Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

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. ​. (shrink)

This paper discusses some of the ways recent models have brought rhetoric into argumentation theory. In particular, it explores the rationale for and role of rhetoric in the strategic maneuvering project of pragma-dialectics and compares it with the author’s own implementation of rhetorical features. A case is made for considering the active ways audiences influence the strategies of arguers and for seeing the role of rhetoric in argumentation as both fundamental and reasonable on its own terms.

We shall investigate the similarities and dissimilarities between old and new dialectic. For the ‘old dialectic’, we base our survey mainly on Aristotle’s Topics and Sophistical Refutations, whereas for the ‘new dialectic’, we turn to contemporary views on dialogical interaction, such as can, for the greater part, be found in Walton’s The New Dialectic. Three issues are taken up: types of dialogue, fallacies, and strategies. Though one should not belittle the differences in scope and outlook that obtain between the old (...) and the new dialectic, the paper will show that in many respects the old dialectic foreshadows the new dialectic. (shrink)

How do Dutch people let each other know that they disagree? What do they say when they want to resolve their difference of opinion by way of an argumentative discussion? In what way do they convey that they are convinced by each other’s argumentation? How do they criticize each other’s argumentative moves? Which words and expressions do they use in these endeavors? By answering these questions this short essay provides a brief inventory of the language of argumentation in Dutch.

According to pragma-dialectical methodology, a party in an argumentative discussion can be assumed to manoeuvre strategically between dialectical and rhetorical objectives. One confrontational form of strategic manoeuvring occurs when a critic charges an arguer with advancing a standpoint that has socially harmful consequences. In special situations this form of manoeuvring can be dialectically sound, for example when the standpoint is advanced in a way that damages the dialectical process. The boundary between fallacious and dialectically sound applications of this form of (...) manoeuvring is examined by looking for the manoeuvring’s soundness conditions. (shrink)

This paper explores applications of concepts from argumentation theory to mathematical proofs. Note is taken of the various contexts in which proofs occur and of the various objectives they may serve. Examples of strategic maneuvering are discussed when surveying, in proofs, the four stages of argumentation distinguished by pragma-dialectics. Derailments of strategies (fallacies) are seen to encompass more than logical fallacies and to occur both in alleged proofs that are completely out of bounds and in alleged proofs that are at (...) least mathematical arguments. These considerations lead to a dialectical and rhetorical view of proofs. (shrink)

A critic may attack an arguer personally by pointing out that the arguer’s position is pragmatically inconsistent: the arguer does not practice what he preaches. A number of authors hold that such attacks can be part of a good argumentative discussion. However, there is a difficulty in accepting this kind of contribution as potentially legitimate, for the reason that there is nothing wrong for a protagonist to have an inconsistent position, in the sense of committing himself to mutually inconsistent propositions. (...) If so, any such charge seems to be irrelevant. The questions to be answered in this essay are: what, if any, is the dialectical rationale for this type of criticism, and in what situations, if any, is this kind of charge dialectically legitimate? It will be shown that these attacks can be dialectically legitimate, in special circumstances, and that they can be seen as strategic␣manoeuvres where a party attempts to reconcile his dialectical and his rhetorical objectives. (shrink)

Van Eemeren and Houtlosser view fallacies as “derailments of strategic maneuvering” that go against a norm for critical reasonableness. What is to happen if such a derailment is perceived to have taken place? Krabbe (2003) and Jacobs (2000) have discussed the possibilities for continuing the argumentative exchange in a constructive way. Starting from their proposals, van Eemeren and Houtlosser argue that the party who observes that something has gone wrong should maneuver in such a way that at the same time (...) the derailment is recognized and a “re-railment” takes place that brings the discussion back on track. This means that then a meta-dialogue or sub-discussion develops in which the alleged fallacy is responded to and readjusted appropriately. Although this solution has something in common with both Krabbe’s proposal and Jacobs’ proposal, it differs from Krabbe’s proposal because it does not result in just retracting or maintaining the alleged fallacy and it differs from Jacobs’ proposal because it does not allow for making a move that has the appearance of a counter-fallacy. (shrink)