Mathematical Rigor: Joint PMA/APMP Group Session at Pacific APA tomorrow

[Richard Zach]

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.