Reliability of mathematical inference

Abstract

Of all the demands that mathematics imposes on its practitioners, one of the most fundamental is that proofs ought to be correct. This is also a demand that is especially hard to fulfill, given the fragility and complexity of mathematical proof. This essay considers some of ways that mathematics supports reliable assessment, which is necessary to maintain the coherence and stability of the practice. * I am grateful to Silvia De Toffoli and Yacin Hamami for helpful comments and suggestions.

Visual Proofs as Counterexamples to the Standard View of Informal Mathematical Proofs?

“…3, where I will discuss their standard view in more detail. In Azzouni's own words: I've suggested in earlier work [5] -in a way related to Avigad's [3] approach to a normative role for formal derivations -that transcribability to a formal derivation has, in the contemporary setting, become a norm for informal rigorous proof. I want to end this section by revisiting considerations that cut against that idea.…”

“…Finally, he states that (iv) this and many other informal rigorous mathematical proofs do not themselves "indicate the existence of formalizations that, in turn, justify why they're true: their content, that is, does nothing of this sort." 2 Because of this-especially because of his statement (iv) -and the fact that the implications of the visual proof can in fact be captured and to a certain degree replicated by a formal analogue, which is also admitted by Azzouni himself, 3 the following appears to be a more appropriate characterization of the normativity thesis:…”

“…1. 1 Note that when the talk is of "Hamami and Avigad's standard view " I am referring to two separate papers, namely [20] and [3]. Insofar as Avigad's work can be understood as an augmentation of Hamami's model of the "standard view of mathematical rigor" which we will see later, and for the purposes of this text, it is convenient to talk about "Hamami and Avigad's derivational account/standard view," as sometimes Azzouni himself does.…”

Rigour and Thought Experiments: Burgess and Norton

“…(Mac Lane 1986, 377) There is a huge range of alternative views about proofs, which I will not pursue. A small slection of views include: Azzouni (2013aAzzouni ( , 2013b, Lakatos (1976), Horgan (1993), Rav (1999), Avigad (2020), Detlefsen (2008) 2 2 I will, however, mention a couple of things in this footnote. Michael Detlefsen is skeptical of the common view, but puts it forward this way: "(i) proper proofs are proofs that either are or can be readily be made rigorous; (ii) proofs that are or can be readily be made rigorous are formalizable; therefore (iii) all proper proofs are formalizable."…”

Groundwork for a Fallibilist Account of Mathematics

“…Other examples areKitcher (1984),Aspray & Kitcher (1988),Avigad (2020) andPrawitz (2012). Dag Prawitz explains:To avoid misunderstandings let me say that I am in no way excluding the possibility of errors about whether something constitutes a conclusive ground.…”