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Thursday, April 01, 2010

Truth Values

Just up on the SEP, by Yaroslav Shramko and Heinrich Wansing, an entry on truth values.

Wednesday, March 24, 2010

Rózsa Péter

Happy Ada Lovelace Day!

Rózsa Péter (1905-1977) was a Hungarian mathematician and early contributor to the theory of (primitive) recursive functions. She received her PhD in 1935 from (what is now) Eötvös Loránd University in Budapest. Her fellow student Laszlo Kálmár had introduced her a few years earlier to the then brand-new work of Gödel, and she proceeded to study the class of (primitive) recursive functions first clearly defined by Gödel in his 1931 incompleteness paper. In a number of articles in the 1930s, she laid the groundwork for the study of hierarchies of sub-recursive functions and clarified the notion of primitive recursive function. I'll just mention four of her contributions on the subject: In her paper, "Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktion" (Math. Ann., 1935) she showed that course-of-values recursion and nested recursion can be reduced to ordinary primitive recursion. In "Konstruktion nichtrekursiver Funktionen" (Math. Ann., 1935), Pétér simplified and expanded on Ackermann's work, and proved that there are multiply recursive but not-primitive recursive functions. In "Über die mehrfache Rekursion" (Math. Ann., 1937), she studied multiple recursion in more detail and showed that the hierarchy of k-recursive functions is proper. In "Zusammenhang der mehrfachen und transfiniten Rekursionen" (JSL, 1950), she proved the equivalence of k-fold recursion and transfinite recursion along ωk. Her early work on primitive recursive function theory is set out in her monograph, Rekursive Funktionen (1951), translated into English as Recursive Functions (1967). She also wrote a popular book on mathematics, Playing with Infinity, which was translated into 14 languages.

Pétér was barred from teaching in 1939 due to her Jewish heritage, but obtained positions at the Budapest Teacher's College in 1945 and at her alma mater in 1955. She was the first female mathematician to be elected to the Hungarian Academy of Sciences. She retired in 1976.

Women in Science (San Diego Supercomputer Center)
Biographies of Women Mathematicians (Agnes Scott College)
MacTutor History of Mathematics Archive

Philosophy of Mathematical Practice Online

If you have access to Oxford Scholarship Online, you can now read Mancosu's excellent collection The Philosophy of Mathematical Practice via the internets.
Contemporary philosophy of mathematics offers us an embarrassment of riches. But anyone familiar with this area will be aware of the need for new approaches that will pay closer attention to mathematical practice. This book provides a unified presentation of this new wave of work in philosophy of mathematics. This new approach is innovative in at least two ways. First, it holds that there are important novel characteristics of contemporary mathematics that are just as worthy of philosophical attention as the distinction between constructive and non constructive mathematics at the time of the foundational debates. Secondly, it holds that many topics that escape purely formal logical treatment — such as visualization, explanation, and understanding — can be nonetheless be subjected to philosophical analysis. The book comprises an introduction and eight sections. Each section consists of a short introduction outlining the general topic followed by a related research article. The eight topics selected represent a broad spectrum of contemporary philosophical reflection on different aspects of mathematical practice: visualization, diagrammatic reasoning and representational systems, mathematical explanation, purity of methods, mathematical concepts, philosophical relevance of category theory, philosophical aspects of computer science in mathematics, philosophical impact of recent developments in mathematical physics.

Tuesday, March 23, 2010

Robin Milner, 1934-2010

Robin Milner died on March 20. He was a leading theoretical computer scientist who developed the LCF theorem prover, the ML programming language, and introduced the π-calculus. He was founding director of the Laboratory for Foundations of Computer Science at the University of Edinburgh and then Professor of Computer Science at Cambridge. Milner was a Fellow of the Royal Society, a Fellow of the ACM, and winner of the Turing Award.

Monday, March 08, 2010

People Who Oscillate

From today's mini-AIR:

The Oscillating Humans Project, announced here, is searching for a living specimen - an exemplar - of an oscillating human.

DEFINITION: For purposes of the project, an Oscillating Human is someone who consistently, repeatedly, over many years, expresses opinions directly opposite to opinions he or she expressed earlier, always ignoring and/or denying the existence of copious, easily found clear documentation of the earlier opinions.

PURPOSE: The exemplary person, once identified, will serve as an example for teachers to use in logic classes. To minimize the chance of lawsuits, the exemplar must be a "public person", with (as stated above) for whom there is copious, easily found, clear documentation of years and years of oscillation.

If you know of an outstanding specimen, please send:

1. The name and a 20-word biographical sketch of the person.
2. Several URLs pointing to clear, unarguable documentation.

Send to: OSCILLATING HUMANS PROJECT: marca AT improbable.com

NOTE: This is an education project. It is NOT an exercise in naming people you don't like. No screeds, please.

Oscillating Humans Literature Review

Published research about this form of human oscillation may be scarce. One of the few apparently relevant items — judging it by its title, if not by its contents, is this British study:

"Oscillation of Human Performance as a Personality Measure," Michael A. Tainsh, Perceptual and Motor Skills, vol. 35, no. 2, October 1972, pp. 677-8.

Truly pertinent citations will be welcomed.

Friday, February 26, 2010

Putting God in Gödel

From Abstruse Goose:

Friday, February 19, 2010

Attack on Logicians at King's College London

Sorry for the long silence...

You may have heard by now, but in case you haven't: The Group in Logic, Language, and Information at King's College is threatened by "budget cuts": looks like the administration is just willfully destroying it by firing several faculty.

Information and links to protest sites etc. given here.

Monday, October 19, 2009

The Development of Modern Logic Online

Leila Haapaaranta's collection The Development of Modern Logic came out earlier this year. It's a handy one-volume compendium to the history of logic in the modern era (full disclosure: I have an article in it). The price tag might still be a bit steep: $150, although that buys you over 1,000 pages of scholarship in an attractive hardback volume! But if you have access to Oxford Scholarship Online, you can now also read the book over the intertubes.

Also online now: JC Beall's Spandrels of Truth.

Videos from Foundational Adventures Conference

Last May, Ohio State had a conference in honor of Harvey Friedman's 60th birthday. Videos of the talks are now available (via Neil Tennant). These include talks by Friedman himself, as well as John Burgess, Sam Buss, Mic Detlefsen, Sol Feferman, Hartry Field, Rohit Parikh, Grisha Mints, Wilfried Sieg, Ted Slaman, Patrick Suppes, and many others.

T-Rex on Hilbert's Infinite Hotel

Today on Dinosaur Comics:

Wednesday, October 14, 2009

Per Lindström, 1936-2009

From the ASL Newsletter, I just learned that Per Lindström died two months ago:
Per (Pelle) Lindström, the Swedish logician, died in Gothenburg, Sweden, on August 21, 2009, after a short period of illness. He was born on April 9, 1936, and spent most of his academic life at the Department of Philosophy, University of Gothenburg, where he was employed first as a lecturer ('docent') and, from 1991 until his retirement in 2001, as a Professor of Logic. Lindström is most famous for his work in model theory. In 1964 he made his first major contribution, the so-called Lindström's test for model completeness (c.f., Chang & Keisler, Model Theory, 3rd ed., Thm. 3.5.9: if a countable set of first-order sentences has only infinite models, is categorical in some infinite power, and is such that the set of its models is closed under unions of chains, then it is model complete). In 1966 he proved the undefinability of well-order in Lω1ω (obtained independently and in more generality by Lopez-Escobar), an early example of the use of recursion theory to obtain model-theoretic results. The same year he also introduced the concept of a Lindström quantifier, which has now become standard in model theory, theoretical computer science, and formal semantics. The paper also contains a characterization of elementary logic among logics with generalized quantifiers, generalizing a result by Mostowski. The proof uses Lindström's version of what is now known as Ehrenfeucht-Fraissé (EF) games, a concept he came up with independently. Another paper from 1966 ("On relations between structures") gives a powerful and extremely general formulation of a preservation/interpolation theorem, again based on EF games. These results were published in the Swedish philosophical journal Theoria and written in an extremely terse style, which had the effect that they escaped the notice of most of the logic community for a while. It was his 1969 paper "On extensions of elementary logic" (also in Theoria), where he presented his famous characterizations of first-order logic---Lindström's Theorem---in terms of properties such as compactness, completeness, and Löwenheim-Skolem properties, that was first recognized as a major contribution to logic. It laid the foundation of what has become known as abstract model theory (c.f., Barwise & Feferman (eds.), Model-Theoretic Logics, 1975). The proof was based on EF games and on a new proof of interpolation, following the line of argument in the papers on relations between structures and Lindström quantifiers. Several other characterizations of first-order logic followed in later papers. Beginning at the end of the 1970's, Lindström turned his attention to the study of formal arithmetic and interpretability. He started a truly systematic investigation of this topic, which had been somewhat dormant since Feferman's pioneering contributions in the late 1950's. In doing so he invented novel technically advanced tools, for example, the so-called Lindström fixed point construction, a far-reaching application of Gödel's diagonalization lemma to define arithmetical formulas with specific properties. His approach to interpretability was based on the study of related lattices, such as the lattice of interpretability types over a fixed extension of Peano Arithmetic (PA), or the lattices of Σn- and Πn -sentences over PA, for some fixed n, and he established many interesting structural properties of these. Other memorable results include the Lindström-Solovay theorem that the interpretability relation between sentences over PA is Π20-complete and the characterization of faithful interpretability over PA as a combination of Π1- and Σ1-conservativity. In the 1990's, he also contributed to the area of provability logic: he gave a simplified proof of the de Jongh-Sambin fixed point theorem and characterized the bimodal logic of PA and PA augmented by the reflection rule: infer a sentence φ from 'φ is provable'.

Pelle Lindström had an exceptionally clear and concise style in writing mathematical logic. His 1997 book, Aspects of Incompleteness, remains a perfect example: it provides a systematic introduction to his work in arithmetic and interpretability. The book is short but rich in material; it also contains some results one cannot find in journal publications, for example, his solution to one of the 102 problems formulated by Harvey Friedman.

Throughout his life, Pelle Lindström also took an active interest in philosophy. He participated in the debate following Roger Penrose's new version of the argument that Gödel's Incompleteness Theorems show that the human mind is not mechanical. He presented his own philosophy of mathematics, which he called 'quasi-realism', in a paper in The Monist in 2000. It is based on the idea that the 'visualizable' parts of mathematics are beyond doubt (and that classical logic holds for them). He counted as visualizable not only the ω-sequence of natural numbers but also arbitrary sets of numbers, the latter visualizable as branches in the infinite binary tree, whereas nothing similar can be said for sets of sets of numbers, for example. Moreover, he made numerous contributions over the years to the Swedish popular philosophy journal Filosofisk Tidskrift---one of these will be published posthumously---on subjects as diverse as the freedom of will, the mind-body problem, utilitarianism, and counterfactuals.

Pelle Lindström will be remembered by the logic community as a great logician, and by his family, friends and colleagues as a remarkable human being.

Monday, October 12, 2009

Reforming Graduate Education

New book out from Princeton UP on the Graduate Education Initiative of the Andrew W. Mellon Foundation, discussed on Inside Higher Ed. Not sure if any philosophy departments participated. In light of previous discussion on differential attrition rates for women in the pipeline, this should be interesting:
Chapter 7 addresses a matter of continuing concern among students, their professors, and administrators. Do marriage and childbearing affect the chances men and women have of completing their degrees and of doing so promptly? Although these questions are not at issue in the GEI, they are important. As a result, we made sure the student survey would yield data on students’ marital status when they entered graduate school and whether they had children at the time. In light of the increasing numbers of women earning PhDs in all fields and their very significant representation in the humanities, having an understanding of the relationships linking gender, marital status, and parenthood and the collective impact of all three on completion and TTD is likely to become increasingly important in the years ahead. Gender differences on average favor men, but we find these differences are due solely to the fact that married men do better than single men and single women. Marriage benefits men but does not do the same for women.

Sunday, October 11, 2009

Women in the Academic Pipeline II

Following up on my previous post, Women in the Academic Pipeline, where I compared rates at which women earned BAs and PhDs in various fields in the US: what does it look like in the faculty ranks? Not surprisingly, the percentages in general go down as you go higher, but there are some interesting (and disturbing) things to notice. First, the data:
Teaching fieldBAPhDLecturer/

Biological sciences62.2%46.5%47.7%±7.5%37.9%±7.3%25.9%±5.9%20.4%±5.4%
Computer and information sciences25.1%22.0%31.9%±4.9%27.1%±11.6%31.6%±12.5%26.8%±14.3%
Mathematics and statistics46.0%28.1%42.3%±5.9%32.9%±13.2%24.5%±11.7%17.8%±7.2%
physical sciences41.7%27.8%31.6%±6.9%29.3%±10.4%19.1%±7.8%8.9%±4.4%
Social sciences50.9%42.6%33.1%±5.4%36.2%±8.7%32.6%±7.2%19.7%±4.5%

This data comes from the U.S. Department of Education, National Center for Education Statistics, 2004 National Study of Postsecondary Faculty (NSOPF:04) and was generated from a table generated using their convenient QuickStats feature. The BA and PhD percentages come from the previous post, for 2003-04 graduates.

The representations of women among Assistant Professors in philosophy (14%) is much lower than expected, and among Associate Professors (24%) much higher than expected. Why? Are the women getting stuck at the Associate Professor rank? In most fields women are better represented in the instructor ranks than in the PhD pool, except in engineering, the physical sciences, and philosophy. And in computer science, the line goes up and not down. Sign something they did in the 90s to increase women representation among faculty worked?

UPDATE: Prompted by Kenny's comment, I computed the errors on those figures, and since they are rather large for some data points (especially pfor philosophy), take these with a grain of salt! And ignore the last paragraph.

Leitgeb's "Untimely Review" of Carnap's Aufbau

Topoi has a series of "untimely reviews", where classic works of philosophy are reviewed as if they had just been published. Hannes Leitgeb did one on Carnap's Aufbau, where he not only pretends that it was just published, but also pretends (as I guess you'd have to if you take the premise seriously) that it wasn't published 80 years ago (philosophy would have looked very different). I would write more and link to it, but I discovered that Chris Pincock blogged this already three months ago (and I missed it/forgot about it), so I'll just send you over to his great blog. Also, read Chris's Philosophy Compass paper on the Aufbau! And: Hannes's serious, substantial, long-awaited paper "New life for Carnap's Aufbau?" is out in Synthese online first (free preprint in the philsci archive). Here many of the things he hints at in the review are spelled out.

Friday, October 09, 2009

Women in the Academic Pipeline

Catarina's comment on the previous post prompted me to find out what the pipeline looks like in philosophy, and so I went to the tables from the Digest of Education Statistics (of the US, tables of Bachelor's, master's, and doctor's degrees conferred by degree-granting institutions, by sex of student and field of study) and made a handy table plus graph:

Biological sciences BA Biological sciences PhD Computer sciences BA Computer sciences PhD Engineering BA Engineering PhD English BA English PhD Mathematics BA Mathematics PhD Philosophy BA Philosophy PhD Physical sciences BA Physical sciences PhD Social sciences BA Social sciences PhD

2006-07 60.1% 49.3% 18.6% 20.6% 16.9% 20.9% 68.3% 59.4% 44.1% 29.8% 31.2% 25.3% 40.9% 31.6% 49.8% 45.1%

2005-06 61.5% 49.2% 20.6% 21.7% 17.9% 20.2% 68.6% 59.3% 45.1% 29.5% 30.9% 26.8% 41.8% 30.0% 50.0% 43.3%

2004-05 61.9% 49.0% 22.2% 19.1% 18.3% 18.7% 68.5% 59.2% 44.7% 28.5% 29.7% 23.9% 42.2% 27.9% 50.5% 42.8%

2003-04 62.2% 46.5% 25.1% 22.0% 18.8% 17.7% 68.9% 60.3% 46.0% 28.1% 29.2% 31.4% 41.7% 27.8% 50.9% 42.6%

2002-03 61.9% 45.8% 27.0% 20.6% 18.7% 17.2% 68.8% 60.5% 45.8% 27.1% 32.2% 26.8% 41.2% 27.6% 51.5% 43.0%

2001-02 60.8% 44.3% 27.6% 22.8% 18.9% 17.3% 68.6% 58.5% 46.7% 29.0% 33.0% 23.6% 42.2% 28.0% 51.7% 43.1%

2000-01 59.5% 44.1% 27.7% 17.7% 18.2% 16.5% 68.4% 60.3% 47.7% 28.8% 31.4% 25.3% 41.2% 26.8% 51.8% 41.4%

1999-01 58.3% 44.1% 28.1% 16.9% 18.5% 15.5% 67.9% 58.8% 47.1% 25.0% 31.5% 30.1% 40.3% 25.5% 51.2% 41.2%

1998-99 56.5% 42.2% 27.1% 18.9% 17.7% 14.3% 67.4% 60.3% 47.8% 26.2% 30.3% 24.5% 39.9% 24.2% 50.5% 41.1%

1997-98 55.1% 42.5% 26.7% 16.3% 16.9% 12.2% 66.9% 59.1% 46.5% 25.7% 31.3% 28.0% 38.4% 25.2% 49.2% 40.8%

Click on the image to see a larger version.
The zig-zaggyness of the philosophy PhD line (dashed red) is probably just caused by the fact that there are relatively few philosophy PhDs awarded each year--under 400 versus between 1,100 and 8,000 for the other fields. Discuss.

NOTE: Evelyn Brister has collected these data for several years on the Knowledge and Experience blog. Be sure to check over there (click on the links on the left side) for additional info and discussion.

UPDATE: More pipeline data, now with faculty by rank!