Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well defined.

Several stages can be identified in Darwin's effort to formulate natural selection. The first stage corresponded, roughly speaking, to the period up to 1844. It was characterized by Darwin's attempt to base his model of geographic speciation on an individualistic dynamics, with species understood as reproductively isolated populations. Toward the end of this period, Darwin's ignorance of the laws of variations and heredity led him to adopt varieties and species as the units of variations. This had the extremely important effect (...) of stimulating him to consider the process of speciation as involving populations. At the end of this period, Darwin also began to regard adaptation as being exclusively toward places in the economy of nature. Thus he faced the problem of integrating the process of natural selection with the process of speciation. Individual variants were the units that fueled the first process, whereas varieties produced new speices. There was no link between adaptation and speciation, except whatever could be supplied by a quasi-historical, developmental idea of optimizing the amount of life.In the second stage, I contend, Darwin's reading of Milne-Edwards crystallized his previous insights into a coherent whole. Milne-Edwards' comments on the advantage of functional specialization could readily be understood in terms of the advantage accruing to the individual, relative to other members of its species, from occupying a different niche. Milne-Edwards' discussion of the division of labor suggested that organisms which moved into unoccupied niches would enjoy reduced competition, and hence a differential advantage in survival and reproduction; thus they would induce the species to do likewise. Rather than base his explanation on an analogy with the artificial economy, Darwin chose the principle of the optimalization of the amount of life per unit area as the overall explanatory principle. The difficulties connected with integrating different levels of description were therefore circumvented, insofar as the problem of diversity and speciation was concerned. Although natural selection considered individuals as the units of selection, and the units of variations were varieties and species, the dynamics of the process understood in terms of natural selection, competition, division of labor and niches could give a plausible account of how individual advantage could be transferred to the species, and how diversity resulted from this mechanism. The problem of the different levels of descriptions was confined to how the properties of variations in individuals (in particular, the frequency of variations and their transmission) were responsible for the assumed variability characteristic of varieties and species. This problem Darwin never solved.A third stage occurred in 1858 with the amalgamation of the tree-of-life vizualization of the process of speciation. Speciation, geographic distribution, and systematics were all then embedded in a conceptual matrix with vast explanatory powers. (shrink)

ArgumentJ. Robert Oppenheimer was a complex person. His work in physics during the 1930s, at Los Alamos during the 1940s, and as governmental advisor in the immediate postwar period, gave him a deep sense of connection with communities that had distinctive purposes. But he found it difficult to conceive an overall creative vision for himself or to devise a compelling objective for the community he belonged to if one had not been formulated at the time he assumed its leadership. I (...) analyze the reasons for his successes: the vision and demands of physics during the 1930s, the make-up of Los Alamos, and the challenges of the postwar atomic world. In each of these enterprises he assumed a distinctive role and came to represent a distinctive persona – but he could not integrate his activities into a coherent whole that might be a model for the intellectual in the new world he had helped to shape. (shrink)

Contingencies of the early nuclear arms race Content Type Journal Article Pages 1-23 DOI 10.1007/s11016-010-9495-z Authors S. S. Schweber, Department of the History of Science, Harvard University, Science Center 371, Cambridge, MA 02138, USA Alex Wellerstein, Department of the History of Science, Harvard University, Science Center 371, Cambridge, MA 02138, USA Ethan Pollock, Department of History, Box N, Brown University, Providence, RI 02912, USA Barton J. Bernstein, History Department, Building 200, Stanford University, Stanford, CA 94305-2024, USA Michael D. Gordin, (...) History Department, 305 Dickinson Hall, Princeton University, Princeton, NJ 08544, USA Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796. (shrink)

Michael Eckert has written a remarkable biography of Arnold Sommerfeld , the “off-scale” theoretical physicist who made his Seminar at the University of Munich the outstanding school of theoretical physics of the first third of the twentieth century. Sommerfeld was the teacher and mentor of a large number of exceptional theoretical physicists who studied with him either as doctoral or post-doctoral studentsSee the Wikipedia entry for Arnold Sommerfeld for a complete listing of all his students by category.; and among these, (...) Peter Debye, Max von Laue, Wolfgang Pauli, Werner Heisenberg, Linus Pauling and Hans Bethe became Nobel laureates.In his prologue, Eckert noted that a scientific biography should serve not only as a vehicle to describe and explain scientific processes in a social and cultural context, but must also present “the ambitions, passions, and moral choices of a life in science”. Eckert was able to meet these demands by drawing on Sommerfeld’s personal and .. (shrink)

ArgumentThe paper is an exploration of the interactions between Einstein and Oppenheimer. It highlights the sharp differences in Einstein's and Oppenheimer's approach to physics, in their presentation of self as iconic figures, and in their relation to the communities they considered themselves part of. To understand their differing approaches to physics it briefly reviews the kinds of unifications that took place in physics during the first two-thirds of the twentieth century and points to the 1961 MIT centennial celebration to demonstrate (...) the potency of Einstein's vision that there might be a fundamental theory from which all known theories could be derived. It also briefly reviews various aspects of the development of theoretical physics and of general relativity in the first two-thirds of the twentieth century, to better understand the context of the sharp, negative remarks that Oppenheimer made about Einstein and about his theory of general relativity in 1965 on the occasion of the tenth anniversary of Einstein's death. To answer the question: “Why the antagonism on Oppenheimer's part?” it looks at Oppenheimer's and Einstein's relation to their Jewish roots, their stance regarding nationalism, and their philosophical commitments. (shrink)

The ArgumentIn the nineteenth century, American legal educators drew on the idea of “legal science” the claim that the study of law was similar to the study of the natural sciences. In this paper, I propose to examine the particular conceptions of “science” that were incorporated into that idea. The primary point of the paper is to argue that in antebellum America, a particular view of the natural sciences dominated public discourse, and it was this conception that was appropriated by (...) contemporaneous legal scientists. Public discussions of natural science in lyceums, surveys, and journals, were carried out in a language grounded in the same religious commitments and the same normative conception of nature that drove the ideology of laissez-faire. Described under the rubric “Protestant Baconianism,” the approach was characterized by commitments to four elements: natural theology; a constrained version of Baconian inductivism; a belief in grand synthesis and proof by analogy; and claims of moral improvement. These elements were typical of the efforts of a particular influential circle of natural scientists and, similarly, of some of the more influential legal scientists in the antebellum period. This article examines the elements of the Protestant Baconian approach that were incorporated into Christopher Columbus Langdell's model of legal science in the 1870s, a model that continues to influence thinking about both legal education and jurisprudence to this day. (shrink)

We say that a theory [Formula: see text] satisfies arithmetic-is-recursive if any [Formula: see text]-computable model of [Formula: see text] has an [Formula: see text]-computable copy; that is, the models of [Formula: see text] satisfy a sort of jump inversion. We give an example of a theory satisfying arithmetic-is-recursive non-trivially and prove that the theories satisfying arithmetic-is-recursive on a cone are exactly those theories with countably many [Formula: see text]-back-and-forth types.

We say that a theory T satisfies arithmetic-is-recursive if any X′-computable model of T has an X-computable copy; that is, the models of T satisfy a sort of jump inversion. We give an example of a...

We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any (...) Borel quotient of the reals reduces to it. (shrink)