Call for Proposals: Group Session at 2023 APA Central and Pacific Meetings

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. Please submit your proposal for a 2- or 3-hour symposium on a topic in the philosophy of mathematics by July 15 August 1, 2022.

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 2023 Central (February 22–25, Denver) or Pacific (April 4–8, San Francisco) meeting.

A proposal requires:

• names, affiliations, and email addresses of organizer and speakers
• a title for the session
• titles and short abstracts (up to 200 words) for the individual talks
• length of session: 2 hours, 3 hours, or either (if you are willing to hold a 2-hour session to accommodate a second session, if possible)
• whether you’d like to hold your session at the Central, Pacific, or either (with a preference if the latter)

Please submit your proposal by July 15 August 1, 2022 by email to rzach@ucalgary.ca.

It is perfectly acceptable to submit a proposal also to the PSA through the PMA; however we will not submit a proposal to the APA if it has previously been selected for submission to, and been accepted by the PSA.

Call for Proposals: Cognate Session at the 2022 PSA Biennial Meeting

The Philosophy of Science Association is inviting selected cognate societies to submit one proposal apiece for a special session during their upcoming biennial meeting in Pittsburgh (November 10-13, 2022). The idea is to seek a broader representation of work in philosophy of science than has traditionally been represented on the regular program of the PSA meeting. Each cognate society whose proposal is approved will be allotted 90 minutes. PSA will provide meeting rooms and AV equipment. Sessions and participants will be listed on the official conference program and on-line schedules. Participants must register for the conference.

If you are interested in having the PMA forward a proposal for a session organized by you, please send, by email to rzach@ucalgary.ca and before June 17, 2022, the following details:

• The title of the proposed session.
• Session topic(s)
• Session chair (can be the session organizer or someone different)
• A list of participants with name, affiliation, and contact email, including any non-presenting co-authors.
• A short descriptive summary of the proposal (100-200 words).
• A description of the topic and a justification of its current importance to the discipline (up to 1-2pp. or 1000 words).

The joint PMA/APMP selection committee will pre-select a proposal to be proposed to the PSA.

Please note that in accordance with current PSA policy:

• No previously published paper may be presented at the PSA meeting.
• No one will be permitted to present more than once at PSA 2022. A scholar may appear as co-author on more than one paper or symposium talk, but may present at PSA 2022 only once (excluding presentations at the poster forum).
• Any individual can be part of only one cognate society proposal in which he or she is a presenting author.

Non-Deductive Methods in Mathematics

The PMA and APMP jointly held a symposium in the APA Eastern Division group sessions program on January 14, 2022, in Baltimore.

Believing the Riemann Hypothesis: Inductive Justification Beyond Enumeration

William D’Alessandro (Munich Center for Mathematical Philosophy)

The Riemann Hypothesis (RH) is one of the most important unproved statements in mathematics, with deep implications for number theory, geometry, analysis and beyond. RH has been open for over a hundred and fifty years and a proof still seems out of reach. But the general consensus is that RH is probably true. Why do mathematicians believe this? Although billions of cases of RH have been checked without any counterexamples having turned up yet, this sort of simple enumerative evidence is viewed as quite weak justification for RH (for reasons having to do with the slow growth rates of functions that could plausibly produce counterexamples). Instead, mathematicians’ confidence in RH is based mainly on the known truth of RH-like principles in various surrogate models of the rational numbers. Understanding the epistemology of this case thus highlights the underappreciated role of models, analogies and similar heuristics in mathematical reasoning.

Beliefs, Degrees of Belief and ‘Probabilistic Proofs’ in Mathematics

Silvia De Toffoli, Princeton University

It is uncontroversial that non-deductive arguments can justify arbitrary high credences in a mathematical proposition. In particular, randomized arguments (aka probabilistic proofs) are used to solve computationally heavy mathematical problems like determining the primality of large numbers (Rabin 1976). It has been argued that this type of argument gives rise to justified belief (Fallis 1997) and even knowledge (Paseau 2015). In my talk, I discuss the thesis that mathematical knowledge can be achieved without proof, and suggest that it is either trivial (as in the case of deference) or mistaken. I argue that the fact that we are justified in holding an arbitrary high credence in a mathematical proposition via randomized arguments does not automatically imply that if we believe it, our full-on belief will be justified. To do so, I consider the proposal that credences and full-on beliefs play different epistemic roles and are subject to different epistemic norms (Buchak 2014). I elaborate this proposal and apply it to the case of mathematics.

Enumerative Induction in Mathematics: The Lure of Small Numbers

Alan Baker (Swarthmore College)

In a 2007 paper, I argue for the following two-part thesis: i. Enumerative induction ought not to increase confidence in universal mathematical generalizations (over an infinite domain). ii. Enumerative induction does not (in general) lead mathematicians to be more confident in the truth of the conclusion of such generalizations. This skepticism about the justificatory role of enumerative induction in mathematics has been challenged by Walsh (2014) and by Waxman (2017), and discussed in some detail in Paseau (2021). In this talk I use several arithmetical case studies, including the Goldbach Conjecture, to further explore how and why the role of enumerative induction in mathematics is potentially problematic. My main argument is that particular positive instances of a given universal arithmetical conjecture will always be in an important sense “small” and thus that any such collection of instances is potentially biased. Size can make an important difference to the expected properties of numbers, so mere induction is insufficient to bridge from evidence about small numbers to a claim about all numbers.

Call for Proposals: Philosophy of Mathematics Group Sessions at the 2022 Central and Pacific APA

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 August 2, 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 Central (February 23–26, Chicago) or Pacific (April 13–16, Vancouver) 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)

Please submit your proposal by August 2, 2021 online here: https://forms.gle/djVziN81SX6RsY5U9

Call for Proposals: Philosophy of Mathematics Group Sessions at the 2022 Eastern APA

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

Joint PMA/APMP Group Session at Pacific APA

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.

Mathematical Rigor

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.

Call for Proposals: Philosophy of Mathematics Group Sessions at APA Divisional Meetings

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, 1942–2020

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.

Book Symposium on John Baldwin, Model Theory and the Philosophy of Mathematical Practice

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)..

Summaries:

Colin McLarty:

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.

Scott Weinstein:

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, 1948–2019

Mic Detlefsen, founding president of the Philosophy of Mathematics Association, has died. The philosophy of mathematics community deeply mourns his loss.

Link to Notre Dame memorial page