Logical Methods in the Humanities Workshop Abstracts Winter 2005

The idea that phenomena are explained by modeling them on exemplars has often been proposed in the literature but never formally worked out. In this talk I will present a computational model of exemplar-based explanation, termed EBE, which is based on Kuhn's idea that "scientists solve puzzles by modeling them on previous puzzle-solutions". I will argue that explanations can be represented by derivation trees which describe each step in linking laws to phenomena. Next, I will develop a matching mechanism which explains novel phenomena out of largest possible subtrees of explanations of previous phenomena such that derivational similarity is maximized.

Drawing on examples from classical and fluid mechanics, I will show that explanatory derivation is "massively redundant": the number of different derivations of a phenomenon grows exponentially with the number of terms in the description of the phenomenon -- even if these derivations are all subsumed under the same general laws. This problem of massive redundancy has been severely underestimated in the philosophy of science. I will argue that the EBE model provides a way to tackle this problem which closely corresponds to human problem solving.

Next, I will provide an excursion into the other end of the scientific spectrum, discussing some examples from the human sciences, in particular from theoretical linguistics and music theory. What counts for the natural sciences also counts for humanities: explanations of linguistic and musical phenomena are massively redundant in that exponentially many derivation trees exist for a linguistic utterance or a musical piece. I urge that EBE suggests a general methodology for the natural and human sciences.

In his article entitled “Aufbau/Bauhaus” and related work, Peter Galison explores the connections between the Vienna Circle and the Dessau Bauhaus. Historically, these groups were related, with members of each group familiar with the ideas and following the progress of the other. Galison argues that the projects of the Vienna Circle and the Bauhaus are related as well, through shared political tendencies and methodological approach. The two main figures that connect the Vienna Circle to the Bauhaus are Rudolf Carnap and Otto Neurath. Yet, on our view, some of the connections that Galison establishes between these figures and the Bauhaus do not fit quite as well as they might. In particular, the connections upon which Galison focuses do not properly capture the common themes between the Bauhaus and Neurath’s philosophical projects.

In this paper, we will examine a few of the historical connections between the Dessau Bauhaus and the Vienna Circle, as well as the philosophical connections that Galison draws between these two groups. By examining in greater depth Neurath’s philosophical commitments, we aim to demonstrate that some of these philosophical connections fail to resonate with Neurath’s projects. And, finally, we develop different connections between Neurath’s projects and the Bauhaus. In our view, these new parallels between Neurath and the Dessau Bauhaus are both substantive and philosophically interesting.

In November and December 1915, Hilbert presented two communications to the Göttingen Academy of Sciences under the common title “The Foundations of Physics”. Versions of each eventually appeared in the Nachrichten of the Academy. Hilbert’s first communication has received significant reconsideration in recent years, following the discovery of proofs of this paper, dated 6 December 1915. The focus has been primarily on the ‘priority dispute’ over the Einstein field equations. However, it is our contention that the discovery of the December proofs makes it possible to see the thematic linkage between the material that Hilbert cut from the first contribution and the published content of the second contribution in 1917. The second contribution has been largely either disregarded or misinterpreted, and our aim is to show that (a) Hilbert’s two contributions should be regarded as part of a wider research program within the overarching framework of “the axiomatic method” (as Hilbert expressly stated was the case), and (b) the second contribution is a fine and coherent piece of work within this framework, whose principal aim is to address the apparent tension between general invariance and causality (in the precise sense of Cauchy determination), pinpointed in theorem 1 of the first contribution. This is not the same problem as that found in Einstein’s “hole argument” – something which, we argue, never confused Hilbert.

The logical empiricists took late nineteenth-century work in the foundations of geometry by Riemann, Helmholtz, Poincare, and Hilbert as an inspiration for their distinctive philosophy of geometry. They also appealed centrally to Einstein's general theory of relativity, which they viewed both as a culmination of the nineteenth-century foundational work in question and as fundamentally in agreement with their philosophical approach. Einstein's famous paper "Geometry and Experience" was paradigmatic here. I examine the main argument of this paper against the late nineteenth-century mathematical background and show that it was fatefully -- but also understandably -- misunderstood by the logical empiricists. At the same time, however, this examination helps us appreciate the extraordinarily rich and intricate interaction between physics, philosophy, and the foundations of geometry which actually led to Einstein's theory.

Twentieth century mathematics is marked by the emergence of ``abstract'' mathematics. Mathematicians often talk of looking at something from an abstract point of view or that a certain piece of mathematics is more abstract than another one. All these elements suggest that abstraction plays a key role in contemporary mathematics and mathematical knowledge in general. However, there is very little in the literature on what this abstraction amounts to and even, how it differs, if at all, from generalization. In this talk, we will propose a simple view of abstraction which allows us to differentiate abstraction from generalization, at least of one kind, and to determine when a mathematical concept is less abstract than another one.

(Joint work with David Corfield)