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\begin{document}

{\bf Open source mathematical software}\footnote{Opinion Column for
the Notices of the AMS, November 2007.}


\vskip .1in

Mathematical software has greatly contributed to mathematical
research, enabling exciting advances in mathematics and providing
extensive data for conjectures.  Perhaps three of the most well-known
applications of computation to mathematical research are resolution of
the {\em four-color conjecture} by Appel and Haken in 1976 (though it
is now reproven with less need for computer verification by
N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas), Thomas
Hales's proof of the {\em Kepler's conjecture}, and the formulation of
the {\em Birch and Swinnerton-Dyer conjecture}, which grew out of
extensive numerical computation.

Open source software, such as \TeX, Mozilla Firefox, and Linux has had a
profound effect on computing during the last decade, and we hope that
open source mathematical software will have a similar positive impact
on mathematics.

\begin{quote}
``I think we need a symbolic standard to make computer manipulations 
easier to document and verify. And with all due respect to the 
free market, perhaps we should not be dependent on commercial 
software here. An open source project could, perhaps, find better 
answers to the obvious problems such as availability, bugs, backward 
compatibility, platform independence, standard libraries, etc. 
One can learn from the success of \TeX\ and more specialized software 
like Macaulay2. I do hope that funding agencies are looking into this.''

\hspace{3ex}-- {\em Andrei Okounkov, 2006 Fields Medalist} 
(see ``Interviews with Three Fields Medalists,''
Notices of the AMS, March 2007, Volume 54 , Number 3 (2007)
405-410).
\end{quote}

The term {\em open source} is defined in \url{http://www.opensource.org/},
but basically it means anyone (including commercial companies or 
the defense department) should be able to inspect open source software, 
modify it, and share it with others.

One key difference between mathematical theorems and software, is that
theorems require little maintenance, whereas {\em mathematical
  software requires substantial and potentially expensive maintenance}
(bug fixes, updates when algorithms or languages change, etc.).
Mathematical research usually generates no direct revenue for
researchers, and likewise open source mathematical software is free to
share and extend, so it rarely generates revenue.  Volunteer effort,
donations, and financial support from the NSF and other organizations
is thus critical to the success of open source mathematical software.

There is a proof in the article by Campbell et al. in {\em The Atlas
  of Finite Groups---Ten Years On} (1998) that describes how many
separate software packages were ``easily used'' to deduce various
mathematical facts---no code is given, and some of the programs are
proprietary software which only runs on hardware many years out of
date.  Such proofs may become increasingly common in mathematics if
something isn't done to reverse this trend.

Suppose Jane is a well-known mathematician who announces she has
proved a theorem. We probably will believe her, but she knows that she
will be required to produce a proof if requested.  However, suppose now
Jane says a theorem is true based partly on the results of software.
The closest we can reasonably hope to get to a rigorous proof (without
new ideas) is the open inspection and ability to use all the computer
code on which the result depends.  If the program is proprietary, this
is not possible.  We have every right to be distrustful, not only due
to a vague distrust of computers but because even the best programmers
regularly make mistakes.

If one reads the proof of Jane's theorem in hopes of extending her
ideas or applying them in a new context, it is limiting to not have
access to the inner workings of the software on which Jane's result
builds.  For example, consider the following quote from the Mathematica
tutorial\footnote{{\tt
    http://reference.wolfram.com/mathematica/tutorial/\\WhyYouDoNotUsuallyNeedToKnowAboutInternals.html}}:
\begin{quote}
  ``Particularly in more advanced applications of Mathematica, it may
  sometimes seem worthwhile to try to analyze internal algorithms in
  order to predict which way of doing a given computation will be the
  most efficient. [...]  But most often the analyses will not be
  worthwhile. For the internals of Mathematica are quite complicated,
  and even given a basic description of the algorithm used for a
  particular purpose, it is usually extremely difficult to reach a
  reliable conclusion about how the detailed implementation of this
  algorithm will actually behave in particular circumstances.''
\end{quote}

No journal would make a statement like the above about the proofs of
the theorems they publish.  Increasingly, proprietary software and the
algorithms used are an essential part of mathematical proofs.  To
quote J. Neub\"user, ``{\em with this situation two of the most basic rules
of conduct in mathematics are violated: In mathematics information is
passed on free of charge and everything is laid open for checking.}''

{\bf Full disclosure:} The second author started a new
mathematics software system in 2005 called SAGE (see {\tt
  http://www.sagemath.org}), which combines Python, GAP, Singular,
PARI, Maxima, SciPy, etc. with several hundred thousand lines of new
code.  SAGE receives contributions from many mathematicians worldwide
that synthesize the latest algorithms from a broad range of topics
into a comprehensive toolkit for mathematical research.

\vskip .3in

\noindent{}D. Joyner, US Naval Academy, Annapolis, {\tt wdj@usna.edu}

\mbox{}

\noindent{}W. Stein, University of Washington, Seattle, {\tt
  wstein@u.washington.edu}


\end{document}