Ian Dove

Written by experts in the field, this volume presents a comprehensive investigation into the relationship between argumentation theory and the philosophy of mathematical practice. Argumentation theory studies reasoning and argument, and especially those aspects not addressed, or not addressed well, by formal deduction. The philosophy of mathematical practice diverges from mainstream philosophy of mathematics in the emphasis it places on what the majority of working mathematicians actually do, rather than on mathematical foundations. The book begins by first challenging the assumption that there is no role for informal logic in mathematics. Next, it details the usefulness of argumentation theory in the understanding of mathematical practice, offering an impressively diverse set of examples, covering the history of mathematics, mathematics education and, perhaps surprisingly, formal proof verification. From there, the book demonstrates that mathematics also offers a valuable testbed for argumentation theory. Coverage concludes by defending attention to mathematical argumentation as the basis for new perspectives on the philosophy of mathematics. ​

Visual arguments—arguments that appeal to visual elements essentially—are legitimate arguments. To show this, I first consider what I call fit arguments—arguments in which the recognition that items fit together suggests that they were once conjoined, perhaps originally. This form of argumentation is a type of abduction or inference to the best explanation. I then consider mathematical visual meta-arguments—arguments in which the validity or soundness of a mathematical argument is confirmed or refuted by appeal to diagrams.

This project has two goals: to analyze the claims of mathematical fallibilism in order to show that they are less controversial than their usual statement implies; to resurrect deductivism with modifications from its premature burial. For the purposes of this project fallibilism is the disjunctive claim that mathematical proofs are insufficient either to justify mathematical knowledge where knowledge has a certainty-clause or to establish conclusively any mathematical truths. The first clause contrasts with the traditional view that mathematical knowledge is certain. The second clause reinforces this uncertainty by undermining the demonstrative force of proofs. Five arguments for fallibilism are considered. First, the distinction between pure and applied math leads to an uncontroversial form of mathematical fallibilism. Moreover, this distinction is shown to fit well with deductivism. Second, mathematics employs non-deductive methods. Non-deductive procedures are also, prima facie, difficult to reconcile with deductivism. The fallibility entailed by non-deductive methods is both limited and uncontroversial. In terms of deductivism, old-style Russellian deductivism is abandoned in favor of a more general notion of deductive proof. Third, there is the possibility of an infinite regress for mathematical justification. For Lakatos the regress originates in the various prospects for mathematical foundations. His favored theory, quasi-empiricism, is justified by comparison to foundational approaches. Modified deductivism is shown to halt the infinite regress as well. Fourth, the fact that informal arguments---i.e., arguments not valid in terms of their logical form---are nearly ubiquitous in mathematics is prima facie evidence in favor of quasi-empiricism. Deductivism is shown to be consistent with informal arguments. Fifth, Quine's naturalism is shown to lead to fallibilism. However, when properly understood this fallibility is shown to be uncontroversial. Moreover, although Quine never officially endorses anything but naturalism, it is shown that naturalism is both consistent with and aided by deductivism. Finally, the modifications to deductivism are unified and the claims of fallibilism are restated in their uncontroversial forms

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.

I argue that a basic similarity analysis of analogical reasoning handles many apparent cases of visual analogy. I consider how the visual and verbal elements interact in analogical cases. Finally, I offer two analyses of visual elements. One analysis is evidential. The visual elements are evidence for their ver-bal counterparts. One is non-evidential: the visual elements link to verbal elements without providing evi-dence for those elements. The result is to make more room for the logical analysis of visual argumentation.