What is experimental mathematics?
David H. Bailey
Lawrence Berkeley National Laboratory (retired) andComputer Science Department, University of California, Davis

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Jonathan M. Borwein, 1951–2016
I 388 published journal articles; another 103 inrefereed conference proceedings.
I ISI Web of Knowledge lists 6,593 citations from351 items; one paper has been cited 666 times.
I His work spanned pure mathematics, appliedmathematics, optimization theory, computerscience, mathematical finance, and experimental
mathematics.
I Borwein sought to do research that is accessible,

and to highlight aspects of his work that a broad
audience (including both researchers and the laypublic) could appreciate.
I More information, including memorials and links tonearly 1700 publications, preprints and talks:
http://www.jonborwein.org.

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What is experimental mathematics?
Experimental mathematics encompasses using a computer to:
I Gain insight and intuition about mathematical phenomena.
I Explore specific mathematical questions.
I Plot or otherwise visually represent mathematical objects.
I Test (and falsify) conjectured hypotheses.
I Discover new mathematical identities and theorems.
I Find rigorous proofs of empirically discovered hypotheses.
In one sense, experimental mathematics is nothing new — all of the famous historicalmathematicians (Newton, Euler, Gauss, Ramanujan, many others) employedexperimentation and computational discovery in their work.
The advent of the computer has fundamentally and permanently changed themethodology of doing mathematics.
But the overall goal remains the same: to find and rigorously establish secure
mathematical knowledge.
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Von Neumann’s warning about the future of mathematics
Experimental mathematics provides a means to escape the trap feared by John von
Neumann when he wrote,
But there is a grave danger that the subject [of mathematics] will develop
along the line of least resistance, that the stream, so far from its source, willseparate into a multitude of insignificant branches, and that the discipline will
become a disorganized mass of details and complexities. In other words, at agreat distance from its empirical source, or after much “abstract” inbreeding,a mathematical subject is in danger of degeneration. . . .
In any event, whenever this stage is reached, the only remedy seems to me tobe the rejuvenating return to the source: the re-injection of more or lessdirectly empirical ideas. I am convinced that this was a necessary condition
to conserve the freshness and the vitality of the subject and that this willremain equally true in the future.
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Tools of experimental mathematics
Experimental mathematicians use these and other tools:
I Symbolic math software, such as Mathematica, Maple and Sage.
I Online facilities such as:
II

The Online Encyclopedia of Integer Sequences: http://www.oeis.org.
The Inverse Symbolic Calculator 2: https://isc.carma.newcastle.edu.au.

I Custom-written software, typically in C++, Python or Fortran.
I High-precision arithmetic software packages, provided in Mathematica, Maple andSage, but also in packages such as MPFR and GMP.
I Software to facilitate highly parallel computations, such as OpenMP and MPI.
I Graphics and visualization software, provided in Mathematica, Maple and Sage,
but also in packages such as Cinderella.
I Specialized math software for particular fields, such as numerical analysis, algebra,topology.

5 / 11Experimental mathematics and outreachExperimental mathematics provides a unique opportunity to involve a much broader
community in doing real mathematical research:
I High school and college students.
I Computer scientists.
I Computer graphics experts.
I Statisticians.
I Data scientists.

We should find ways to involve computational methods in every aspect ofmathematical teaching and research.

6 / 11π and experimental mathematics
π continues to excite millions worldwide,
leading many to pursue careers in math,science and engineering.

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A random walk on the first 100 billion base-4 digits of πThis dataset can be explored online: http://gigapan.com/gigapans/106803

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Winning the battle, but losing the war
Mathematicians and scientists may be winning battles to publish papers and obtain
grants, but we are losing the war for the hearts and minds of the public:
I 51% in U.S. (54% in Australia) either do not believe in climate change, or do not believe

there is any human connection.
I 42% in U.S. (23% in Australia) believe that humans were created within past 10,000years.
I 38% in U.S. (32% in Australia) do not believe in evolution.
I 32% in U.S. (24% in Australia) do not believe vaccinations are safe.
I 48% in U.S. (34% in Australia) believe humans are being visited by extraterrestrial UFOs.
I 6% in U.S. believe NASA faked the Apollo moon landings.
I Some even dispute the value of π. (I frequently receive such email.)

Anti-science movements arise from both sides of the political spectrum:
I From the left: anti-vaccination and anti-fluoridation.
I From the right: anti-climate change and anti-evolution.
9 / 11Carl Sagan’s warning (The Demon Haunted World, 1995)
I have a foreboding of an America in my children’s or my grandchildren’s time — when
the United States is a service and information economy; when nearly all the key
manufacturing industries have slipped away to other countries; when awesome
technological powers are in the hands of a very few, and no one representing the public
interest can even grasp the issues; when the people have lost the ability to set theirown agendas or knowledgeably question those in authority; when, clutching ourcrystals and nervously consulting our horoscopes, our critical faculties in decline,
unable to distinguish between what feels good and what’s true, we slide, almost
without noticing, back into superstition and darkness. ...
We’ve arranged a global civilization in which most crucial elements ... profoundly
depend on science and technology. We have also arranged things so that almost noone understands science and technology. This is a prescription for disaster. We might
get away with it for a while, but sooner or later this combustible mixture of ignorance
and power is going to blow up in our faces.
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How can we turn the tide?
I Start a blog.
I Visit schools or give lectures.
I Write books for the general public.
I Write articles for science news forums.
I Write expository articles for scientific journals.
I Pursue research topics that have potentially wide appeal.
I Recognize communication skills in hiring, promotion andresearch grant decisions.
I Find ways to utilize computers and otherwise make teaching

and research much more engaging and interesting.

This talk is available at http://www.davidhbailey.com/dhbtalks/dhb-expmath-2017.pdf

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