The Limits of Mathematics

This condensed version of chao-dyn/9509010 will be the main hand-out for a course on algorithmic information theory to be given 22-29 May 1996 at the Rovaniemi Institute of Technology, Rovaniemi, Finland (see announcement at http://www.rotol.fi/ ).Comment: LaTeX, 45 page

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

Philosophy of Computer Science: An Introductory Course

There are many branches of philosophy called “the philosophy of X,” where X = disciplines ranging from history to physics. The philosophy of artificial intelligence has a long history, and there are many courses and texts with that title. Surprisingly, the philosophy of computer science is not nearly as well-developed. This article proposes topics that might constitute the philosophy of computer science and describes a course covering those topics, along with suggested readings and assignments

Data types with symmetries and polynomial functors over groupoids

Polynomial functors are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect is often prominent and justifies the terminology. For example, Tambara's theorem states that the category of finite polynomial functors is the Lawvere theory for commutative semirings. In this talk I will explain how an upgrade of the theory from sets to groupoids is useful to deal with data types with symmetries, and provides a common generalisation of and a clean unifying framework for quotient containers (cf. Abbott et al.), species and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan. The multi-variate setting also includes relations and spans, multispans, and stuff operators. An attractive feature of this theory is that with the correct homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits, etc. - the groupoid case looks exactly like the set case. After some standard examples, I will illustrate the notion of data-types-with-symmetries with examples from quantum field theory, where the symmetries of complicated tree structures of graphs play a crucial role, and can be handled elegantly using polynomial functors over groupoids. (These examples, although beyond species, are purely combinatorial and can be appreciated without background in quantum field theory.) Locally cartesian closed 2-categories provide semantics for 2-truncated intensional type theory. For a fullfledged type theory, locally cartesian closed \infty-categories seem to be needed. The theory of these is being developed by D.Gepner and the author as a setting for homotopical species, and several of the results exposed in this talk are just truncations of \infty-results obtained in joint work with Gepner. Details will appear elsewhere.Comment: This is the final version of my conference paper presented at the 28th Conference on the Mathematical Foundations of Programming Semantics (Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer Science. 16p

Pragmatic Holism

The reductionist/holist debate seems an impoverished one, with many participants appearing to adopt a position first and constructing rationalisations second. Here I propose an intermediate position of pragmatic holism, that irrespective of whether all natural systems are theoretically reducible, for many systems it is completely impractical to attempt such a reduction, also that regardless if whether irreducible `wholes' exist, it is vain to try and prove this in absolute terms. This position thus illuminates the debate along new pragmatic lines, and refocusses attention on the underlying heuristics of learning about the natural world