/5.0)
Early Modern: 17th C (2/2-11, 4.0/4.0)
Early Modern: 18th C (1/1-7, 4.0/4.0)
Kant and German Idealism (4/9-18, 3.5/3.5)
19th C Continental Philosophy after Hegel (4/18-27, 3.0/3.25)
American Pragmatism (1/1-3, 4.0/4.0)
20th Century Continental (4/17-33, 3.0/3.5)
Feminist Philosophy (2/3-13, 4.0/4.0)
University of Western Ontario
Overall rank: 2 (2.7)
Canada rank: 2 (3.1)
Philosophy of Language (4/24-36, 3/3)
Applied Ethics (3/19-42, 3.0/3.25)
Philosophy of Science (3/4-12, 4.0/4)
Philosophy of Physics (3/4-10, 4.0/4.0)
Philosophy of Social Science (3/7-15, 3.5/3.75)
Decision, Rational Choice, and Game Theory (5/10-19, 3.0/3.0)
Philosophy of Mathematics (4/13-22, 3.5/3.5)
Mathematical Logic (4/20-24, 3.0/3.0)
Medieval Philosophy (5/15-25, 3.0/3.5)
Early Modern: 17th C (3/12-33, 3.5/4)
Early Modern: 18th C (2/8-14, 3.5/4.0)
Feminist Philosophy (4/23-27, 3.0/3.0)
McGill University
Overall rank: 3 (2.4)
Canada rank: 3 (2.8)
Philosophy of Art (3/7-13, 4.0/4.25)
Philosophy of Mathematics (5/23-32, 3.0/3.75)
Ancient Philosophy (5/13-21, 3.0/3.0)
Medieval Philosophy (5/15-25, 3.0/3.25)
Early Modern: 17th C (3/12-33, 3.5/3.5)
Early Modern: 18th C (3/15-39, 3.0/3.0)
Kant and German Idealism (5/19-32, 3.0/3.5)
University of British

Overall rank: 4 (2.2)
Canada rank: 4 (2.6)
Philosophy of Art (4/14-21, 3.5/4)
Philosophy of Science (4/13-29, 3.5/3.5)
Philosophy of Biology (3/8-16, 3.5/3.5)
Philosophy of Social Science (4/16-34, 3.0/3.0)
History of Analytic Philosophy (4/18-37, 3.0/3.25)
University of Alberta
Overall rank: 5 (2.1)
Canada rank: 4 (2.6)
Philosophy of Art (5/22-28, 3.0/2.75)
Feminist Philosophy (2/3-13, 4.0/4.0)

Overall rank: 6 (2.0)
Canada rank: 6 (2.5)
Political Philosophy (3/14-27, 3.5/3.75)
Applied Ethics (3/19-42, 3.0/3.25)
Feminist Philosophy (2/3-13, 4.0/4.0)
Simon Fraser University
Overall rank: 6 (2.0)
Canada rank: 7 (2.4)
Philosophical Logic (4/22-36, 3.0/3)
University of Calgary
Overall rank: 6 (2.0)
Canada rank: 7 (2.4)
Philosophical Logic (4/22-36, 3.0/3)
Philosophy of Action (incl. Free Will) (4/13-19, 3.0/3)
Philosophy of Biology (4/17-23, 3.0/3.0)
York University
Overall rank: 9 (1.8)
Canada rank: 10 (1.9)
Philosophy of Law (4/21-33, 3.0/3)
American Pragmatism (3/7-10, 3.0/2.75)
Tri-University (Guelph, McMaster, Laurier)
Overall rank: 9 (1.8)
Canada rank: 9 (2.1)
Philosophy of Law (4/21-33, 3.0/3)
Early Modern: 18th C (3/15-39, 3.0/3.0)
History of Analytic Philosophy (incl. Wittgenstein) (4/18-36, 3.0/3.0)
University of Waterloo
Overall rank: 11 (1.7)
Canada Rank: 10 (1.9)

Hilbert in Kyoto

I just spent a wonderful week in Kyoto at the invitation of Susumu Hayashi. Susumu’s been working on Hilbert’s notebooks, and he, Mariko Yasugi, Wilfried Sieg, Koji Nagatogawa, and I have had several days of interesting discussions about them. The last two days there was a workshop on Hilbert and computability, and it was a pleasure to see and talk to Yasuo Deguchi, Anton Setzer, Toshi Arai, and many others. Many thanks to Susumu and his students, and in particular to Koji, without whose help and translation services Wilfried and might have gotten lost, starved to death, and certainly wouldn’t have had as good a time.

If you read German, check out Susumu’s students’ compilation of Hilbert’s maxims from the notebooks.