And so on . . . : reasoning with infinite diagrams

Abstract

Abstract. This paper presents examples of infinite diagrams (as well as infinite limits of finite diagrams) whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized and a "pre" form of this thesis that every proof can be presented in everyday statements-only form.

Signs as a Theme in the Philosophy of Mathematical Practice

“…In this spirit, there are a number of recent case studies describing the advantages of particular mathematical signs in particular contexts, both historical and contemporary. To take just a few examples, De Toffoli (2017Toffoli ( , 2023 discusses the use of commutative diagrams in homological algebra, and of arrow diagrams for representing closed connected surfaces; Feferman (2012) explores the use of some diagrams that contain ellipses (i.e., '. .…”

Mathematics Majors’ Perceptions of the Admissibility of Graphical Inferences in Proofs

“…This was the case even though the visual argument was written by a famous mathematician and is generally regarded as mathematically correct. Recently, several philosophers have argued that visual arguments should be perfectly convincing and hence ought to be acceptable in a proof (e.g., Azzouni 2013;Feferman 2012;Kupla 2009). However, as these authors are arguing against the status quo-i.e., that such arguments are usually regarded as unacceptable-the existence of these essays in support of visual arguments offers further evidence for the viewpoint that diagrammatic reasoning is not currently permissible in a proof.…”

Perceiving Necessity

“…6 Image created by Maria Miller, reproduced with permission from http://www. homeschoolmath.net/teaching/rational-numbers-countable.php 7 Further on seeing truths in diagrams that represent continuation to infinity in (Feferman, 1998(Feferman, , 2012. 8 Advanced players will notice that the three distinct implications just listed also have necessary connections between them.…”