Friday, 14 November 2014

In memoriam: Alexander Grothendieck

By Catarina Dutilh Novaes

Alexander Grothendieck, who is viewed by many as the greatest mathematician of the 20th century, has passed away yesterday after years of living in total reclusion. (To be honest, I did not even know he was still alive!) He was a key figure in the development of the modern theory of algebraic geometry, among others, but to philosophers and logicians he is perhaps best known as one of the major forces behind the establishment of category theory as a new foundational framework for mathematics.

So far, I’ve only seen obituaries in French (Libération and Le Monde), as Grothendieck (of German origin) lived almost all of his life in France (I expect that soon obituaries in English will be available too). His life story is almost as remarkable as his mathematical achievements: his father died in Auschwitz in 1942, while he was sent to a concentration camp in France with his mother. In all of his adult life, he was as passionate about pacifism as he was about mathematics (and perhaps even more), and continuously engaged in a number of activist initiatives like lecturing on category theory in a forrest of bombed Vietnam to protest against the war. He seems to have basically stopped doing any mathematics in the 1970s, but the influence of his work in the field is bound to remain colossal in many years or even centuries to come.

Terence Tao on prime numbers on the Colbert Report

H/T Tina Eliassi-Rad

(And comments that pick on Tao for (sort of) calling 27 a prime will be summarily deleted.)

Wednesday, 5 November 2014

Final CfP: 15th Congress of Logic, Methodology, and Philosophy of Science (CLMPS 2015)


University of Helsinki, Finland, 3-8 August 2015

Submission deadline: 30 November 2014

The Congress of Logic, Methodology and Philosophy of Science (CLMPS) is organized every four years by the Division of Logic, Methodology and Philosophy of Science (DLMPS). The Philosophical Society of Finland, the Academy of Finland Centre of Excellence in the Philosophy the Social Sciences (TINT) and the Division of Theoretical Philosophy (Department of Philosophy, History, Culture and Art Studies) are proud to host the 15th Congress of Logic, Methodology and Philosophy of Science (CLMPS 2015).
CLMPS 2015 is co-located with the European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium 2015 (the abstract submission for Logic Colloquium 2015 opens in early 2015).

CLMPS 2015 will host six plenary lectures and 40 invited lectures 

CLMPS 2015 Plenary Lectures

Rachel Ankeny
Steve Awodey
Patricia Blanchette
Harold Kincaid
Eleanor Knox
Johan van Benthem

CLMPS 2015 Invited Lectures

Anna Alexandrova
Sergei Artemov
Verónica Becher
Giovanni Boniolo
Mieke Boon
Cameron Buckner
Hasok Chang
Ellen Clarke
Mark Colyvan
Eleonora Cresto
William Demopoulos
Zoe Drayson
Solomon Feferman
Maria Carla Galavotti
Paul Griffiths
Sven Ove Hansson
Stephan Hartmann
Jenann Ismael
Dov Jaron
Juliette Kennedy
Rebecca Kukla
Barry Loewer
Edwin Mares
Michela Massimi
Gordon McBean
Kerry McKenzie
Uskali Mäki
Jennifer Nagel
Dean Rickles
Gillian Russell
Teddy Seidenfeld
Orly Shenker
Paul Thom
Moshe Y. Vardi
Michael Weisberg
Philip Welch
Charlotte Werndl
Alison Wylie
Sang Wook Yi
Kevin Zollman

CLMPS 2015 calls for contributed papers, contributed symposia, and affiliated meetings in 17 thematic sections:

A. Logic
A1. Mathematical Logic
A2. Philosophical Logic
A3. Computational Logic and Applications of Logic
A4. Historical Aspects of Logic

B. General Philosophy of Science
B1. Methodology
B2. Formal Philosophy of Science and Formal Epistemology
B3. Metaphysical Issues in the Philosophy of Science
B4. Ethical and Political Issues in the Philosophy of Science
B5. Historical Aspects in the Philosophy of Science

C. Philosophical Issues of Particular Disciplines
C1. Philosophy of the Formal Sciences (incl. Logic, Mathematics, Statistics, Computer Science)
C2. Philosophy of the Physical Sciences (incl. Physics, Chemistry, Earth Science, Climate Science)
C3. Philosophy of the Life Sciences
C4. Philosophy of the Cognitive and Behavioural Sciences
C5. Philosophy of the Humanities and the Social Sciences
C6. Philosophy of the Applied Sciences and Technology
C7. Philosophy of Medicine
C8. Metaphilosophy

In addition, some submitted abstracts will be invited to the International Union of History and Philosophy of Science (IUHPS) Joint Commission Symposium Sessions if the programme committee considers the abstracts well suited for IUHPS themes.

Contributed papers: Please submit an abstract of 300 words prepared for anonymous review. Accepted contributed papers will be allocated in total 30 minutes (20 min for the presentation + 10 min for the discussion).

Contributed symposia: Please submit an abstract of max. 1700 words prepared for anonymous review.

The abstract should include:

a. a general description of the format and the topic of the proposed symposium and its significance (up to 500 words)
b. a 300-word abstract of each paper (3-4 papers)

Each accepted contributed symposia will be allocated a full two-hour session.

Affiliated Meetings: Affiliated meetings are half-day to full day symposia that run parallel to the CLMPS 2015 programme, and belong to the congress programme. Please consult the CLMPS 2015 submission guidelines for further information.

 Rules for multiple presentations
+     Maximally one contributed individual paper
+     One is allowed to present a second paper of which one is a co-author, but then the main author of this paper must submit the paper and be registered as a participant.
+     If one participates in a contributed symposia proposal, affiliated meeting proposal or is an invited speaker, one is not allowed to submit an individual contributed paper in which one is the main author (it is possible to be a co-author of a contributed paper, but then the main author of this paper must submit the paper and be registered as a participant).

Abstracts should be submitted by using the CLMPS 2015 registration form.
Authors are kindly asked to consult the detailed submission guidelines before submitting.

All questions about submissions should be directed to the congress sectary, Ms. Päivi Seppälä
( The members of the programme committee, DLMPS committees and the local organising committee are listed here.

Hannes Leitgeb (Chair of the Programme Committee)
Ilkka Niiniluoto (Chair of the Local Organizing Committee)

Important dates

30 November, 2014
Deadline for abstract submissions

15 January, 2015
Congress registration opens

30 January, 2015
Notifications of acceptance

March 2015
Preliminary programme announced

31 May, 2015
Deadline for early registrations

13 July, 2015
No more refunds for registration cancellations

3-8 August, 2015
CLMPS 2015, University of Helsinki

Saturday, 18 October 2014

FINAL CFP and *EXTENDED DEADLINE*: SoTFoM II 'Competing Foundations?', 12-13 January 2015, London.

The focus of this conference is on different approaches to the foundations of mathematics. The interaction between set-theoretic and category-theoretic foundations has had significant philosophical impact, and represents a shift in attitudes towards the philosophy of mathematics. This conference will bring together leading scholars in these areas to showcase contemporary philosophical research on different approaches to the foundations of mathematics. To accomplish this, the conference has the following general aims and objectives. First, to bring to a wider philosophical audience the different approaches that one can take to the foundations of mathematics. Second, to elucidate the pressing issues of meaning and truth that turn on these different approaches. And third, to address philosophical questions concerning the need for a foundation of mathematics, and whether or not either of these approaches can provide the necessary foundation.

Date and Venue: 12-13 January 2015 - Birkbeck College, University of London.

Confirmed Speakers: Sy David Friedman (Kurt Goedel Research Center, Vienna),
Victoria Gitman (CUNY), James Ladyman (Bristol), Toby Meadows (Aberdeen).

Call for Papers: We welcome submissions from scholars (in particular, young scholars, i.e. early career researchers or post-graduate students) on any area of the foundations of mathematics (broadly construed). While we welcome submissions from all areas concerned with foundations, particularly desired are submissions that address the role of and compare different foundational approaches. Applicants should prepare an extended abstract (maximum 1,500 words) for blind review, and send it to sotfom [at] gmail [dot] com, with subject `SOTFOM II Submission'.

Submission Deadline: 31 October 2014

Notification of Acceptance: Late November 2014

Scientific Committee: Philip Welch (University of Bristol), Sy-David Friedman (Kurt Goedel Research Center), Ian Rumfitt (University of Birmigham), Carolin Antos-Kuby (Kurt Goedel Research Center), John Wigglesworth (London School of Economics), Claudio Ternullo (Kurt Goedel
Research Center), Neil Barton (Birkbeck College), Chris Scambler (Birkbeck College), Jonathan Payne (Institute of Philosophy), Andrea Sereni (Universita Vita-Salute S. Raffaele), Giorgio Venturi (CLE, Universidade Estadual de Campinas)

Organisers: Sy-David Friedman (Kurt Goedel Research Center), John Wigglesworth (London School of Economics), Claudio Ternullo (Kurt Goedel Research Center), Neil Barton (Birkbeck College), Carolin Antos-Kuby (Kurt Goedel Research Center)

Conference Website: sotfom [dot] wordpress [dot] com

Further Inquiries: please contact
Carolin Antos-Kuby (carolin [dot] antos-kuby [at] univie [dot] ac [dot] at)
Neil Barton (bartonna [at] gmail [dot] com)
Claudio Ternullo (ternulc7 [at] univie [dot] ac [dot] at)
John Wigglesworth (jmwigglesworth [at] gmail [dot] com)

The conference is generously supported by the Mind Association, the Institute of Philosophy, British Logic Colloquium, and Birkbeck College.

Wednesday, 15 October 2014

Visiting Fellowships at the MCMP (Academic Year 2015/2016)

The Munich Center for Mathematical Philosophy invites applications for visiting fellowships for one to three months in the academic year 2015/16 (15 October 2015 to 15 February 2016 or 15 April to 15 July 2016) intended for advanced Ph.D. students (“Junior Fellowships") and postdocs or faculty (“Senior Fellowships"). Candidates should work in general philosophy of science, the philosophy of one of the special sciences, formal epistemology, or social epistemology and have a commitment to interdisciplinary and collaborative work. To apply, send your application (ideally everything in one pdf file) to with the subject “Junior Fellowship Application” or “Senior Fellowship Application”. Candidates should include a letter of interest (which also indicates the period of the planned stay), a CV, and a project outline of no more than 1000 words. Candidates for a Junior Fellowship should additionally supply one letter of recommendation. We offer a tax-free stipend of 800 Euro/month for junior fellows and 1200 Euro/month for senior fellows to partly cover additional expenses such as housing and transportation to and from Munich. It is also possible to stay for a longer period (e.g. if you are on a sabbatical), but stipends will be for maximally three months.

We also encourage groups of two to four researchers, which may also include scientists, to jointly apply for fellowships (“Research Group Fellowships") to work on an innovative collaborative project from the above-mentioned fields which is of relevance for the research done at the MCMP and which ideally includes a member of the MCMP as a collaborator. To apply, send your application (if possible everything in one pdf file) to with the subject “Research Group Fellowship Application”. Interested groups should include a letter of interest (which also indicates the period of the planned stay), a CV of each group member, and a project outline of no more than 2000 words that also includes information about the intended output of the project. We offer a tax-free stipend of 800 Euro/month for junior group members and 1200 Euro/month for senior group members to partly cover additional expenses such as housing and transportation to and from Munich. It is also possible to stay for a longer period, but stipends will be for maximally three months.

The deadline for applications is 15 February 2015. Decisions will be made by 1 March 2015. For further information, please contact

3-year Doctoral Fellowship on “The Evolution of Unpopular Norms and Bullying"

The Munich Center for Mathematical Philosophy seeks applications for a Doctoral Fellowship. The successful candidate will work on the project "The Evolution of Unpopular Norms and Bullying” (project summary below), which is funded by the German Research Council (DFG) and part of the DFG Priority Programme “New Frameworks of Rationality”. The fellowship is open for candidates with a masters degree in philosophy or a related social science. The funding is for three years, and the fellowship should be taken up by January 1, 2015. However, a later starting date is also possible. (Please let us know if you wish to start at a later date.)

Applications (including a cover letter that addresses, amongst others, one's academic background and research interests, a CV, a list of publications (if applicable), a sample of written work of no more than 5000 words (ideally in English, but German is also acceptable), and a description of a planned research project of 1000-1500 words) related to the above mentioned project should be sent by email (ideally everything requested in one PDF document) by November 20, 2014. Hard copy applications are not accepted. Additionally, two confidential letters of reference addressing the applicant's qualifications for academic research should be sent to the same address from the referees directly.

The MCMP hosts a vibrant research community of faculty, postdoctoral fellows, doctoral fellows, master students, and visiting fellows. It organizes at least two weekly colloquia and a weekly internal work in-progress seminar, as well as various other activities such as workshops, conferences, summer schools, and reading groups. Several of our research projects are conducted in collaboration with scientists. The successful candidate will partake in all of MCMP's academic activities and enjoy its administrative facilities and financial support. The official language at the MCMP is English and fluency in German is not mandatory.

We especially encourage female scholars to apply. The LMU in general, and the MCMP in particular, endeavor to raise the percentage of women among its academic personnel. Furthermore, given equal qualification, preference will be given to candidates with disabilities.

Contact for informal inquiries:

Professor Stephan Hartmann (

Abstract: The Evolution of Unpopular Norms and Bullying

Although there is growing research about the relationship between individual and collective rationality, there has been relatively little work investigating the irrational behavior. The purpose of this project is to employ agent-based models to explain the evolution of norms that are collectively irrational. Unpopular social norms (e.g. feet-binding in China) are the most well-known examples, and this project will contribute to the small (but growing) literature on the emergence of unpopular norms. In addition to unpopular norms, the investigators plan to focus on an unexplored application: bullying. The project is jointly led by Stephan Hartmann (MCMP) and Conor Mayo-Wilson (University of Washington and MCMP).

For a more detailed description of the project, click here (PDF, 146 kb).

Tuesday, 14 October 2014

Call for Papers: Formal Epistemology Workshop 2015

May 20-22, 2015 (Wednesday to Friday)
Washington University in St. Louis

Keynote speakers:

Tom Kelly (Princeton), Jeff Horty (University of Maryland, College Park)

The Formal Epistemology Workshop will be held in connection with the 2015 meeting of the St. Louis Annual Conference on Reasons and Rationality (SLACRR), which will take place immediately before, from May 17-19, 2015.

There will be conference sessions all day on May 20 & 21, and in the morning on May 22.

Contributors are invited to send full papers as PDF files (suitable for presenting as a 40 minute talk) to by Friday, January 16, 2015. Papers should be accompanied by abstracts of up to 300 words. Identifying information about the author(s) (including obvious self-citations) should be removed from the body of the paper, but the name (and any other relevant information) should be included in the text of the e-mail.

Submissions should be prepared for anonymous review. Initial evaluation will be done anonymously. The final program will be selected with an eye towards maintaining diversity, so graduate students, people outside the tenure track, women, and members of underrepresented minorities are particularly encouraged to submit papers. We also welcome submissions from researchers in related areas, such as economics, computer science, and psychology. Past programs can be viewed here:

Submitting the same paper to both FEW and SLACRR is permitted (though the organizers will coordinate the paper selection in order to ensure that the same paper doesn’t get presented at both conferences).

Final selection of the contributed talks will be made by March 31, 2015.

There will be childcare available for conference participants who bring their children. It will be provided on site by a local certified childcare provider.

Organizers: Kenny Easwaran (Texas A&M), Julia Staffel (Washington University in St. Louis), Mike Titelbaum (UW Madison)

Conference webpage:

Thursday, 9 October 2014

The upside down world paradox

By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)

As most kids (I suspect), my daughters sometimes play ‘upside down world’, especially when I ask them something to which they should say ‘yes’, but instead they say ‘no’ and immediately regret it: ‘Upside down world!’ The upside down world game basically functions as a truth-value flipping operator: if you say yes, you mean no, and if you say no, you mean yes.

My younger daughter recently came across the upside down world paradox: if someone asks you ‘are you playing upside down world?’, all kinds of weird things happen to each of the answers you may give. If you are not playing upside down world, you will say no; but if you are playing upside down world you will also say no. So the ‘no’ answer underdetermines its truth-value, a bit like the no-no paradox. Now for the ‘yes’ answer: if you are playing upside down world and say ‘yes’, then that means ‘no’, and so you are not playing the game after all if you are speaking truthfully. But then your ‘yes’ was a genuine yes in the first place, and so you are playing the game and said yes, which takes us back to the beginning. (In other words, 'no' is the only coherent answer, but it still doesn't say anything about whether you are actually playing the game or not.)

I do not think the upside down paradox is of particular theoretical interest, but what struck me is that it arose in a fairly mundane situation, and was viewed as paradoxical by a 7-year old (who is admittedly the daughter of a philosopher of logic, fair enough…). She didn’t call it a paradox at first; she just said that this was a really difficult question to answer (‘are you playing upside down world?’); whatever you said, strange things happened. So this may well be a modest example of how Liar-like paradoxes may emerge even in everyday situations. (Hum, maybe I should write a paper with her, following the example of Veronique and Peter Eldridge-Smith on Pinocchio’s paradox.)

Monday, 22 September 2014

Winter School on Paradoxes and Dilemmas -- Groningen

On January 26th-27th 2015, the Faculty of Philosophy of the University of Groningen will host a short Winter School aimed at advanced undergraduate students and early-stage graduate students. The theme of the winter school is Paradoxes and Dilemmas, and it will consist of 6 tutorials where the topic will be discussed from different viewpoints: theoretical philosophy, practical philosophy, and the history of philosophy. As such, the Winter School may be of interest to at least some of the M-Phi readers; for further details, check the site of the Winter School.


  • Catarina Dutilh Novaes: ‘Paradoxes: at the heart of philosophy’
  • Barteld Kooi: ‘Epistemic paradoxes: is the concept of knowledge incoherent?’
  • Han -Thomas Adriaenssen : ‘Divine foreknowledge versus free will? Theology and modality in the Middle Ages’
  • Sander de Boer: 'So what were these Aristotelian forms supposed to do again? Late Medieval and Early Modern metaphysics'
  • Frank Hindriks: ‘Trolleyology: The Philosophy and Psychology of a Moral Dilemma’
  • Marc Pauly: ‘Philosophical Dilemmas in Public Policy: Ontology meets Ethics'

The Faculty is offering up to three EUR 300 scholarships for the best students enrolling in the winter school, and who express serious interest in later applying for the Research Masters’ program. Moreover, participants who are then accepted in the Research Masters’ program for the year 2015/2016 will have their registration fee for the winter school reimbursed.
To apply for the scholarships, send a short CV (max 2 pages) and a letter (max 1 page) stating your interest in the Faculty of Philosophy in Groningen and the Research Masters’ program in particular, to winterschoolphilosophy 'at' with 'Application for winter school scholarship' as subject. Deadline to apply for the scholarships: December 1st 2014. Preference will be given to members of underrepresented groups in philosophy (women, people of color, persons with disabilities etc.).

To register, send an email with your name, affiliation and status (undergraduate, graduate) to winterschoolphilosophy 'at' with 'Registration for winter school' as subject, no later than December 15th 2014. As the number of spots is limited, you are encouraged to register early.

Friday, 19 September 2014

Review of T. Parsons' Articulating Medieval Logic

By Catarina Dutilh Novaes
(Cross-posted at NewAPPS)

I was asked to write a review of Terry Parsons' Articulating Medieval Logic for the Australasian Journal of Philosophy. This is what I've come up with so far. Comments welcome!

Scholars working on (Latin) medieval logic can be viewed as populating a spectrum. At one extremity are those who adopt a purely historical and textual approach to the material: they are the ones who produce the invaluable modern editions of important texts, without which the field would to a great extent simply not exist; they also typically seek to place the doctrines presented in the texts in a broader historical context. At the other extremity are those who study the medieval theories first and foremost from the point of view of modern philosophical and logical concerns; various techniques of formalization are then employed to ‘translate’ the medieval theories into something more intelligible to the modern non-historian philosopher. Between the two extremes one encounters a variety of positions. (Notice that one and the same scholar can at times wear the historian’s hat, and at other times the systematic philosopher’s hat.) For those adopting one of the many intermediary positions, life can be hard at times: when trying to combine the two paradigms, these scholars sometimes end up displeasing everyone (speaking from personal experience).

Terence Parsons’ Articulating Medieval Logic occupies one of these intermediate positions, but very close to the second extremity; indeed, it represents the daring attempt to combine the author’s expertise in natural language semantics, linguistics, and modern philosophy with his interest in medieval logical theories (which arose in particular from his decade-long collaboration with Calvin Normore, to whom the book is dedicated). For scholars of Latin medieval logic, the fact that such a distinguished expert in contemporary philosophy and linguistics became interested in these medieval theories only confirms what we’ve known all along: medieval logical theories have intrinsic systematic interest; they are not only curious museum pieces.

Despite not being the first to employ modern logical techniques to analyze medieval theories, Parsons' approach is quite unique (one might even say idiosyncratic). It seems fair to say that nobody has ever before attempted to achieve what he wants to achieve with this book. A passage from the book’s Introduction is quite revealing with respect to its goals:

Tuesday, 16 September 2014

What makes a mathematical proof beautiful?

(Cross-posted at NewAPPS)

In December, I will be presenting at the Aesthetics in Mathematics conference in Norwich. The title of my talk is Beauty, explanation, and persuasion in mathematical proofs, and to be honest at this point there is not much more to it than the title… However, the idea I will try to develop is that many, perhaps even most, of the features we associate with beauty in mathematical proofs can be subsumed to the ideal of explanatory persuasion, which I take to be the essence of mathematical proofs. 

As some readers may recall, in my current research I adopt a dialogical perspective to raise a functionalist question: what is the point of mathematical proofs? Why do we bother formulating mathematical proofs at all? The general hypothesis is that most of the defining criteria for what counts as a mathematical proof – and in particular, a good mathematical proof – can be explained in terms of the (presumed) ultimate function of a mathematical proof, namely that of convincing an interlocutor that the conclusion of the proof is true (given the truth of the premises) by showing why that is the case. (See also this recent edited volume on argumentation in mathematics.) Thus, a proof seeks not only to force the interlocutor to grant the conclusion if she has granted the premises; it seeks also to reveal something about the mathematical concepts involved to the interlocutor so that she also apprehends what makes the conclusion true – its causes, as it were. On this conception of proof, beauty may well play an important role, but its role will be subsumed to the ideal of explanatory persuasion.

There is a small but very interesting literature on the aesthetics of mathematical proof – see for example this 2005 paper by my former colleague James McAllister, and a more recent paper on Kant’s conception of beauty in mathematics applied to proof by Angela Breitenbach, one of the organizers of the meeting in Norwich. (If readers have additional literature suggestions, please share them in comments.) But perhaps the locus classicus for the discussion of what makes a mathematical proof beautiful is G. H. Hardy’s splendid A Mathematician’s Apology (a text that is itself very beautiful!). In it, Hardy identifies and discusses a number of features that should be present for a proof to be considered beautiful: seriousness, generality, depth, unexpectedness, inevitability, and economy. And so, one way for me to test my dialogical hypothesis would be to see whether it is possible to provide a dialogical rationale for each of these features that Hardy discusses. My prediction is that most of them can receive compelling dialogical explanations, but that there will be a residue of properties related to beauty in a mathematical proof that cannot be reduced to the ideal of explanatory persuasion. (What this residue will be I do not yet know).

As I mentioned, this is still very much work in progress, but for now I would like to sketch what a dialogical account of beauty in a mathematical demonstration might look like for a specific feature. Now, a fascinating desideratum for a mathematical proof, which has been discussed in detail recently by Detlefsen and Arana, is the ideal of purity:
Throughout history, mathematicians have expressed preference for solutions to problems that avoid introducing concepts that are in one sense or another “foreign” or “alien” to the problem under investigation. (Detlefsen & Arana 2011, 1)
A mathematical proof is said to be pure if it does not rely on concepts that are not present in the statement of the conclusion of the proof (the theorem). Many famous mathematical proofs are not pure in this sense, such as Wiles’ proof of Fermat’s Last Theorem, which utilizes incredibly sophisticated and complex mathematical machinery to prove a theorem the statement of which can be understood with knowledge of standard high school level mathematics. (The impurity of Wiles’ proof is one of the motivations often given to seek for alternative proofs of FLT, as described in this guest post by Colin McLarty.) Now, I take it to be fairly obvious that purity concerns can be readily understood as aesthetic concerns, in particular related to simplicity (which is one of the features widely associated with beauty). 

What would a dialogical account of the purity desideratum look like? Going back to the idea that the function of a proof is that of eliciting persuasion by means of understanding in an interlocutor (hence the stress on the explanatory dimension), it is clear that, in general, the less complex the mathematical machinery of a proof, the less it will demand of the interlocutor being persuaded in terms of cognitive investment. Moreover, if it relies on simpler machinery, the proof will most likely reach a larger audience, i.e. be persuasive for a larger number of people (those possessing mastery of the concepts used in it). In particular, a proof that only uses concepts already contained in the formulation of the theorem will be at least in theory comprehensible to anyone who can understand the statement of the conclusion. Thus, a pure proof maximizes its penetration among potential audiences, as it only excludes those who do not even grasp the statement of the theorem in the first place. In other words, purity sets the lower bound of cognitive sophistication required from an interlocutor precisely at the right place. (Naturally, I can also be convinced of the truth of a theorem even if I do not understand the proof myself, i.e. by relying on the expertise of the mathematical community as a whole.)

As I said, these are only tentative ideas at this point, so I look forward to feedback from readers. In particular, I would like to hear from practicing mathematicians their answers to the question in the title: what makes a mathematical proof beautiful? Do you agree with Hardy's list? (I could definitely use some input so as to render my investigation more in sync with actual practices!)

Wednesday, 10 September 2014


On behalf of the M-Phi contributors, I want to sincerely apologize to our readers for the misguided and inappropriate post that was online at M-Phi for four days (now taken down, as well as all other posts referencing the Oxford events). The moderation structure of the blog was such that none of us could do anything to take it down, except for pleading with the author to do so.

[UPDATE (Sep. 12th): It has been brought to my attention that we owe an apology not only for the most recent post, but also for at least some of the content of the other posts pertaining to the Oxford events, which had been posted a few months ago (now also deleted). So, for those too, our apologies. We are also looking into additional ways to make amends with the people negatively affected.]

The structure and moderation of the blog will change completely now; Jeffrey Ketland will no longer be a contributor (of his own initiative). The exact details still need to be discussed, but we hope to come back with something more concrete within a week or so.

Again, our apologies, to our readers and to those who were negatively affected by the post.

(And thank you Jeff, for all your otherwise very good work here at M-Phi over the years.)

UPDATE: the opinions of those who felt negatively affected by the posts are most welcome in comments below (or in private to me by email).

Tuesday, 9 September 2014

A break

This is a short note just to say that I will not be contributing posts to M-Phi for the time being.

UPDATE: In view of recent events here at M-Phi, some important changes will take place regarding the management of the blog. We will talk more concretely about them in the near future, but for now let me say that we will do our utter best to restore the readers' trust in the blog, which may have been affected by recent developments.

Wednesday, 27 August 2014

CFP: SoTFoM II 'Competing Foundations?', 12-13 January 2015, London.

The focus of this conference is on different approaches to the foundations
of mathematics. The interaction between set-theoretic and category-theoretic
foundations has had significant philosophical impact, and represents a shift
in attitudes towards the philosophy of mathematics. This conference will
bring together leading scholars in these areas to showcase contemporary
philosophical research on different approaches to the foundations of
mathematics. To accomplish this, the conference has the following general
aims and objectives. First, to bring to a wider philosophical audience the
different approaches that one can take to the foundations of mathematics.
Second, to elucidate the pressing issues of meaning and truth that turn on
these different approaches. And third, to address philosophical questions
concerning the need for a foundation of mathematics, and whether or not
either of these approaches can provide the necessary foundation.

Date and Venue: 12-13 January 2015 - Senate House, University of London.

Confirmed Speakers: Sy David Friedman (Kurt Gödel Research Center, Vienna),
Victoria Gitman (CUNY), James Ladyman (Bristol), Toby Meadows (Aberdeen).

Call for Papers: We welcome submissions from scholars (in particular, young
scholars, i.e. early career researchers or post-graduate students) on any
area of the foundations of mathematics (broadly construed). Particularly
desired are submissions that address the role of and compare different
foundational approaches. Applicants should prepare an extended abstract
(maximum 1’500 words) for blind review, and send it to sotfom [at] gmail
[dot] com, with subject `SOTFOM II Submission'.

Submission Deadline: 15 October 2014

Notification of Acceptance: Early November 2014

Scientific Committee: Philip Welch (University of Bristol), Sy-David
Friedman (Kurt Gödel Research Center), Ian Rumfitt (University of
Birmigham), John Wigglesworth (London School of Economics), Claudio Ternullo
(Kurt Gödel Research Center), Neil Barton (Birkbeck College), Chris Scambler
(Birkbeck College), Jonathan Payne (Institute of Philosophy), Andrea Sereni
(Università Vita-Salute S. Raffaele), Giorgio Venturi (Université de Paris
VII, “Denis Diderot” - Scuola Normale Superiore)

Organisers: Sy-David Friedman (Kurt Gödel Research Center), John
Wigglesworth (London School of Economics), Claudio Ternullo (Kurt Gödel
Research Center), Neil Barton (Birkbeck College), Carolin Antos-Kuby (Kurt
Gödel Research Center)

Conference Website: sotfom [dot] wordpress [dot] com

Further Inquiries: please contact
Carolin Antos-Kuby (carolin [dot] antos-kuby [at] univie [dot] ac [dot] at)
Neil Barton (bartonna [at] gmail [dot] com)
Claudio Ternullo (ternulc7 [at] univie [dot] ac [dot] at)
John Wigglesworth (jmwigglesworth [at] gmail [dot] com)

The conference is generously supported by the Mind Association, the Institute of Philosophy, and Birkbeck College.

Monday, 25 August 2014

What’s the big deal with consistency?

(Cross-posted at NewAPPS)

It is no news to anyone that the concept of consistency is a hotly debated topic in philosophy of logic and epistemology (as well as elsewhere). Indeed, a number of philosophers throughout history have defended the view that consistency, in particular in the form of the principle of non-contradiction (PNC), is the most fundamental principle governing human rationality – so much so that rational debate about PNC itself wouldn’t even be possible, as famously stated by David Lewis. It is also the presumed privileged status of consistency that seems to motivate the philosophical obsession with paradoxes across time; to be caught entertaining inconsistent beliefs/concepts is really bad, so blocking the emergence of paradoxes is top-priority. Moreover, in classical as well as other logical systems, inconsistency entails triviality, and that of course amounts to complete disaster.

Since the advent of dialetheism, and in particular under the powerful assaults of karateka Graham Priest, PNC has been under pressure. Priest is right to point out that there are very few arguments in favor of the principle of non-contradiction in the history of philosophy, and many of them are in fact rather unconvincing. According to him, this holds in particular of Aristotle’s elenctic argument in Metaphysics gamma. (I agree with him that the argument there does not go through, but we disagree on its exact structure. At any rate, it is worth noticing that, unlike David Lewis, Aristotle did think it was possible to debate with the opponent of PNC about PNC itself.) But despite the best efforts of dialetheists, the principle of non-contradiction and consistency are still widely viewed as cornerstones of the very concept of rationality.

However, in the spirit of my genealogical approach to philosophical issues, I believe that an important question to be asked is: What’s the big deal with consistency in the first place? What does it do for us? Why do we want consistency so badly to start with? When and why did we start thinking that consistency was a good norm to be had for rational discourse? And this of course takes me back to the Greeks, and in particular the Greeks before Aristotle.

Variations of PNC can be found stated in a few authors before Aristotle, Plato in particular, but also Gorgias (I owe these passages to Benoît Castelnerac; emphasis mine in both):

You have accused me in the indictment we have heard of two most contradictory things, wisdom and madness, things which cannot exist in the same man. When you claim that I am artful and clever and resourceful, you are accusing me of wisdom, while when you claim that I betrayed Greece, you accused me of madness. For it is madness to attempt actions which are impossible, disadvantageous and disgraceful, the results of which would be such as to harm one’s friends, benefit one’s enemies and render one’s own life contemptible and precarious. And yet how can one have confidence in a man who in the course of the same speech to the same audience makes the most contradictory assertions about the same subjects? (Gorgias, Defence of Palamedes)
You cannot be believed, Meletus, even, I think, by yourself. The man appears to me, men of Athens, highly insolent and uncontrolled. He seems to have made his deposition out of insolence, violence and youthful zeal. He is like one who composed a riddle and is trying it out: “Will the wise Socrates realize that I am jesting and contradicting myself, or shall I deceive him and others?” I think he contradicts himself in the affidavit, as if he said: “Socrates is guilty of not believing in gods but believing in gods”, and surely that is the part of a jester. Examine with me, gentlemen, how he appears to contradict himself, and you, Meletus, answer us. (Plato, Apology 26e- 27b)
What is particularly important for my purposes here is that these are dialectical contexts of debate; indeed, it seems that originally, PNC was to a great extent a dialectical principle. To lure the opponent into granting contradictory claims, and exposing him/her as such, is the very goal of dialectical disputations; granting contradictory claims would entail the opponent being discredited as a credible interlocutor. In this sense, consistency would be a derived norm for discourse: the ultimate goal of discourse is persuasion; now, to be able to persuade one must be credible; a person who makes inconsistent claims is not credible, and thus not persuasive.
As argued in a recent draft paper by my post-doc Matthew Duncombe, this general principle applies also to discursive thinking for Plato, not only for situations of debates with actual opponents. Indeed, Plato’s model of discursive thinking (dianoia) is of an internal dialogue with an imaginary opponent, as it were (as to be found in the Theaetetus and the Philebus). Here too, consistency will be related to persuasion: the agent herself will not be persuaded to hold beliefs which turn out to be contradictory, but realizing that they are contradictory may well come about only as a result of the process of discursive thinking (much as in the case of the actual refutations performed by Socrates on his opponents).
Now, as also argued by Matt in his paper, the status of consistency and PNC for Aristotle is very different: PNC is grounded ontologically, and then generalizes to doxastic as well as dialogical/discursive cases (although one of the main arguments offered by Aristotle in favor of PNC is essentially dialectical in nature, namely the so-called elenctic argument). But because Aristotle postulates the ontological version of PNC -- a thing a cannot both be F and not be F at the same time, in the same way -- it is difficult to see how a fruitful debate can be had between him and the modern dialethists, who maintain precisely that such a thing is after all possible in reality.
Instead, I find Plato’s motivation for adopting something like PNC much more plausible, and philosophically interesting in that it provides an answer to the genealogical questions I stated earlier on. What consistency does for us is to serve the ultimate goal of persuasion: an inconsistent discourse is prima facie implausible (or less plausible). And so, the idea that the importance of consistency is subsumed to another, more primitive dialogical norm (the norm of persuasion) somehow deflates the degree of importance typically attributed to consistency in the philosophical literature, as a norm an sich.
Besides dialetheists, other contemporary philosophical theories might benefit from the short ‘genealogy of consistency’ I’ve just outlined. I am now thinking in particular of work done in formal epistemology by e.g. Branden Fitelson, Kenny Easwaran (e.g. here), among others, contrasting the significance of consistency vs. accuracy. It seems to me that much of what is going on there is also a deflation of the significance of consistency as a norm for rational thought; their conclusion is thus quite similar to the one of the historically-inspired analysis I’ve presented here, namely: consistency is over-rated.

Servus, New York! Invitation to the MCMP Workshop "Bridges" (2 and 3 Sept, 2014)

MCMP Workshop "Bridges 2014"

New York City, 2 and 3 Sept, 2014

The Munich Center for Mathematical Philosophy (MCMP) cordially invites you to "Bridges 2014" in the German House, New York City, on 2 and 3 September, 2014. The 2-day trans-continental meeting in mathematical philosophy will focus on inter-theoretical relations thereby connecting form and content of this philosophical exchange. The workshop will be accompanied by an open-to-public evening event with Stephan Hartmann and Branden Fitelson on 2 September, 2014 (6:30 pm).


Lucas Champollion (NYU)
David Chalmers (NYU)
Branden Fitelson (Rutgers)
Alvin I. Goldman (Rutgers)
Stephan Hartmann (MCMP/LMU)
Hannes Leitgeb (MCMP/LMU)
Kristina Liefke (MCMP/LMU)
Sebastian Lutz (MCMP/LMU)
Tim Maudlin (NYU)
Thomas Meier (MCMP/LMU)
Roland Poellinger (MCMP/LMU)
Michael Strevens (NYU)

Idea and Motivation

We use theories to explain, to predict and to instruct, to talk about our world and order the objects therein. Different theories deliberately emphasize different aspects of an object, purposefully utilize different formal methods, and necessarily confine their attention to a distinct field of interest. The desire to enlarge knowledge by combining two theories presents a research community with the task of building bridges between the structures and theoretical entities on both sides. Especially if no background theory is available as yet, this becomes a question of principle and of philosophical groundwork: If there are any – what are the inter-theoretical relations to look like? Will a unified theory possibly adjudicate between monist and dualist positions? Under what circumstances will partial translations suffice? Can the ontological status of inter-theoretical relations inform us about inter-object relations in the world? Our spectrum of interest includes: reduction and emergence, mechanistic links between causal theories, belief vs. probability, mind and brain, relations between formal and informal accounts in the special sciences, cognition and the outer world.

Program and Registration

Due to security regulations at the German House registering is required (separately for workshop and evening event). Details on how to register and the full schedule can be found on the official website:

Sunday, 24 August 2014

Extending a theory with the theory of mereological fusions

"Arithmetic with fusions" (draft) is a joint paper with my graduate student Thomas Schindler (MCMP).  The abstract is:
In this article, the relationship between second-order comprehension and unrestricted mereological fusion (over atoms) is clarified. An extension $\mathsf{PAF}$ of Peano arithmetic with a new binary mereological notion of ``fusion'', and a scheme of unrestricted fusion, is introduced. It is shown that $\mathsf{PAF}$ interprets full second-order arithmetic, $Z_2$.
Roughly this shows:
First-order arithmetic + mereology = second-order arithmetic.
This implies that adding the theory of mereological fusions can be a very powerful, non-conservative, addition to a theory, perhaps casting doubt on the philosophical idea that once you have some objects, then having their fusion also is somehow "redundant". The additional fusions can in some cases behave like additional "infinite objects"; positing their existence allows one to prove more about the original objects.

Friday, 22 August 2014

L. A. Paul on transformative experience and decision theory II

In the first part of this post, I considered the challenge to decision theory from what L. A. Paul calls epistemically transformative experiences.  In this post, I'd like to turn to another challenge to standard decision theory that Paul considers.  This is the challenge from what she calls personally transformative experiences.  Unlike an epistemically transformative experience, a personally transformative experience need not teach you anything new, but it does change you in another way that is relevant to decision theory---it leads you to change your utility function.  To see why this is a problem for standard decision theory, consider my presentation of naive, non-causal, non-evidential decision theory in the previous post.

Tuesday, 19 August 2014

Is the human referee becoming expendable in mathematics?

(Cross-posted at NewAPPS)

Mathematics has been much in the news recently, especially with the announcement of the latest four Fields medalists (I am particularly pleased to see the first woman, and the first Latin-American, receiving the highest recognition in mathematics). But there was another remarkable recent event in the world of mathematics: Thomas Hales has announced the completion of the formalization of his proof of the Kepler conjecture. The conjecture: “what is the best way to stack a collection of spherical objects, such as a display of oranges for sale? In 1611 Johannes Kepler suggested that a pyramid arrangement was the most efficient, but couldn't prove it.” (New Scientist)

The official announcement goes as follows:
We are pleased to announce the completion of the Flyspeck project, which has constructed a formal proof of the Kepler conjecture. The Kepler conjecture asserts that no packing of congruent balls in Euclidean 3-space has density greater than the face-centered cubic packing. It is the oldest problem in discrete geometry. The proof of the Kepler conjecture was first obtained by Ferguson and Hales in 1998. The proof relies on about 300 pages of text and on a large number of computer calculations.
The formalization project covers both the text portion of the proof and the computer calculations. The blueprint for the project appears in the book "Dense Sphere Packings," published by Cambridge University Press. The formal proof takes the same general approach as the original proof, with modifications in the geometric partition of space that have been suggested by Marchal.
So far, nothing very new, philosophically speaking. Computer-assisted proofs (both at the level of formulation and at the level of verification) have attracted the interest of a number of philosophers in recent times (here’s a recent paper by John Symons and Jack Horner, and here is an older paper by Mark McEvoy, which I commented on at a conference back in 2005; there are many other papers on this topic by philosophers).  More generally, the question of the extent to which mathematical reasoning can be purely ‘mechanical’ remains a lively topic of philosophical discussion (here’s a 1994 paper by Wilfried Sieg on this topic that I like a lot). Moreover, this particular proof of the Kepler conjecture does not add anything substantially new (philosophically) to the practice of computer-verifying proofs (while being quite a feat mathematically!). It is rather something Hales said to the New Scientist that caught my attention (against the background of the 4 years and 12 referees it took to human-check the proof for errors): "This technology cuts the mathematical referees out of the verification process," says Hales. "Their opinion about the correctness of the proof no longer matters."

Now, I’m with Hales that ‘software intensive mathematics’ (to borrow Symons and Horner’s terminology) is of great help to offload some of the more tedious parts of mathematical practice such as proof-checking. But there are a number of reasons that suggest to me that Hales’ ‘optimism’ is a bit excessive, in particular with respect to the allegedly expendable role of the human referee (broadly construed) in mathematical practice, even if only for the verification process.

Indeed, and as I’ve been arguing in a number of posts, proof-checking is a major aspect of mathematical practice, basically corresponding to the role I attribute to the fictitious character ‘opponent’ in my dialogical conception of proof (see here). The main point is the issue of epistemic trust and objectivity: to be valid, a proof has to be ‘replicable’ by anyone with the relevant level of competence. This is why probabilistic proofs are still thought to be philosophically suspicious (as argued for example by Kenny Easwaran in terms of the notion of ‘transferability’). And so, automated proof-checking will most likely never replace completely human proof-checking, if nothing else because the automated proof-checkers themselves must be kept ‘in check’ (lame pun, I know). (Though I am happy to grant that the role of opponent can be at least partially played by computers, and that our degree of certainty in the correctness of Hales’ proof has been increased by its computer-verification.)

Moreover, mathematics remains a human activity, and mathematical proofs essentially involve epistemic and pragmatic notions such as explanation and persuasion, which cannot be taken over by purely automated proof-checking. (Which does not mean that the burden of verification cannot be at least partially transferred to automata!) In effect, a good proof is not only one that shows that the conclusion is true, but also why the conclusion is true, and this explanatory component is not obviously captured by automata. In other words, a proof may be deemed correct by computer-checking, and yet fail to be persuasive in the sense of having true explanatory value. (Recall that Smale’s proof of the possibility of sphere eversion was viewed with a certain amount of suspicion until models of actual processes of eversion were discovered.)

Finally, turning an ‘ordinary’ mathematical proof* into something that can be computer-checked is itself a highly theoretical, non-trivial, and essentially informal endeavor that must itself receive a ‘seal of approval’ from the mathematical community. While mathematicians hardly ever disagree on whether a given proof is or is not valid once it is properly scrutinized, there can be (and has been, as once vividly described to me by Jesse Alama) substantive disagreement on whether a given formalized version of a proof is indeed an adequate formalization of that particular proof. (This is also related to thorny issues in the metaphysics of proofs, e.g. criteria of individuation for proofs, which I will leave aside for now.) 

A particular informal proof can only be said to have been computer-verified if the formal counterpart in question really is (deemed to be) sufficiently similar to the original proof. (Again, the formalized proof may have the same conclusion as the original informal proof, in which case we may agree that the theorem they both purport to prove is true, but this is no guarantee that the original informal proof itself is valid. There are many invalid proofs of true statements.) Now, evaluating whether a particular informal proof is accurately rendered in a given formalized form is not a task that can be delegated to a computer (precisely because one of the relata of the comparison is itself an informal construct), and for this task the human referee remains indispensable.

And so, I conclude that, pace Hales, the human mathematical referee is not going to be completely cut out of the verification process any time soon. Nevertheless, it is a welcome (though not entirely new) development that computers can increasingly share the burden of some of the more tedious aspects of mathematical practice: it’s a matter of teamwork rather than the total replacement of a particular approach to proof-verification by another (which may well be what Hales meant in the first place).

* In some research programs, mathematical proofs are written directly in computer-verifiable form, such as in the newly created research program of homotopy type-theory.