“Experimental mathematics” has emerged in the past 25 years or so as a competing paradigm for research in the mathematical sciences. Challenges in 21st Century Experimental Mathematical Computation, an exciting workshop held at ICERM (the Institute for Computational and Experimental Research in Mathematics), July 21–25, explored emerging challenges of experimental mathematics in the rapidly changing era of modern computer technology. We summarize the workshop findings in this article; information about the research presentations can be found here.
Despite several more precise definitions that have been offered for “experimental mathematics,” we prefer the informal one given in the book The Computer as Crucible (Jonathan Borwein and Keith Devlin, AK Peters, 2008):
“Experimental mathematics is the use of a computer to run computations––sometimes no more than trial-and-error tests––to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search.”
“Experimental mathematics” is distinguished from “computational mathematics” and “numerical mathematics” in that the latter two generally encompass methods for applied mathematics, whereas “experimental mathematics” refers to advancing the state of the art in mathematical research per se.
While the overall approach and philosophy of experimental mathematics have not changed greatly in the past 25 years, its techniques, scale, and sociology have changed dramatically. The field has benefited immensely from advances in computer technology, including those predicted by Moore’s law, but the increases in speed brought by algorithmic progress have often outpaced Moore’s law, notably in such areas as linear programming, linear system solving, and integer factorization.
Software available to experimental mathematicians has also advanced impressively. Along with improvements in earlier versions of commercial products like Maple, Mathematica, and MATLAB, many new “freeware” packages are now in use, including the open-source Sage, numerous high-precision computation packages, and an impressive array of software tools and visualization facilities.
With all these tools and facilities, many new results have been published, ranging from new formulas for mathematical constants, such as pi, log(2), and zeta(3), to computer-verified proofs of the Kepler conjecture. Whereas it was once considered atypical or even improper to mention computations in a published paper, now it is commonplace. Several journals, such as Experimental Mathematics and Mathematics of Computation, are devoted almost exclusively to mathematical research involving computations.
Yet many challenges remain as researchers push the envelope in mathematical computing. Among the most critical issues are the following:
Adapting codes to new platforms. The emergence of powerful, advanced-architecture platforms, particularly those incorporating such features as highly parallel, multi-core, or many-core designs, present daunting challenges to researchers, who must adapt their codes to these new architectural innovations or risk being left behind in the scientific computing world.
Ensuring reliability and reproducibility. Reproducibility means ensuring, for example, that the results of floating-point computations are numerically reproducible, or that the results of a symbolic computation are reliable (complications can arise when two expressions are compared to determine whether they are mathematically equivalent). Many users implicitly trust results obtained with these tools, losing sight of the fact that they are far from infallible. One of the approaches to increased reliability should be stronger interactions with the cousin discipline of formal proof systems (as used by Thomas Hales to complete, in 2014, a multi-year computer-verified proof of the Kepler conjecture on stacking spheres), but huge efficiency issues have to be addressed.
Managing the exploding scale of data. The size of datasets used in the field has increased at least as fast as Moore’s law growth. Algorithmic progress is thus necessary in, for example, tools that aid in the quest for structure in large numerical or symbolic datasets.
Large-scale software maintenance. The rapidly increasing size of many of the software tools used in the field means that mathematicians must confront the challenge of large-scale software maintenance. This includes the discipline, unfamiliar to many research mathematicians, of strict version control, collaborative protocols for checking out and updating software, validation tests, issues of worldwide distribution and support, and persistence of the code base.
Changing sociological and community issues. Numerous recently published results arise from Internet-based collaborations, with research ideas, computer code, and working manuscripts often circling the globe multiple times in a single day. One example is the PolyMath project, whereby loosely knit Internet-based teams of mathematicians have addressed and, in several cases, “solved” or progressed toward the solution of interesting unsolved mathematical problems. Further progress will require improved tools and platforms for such collaborations, as well as an international “clearing house” that will collect, validate, and coordinate such activities.
Education. Computer-based tools are also being introduced into mathematical education, permitting students to see mathematical concepts emerge from hands-on experimentation and thus attracting to the field a cadre of 21st-century computer-savvy students. This is not the first time that technology has promised to reinvent mathematical education, but it is clear that much additional thought is needed on how computation can be best incorporated into education.
Other issues. The workshop discussion highlighted the fact that much of the published work to date in experimental mathematics has focused on a few areas that are particularly amenable to computational exploration––among them finite group theory, combinatorics and graph theory, number theory, evaluation of series and integrals. How can we expand the scope of questions that have been examined with these methodologies, not just to other areas of mathematics but to other fields as well?
All this also raises the question of how such work can be paid for. Unlike the case in the “hard sciences,” the majority of published mathematical research (pure and applied) is completed without direct research funding, by academic mathematicians or others as they have time alongside their teaching or other formal duties. But some of the work described here, particularly that involving substantial software development and maintenance, cannot be done so informally. Nor does a royalty model work, as it has for traditional publications––the development costs are too great and the academic rewards too small.
It is clear that researchers in experimental mathematics need to work more vigorously with government funding agencies to find ways to provide this funding. Perhaps this may be done more easily if projects can be pursued in collaboration with researchers in other disciplines, particularly in fields such as computer science that have typically been somewhat more generously funded.
*The full report, by D.H. Bailey, J.M. Borwein, U. Martin, B. Salvy, and M. Taufer, “Opportunities and Challenges in 21st Century Experimental Mathematical Computation,” August 26, 2014, is available at http://www.davidhbailey.com/dhbpapers/icerm-2014.pdf.