A Foundational View on Integration Problems

The integration of reasoning and computation services across system and language boundaries is a challenging problem of computer science. In this paper, we use integration for the scenario where we have two systems that we integrate by moving problems and solutions between them. While this scenario is often approached from an engineering perspective, we take a foundational view. Based on the generic declarative language MMT, we develop a theoretical framework for system integration using theories and partial theory morphisms. Because MMT permits representations of the meta-logical foundations themselves, this includes integration across logics. We discuss safe and unsafe integration schemes and devise a general form of safe integration

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Curriculum Guidelines for Undergraduate Programs in Data Science

The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in Data Science. The group consisted of 25 undergraduate faculty from a variety of institutions in the U.S., primarily from the disciplines of mathematics, statistics and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in Data Science

Computational reverse mathematics and foundational analysis

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert's program due to Simpson [1988], and predicativism in the extended form due to Feferman and Sch\"{u}tte. Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only $\omega$-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a $\Pi^1_1$ complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than $\Pi^1_1\text{-}\mathsf{CA}_0$.Comment: Submitted. 41 page

"Revolution? What Revolution?" Successes and limits of computing technologies in philosophy and religion

Computing technologies like other technological innovations in the modern West are inevitably introduced with the rhetoric of "revolution". Especially during the 1980s (the PC revolution) and 1990s (the Internet and Web revolutions), enthusiasts insistently celebrated radical changes— changes ostensibly inevitable and certainly as radical as those brought about by the invention of the printing press, if not the discovery of fire.\ud These enthusiasms now seem very "1990s�—in part as the revolution stumbled with the dot.com failures and the devastating impacts of 9/11. Moreover, as I will sketch out below, the patterns of diffusion and impact in philosophy and religion show both tremendous success, as certain revolutionary promises are indeed kept—as well as (sometimes spectacular) failures. Perhaps we use revolutionary rhetoric less frequently because the revolution has indeed succeeded: computing technologies, and many of the powers and potentials they bring us as scholars and religionists have become so ubiquitous and normal that they no longer seem "revolutionary at all. At the same time, many of the early hopes and promises instantiated in such specific projects as Artificial Intelligence and anticipations of virtual religious communities only have been dashed against the apparently intractable limits of even these most remarkable technologies. While these failures are usually forgotten they leave in their wake a clearer sense of what these new technologies can, and cannot do