Why Mathematica violates basic rules of math conduct
By Christopher Dawson | November 19, 2007, 12:17am PST
Summary: In an editorial published by the American Mathematical Society, mathematicians from the US Naval Academy and the University of Washington noted that, because so much mathematical research is now conducted using proprietary software like Mathematica, much of the ability to meaningfully review such research is being lost. As the authors point out,
There is a [...]
In an editorial published by the American Mathematical Society, mathematicians from the US Naval Academy and the University of Washington noted that, because so much mathematical research is now conducted using proprietary software like Mathematica, much of the ability to meaningfully review such research is being lost. As the authors point out,
There is a proof in the article by Campbell et al. in 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facts—no code is given, and some of the programs are proprietary software that runs only on hardware many years out of date. Such proofs may become increasingly common in mathematics if something isn’t done to reverse this trend.
Similarly,
…suppose now Jane [an imaginary mathematician] 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…To quote J. Neubüser, “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.”
Their solution, of course, is open source alternatives to proprietary software like Mathematica, Maple, and SAS.
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Chris Dawson is a freelance writer and consultant with years of experience in educational technology and web-based systems. In 2011, he became the Vice President of Marketing for WizIQ, Inc., a virtual classroom and learning network SaaS provider.
Disclosure
Christopher Dawson
Christopher Dawson is the Vice President of Marketing for WizIQ, Inc., by day and a freelance writer and educational technology consultant by night. Well, most of his colleagues at WizIQ are based in India, so really he's working with them whenever he can stay awake. He has worked for his local school district as a teacher and technology director, for the Johns Hopkins University School of Public Health, and for Biogen, Inc. (now Biogen-IDEC, Inc.).
He has also consulted with STATNet and Cytyc Corporation and retains close ties with X2 Development Corporation (now owned by Follett Software, the supplier of the student information system he administered for several years). Follett is paying him a monthly honorarium to act as a presenter for their "SIS Voices for Student Achievement" community (he produces occasional blog posts and hosts a monthly webinar on the use of student information systems to inform data-driven instruction and school-wide change.
He regularly purchases and/or recommends Dell hardware. This is because Dell makes good hardware and has truly committed itself to education in innovative ways, particularly with their "Connected Classroom" initiative. It isn't because he has dealings with the company through his role at WizIQ (which he does) or because they have provided him with long-term loans of a variety of equipment for in-depth testing (which they have).
Intel (reference designer for the Classmate PCs he has implemented in his local schools) has provided him with long-term loans of Classmate PCs for testing, as have Dell and Lenovo with their educational offerings. He may report on any of these companies as his experiences with them have direct bearing on educational technology; positive reports are not necessarily an endorsement and he receives no direct financial compensation from these companies or any others.
Intel paid all expenses for his attendance at the 2009 Intel Classmate PC Ecosystem Summit which he attended as the sole representative of the technology press. He was invited to attend in 2010 but his wife would have killed him if he spent 3 days in Vegas geeking out and left her home alone with a new baby. Acer provided him with a 50% discount on an Aspire One netbook in early 2009 after he tested it for 30 days through their educational seed program. He liked the netbook at the time but it has since broken and sits unused in his office. Canonical sent him Ubuntu lanyards, t-shirts, and mousepads for his kids. He stole one of the lanyards and proudly hangs his keys from it and occasionally features his 8-year old wearing an oversized Ubuntu t-shirt on his Facebook profile. Gunnar Optiks sent him a pair of computer glasses to evaluate for a holiday gift guide. He is wearing them now as he types this because they never asked for them back and they rock out loud. Seriously - they work brilliantly and make it much easier to spend 20 hours a day staring at an LCD. If they ever asked for them back, he would fork over the $99 and buy a pair.
Microsoft gave him 2 free copies of Office 2010 professional, a desktop clock, and a useless book on Office 2010 when he attended the launch of Office/Sharepoint 2010. He occasionally uses the SharePoint lanyard they gave him instead of the Ubuntu lanyard for his keys, but feels dirty afterwards.
Adobe provided him with a pre-release version of the CS5 Master Collection for evaluation and ultimately provided a full, licensed copy for ongoing testing of educational applications of this admittedly expensive software. Like the Gunnars, if the license expires or they come out with CS6, he'd actually go out and buy it himself. Which is saying something, because he's actually pretty cheap.
Any other companies wishing to send him cool things to evaluate, wear, or otherwise adorn his kids are more than welcome to; he promises to disclose it here if he keeps any of the stuff.
Finally, because WizIQ is a virtual classroom and learning network provider, Chris, as VP of Marketing, frequently interacts with, seeks out deals with, and directly or indirectly competes with a whole lot of LMS, SIS, and other Education 2.0 companies. In general, he'll limit his reporting about these companies to news that does not impact his relationship with them or with WizIQ. If he reports on them, it's because what they are doing is newsworthy or worth the attention of his readers and not because he's trying to broker some deal, damage competition, or otherwise advance his position in his day job. LMS and SIS companies, along with other online learning communities, are a pretty important part of Ed Tech. If he stops reporting on them completely, there won't be a whole lot left. He'll be sure to call out any overt conflicts of interest if they are unavoidable.
Finally, Follett Software Company pays him a little tiny honorarium every month to present on their SIS Voices webinars and to write the occasional blog or discussion thread for them. Since Follett recently bought X2 (maker of an awesome web-based SIS that Chris just happened to have used, served in advisory groups for, and frequently reported on), this is probably also worth disclosing.
Biography
Christopher Dawson
Christopher Dawson grew up in Seattle, back in the days of pre-antitrust Microsoft, coffeeshops owned by something other than Starbucks, and really loud, inarticulate music. He escaped to the right coast in the early 90's and received a degree in Information Systems from Johns Hopkins University. While there, he began a career in health and educational information systems, with a focus on clinical trials and related statistical programming and database modeling. This focus led him to several positions at Johns Hopkins, a couple-year stint in private industry, teaching high school math and technology, and 2 years as the technology director for his local school district. Most recently, he started his own consulting business and is now the Vice President of Marketing for WizIQ, Inc., a virtual classroom and learning network provider.
He lives with his wife, five kids (yes, 5), 2 dogs, and a hateful cat in a small town in north-central Massachusetts. Although he is no longer teaching, his roles with WizIQ and ZDNet allow him to continue helping students and teachers add value to education with technology rather than merely adding to the bottom line.
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Information, especially in science, should be available for everyone interested to disect and work with. Using proprietary apps which do not produce a usable "paper trail" of sorts are useless in science. If other scientists can't reproduce something without a proprietary tool, then that is "proprietary science", which would be a very sad place for the world to go.
...be prepared to explain how to do the work without the proprietary software in question. I wouldn't accept a theorem as proved just because Mathematica says it is, but proper mathematical software (and I've never used Mathematica) should be able to output all of the required math for human inspection and/or independent implementation.
I've used SAS for many years and have no apologies for doing so (even though the SAS Institute does have something of a greed problem). The programming interface is relatively easy to understand, SAS code is easily translated into the language of one's choice, and datasets can be readily converted to CSV.
Reproducibility is the hallmark of good science. As long as other researchers have the data required to reproduce the results with whatever software seems good to them (or none at all), I don't see any problem with using proprietary software as a research aid.
but proper mathematical software (and I've never used Mathematica) should be able to output all of the required math for human inspection and/or independent implementation.
I agree with this and your statement of:
Reproducibility is the hallmark of good science.
Using commercial software whose code cannot be inspected is not the problem. As long as the software outputs all of the work for manual inspection, the source code should be irrelevant.
My first venture into software development was writing applications in QBASIC to do my math homework in high school. I had it output all of the work so I could quickly transcribe it. My teacher was none the wiser. I told him about it at the end of the year. He actually gave me bonus points on the final exam since I wrote the software.
So, if Mathematica does not reveal how it came to the final results, it should be changed to do so (I've also never used it). This will allow anyone, including the users, to examine the complete workflow.
So it's 2107. Nobody remembers Lisp, and the details of all the hardware that runs Lisp binaries have been forgotten. OK, maybe it's more like 2307...we're still sending email via SMTP after a quarter of a century...
While probably someone could work out how to make Macsyma run, it would probably be easier to re-prove the theorem in question with modern tools, rather than from stuff from the first century of the era of computation.
But if you kept a TRACE of Macsyma's proof, you could actually check what Macsyma was doing.
Of course, there are still issues. A simplification step for a 20-page formula is not necessarily going to be a very easy thing to follow -- you'd need a whole proof just of the single step. So one issue is at what level of detail do you preserve the trace?
I think you've hit the nail on the head -- the guiding principle is reproducibility. If your steps are too course-grained for someone to reproduce, you'd need to break it down. You can do that with Macsyma, with some care. I don't immediately see how you could automate it, as it's a judgement call how to break up the process in a way that is not too course-grained or too fine-grained.
Another issue is bugs. As much as I hate to admit it, (having worked on it back in the '70s) Macsyma has bugs in some areas. Some of them may even be as yet undetected -- or newly introduced. Even the chips on which it runs may have bugs.
For any computer proof to be considered truly solid, some sort of verification and validation should be done. Reproducing the methodology in another framework is one approach to this.
Anyway, I don't think that Mathematica inherently violates the rules of math conduct. I think it's the mathematician's responsibility to use it in a way that does not violate the rules.
At the same time, I think the rules need to evolve. From a practical standpoint, it's not feasible to publish huge proofs as part of a journal article. And there's a need to establish just what operations ARE well-understood -- how can you describe the steps in a proof in ways that are expressions of mathematics, rather than expressions of the tool?
LISP is typically interpreted, which makes it much more likely that source code is available for programs written in it (there are also open source LISP interpreters).
I think we're in basic agreement that computer programs are tools which can be used in both appropriate and inappropriate ways. The key is to provide enough data to allow the results to be reproduced independently, to include replacement of any software utilized.
Not only mathematics should be passed on free, I think knowledge should be free as well.
Why do people continually rehash this same tired nonsense? Information wants to be free. Yeah, right. No information is free - researchers have to be paid, even you local "free" library costs real money to operate. No information was, is, or ever will be free. There is always an expense to gather and disseminate information.
When most research is done using money coming from everyone's taxes, why should we pay again to see the results of such research? If the information were the result of private funded research you would be right. But you are not.
Maxima (originally DOE Macsyma) was developed with a variety of funding sources, including a number of private companies.
The reason that there is a free version is precisely because of the public funding involved, specifically the Department of Energy, and the constraints imposed by that funding source.
There have been commercial versions as well, but I'm not aware of any that are currently available.
Information, especially in science, should be available for everyone interested to disect and work with. Using proprietary apps which do not produce a usable "paper trail" of sorts are useless in science. If other scientists can't reproduce something without a proprietary tool, then that is "proprietary science", which would be a very sad place for the world to go.
What are the open source alternatives that are out there?
"What are the open source alternatives that are out there?"
The almighty pencil and paper is the only open source alternative I have seen.
I use Octave.
A quick look at sf.net found a number of alternatives from MathStudio and Crystal Math to Octave(a matlab clone) and eclipse and java plugins for math.
They have a seperate topic for just Math programs.
If your looking for open source always check sf.net and eclipse.org first.
LLAP
\\//_
Yea open source lets you examine the process that a result is derived by.
Dear Colleague,
Re your comment about Mathematica and the (implied) need for "free" software. Please do not take this in the wrong way. Do you cash your teacher's paycheck? Do you see the connection between a reward for a service and a reward for a product? You don't work for nothing and Mathematica is entitled to what it can get -- just as you are. Having said that, I understand the issue you are raising -- but to raise it withoug offering a solution or an alternative is just blogging for the sake of blogging.
Regards,
Harvey
not as in "free beer". Bassically "open source" software doesn't mean free/gratis software, it means everyone is free to see the source code.
So what is needed is better use of markup languages - mathML for example, which can be used to encapsulate the semantic content of the proof in an open manner - the computational engines used to operate on the markup need to be therefore, provable in a formal sense and verfified as such by standards bodies such as NIST in cahoots with the AMS.
Ohterwise it aint proof!
An extremely excellent point. If anyone has noticed, there are two entirely free alternatives to "MATLAB", the local one called Octave and one that the French Government calls "SciLab". Google them and be surprised.
Joe Ouellette
Woodland Hills, CA
Mathematicians violate the basic rules of math if they are not providing sufficient evidence to prove their theorems. Software is a tool, it is up to the user to use it responsibly. (guns don't kill people, people kill people.....)
If the mathematical community accepts proofs that cannot be verified by independant means, then they are also the ones to blame.
I believe the Problem here is that SOME math theorems have been testable only via
computer analysis.
For example, the Four Color Theorem. No human has ever come up with a logical
proof, but computer analysis, checking each and every combination, solved the
problem.
Even more to the point: The article I read about the ancient "can you re-arrange a
circle into a square of equal surface area" problem.
I think it's called "Squaring the Circle".
Well, the original challenge was supposed to be to solve the problem with string,
scissors, and a pencil.
The computer solution, however, basically made a jigsaw puzzle out of the circle
and re-assembled the pieces to get a square.
Now what ? Are you going to print that out? The number of steps required would
be staggering.
And you can't do it by hand, either. Some of the "jigsaw pieces" would have to be
smaller than individual molecules, IIRC.
Now, the original point of this article MAY have some merit. If you solve a
mathematical problem, the CODE you used IS the proof.
And let me end with a note of suspicion.
I once played a computer chess game with a friend. And I SAW the program cheat!
I tried the "take back move" command and watched, as again, the game
deliberately moved a pawn straight to the left...
A bug in the program ? Probably.
What if that happens in Mathematica? Who exactly is smart enough to keep all that
code in his head?
the meaning of "free". "free" as in publicly ("freely") available - perhaps at a cost, but not propritary.
because if we use the mathematics in our life we make our lives complete.and we learn more solution......................thats all thank you......................
