WORKSHOP 2
Saturday, February 16, 2012 at UCI
Speakers:, John Mumma (Cal State San Bernardino), Dorothy Edgington (Birkbeck College, London), Michael Rescorla (UC Santa Barbara)
Location: UCI's Social & Behavioral Sciences Gateway (SBSG), Room 1517 | maps & directions

Speaker: Speaker: John Mumma (Cal State San Bernardino)

Title: "Parallelism and diagrams"

Abstract: I examine the geometric concept of parallelism in the context of Euclid's diagrammatic proof method, where this method is understood in terms of the formal system E presented in (Avigad et al. 2009).  Euclid's version of the notorious parallel postulate does not actually reference parallelism, but instead provides a condition for the intersection of lines.  I first explain how this version is naturally motivated when one does geometry with diagrams.  I then consider the epistemological question of how parallelism ought to be represented when one does geometry with diagrams.  This leads to a proposal for modifying E whereby the relation is no longer represented as the negation of the intersection relation between infinite lines.  I close the talk by discussing how the modification fits with Euclid's treatment of parallelism in the Elements, and with Saccheri's investigations into the parallel postulate in Euclides Vindicatus (1733).  

References: J. Avigad, E. Dean, and J. Mumma (2009).  A formal system for Euclid’s Elements. Review of Symbolic Logic, 2:700–768.

Speaker: Dorothy Edgington (Birkbeck)

Title: "Conditionals and the Frege-Geach Problem"

Abstract: Background. How do you assess a conditional, if A, C? One answer: you suppose that A, and consider how likely it is that C, under that supposition; i.e., conditionals are assessed by the conditional probability of consequent given antecedent. Now a conditional probability cannot be equated with an unconditional probability of the truth of a proposition. So on the above (suppositional) view, we should not think of conditionals as propositions, believed to the extent that we think they are true. To assert a conditional is not to assert that a proposition is true; it is to conditionally assert the consequent, on the supposition of the antecedent.

Foreground. Geach, attributing the point to Frege, objected to non-propositional accounts of particular areas of discourse (principally, but not exclusively, ethical discourse) as follows: a non-standard account of what one is doing when one utters a sentence S of a given kind leaves us with no account of how to understand S when it occurs as a component of a longer sentence—how it contributes to the meaning of the longer sentence—when it occurs, for example, as the antecedent of a conditional, or as a component in a disjunction.

Thus, on the suppositional view outlined above, there is a problem about embedded conditionals. There continues to be a lot of work on this problem, which I will assess.

 

Speaker: Michael Rescorla (UC Santa Barbara)

Title: "The Representational Foundations of Computation."

Abstract: I will discuss how computability theory, including both classical recursion theory and computable analysis, can illuminate the concept of representation. A Turing machine manipulates syntactic items inscribed on a machine tape. For a Turing machine to compute over a non-syntactic domain, syntactic items manipulated by the Turing machine must represent elements of the non-syntactic domain. Thus, as many philosophers have noted, representation is central to computability theory. Less widely appreciated is that computability offers numerous detailed insights into representation. Those insights extend far beyond the mathematical realm, encompassing mental representation of many important empirical phenomena. Building on previous discussions by Jerry Fodor, Saul Kripke, and Stewart Shapiro, I will elucidate the foundational connections between computation and representation. As I will discuss, computability theory studies computational properties of the representation relation, placing particular emphasis upon the interplay between computation and the "modes of presentation" under which we represent entities. By embedding intensionality within a systematic mathematical framework, computability theory provides diagnostic and classificatory tools that can strikingly illuminate modes of presentation. We have only begun to mine computability theory for its philosophical payoff.
 

 

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