This paper focusses on Carnap's notes for an "Attempt at a Metalogic" which Carnap wrote on a famous "sleepless night" in January 1931 after learning of Goedels incompleteness results. This draft, Carnap tells us, represents the original kernel of his Logical Syntax of Language. I discuss the context of this work in the relationship between Carnap and Goedel, and conclude with a somewhat surprising anti-Quinean twist at the end.
Contemporary work in scientific explanation has pursued to a great extent the project of a single unified account of the nature of explanation. Unfortunately the drive towards unification has also left by the wayside an important number of phenomena. In particular, many theories of scientific explanation do not address mathematical explanation, either because they rule mathematical explanations out of court from the outset or because they hold that their account of explanation automatically takes care of mathematical explanation. In this paper we begin by providing evidence for the claim that mathematicians seek explanations in their ordinary practice and cherish different types of explanations. We go on to suggest that a fruitful approach to the topic of mathematical explanation would consist in providing a taxonomy of recurrent types of mathematical explanation and then trying to see whether these patters are heterogeneous or can be subsumed under a general account. We maintain that mathematical explanations are heterogeneous. However, neither giving the taxonomy nor arguing for the previous claim is what we have set for ourselves in this paper. Rather, we would like to provide a single case study of how to use mathematical explanations as found in mathematical practice to test theories of mathematical explanation. This can be seen, as it were, as a case study of how to show that the variety of mathematical explanations cannot be easily reduced to a single model. The case study will focus on Steiner's theory of mathematical explanation and Pringsheim's explanatory proof of Kummer's convergence criterion in the theory of infinite series.
Central to the traditional mathematical inquiry about knots is
its intellectual motivating theme: a spatial sense of what matters, and
does not matter, about knottedness: what makes this knottedness
different from that; what different knottednesses there are.
The mathematical challenges in getting a precise understanding of these knottedness matters start at a ``representational'' level that conceptually precedes precise definitions, proposition and proof. The mathematical challenge is primarily articulative: finding ways of expressing knottedness types, ways that allow one to connect these types to given knots. Even proof, though necessary, plays only a secondary role; the ontology one might attribute based on the precise form of definitions, only a tertiary one.
This paper investigates Hobbes's stated criteria for rigorous demonstration and contrasts them with his mathematical practice, as exemplified in various efforts at circle quadrature and failed attempts to solve other notable problems. I argue that Hobbes was misled into thinking that not only must every geometric problem must be solvable, but also that he had hit upon a method that would deliver quick and easy solutions to any geometric problem. The sources of this mistake are to be found in Hobbes's methodological doctrine that all proper geometric demonstrations must proceed from causes, together with his belief that his own program for geometry was based upon definitions that adequately expressed geometric causes and thus offered the prospect of solving every problem. I also argue that Hobbes's repeated and emphatic rejection of analytic geometry is a consequence of his belief that analytic techniques are essentially arithmetical rather than geometrical, and are consequently irrelevant to the solution of purely geometric problems.