The Philosophy of Mathematics Association is an affiliated group of the American Philosophical Association and as such is invited to organize sessions in the group program at APA divisional meetings. The PMA has held such a group session at the 2020 Eastern meeting and the 2021 Pacific meeting, and is hoping to make philosophy of mathematics symposia a regular component of APA divisional meetings. Please submit your proposal for a 2- or 3-hour symposium on a topic in the philosophy of mathematics by June 1, 2021.
Proposals will be vetted by a joint committee of the PMA and the Association for the Philosophy of Mathematical Practice (APMP), and successful proposals will be scheduled for inclusion at the 2022 APA Eastern division meeting (January 4–8, 2022, Montreal). A separate call for proposals will be issued for group sessions at the 2022 Central and Pacific division meetings, although you can now indicate if you would like your proposal to be considered then if it is not selected for the Eastern division meeting.
A proposal requires:
- names, affiliations, and email addresses of organizer and speaker
- short abstracts (up to 200 words) for the session/talks
- confirmation that organizers and speakers commit to attending the meeting
- length of session (2 or 3 hours; select both if you are willing to hold a 2-hour session to accommodate a second session, if possible)
- whether you would like your proposal to be considered for the 2022 Central (February 23–26, Chicago) or Pacific (April 13–16, Vancouver) meeting if it is not selected for the Eastern.
Please submit your proposal by June 1, 2021 online here: https://forms.gle/djVziN81SX6RsY5U9
The joint PMA/APMP Group Session on Mathematical Rigor is taking place tomorrow, Saturday April 10, 11:30am-2pm PDT at the Pacific APA meeting.
Silvia De Toffoli, Princeton University
Mathematical rigor is often spelled out invoking the notions of formal proof and formalizability. For example, according to Burgess (Rigor and Structure, OUP, 2015), a rigorous proof is a proof that contains enough details to convince (for the right reasons) the relevant audience that a formal proof exists. This is a clear and general definition. It implies, however, that in order to evaluate whether a proof is acceptable as rigorous, we have to wade into its context and clear the murky waters by asking who is the “relevant audience” and what is “enough” to convince such relevant audience for the right reasons. These problems arise because it is not easy to specify what relation holds between traditional proofs and formal proofs – if it is at all possible. In particular, one vexed issue concerns the compatibility of rigor and intuition. In this symposium, we will explore these issues. The focus will be on the nature and role of rigor in mathematics, and the relationship between rigor and formalization on the one hand and intuition on the other. Here are also the abstracts for the three contributions:
Juliette Kennedy, “Beyond Kreisel’s notion of informal rigour”
In his 1967 paper “Informal Rigour and Completeness Proofs” Kreisel presents a conceptual analysis based on informal methods in the form of the so-called squeezing arguments. In this talk we first explore Kreisel’s notion of informal rigour, in connection with completeness theorems and beyond. We will then probe the notion of informal rigour, outlining a methodology going beyond Kreisel’s. We ask: what is the relation between formalisation and rigour, and what is the place of informal rigour in foundational practice? Is formalisation just one source of rigour for the mathematician, and if not—as seems to be—what does informal rigour look like for the working mathematician?
Jeremy Avigad, “Intuition and Rigor”
Discussions of mathematics often presuppose that there is a tension between intuition and rigor. Important historical developments often seem to favor one over the other, and proofs are often thought to be intuitive at the expense of rigor or rigorous at the expense of intuition. I will argue that the relationship between intuition and rigor is subtle, and that the two provide complementary means of upholding common mathematical values.
Yacin Hamami, “Rigor Judgments in Mathematical Practice”
How are mathematical proofs judged to be rigorous in mathematical practice? Traditional answers to this question have usually considered that judging the rigor of a mathematical proof proceeds through some sort of comparisons with the standards of formal proof. Several authors have argued, however, that this kind of view is implausible (see, e.g., Robinson, 1997; Detlefsen, 2009; Antonutti Marfori, 2010), and have thus called for the development of a more realistic account of rigor judgments in mathematical practice. In this talk, I will sketch a framework aiming to move forward in this direction. My starting point is the observation that judging a mathematical proof to be correct or rigorous amounts to judging the validity of each of the inferences that comprise it. Accordingly, the framework focuses on the processes by which mathematical agents identify and judge the validity of inferences when processing the text of an ordinary mathematical proof. From the perspective of the resulting framework, I will then discuss what is sometimes called the standard view of mathematical rigor (Hamami, 2019), by examining whether there is any ground supporting the thesis that whenever a proof has been judged to be rigorous in mathematical practice it can be routinely translated into a formal proof.
The Philosophy of Mathematics Association is an affiliated group of the American Philosophical Association and as such is invited to organize sessions in the group program at APA divisional meetings. The PMA has held such a group session at the 2020 Eastern meeting, and is hoping to make philosophy of mathematics symposia a regular component of APA divisional meetings. Please submit your proposal for a 2- or 3-hour symposium on a topic in the philosophy of mathematics by July 30, 2020.
Proposals should be submitted online at https://forms.gle/L99aE6s1GtJWYCMy5
Proposals will be vetted by a joint committee of the PMA and the Association for the Philosophy of Mathematical Practice (APMP), and successful proposals will be scheduled for inclusion at a 2021 APA divisional meeting. A proposal requires:
- names, affiliations, and email addresses of organizer and speaker
- short abstracts (up to 200 words) for the session
- confirmation that organizers and speakers commit to attending the meeting
- whether the session shall be held at the 2021 Central (February 24-27, New Orleans) or Pacific (March 31-April 4, Portland) meeting
During the COVID crisis it is of course not easy to predict whether these meetings will take place, or whether they will take place face-to-face or in some online format. The APA is currently planning to hold in-person meetings in 2021, but is also considering alternative formats. In case the meetings do not take place, PMA and APMP will ensure that the online versions of the symposia are advertised widely.
Mark Steiner, Professor emeritus of Philosophy at Hebrew University in Jerusalem, died April 6, 2020 of complications from COVID-19. He was a leading philosopher of mathematics, who was especially well known for his books The Applicability of Mathematics as a Philosophical Problem (1998, Harvard University Press) and Mathematical Knowledge (1975, Cornell University Press) and his pioneering work on explanation in mathematics.
The PMA held a session in the Group Program at the 2020 Eastern Division Meeting of the American Philosophical Association. The session was organized by Juliette Kennedy, and consisted of a book symposium on John Baldwin’s Model Theory and the Philosophy of Mathematical Practice. The participants were John Baldwin (University of Illinois at Chicago), Timothy Bays (Notre Dame University), Colin McLarty (Case Western) and Scott Weinstein (University of Pennsylvania)..
This book can advance philosopher’s understanding of both structure and mathematicians. As to structure, John gives a great look at current working methods, which differ markedly from the usual philosophic ideas for an obvious reason: Mathematicians ideas cannot just be defended in philosophic debate. They must produce results. And they do. John describes how a lot of these developed and what they achieve. This leads to the point about mathematicians. Philosophy of mathematics, and philosophy of science more generally, too often neglect the raw experience of desperately desiring scientific answers. Mathematical proof is so different from philosophic argument that we sometimes treat mathematical knowledge as something that just follows from the axioms. As if you just identify which axioms are relevant, put them in the right order, and then the answer is in front of you. John shows the living experience of the model theory community seeking, sometimes finding, and always reacting to, new concepts and methods to answer various standing questions. He shows how this goes at stages before anyone knows whether it will work. Philosophy of mathematics can benefit immensely from absorbing these insights. I will say, though, that textbooks and research talks and papers all show mathematicians use model theory, per se, much less often than they use categories and functors to describe structure.
John Baldwin heralds the significance of the “dividing lines” at the very beginning of Model Theory and the Philosophy of Mathematical Practice.
After the paradigm shift there is a systematic search for a finite set of syntactic conditions which divide first order theories into disjoint classes such that models of different theories in the same class have similar mathematical properties. In this framework one can compare different areas of mathematics by checking where theories formalizing them lie in the classification.
For John, the interest of the “syntactic” character of the dividing lines consists primarily in the fact that they are absolute for transitive models of set theory, and in consequence, that many of the central results of contemporary model theory do not rely on theories stronger than ZF. But, a number of the dividing lines, for example, NOP, are syntactic in a more robust sense – they are Π02 properties familiar from combinatorics. We explore whether this aspect of the dividing lines may have any interest from the point of view of the philosophy of mathematical practice. On the one hand, we observe that for every formal system F , there is an NOP theory T , such that T cannot be established to be NOP in F . On the other hand, all “naturally occurring” NOP theories (of which we are aware) can be proven to be NOP in primitive recursive arithmetic. We suggest investigating whether there is a recursively axiomatizable theory T that arises naturally in the course of mathematical practice such that T is NOP, but T is not provably NOP in primitive recursive arithmetic. If not, why? Might this tell us something interesting about mathematical practice? Or about the nature of mathematics itself? If so, might the additional strength required to establish that such a theory is NOP reflect some interesting phenomenon that helps to clarify our understanding of the mathematical topic it formalizes?
Slides of John Baldwin’s Responses.
Mic Detlefsen, founding president of the Philosophy of Mathematics Association, has died. The philosophy of mathematics community deeply mourns his loss.
Call for papers
Robert Thomas, editor of Philosophia Mathematica, writes:
Oxford University Press has given the impression this year that they would not be continuing personal subscriptions to Philosophia Mathematica since in 2018 the journal will be online only, what institutions overwhelmingly want. This decision was in fact made, but it has been reversed and at favorable rates as follows:
|Euro zone||North America||GB & rest of world|
|Unaffiliated||78 euros||99 US dollars||62 pounds|
|ASL members||39 euros||50 US dollars||31 pounds|
CSHPM members, anywhere outside Canada, 50 US dollars, paid to the Canadian Society for History and Philosophy of Mathematics.
Two things to note. Membership in CSHPM is so cheap at 23 US dollars (Cdn$30 in Canada), that it could easily pay to join CSHPM for the sole purpose of saving money on a PM subscription. See www.cshpm.org/join/
Subscriptions by ASL members and others are arranged through OUP:
UK Journals Customer Service, Tel: + 44 (0)1865 353907, Fax: + 44 (0)1865 353485.
USA Journals Customer Service, Tel: + 1 919-677-0977, + 1 800-852-7323 (toll-free in USA and Canada), Fax: + 1 919-677-1714.
12-14 October 2016
Munich Center for Mathematical Philosophy, LMU Munich
In the course of the last century, different general frameworks for the foundations of mathematics have been investigated. The orthodox approach to foundations interprets mathematics in the universe of sets. More recently, however, there have been other developments that call into question the whole method of set theory as a foundational discipline. Category-theoretic methods that focus on structural relationships and structure-preserving mappings between mathematical objects, rather than on the objects themselves, have been in play since the early 1960s. But in the last few years they have found clarification and expression through the development of homotopy type theory. This represents a fascinating development in the philosophy of mathematics, where category-theoretic structural methods are combined with type theory to produce a foundation that accounts for the structural aspects of mathematical practice. We are now at a point where the notion of mathematical structure can be elucidated more clearly and its role in the foundations of mathematics can be explored more fruitfully.
The main objective of the conference is to reevaluate the different perspectives on mathematical structuralism in the foundations of mathematics and in mathematical practice. To do this, the conference will explore the following research questions: Does mathematical structuralism offer a philosophically viable foundation for modern mathematics? What role do key notions such as structural abstraction, invariance, dependence, or structural identity play in the different theories of structuralism? To what degree does mathematical structuralism as a philosophical position describe actual mathematical practice? Does category theory or homotopy type theory provide a fully structural account for mathematics?
- Prof. Steve Awodey (Carnegie Mellon University)
- Dr. Jessica Carter (University of Southern Denmark)
- Prof. Gerhard Heinzmann (Université de Lorraine)
- Prof. Geoffrey Hellman (University of Minnesota)
- Prof. James Ladyman (University of Bristol)
- Prof. Elaine Landry (UC Davis)
- Prof. Hannes Leitgeb (LMU Munich)
- Dr. Mary Leng (University of York)
- Prof. Øystein Linnebo (University of Oslo)
- Prof. Erich Reck (UC Riverside)
Call for Abstracts
We invite the submission of abstracts on topics related to mathematical structuralism for presentation at the conference. Abstracts should include a title, a brief abstract (up to 100 words), and a full abstract (up to 1000 words), blinded for peer review. Authors should send their abstracts (in pdf format), together with their name, institutional affiliation and current position to firstname.lastname@example.org. We will select up to five submissions for presentation at the conference. The conference language is English.
Dates and Deadlines
Submission deadline: 30 June, 2016
Notification of acceptance: 31 July, 2016
Registration deadline: 1 October, 2016
Conference: 12 – 14 October, 2016
For further details on the conference, please visit: http://www.mathematicalstructuralism2016.philosophie.uni-muenchen.de/
The Emergence of Structuralism and Formalism, June 24- 26, 2016
PRAGUE, CZECH REPUBLIC
organized by Catholic Theological Faculty, Charles University and Institute of Philosophy, Czech Academy of Sciences, v.v.i.
The focal question of the workshop is how the nature of mathematics is regarded by representatives of formalism and structuralism. The conference language is English. To submit a proposal, please send a proposal of your paper to email@example.com
Proposals for papers should be prepared for anonymous review. Proposals should include title and abstract of the paper (maximum 500 words).
If you have inquiries about the conference or about the submission process, please write to firstname.lastname@example.org
SUBMISSION DEADLINE: April 30. 2016
Notification of acceptance on May 10. 2016.
The scheduled length of lectures is 30 minutes including approx. 10 minutes for discussion. Selected contributions will be published.