“Deep Disagreement in Mathematics” by Andrew Aberdein

ABSTRACT: Disagreements that resist rational resolution, often termed “deep disagreements”, have been the focus of much work in epistemology and informal logic, but have not yet attracted the attention of philosophers of mathematics. In this paper, I link the question of whether there can be deep disagreements in mathematics to a more familiar debate over whether there can be revolutions in mathematics. I propose an affirmative answer to both questions, using the controversy over Shinichi Mochizuki’s work on the abc conjecture as a potential example of both phenomena. I conclude by investigating the prospects for the resolution of mathematical deep disagreements in virtue theoretic approaches to informal logic and mathematical practice.

“Rejection, Disagreement, Controversy and Acceptance in Mathematical Practice: Episodes in the Social Construction of Infinity” by Paul Ernest (University of Exeter)

ABSTRACT: The unit of analysis of social constructionism is conversation. This is the social mechanism whereby new mathematical claims are scrutinised and critiqued. Minimally, conversation is based on the two roles of proponent and critic. The proponent  puts forward a proposal, which is reacted to and evaluated by those in the role of critic. There is a continuum of contexts in which such conversations take place from casual face-to face interactions between mathematicians at the chalkboard, all the way to the formal responses of referees and editors to submitted journal papers. Such responses vary from unconditional acceptance, partial acceptance through to outright rejection. There may be disagreements between proponents and critics, among those in the joint role of critic, and broader, community-wide disagreements and controversies, according to specific mathematical proposal and the critical judgements of it.

 This paper looks briefly at four episodes in the social construction of infinity, illustrating the range of conversations and outcomes within the social practice of mathematics. 

  1. Cantor’s founding contributions to the theory of transfinite numbers and set theory and the resultant controversy, disagreement,  and final acceptance by the mathematical community.
  2. The Intuitionists’ and constructivists’ rejection of completed infinities and the resultant  controversy and disagreement, leading to the rejection of their negative theses, but acceptance of constructive mathematics within the pale of classical mathematics.
  3. Sergeyev’s introduction of the infinite element ‘Grossone’ in theory, calculation and applied mathematics and the continuing controversy and disagreement  about its validity and utility.
  4. Saburou Saitoh’s controversial publication of the ‘Division by Zero Calculus’ and its rejection by the mathematical community. 
“The ethics of inclusion in mathematics” by Eugénie Hunsicker (University of Loughborough)

ABSTRACT:  Much discussion of ethics in mathematics has centered on the ethics associated with the uses of mathematics in the community, such as in military applications.  However, another dimension of ethics relates to our treatment of people within our community or who wish to join our community.  In this talk, I will discuss evidence for poor practice around inclusion in mathematics, and refer to some literature that has investigated aspects of inclusion in a variety of fields to propose research projects related to the ethics and mechanisms of inclusion or exclusion in the global mathematics community.

“Mapping disagreements in mathematics” by Mikkel Moseholm and Henrik Kragh Sørensen

ABSTRACT: At first, it seems difficult to imagine, how research mathematicians might disagree over ‘deep’ epistemic issues. And moreover, it seems difficult to get access to such disagreements which are, presumably, often discussed in informal situations rather than in print.

However, using methods from machine-learning augmented digital humanities, we can begin to process large online fora, where such discussions take place, such as mailing lists and MathOverflow.

Based on a list of relevant search terms, we have generated an extract of threads from both the mailing list FOM (Foundations of Mathematics) and from MathOverflow with the highest frequencies of said terms. We analyzed the resulting threads through comparative close-reading and topic modelling in order to identify possible disagreements and to tentatively classify topics on which research mathematicians disagree. The concept of ‘deep disagreements’ is suggested as a possible analytical tool to elucidate the nature of (some) disagreements in mathematical practice.

Our project consists of three parts:

  1. A philosophical discussion of deep disagreements with special attention to how this concept may be applicable to disagreements in mathematics,
  2. A mapping and classification of topics of discussion in online mathematical conversations, using computational methods and qualitative approaches on corpora of online discussions, and
  3. A discussion, based on closer reading of key instances identified by pt. 2, of particular instances of deep epistemic disagreements in mathematics.

Through these steps, we are able to provide a preliminary characterization of (some) disagreements as they occur in mathematical practice. In our presentation, we will outline these three parts and give examples.

“Disagreement of peer reviewers in mathematics” by Benedikt Löwe (Amsterdam, Cambridge & Hamburg)

ABSTRACT: Disagreeing judgments of peer reviewers have been discussed as an indicator of a lack of intersubjectivity of the quality concept of a discipline. According to the standard view, since mathematics is considered to be remarkably intersubjectively stable, the naive expectation is that mathematical peer reviewers should rarely disagree. We discuss why this expectation is not reflected in the empirical data.

“Applied Mathematics” by Graham Priest (CUNY Graduate Center & University of Melbourne)

ABSTRACT: As far as disputes in the philosophy of pure mathematics goes, these are usually between classical mathematics, intuitionist mathematics, paraconsistent mathematics, and so on. My own view is that of a mathematical pluralist: all these different kinds of mathematics are equally legitimate. Applied mathematics is a different matter. In this, a piece of pure mathematics is applied in an empirical area, such as physics, biology, or economics. There can then certainly be disputes about what the correct pure mathematics to apply is. Such disputes may be settled by the standard criteria of scientific theory selection (adequacy of empirical predications, simplicity, etc.) But what, exactly is it to apply a piece of pure mathematics? How is mathematics applied? By and large, philosophers of mathematics have cared more about pure mathematics than applied mathematics, and not a lot of thought has gone into this question. In this talk I will address the issue and some of its ramifications.

“Mathematical Consensus” by Roy Wagner (ETH Zurich)

ABSTRACT: One of the distinguishing features of mathematics is the exceptional level of consensus among professional mathematicians. While this fact is brought up quite often, and some practical or metaphysical explanations have been suggested, I did not find in the literature sufficient analysis of what mathematicians actually agree on, how they achieve this agreement in practice, and under what conditions. The talk is therefore meant as a programmatic intervention in the hope of motivating more research into this question.

It is commonplace to say that mathematicians agree on the validity of arguments (rather than, say, their importance, elegance, or notions of truth not reducible to provability). But even if we restrict attention to the validity of arguments, mathematicians sometimes find it hard to reach agreement. This issue persists even if we exclude the problem of context (e.g. textbook, classroom, levels of expertise) and focus on research communication between experts. In practice, agreement depends on a rather intricate “negotiation”. Still, as some contemporary case studies suggest, there are cases where mathematicians fail to agree on the validity of some difficult proofs even after such process of negotiation.

To engage with this problematic, the talk will be divided into three parts.

In the first part, I will review the process of “negotiation” by which mathematicians achieve agreement about the validity of proofs. The process of isolating problematic aspects of a proof and revising them (by breaking them down into smaller components, translating them to a different “language” or modality, or arguing by analogy) most often does generate consensus. I will argue, however, that at this point a new kind of disagreement may arise: mathematicians may fail to agree whether the original proof and the re-negotiated proof are effectively the same or substantially different, and so may disagree whether the original proof is valid or not.

In the second part, I will briefly historicize the phenomenon of consensus. I will show that in earlier European mathematics (going as far as the early 20th centuries), consensus about the validity of arguments was substantially weaker than it is today, even if we exclude the famous foundations debates. Moreover, in various other mathematical cultures, the question of consensus had different forms and objectives. This means that contemporary consensus about the validity of mathematical proofs should be explained by recent historical changes in mathematical practice.

In the final part of the talk, I will try to explain what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus concerning the validity of proof occurs in the decades around the turn of the 20th century, it makes sense to explain this consensus by the concurrent logification and formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which aspects of formalization do enter mathematical practice so as to account for contemporary forms of mathematical consensus.

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