The Uses of Argument in Mathematics

Stephen Toulmin once observed that `it has never been customary for philosophers to pay much attention to the rhetoric of mathematical debate'. Might the application of Toulmin's layout of arguments to mathematics remedy this oversight? Toulmin's critics fault the layout as requiring so much abstraction as to permit incompatible reconstructions. Mathematical proofs may indeed be represented by fundamentally distinct layouts. However, cases of genuine conflict characteristically reflect an underlying disagreement about the nature of the proof in question.Comment: 10 pages, 5 figures. To be presented at the Ontario Society for the Study of Argumentation Conference, McMaster University, May 2005 and LOGICA 2005, Hejnice, Czech Republic, June 200

The Informal Logic of Mathematical Proof

Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is that a more nuanced understanding of mathematical proof and discovery may be achieved by paying attention to the aspects of mathematical argumentation which can be captured by informal, rather than formal, logic. Two accounts of argumentation are considered: the pioneering work of Stephen Toulmin [The uses of argument, Cambridge University Press, 1958] and the more recent studies of Douglas Walton, [e.g. The new dialectic: Conversational contexts of argument, University of Toronto Press, 1998]. The focus of both of these approaches has largely been restricted to natural language argumentation. However, Walton's method in particular provides a fruitful analysis of mathematical proof. He offers a contextual account of argumentational strategies, distinguishing a variety of different types of dialogue in which arguments may occur. This analysis represents many different fallacious or otherwise illicit arguments as the deployment of strategies which are sometimes admissible in contexts in which they are inadmissible. I argue that mathematical proofs are deployed in a greater variety of types of dialogue than has commonly been assumed. I proceed to show that many of the important philosophical and pedagogical problems of mathematical proof arise from a failure to make explicit the type of dialogue in which the proof is introduced.Comment: 14 pages, 1 figure, 3 tables. Forthcoming in Perspectives on Mathematical Practices: Proceedings of the Brussels PMP2002 Conference (Logic, Epistemology and the Unity of the Sciences Series), J. P. Van Bendegem & B. Van Kerkhove, edd. (Dordrecht: Kluwer, 2004

Rationality and Reasonableness in Ethics

Stephen Toulmin is the Henry Luce Professor of International Relations and Anthropology at the University of Southern California. This talk was delivered at Sacred Heart University on January 30, 1997, sponsored by the Hersher Institute for Applied Ethics and the Center for Christian-Jewish Understanding

The Cult of Empiricism in Psychology, and Beyond

At some stage in it\u27s development, any field of intellectual discussion or scientific speculation may reach a point at which it begins to generate large numbers of empirical questions, that is, questions whose answers must refer to carefully documented observations, or even to controlled experiments. In physics, this happened most strikingly in the course of the seventeenth century; in biology, the comparable stage was not reached until around 1770, rising to its peak in the course of the nineteenth century (Toulmin, 1972; Toulmin & Goodfield, 1962); whereas in psychology, it has become customary-though a trifle arbitrary-to argue that this happened just one hundred years ago, with the establishment of Wilhelm Wundt\u27s pioneer psychological laboratory in Leipzig in 1879

La Viena de Wittgenstein

Madrid: Taurus, 1974

Collision, Collusion and Coincidence: Pop Art’s Fairground Parallel

This article looks at parallel methods, motivations and modes of consumption between formative British pop art and British fairground art. I focus on two strands, the emergent critical work of the Independent Group and the school of artists based at the Royal College of Art under the nominal leadership of Peter Blake. I use iconographical and iconological methods to compare the content of the art, and then examine how pop art tried to create both a critical and playful distancing from established rules and practices of the artistic canon. I focus on non-institutional cultural groupings and diffuse production and consumption models

Reactivation of tectonics, crustal underplating, and uplift after 60 Myr of passive subsidence, Raukumara Basin, Hikurangi-Kermadec fore arc, New Zealand: implications for global growth and recycling of continents

We use seismic reflection and refraction data to determine crustal structure, to map a fore-arc basin containing 12 km of sediment, and to image the subduction thrust at 35 km depth. Seismic reflection megasequences within the basin are correlated with onshore geology: megasequence X, Late Cretaceous and Paleogene marine passive margin sediments; megasequence Y, a similar to 10,000 km(3) submarine landslide emplaced during subduction initiation at 22 Ma; and megasequence Z, a Neogene subduction margin megasequence. The Moho lies at 17 km beneath the basin center and at 35 km at the southern margin. Beneath the western basin margin, we interpret reflective units as deformed Gondwana fore-arc sediment that was thrust in Cretaceous time over oceanic crust 7 km thick. Raukumara Basin has normal faults at its western margin and is uplifted along its eastern and southern margins. Raukumara Basin represents a rigid fore-arc block > 150 km long, which contrasts with widespread faulting and large Neogene vertical axis rotations farther south. Taper of the western edge of allochthonous unit Y and westward thickening and downlap of immediately overlying strata suggest westward or northwestward paleoslope and emplacement direction rather than southwestward, as proposed for the correlative onshore allochthon. Spatial correlation between rock uplift of the eastern and southern basin margins with the intersection between Moho and subduction thrust leads us to suggest that crustal underplating is modulated by fore-arc crustal thickness. The trench slope has many small extensional faults and lacks coherent internal reflections, suggesting collapse of indurated rock, rather than accretion of > 1 km of sediment from the downgoing plate. The lack of volcanic intrusion east of the active arc, and stratigraphic evidence for the broadening of East Cape Ridge with time, suggests net fore-arc accretion since 22 Ma. We propose a cyclical fore-arc kinematic: rock moves down a subduction channel to near the base of the crust, where underplating drives rock uplift, oversteepens the trench slope, and causes collapse toward the trench and subduction channel. Cyclical rock particle paths led to persistent trench slope subsidence during net accretion. Existing global estimates of fore-arc loss are systematically too high because they assume vertical particle paths. Citation: Sutherland, R., et al. (2009), Reactivation of tectonics, crustal underplating, and uplift after 60 Myr of passive subsidence, Raukumara Basin, Hikurangi-Kermadec fore arc, New Zealand: Implications for global growth and recycling of continents, Tectonics, 28, TC5017, doi: 10.1029/2008TC002356