\documentclass[12pt]{article} %\usepackage[active]{srcltx} \usepackage{amssymb} \usepackage{amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{url} \usepackage{xspace} \newcommand{\SAGE}{{\sf SAGE}\xspace} \newcommand{\sage}{\SAGE} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\GF}{GF} %%%% Theoremstyles \theoremstyle{plain} \newtheorem*{theorem}{Theorem} \newtheorem*{conjecture}{Conjecture} \begin{document} \title{Open Source Mathematical Software:\\A White Paper %In an email, Emil Volcheck %(Chair of the ACM SIGSAM executive committee) %mentioned that Bob Grafton, an NSF program officer who works %with symbolic computation funding, requested a ``white paper'' on %suggested NSF funding of CAS's. This paper is to address that request. \\ %{\small {\it (12th draft version)}} } \author{ D. Joyner\thanks{Math Dept, USNA, Annapolis, MD 21402, wdjoyner@gmail.com}\hspace{1ex} and W. Stein\thanks{Math Dept, Univ Washington, Seattle, WA 98195, wstein@gmail.com}} \date{1-15-2008} \maketitle \tableofcontents \newpage Open source software has had a profound effect on computing during the last decade, especially on web servers (Apache), web browsers (Firefox), operating systems (Linux and OS X), and programming languages (GCC, Java, Python, Perl, etc.). The purpose of this paper is to put forward the case that open source development methodologies might also have a positive effect on mathematical software, especially if the National Science Foundation (NSF) increases their support of open source mathematical software development. We argue that careful funding of open source mathematical software may lead to a lower total cost of ownership in the research and education community, and to more efficient and trustworthy mathematical software. \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 \cite{MP}). \end{quote} By {\it mathematical software} we mean a collection of specific implementations of algorithms and data structures for computation with mathematical objects such as matrices, functions, groups, rings, graphs, differential equations, and so on. The term {\em open source} is defined in \cite{osi}: \begin{quote} ``Open source is a development method for software that harnesses the power of distributed peer review and transparency of process. The promise of open source is better quality, higher reliability, more flexibility, lower cost, and an end to predatory vendor lock-in.'' \end{quote} Anyone should be able to inspect open source software, modify it, and share it with others. In Section~\ref{sec:contrib} we discuss some selected contributions of mathematical software to mathematics research. In Section~\ref{sec:principles} we lay out basic values, priorities and principles for mathematical software. Section~\ref{sec:sage} discusses \SAGE, which is mathematical software created to address many ease of use and functionality issues with existing mathematical software. In Section~\ref{sec:which} we briefly discuss some criteria that NSF may use in deciding which software to consider funding, and give a list of software that satisfies these criteria. Finally, Section~\ref{sec:edu} discusses open source software in the context of mathematics education. Througout this paper we will focus about equally on the open source software GAP and \sage, because those are the systems the authors are most familiar with. %\vspace{2ex} %\noindent{\bf Key Points:} %\begin{enumerate} %\item %Mathematical software leads to {\bf important advances} in pure and applied %mathematics. %\item %Funding research that critically depends on %{\bf closed source software} may be counterproductive. %\item %Funding research and development of open source mathematical software %may lower the {\bf total cost of ownership}. %\item %Open source mathematical software is more likely to remain {\bf publicly %accessible} than software built using closed source software. %\end{enumerate} \vspace{2ex} \noindent{\bf Acknowledgment:} Many people have offered their generous suggestions (almost all of which of which were included in one form of another) but we are especially grateful to ACM SIGSAM and especially Emil Volcheck, in his capacity as chair, for support and advice. We thank Craig Citro, Richard Fateman, Dan Grayson, and Jaap Spies for their very careful reading and helpful suggestions. \section{Some contributions of the software GAP}\label{sec:contrib} 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 (see \cite{AH}, \cite{AHK}, though it is now reproven in \cite{RSST}), Hales's proof \cite{H} of the {\em Kepler's conjecture}, and the formulation of the {\em Birch and Swinnerton-Dyer conjecture} \cite{birch:bsd} which grew out of extensive numerical computation. To give a concrete flavor of how open source mathematical software contribute to research, we will focus in the rest of this section on the 20-year old open source mathematical software package GAP. The GAP web page lists {\em 780 publications that reference GAP}. We now give a few specific illustrations of these applications of GAP to mathematical research. \subsection{Riemann Surfaces} Max Dehn once said that there is no difference between combinatorial group theory and the theory of the fundamental group of two-dimensional topological surfaces. Therefore, it is not surprising that GAP -- primarily a group theory package -- is very helpful in the investigation of Riemann surfaces and curves with ``large'' automorphism groups. Thomas Breuer's Ph.D. thesis \cite{B1} initiated a computational program to exploit this connection using GAP. His thesis focues on the following two problems: \begin{itemize} \item[(1)] Classify all groups of automorphisms of compact Riemann surfaces $X$ of fixed genus $g(X)\ge 2$ up to equivalence of the natural action on ${\cal H}^1(X)$. \item[(2)] Classify all characters of a given finite group $G$ that arise from the natural action of groups of automorphisms (isomorphic to $G$) of compact Riemann surfaces $X$ on ${\cal H}^1(X)$. \end{itemize} Heavy use of calculations with GAP help to present a classification as required in (1) for genus $2\le g\le 48$. (Only classifications for $2\le g \le 5$ had been achieved earlier.) Considering (2), for example, he obtained full classifications for arbitrary dihedral and abelian groups. The work \cite{MSV} using the GAP package BRAID was inspired by Thomas Breuer's earlier work. Let $G$ be a finite group. By Riemann's Existence Theorem, braid orbits of generating systems of $G$ with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group $G$. Using BRAID for the computation of braid orbits, it is possible to classify irreducible families of indecomposable rational functions with exceptional monodromy group. Other papers by Magaard, Shpectorov, and V\"olklein use BRAID to deal with other questions in the theory of monodromy groups of Riemann surfaces. This is not the only use of GAP in Riemann surface theory. Indeed, Paul Bailey, in his Ph.D. thesis \cite{Ba}, computed a Hurwitz space using GAP. Then he and his advisor Mike Fried wrote a paper \cite{BaF} based on a different Hurwitz space, where they also used GAP to compute the number of components and their genus. Joyner and Ksir \cite{JK} used it to understand which representations of $\SL_2(\GF(p))$ occur in the decomposition of $H^1(X(p),{\mathbb{C}})$, where $X(p)$ is a modular curve and $p>5$ is a prime. \subsection{Character theory} Knutson's 1973 book \cite{K} contains the following conjecture: \begin{conjecture} Let $G$ be a finite group, $\rho$ the right regular representation of $G$ and $\pi$ any irreducible representation. There is a $\pi'$ in the Grothendieck ring $R(G)$ such that $({\rm tr}\, \pi)({\rm tr}\, \pi')={\rm tr}\, \rho$. \end{conjecture} This was disproved by Thomas Breuer using GAP by examining character tables of finite groups. He found that the group $\SL_2(\GF(13))$ yielded a counterexample \cite{B2}. This discovery could, in principle, have been done in 1973 but the search would likely have taken too long. With GAP, it is a matter of knowing enough representation theory to figure out what to program and then have GAP (fairly quickly) compute the necessary data. This is illustrative of GAP's ability to dispose of such character-theoretic questions with relative ease. \subsection{Coding theory} The open source GAP error-correcting codes package GUAVA was originally written in the 1990's by Jasper Cramwinckel, Erik Roijackers, and Reinald Baart as a final project at Delft University of Technology, Department of Pure Mathematics, under the direction of Professor Juriaan Simonis \cite{GU}. Regarding a code $C$ as a vector subspace of $F^n$ ($F$ a finite field) with a fixed basis, the key ``parameters'' of $C$ are its length $n$, its dimension $k$, and the smallest number $d$ of nonzero coordinates of any of the nonzero vectors in $C$. For a given $n$, the larger $k$ is the more ``information'' $C$ can be used to transmit, and the larger $d$ is the more errors $C$ can ``correct''. A main problem in coding theory is to find, for a given $F$, $n$, and $k$, the code $C$ that has the largest possible $d$. Using GUAVA and some additional GAP programming, a code $C$ over $\GF(8)$ with previously unknown parameters $[49,11,28]$ was discovered by the first author \cite{J}. This illustrates the wide variety of applications that GAP has to not only pure mathematics but also in applied mathematics. Of even more significance, GUAVA was also useful in the classification of self-dual codes \cite{FGHP}. Coding theory is not the only applied area of mathematics on which GAP has had an impact. For example, several applications of GAP to chemistry (via the theory of crystallographic groups) are indicated on the GAP web page under ``Examples''. \subsection{Sporadic groups} Sporadic groups are important objects in the classification of all finite simple groups. GAP has been used by many to investigate the sporadic groups, especially the large ones, which are difficult to analyze ``by hand''. Two examples are A. Hulpke and S. Linton \cite{HL}, who investigate the group $\text{Co}_3$, and S. Linton, R. Parker, P. Walsh, and R. Wilson \cite{LPWW}, who investigate the Monster simple group. %It is interesting to note that the paper %\cite{HL} used, in addition to the ``core'' GAP functionality, the GAP %package GUAVA referred to above. \section{Rigor and design principles}\label{sec:principles} In mathematics, correctness and verifiability rule with an iron fist. This holds true for scholarly papers, and should hold as well for the mathematical software that supports them. In this section we expand on this idea and its implications for software design. \subsection{Rigor in research and software} If we (the collective ``we'' -- the society of mathematicians) are to welcome results of mathematical software into the field as the partner of our theorems then we must also subject the implementations to the same level of rigor. Mathematical theorems are usually not {\em formally} proved correct by writing them out as logical statements that rely on basic axioms; likewise, we do {\em not} advocate that mathematical software must be {\em formally} proved correct. Formal proof of correctness of programs is often not feasible except in fairly trivial cases. Instead we require that any mathematician should be able to inspect any part of a calculation, in the same way that a mathematician can inspect any part of a proof in a research paper, without having to sign a nondisclosure agreement. That proofs are openly published is critical to the development of new mathematics. Mathematical research usually builds on a careful analysis of the reasoning of several key, important papers in the field -- they extend the other paper's arguments, improve the analysis, or provide a simpler proof of an important result. The development of mathematical software also has the potential to greatly benefit from this open model. Implementations of new algorithms often are built on older ones, and the algorithms of today form the basis of tomorrow's algorithm. \begin{quote} ``I think, fundamentally, open source does tend to be more stable software. It's the right way to do things. I compare it to science versus witchcraft. In science, the whole system builds on people looking at other people's results and building on top of them. In witchcraft, somebody had a small secret and guarded it -- but never allowed others to really understand it and build on it. Traditional software is like witchcraft. In history, witchcraft just died out. The same will happen in software. When problems get serious enough, you can't have one person or one company guarding their secrets. You have to have everybody share in knowledge.'' \hspace{3em} -- Linus Torvalds, author of the Linux operating system. \end{quote} There is a legitimate concern that much code that might have been made publicly available will instead be made part of commercial programs, thereby diminishing their usefulness to the general public. Most commercial software requires that copyright of code they include be signed over to them, just as assignment of copyright is required when publishing an article in most mathematical journals. Commercial mathematics software closely resembles a journal that only publishes the statements of theorems but not their proofs! Increasingly, implementations of mathematical algorithms whose development was funded by taxpayers, is thus absorbed into commercial programs. If mathematical research were published in exactly this way, the NSF would likely refuse to fund it. It is not desirable for mathematical papers to be supported by computations that cannot be inspected or verified except possibly by employees of the commercial software that was used. Consider the following scenario. A well-known mathematician, call her Jane, says a theorem is true and she can prove it. We probably will believe her, but she knows that she will be required to produce a proof if required. However, suppose now Jane says a theorem is true based on the results of mathematical software. In this case, something else enters into the balance -- the mathematical 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, then verification by any researcher is usually not possible, and we mathematicians have every right to be distrustful. This is not only due to a vague distrust of computers but because even the best programmers regularly make mistakes. Moreover, if one reads the proof of Jane's theorem in hopes of extending her ideas or applying them in a new context, it could be very frustrating to not have access to the inner workings of the software on which Jane's result builds. Even if not all mathematicians value open source mathematical software as being {\it a priori} more trustworthy than proprietary software, having an important algorithm in both a proprietary or commercial program and an open source program is an advantage. Often in practice the implementations are different, so if the commercial and open source programs output the same answer, the result of the calculation is perhaps more convincing. %If in addition a description of the %implementation has been published in a research journal such as the %Mathematics of Computation then an even greater degree of confidence %is given the result. Non-verifiable results are data, possibly even interesting data, but {\it not rigorous mathematics}. We are not saying such data is not useful; only that it is not rigorous mathematics. Here are a few examples where such data could be important and the type of software used (open or closed source) is not immediately relevant: \begin{itemize} \item[(a)] Suppose one is searching for a counterexample to a conjecture. Often times, it takes just one data point to disprove a conjecture and perhaps the data point itself, if computed using proprietary software, can be independently checked. For example, if one is searching for a factorization of, say, a large integer into proper divisors, once it is found by even a proprietary system, the result can be quickly verified. \item[(b)] Suppose a mathematician is trying to model the banking industry using very complicated economic principles. Her mathematical model looks correct but she needs to run simulations to see if her parameters are reasonable. Once she has more data, she can focus on using mathematics to improve her model. The data itself is not mathematics, but it is used to help her determine what mathematical tools to use. For her purpose, the data could be gathered using commercial or open source software. \end{itemize} In our opinion, when possible, open source software is to be preferred. For example, what if in case (a) the system was closed source and next year someone wants to test a slightly different conjecture? We close this section with a quote by Wolfram Research from the documentation of Mathematica that illustrates the opposing perspective that understanding how the mathematical software we use actually works is usually a waste of time. \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. And there are indeed occasionally major improvements that you will be able to make in specific computations as a result of such analyses. 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.'' \hspace{1em}-- \url{http://reference.wolfram.com/mathematica/tutorial/WhyYouDoNotUsuallyNeedToKnowAboutInternals.html} \end{quote} Increasingly, proprietary software and the algorithms used are an essential part of mathematical proofs. To quote Joachim Neub\"user \cite{N}, the ``father'' of GAP, ``{\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.}'' \subsection{Design principles} In trying to design mathematical software based on objective {\it principles}, as opposed to what features will increase the customer base the fastest, and make the most money, we want software that is \begin{enumerate} \item {\bf open source} and written in a readable and accessible language, for reliability and extensibility, \item {\bf easy to use and free} for greater availability, thus fostering cooperation and collaboration, \item {\bf maintainable} so that researchers are confident to build their work on the software, and \item {\bf scientifically motivated} to address important research problems. \end{enumerate} We elaborate on some of the above points. With open source software, ideally it is possible for {\it anyone} who is sufficiently comfortable with the implementation programming language to find and fix bugs by analyzing the code, especially if the implementation language is easy to learn and readable. This increases reliability because it exposes the flaws (if any) in the source code to more eyes, and is critical to long-term maintainability. By {\it extensibility} we mean that the source code can be analyzed, inspected, modified and improved. When you read the proof of a theorem instead of just the statement, you much more deeply understand what the theorem really asserts, you are better equipped to prove a similar theorem, and you learn {\it techniques} that can be applied to proving new theorems in related (and sometimes unrelated) areas. Likewise, with open source mathematical software, something analogous happens. A key difference between mathematical theorems and software, is that theorems require little maintenance (other than perhaps submitting an errata list to the publisher for typos), whereas {\em mathematical software requires substantial and potentially expensive maintenance} (bug fixes, changes in the underlying interpreter/compiler, updates when the underlying algorithms are improved, and so on). Mathematical research usually generates no direct revenue for researchers, and likewise open source mathematical software does not directly generate revenue. The financial support of the NSF (and other organizations) is thus critical to the success of open source mathematical software. \subsection{\SAGE} \label{sec:sage} The second author (Stein) created \SAGE \cite{sage} in 2005 to meet the basic principles outlined above. Since its inception \SAGE has received substantial NSF funding (substantial computer hardware, two workshops, student support, VIGRE-funded undergraduate projects, etc.), and the NSF recently generously funded a three-year \SAGE postdoc. Instead of reinventing the wheel, \SAGE combines the best open source mathematical software (e.g., GAP, Pari, Maxima, SciPy, and Singular) together in a coherent package, provides a unified interface to it, and implements a wide range of important new functionality. The main guiding principles of \SAGE development are as follows: \begin{itemize} \item {\bf Useful}: \SAGE's intended audience is mathematics students (from high school to graduate school), teachers, research mathematics, and engineers. The aim is to provide software that can be used to explore and experiment with mathematical constructions in algebra, geometry, number theory, calculus, cryptography, numerical computation, and many other areas. \item {\bf Efficient:} Be very fast, often as good or better than anything available in any other general purpose system. \sage uses highly-optimized robust components including GMP, PARI, GAP, SciPy, Singular and NTL; these are very fast at certain operations. \item {\bf Free and open source:} The source code must be freely available and readable so users can understand what the system is really doing and more easily extend it. If you use \sage to do computations in a paper you publish, you can rest assured that your readers will always have free access to \sage and all its source code, and you are even allowed to {\em archive and re-distribute} the version of \sage used. \item {\bf Easy to read and compile:} \SAGE source code should be easy to read and to compile from source code. This provides more flexibility for users to modify the system. \item {\bf Cooperation:} Provide robust {\em interfaces} to most mathematical software (open source and proprietary), including Kash/Kant, MATLAB, Magma, Maple, and Mathematica. \sage is meant to unify existing mathematics software. \item {\bf Well documented and tested:} Tutorial, programming guide, reference manual, with numerous examples and discussion of background mathematics. To be accepted, code must have carefully formatted doctests and pass a refereeing process. \item {\bf Extensible:} Be able to define new data types or derive from built-in types, and use code written in a range of languages. \item {\bf User friendly}: Easy to understand what functionality is provided for a given object and view documentation and source code. Also easy to attain a high level of user support. \end{itemize} \section{Which mathematical software should NSF fund?} \label{sec:which} Research projects that mainly involve creating contributions to proprietary systems should only be funded by the NSF if they can be justified based on the total cost of ownership to the proposer {\it and others in his research area}. For example, given the choice, funding a student or researcher to implement a needed function in an open source mathematical software package is likely preferable to funding dozens of researchers over the course of several years to purchase a commercial software program to perform the same function. Here is an example of this idea. In the 1980's Jeffrey Leon began working on a ``partition backtracking'' programs written in C which performs certain difficult computations in combinatorics, design theory, and error-correcting codes very well. Though Leon soon afterwards ceased maintaining his code, his programs were used both by GUAVA and by MAGMA's error-correcting codes package. For years, use of Leon's code was subject to the following: ``Copyright (C) 1992 by Jeffrey S. Leon. This software may be used freely for educational and research purposes. Any other use requires permission from the author.'' As bugs were reported to the MAGMA version, presumably they were fixed. As bugs were discovered in the GUAVA version, this license became an increasingly serious problem. It is hard to find volunteers to maintain someone else's abandoned code, since the original author can do what he wants with it at any time. Leon was convinced, through a fortuitous series of events, to release his code in April 2007 under an open source license that allows others to modify and improve his code. Immediately afterwards (within hours), Robert Miller, (graduate student, University of Washington) and Tom Boothby (undergraduate, University of Washington), began fixing all of the bugs in Leon's code as part of Jim Morrow's NSF-funded REU at University fo Washington. Within a month, all the known bugs were fixed. The ``openness'' of the new license (the GNU Public License), combined with the research and educational necessity to have more rigorously verifiable data for Robert Miller's Ph.D. research (which Leon's code was used to produce), were the main motivations for Robert Miller to volunteer his work as he did (and continues to do). This case illustrates that (a) great open source software can be maintained with a little NSF support (NSF provided support for Robert Miller and Tom Boothby for this work), (b) there is a direct research and educational benefit to this support, (c) the students gain self-esteem and a greater sense of service in the knowledge that their work can now be used by everyone for free. (In our experience, it is remarkable how well ``service to the scientific/educational community'' motivates so many students.) The following is an alphabetical list of some open source mathematical software projects. \begin{itemize}\setlength{\itemsep}{0ex} \item Axiom (and OpenAxiom and FriCAS) \item CoCoA version 5 \item GAP \item GINAC \item Macaulay2 \item Magnus \item Maxima \item Octave \item Pari \item R \item \sage \item scipy \item Singular \item YACAS \end{itemize} We next sketch some details about these systems. \begin{itemize} \item Axiom is a general purpose computer algebra system. It is useful for research and development of mathematical algorithms. It defines a strongly typed, mathematically correct type hierarchy. It has a programming language and a built-in compiler. Axiom is free and open source (distributed under a BSD-like license). See also the Axiom developer page, \url{http://wiki.axiom-developer.org/FrontPage/}, the FriCAS page \url{http://www.math.uni.wroc.pl/~hebisch/fricas.html}, and the OpenAxiom page \url{http://www.open-axiom.org/index.html}. For example, Axiom is a leader in ``reverse computation'' (guessing formulas for a sequence given its first few terms; see Martin Rubey's package GUESS \cite{R}). \item CoCoALib 5 (``{\it Co}mputations in {\it Co}mmutative {\it A}lgebra'') Its web page is: \url{http://cocoa.dima.unige.it/} (at the time of this writing, CoCoALib 5 is still in development version). \item GAP is a system for computational discrete algebra, with particular emphasis on computational group theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, error-correcting codes, and more. GAP is included in \sage and is distributed freely under the GNU General Public License.\url{http://www.gap-system.org/} \item GINAC ("Ginac Is Not A Cas") does not try to provide extensive algebraic capabilities and a simple programming language but instead accepts a given language (C++) and extends it by a set of algebraic capabilities. See the tutorial for more details. It is free software distributed under the GPL. \url{http://www.ginac.de/}. \item Macaulay2 is mathematical software for algebraic geometry and commutative algebra that has {\em received substantial NSF funding.} It is distributed under the terms of the GNU General Public License, version 2. \url{http://www.math.uiuc.edu/Macaulay2/}. \item The Magnus Computational Group Theory Package is an innovative symbolic algebra package providing facilities for doing calculations in and about infinite groups. It is distributed under the terms of the GNU General Public License, version 2. \url{http://zebra.sci.ccny.cuny.edu/web/nygtc/software/aboutmagnus.html}. \item Maxima is general purpose mathematical software, with abilities such as symbolic integration, 3D plotting, and an ODE solver. Maxima is a descendant of DOE (Department of Energy) Macsyma, which had its origins in the late 1960s at MIT. Macsyma was the first of a new breed of computer algebra systems, leading the way for programs such as Maple and Mathematica. This particular variant of Macsyma was maintained by William Schelter from 1982 until he passed away in 2001. In 1998 he obtained the copyright from the Department of Energy and made the source code open source. \url{http://maxima.sourceforge.net/}. Maxima is included in \sage. \item GNU Octave is primarily intended for numerical computations. It provides a convenient command line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with Matlab. It may also be used as a batch-oriented language. You may redistribute Octave and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. \url{http://www.octave.org}. \item PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves...), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers, etc., and many transcendental functions. It is distributed under the GNU General Public License. \url{http://pari.math.u-bordeaux.fr/} and is included in \sage. \item The R Project for Statistical Computing is a system for statistical computation and graphics. It consists of a language plus a run-time environment with graphics, a debugger, access to certain system functions, and the ability to run programs stored in script files. See also CRAN for some mirror sites and extensions. It is open source software distributed under a GNU-style copyleft, and an official part of the GNU project. \url{http://www.r-project.org/}. \item \sage was mentioned above already. \url{http://www.sagemath.org/}. \item SciPy (pronounced ``Sigh Pie'') is free open-source software for numerical mathematics, science, and engineering that is included in \SAGE. It is written in C, Python and Fortran. The commercial company Enthought maintains SciPy and uses it to provide its clients with solutions to industrial problems. \url{http://www.scipy.org/}. \item Singular is a Computer Algebra System for polynomial computations with special emphasis on the needs of commutative algebra, algebraic geometry, and singularity theory. It is distributed freely under the GNU General Public License. \url{http://www.singular.uni-kl.de/}, and is included in \sage. \item YACAS (``{\it Y}et {\it A}nother {\it C}omputer {\it A}lgebra {\it S}ystem'') is a a program for symbolic manipulation of mathematical expressions. The system has a library of scripts that implement many of the symbolic algebra operations; new algorithms can be easily added to the library. (Unlike Maxima, YACAS also function as a {\it library} and is written in C++.). It is open source software. \url{http://yacas.sourceforge.net/}. \end{itemize} \section{Educational impact}\label{sec:edu} Mathematical education is incredibly important to mathematical research and to a healthy economy, so we conclude this paper with some remarks on the educational impact of the principles discussed in Section~\ref{sec:principles}. Through many programs, such as CSUMS, CCLI and VIGRE, the NSF generously supports educational training and development in U.S. universities and colleges. Much of this support involves development of software ``modules'' (worksheets for particular software, with some mathematical explanation, and accompanying exercises). When the worksheets developed using NSF funds use proprietary software, then not only must all the students have access to the system (at a considerable cost to either the student or the school), but any other school that wishes to make use of this government-funded work must also purchase this software. Now, contrast this with the situation for free software. When worksheets developed using NSF funds use a free and cross-platform program, then not only can all the students have equal (free) access to the system, but any other school (no matter what their software budget is) that wishes to make use of this government-funded work can also do so. The total cost of ownership and the greater accessibility of such worksheets make a compelling case that the NSF should fund educational projects that involve free and open source software. %Another illustration of the educational impact: Ted Kosan is currently developing teaching materials for computer technology for distance learning courses taught to high-schoolers in Ohio. He selected \SAGE. In his own words \cite{Ko}: \begin{quotation} ``As part of this process, I have been searching for open source mathematics software that has a modern programming environment intimately integrated into it. I also wanted it to be alive and well 20 or 30 years from now. After a long and grueling search I have finally decided to go with \SAGE. I think that \SAGE's design philosophy is brilliant and the balance it strikes between mathematics and computing is exceptional.'' \end{quotation} % There is one other impact worth mentioning - the contribution to % research in computer algebra. The development of \SAGE, % Axiom, GAP, MAGMA, SciPy, and even commercial % mathematical software has forced one to address the following % questions: % % \begin{itemize} % \item % What types of data structures % should be implemented in an interpreted computer language % designed to solve problems in group theory? % \item % How should the data structure % representing a homomorphism of abelian groups be related (if at all) % to homomorphisms of (slightly more general) polycyclic groups, % or to (much more generally) finitely presented groups? % \item % Once a property of an object (say, a group or a linear code) % is computed, how is this best stored in memory? % \end{itemize} % % Generally, how does one best to think of mathematics from % the computer scientist's point of view? % % These types of important questions must be carefully thought out by % mathematical software developers. Only by continued NSF funding can % these issues be properly worked out, thereby leading to more efficient % implementations, faster computations, and therefore more new % computational discoveries. % % %\section{Impact in Industry of open source mathematical software} % %Both the writers being in the academic world, we are %not experts on the commercial impact of mathematical software. \section{Conclusions} Mathematical software leads to important advances in mathematics (Kepler's conjecture, the classification of automorphism groups of Riemann surfaces, the theory of error-correcting codes, and so on). We have discussed how greater funding can lead to new and better research. Funding open source mathematical software has the potential to lead to a lower total cost of ownership in the research and educational community. 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