Forgot your password?
typodupeerror
Math

Open Source Math 352

Posted by CmdrTaco
from the wouldn't-it-be-nice dept.
An anonymous reader writes "The American Mathematical society has an opinion piece about open source software vs propietary software used in mathematics. From the article : "Increasingly, proprietary software and the algorithms used are an essential part of mathematical proofs. 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.'""
This discussion has been archived. No new comments can be posted.

Open Source Math

Comments Filter:
  • Lol (Score:5, Funny)

    by Matt867 (1184557) on Sunday November 18, 2007 @12:12PM (#21398025)
    Thanks for the article, now some crazed company is going to try to copyright math.
  • It's all... (Score:2, Insightful)

    by Shikaku (1129753)
    about the money.
    • by mosel-saar-ruwer (732341) on Sunday November 18, 2007 @02:55PM (#21399261)
      In mathematics information is passed on free of charge and everything is laid open for checking.'

      I'm not going to disagree with the "laid open" part, but the "free of charge" nonsense is just typical marxist university professor hypocrisy.

      Let's price some math texts:

      Atiyah & MacDonald, Commutative Algebra; $57.54, http://www.amazon.com/dp/0201407515/ [amazon.com]

      Eisenbud, Commutative Algebra; $41.30, http://www.amazon.com/dp/0387942696/ [amazon.com]

      Hartshorne, Algebraic Geometry; $59.10, http://www.amazon.com/dp/0387902449/ [amazon.com]

      Elements de Geometrie Algebrique; out of print, http://www.amazon.com/dp/3540051139/ [amazon.com]

      Rudin, Real and Complex Analysis; $142.50, http://www.amazon.com/dp/0070542341/ [amazon.com]

      Rudin, Functional Analysis; $137.16, http://www.amazon.com/dp/0070542368/ [amazon.com]

      Dym & McKean, Fourier Series and Integrals; $85.00, http://www.amazon.com/dp/0122264517/ [amazon.com]

      Sugiura, Unitary Representations and Harmonic Analysis, 2nd Edition; Out of Print, http://www.abebooks.com/servlet/SearchResults?an=Sugiura&tn=Representations [abebooks.com][Someone wants $495.00 [amazon.com] for the first edition.]

      Or try a few titles which might be a little more familiar to Slashdotters:

      Knuth, The Art of Computer Programming, Volumes 1-3 Boxed Set; $145.00, http://www.amazon.com/dp/0201485419/ [amazon.com]

      Sedgewick, Algorithms in C++, Parts 1-5; $93.00, http://www.amazon.com/dp/020172684X/ [amazon.com]

      Cormen, Leiserson, Rivest & Stein, Introduction to Algorithms; $61.88, http://www.amazon.com/dp/0262032937/ [amazon.com]

      Aho, Ullman & Hopcroft, Data Structures and Algorithms; $53.20, http://www.amazon.com/dp/0201000237/ [amazon.com]

      McLachlan, Discriminant Analysis and Statistical Pattern Recognition; $90.40, http://www.amazon.com/dp/0471691151/ [amazon.com]

      Haykin, Neural Networks: A Comprehensive Foundation; $120.12, http://www.amazon.com/dp/0132733501/ [amazon.com]

      Duda, Hart & Stork, Pattern Classification; $117.00, http://www.amazon.com/dp/0471056693/ [amazon.com]

      Fukunaga, Introduction to Statistical Pattern Recognition; $74.40, http://www.amazon.com/dp/0122698517/ [amazon.com]

      Bishop, Neural Networks for Pattern Recognition; $82.81, http://www.amazon.com/dp/0198538642/ [amazon.com]

      Bishop, Pattern Recognition and Machine Learning; $66.54, http://www.amazon.com/dp/0387310738/ [amazon.com]

      Higgins, Sampling Theory in Fourier and Signal Analysis: Volume I; $171.60, http://www.amazon.com/dp/0198596995/ [amazon.com]

      Higgins & Sten, Sampling Theory in Fourier and Signal Analysis: Volume II; $264.00, http://www.amazon.com/dp/0198534965/ [amazon.com]

      Princeton, which has the finest mathematics department in the world [or at least had the finest mathematics department in the world, before Harold Shapiro & Shirley Tilghman decided they wanted to turn the

      • by William Stein (259724) <wstein@gmail.com> on Sunday November 18, 2007 @03:39PM (#21399633) Homepage
        > In mathematics information is passed on free of charge and everything is laid open for checking.'

        > I'm not going to disagree with the "laid open" part, but the "free of charge" nonsense
        > is just typical marxist university professor hypocrisy.

        Taken out of context the quote might not make sense to you. The full quote from Neubuser is:

        You can read Sylow's Theorem and its proof in Huppert's book in the
        library [...] then you can use Sylow's Theorem for the rest of your
        life free of charge, but for many computer algebra systems license
        fees have to be paid regularly [...]. You press buttons and you get
        answers in the same way as you get the bright pictures from your
        television set but you cannot control how they were made in either
        case.

        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. Not applying
        these rules to computer algebra systems that are made for mathematical
        research [...] means moving in a most undesirable direction.
        Most important: Can we expect somebody to believe a result of a
        program that he is not allowed to see? Moreover: Do we really want to
        charge colleagues in Moldova several years of their salary for a
        computer algebra system?


        When Neubuser says that mathematics is "free of charge" he means that
        one can use theorems one reads without having to pay to use those theorems.
        He is of course not at all claiming that publishers do not charge for
        books and papers that contain mathematics. Put simply, if I want to use
        the "FactorN" function in Mathematica, I have to pay for the privilege
        every time I use it. If I want to use the theorem that every integer
        factors uniquely as a product of primes, then I never have to pay, even if
        I am using that theorem in a published proof.

          -- William

      • by Cowculator (513725) on Sunday November 18, 2007 @05:37PM (#21400661) Homepage
        There's a growing trend in math (and maybe other disciplines, for all I know) away from non-free publishing.

        Prominent mathematicians have been complaining [ams.org] for [berkeley.edu] years [ams.org] (more links here [umd.edu]) about overpriced journals, and entire editorial boards of some journals have resigned in protest (see a list of mass resignations and similar changes here [ucsb.edu]). There are now plenty of entirely free journals in combinatorics [combinatorics.org], topology [www.emis.de], and other fields, and pretty much everything that gets published these days is either available on the author's website or on the arXiv [arxiv.org].

        So modern research tends to be free, but what about all the books you need to read before you understand this research? Sure, a copy of Rudin may be expensive and there's not much we can do about that, but maybe you can learn from the free analysis course notes at MIT's OCW [mit.edu] site. You complain that EGA is out of print, but basically everything Grothendieck wrote is available for free [jussieu.fr], and you can even get them along with tons of other old French publications through NUMDAM [numdam.org]. (There's even a project to transcribe SGA [leidenuniv.nl] into LaTeX.) Lots of other books are free to download legally [gatech.edu] (and this is by no means a complete list), even though many are commercially published as well.

        Finally, you can complain all you want about university tuition, but I really doubt that free tuition is going to open up mathematics to the masses. Ultimately the very top students who can't afford it are getting scholarships and grants to cover their education (and I do know some people who got free rides at Princeton because they couldn't afford it -- that school is definitely more generous than most), and since most other people couldn't get into Princeton anyway the tuition is never even an issue for them. The best way to make mathematics more accessible is to give everyone access to free textbooks and current research, and the "marxist university professors" you deride have been gradually moving in that direction for years now.

        By the way, what do you think has been done to damage the Princeton math department's reputation? Whatever you think Shapiro and Tilghman have done to the university, nobody in their right mind would deny that it's one of the top few in the world and I doubt most people would openly proclaim any one department to be the best anyway.
  • by Ckwop (707653) * <Simon.Johnson@gmail.com> on Sunday November 18, 2007 @12:19PM (#21398061) Homepage

    I am no a mathematician but surely if you're going to submit a computer aided proof you must submit a full copy of the program. The are all manor of subtle mistakes that can be made in a program that could cause serious problems with a proof.

    Suppose you inspect the source and find it to be faultless, how can you trust [cryptome.org] the compiler. And if you hand compile the compiler, how can you trust the CPU [wikipedia.org]? Surely it's turtles all the way down.

    In many ways, establishing the correctness of a computer-aided proof is very much like security engineering. You want to verify that the whole software stack is operating correctly before you can trust the result. Having the source-code is a pre-requisite to this exercise.

    Changing to topic slightly, I was particularly heartened to see that the open-source mathematics framework being developed one of the authors of the article involves the use of Python.

    My immediate thought when seeing the title to the article was "Python is the answer." When some problem or algorithm intrigues me the first thing that happens is that I reach for the Python interpreter.

    Python seems to deftly marry precision with looseness. When code is laid out in Python I find it is easier to see what it's trying to do than other languages. It's aesthetic qualities aside, it supports a number of features out of the box which I imagine would be ideal of mathematicians. To list a few, it's treating of lists and tuples as first class objects, support for large integers, complex numbers, it's ability to integrate with C for high-performance work.

    I often think of Python as "basic done right" and it's ideal for mathematicians (or anybody) who don't want to think about programming but the problem at hand.

    Simon

    • by snarkh (118018) on Sunday November 18, 2007 @12:51PM (#21398313)
      I have seen from personal experience, how a compiler error (some sort of incorrect optimization) led to a subtle difference in the results of a simple classification task.

      The insidious thing about that particular result was that it looked very similar to the correct. In fact the difference would not have been found if two people did not run different versions of code independently (and more or less coincidentally) arriving to slightly different error rates.
      • Re: (Score:3, Informative)

        by jelle (14827)
        From your description, it sound as if you found that the code returned different results at different optimization settings for the compiler, but did not pinpoint what instruction sequence exactly caused the difference.

        Unless you were using an experimental compiler, that usually means a bug in the code, not a bug in the compiler. Run the code with valgrind, you'll probably find out-of-bound addressing, or uninitialized reads (the signs of the problem being in the code, not the compiler)... Or if you use thr
    • by nwbvt (768631) on Sunday November 18, 2007 @12:52PM (#21398323)
      I used Python fairly extensively in my number theory course back in college, it did the job fairly well. Its support for large integers was especially important for that class. And the fact that it was very familiar to me (I was a double major in CS and math), it was very easy for me to crank out an algorithm in it. However, most of the book's examples were in Mathematica, which I ended up getting as well. It was a neat tool, but now that my student license has expired and I don't feel like spending a few grand on another license, everything I wrote in that is useless. However I can still pull out my old Python programs and see what it was I was doing.
    • by ndevice (304743)
      ironically, it's all a house of cards
    • by Dunbal (464142) on Sunday November 18, 2007 @01:03PM (#21398415)
      The are all manor of subtle mistakes that can be made in a program that could cause serious problems with a proof.

      No mistakes. After all, the Ultimate Answer really is 42. My program proves it!

      #define MYANSWER "42"

      int main()
      {
            printf("The result is: %s.", MYANSWER);
      }


      No, you CAN'T have the source code... but look, my program proves it! LOOK AT THE PROGRAM!
    • by otomo_1001 (22925)
      Not to troll or anything, but every one of your reasons for using Python is why I use Ruby.

      Some *very* recent others that make me like it:
      * I can now use versions of Ruby that work with dtrace on Leopard and Solaris/Opensolaris (haven't tried FreeBSD yet).
      * Ruby on Rails, yes despite the hype I like it. Though there are annoyances.
      * I can also build Ruby (and Python) programs in osx without Coacoa/Objective C. Supported too, yay.
      * (Not recent, but the reason I prefer Ruby to Python) Whitespace is optional,
      • by aldheorte (162967)
        Second that on Ruby. I think Ruby is where the brain share and community is going, nothing against Python per se.

        You have to be careful with Python and Ruby though. For example, I wrote a symbolic math interpreter for simplifying algebraic equations in Ruby. I then realized that I had reinvented LISP.

        I do not actually program LISP, but in the end, LISP rules all as a programming language, especially when pure math is considered.
    • by poopdeville (841677) on Sunday November 18, 2007 @01:12PM (#21398495)
      I am no a mathematician but surely if you're going to submit a computer aided proof you must submit a full copy of the program. The are all manor of subtle mistakes that can be made in a program that could cause serious problems with a proof.

      I am a mathematician. Your referees might ask to inspect the source code. This is akin to a biologist being asked to produce her raw data. But it's pointless anyway. Because...

      In many ways, establishing the correctness of a computer-aided proof is very much like security engineering. You want to verify that the whole software stack is operating correctly before you can trust the result. Having the source-code is a pre-requisite to this exercise.

      The AMS isn't worried about the correctness of these "proofs." They aren't proofs. It is logically possible for one of these programs to return the wrong answer, even if the program is correctly implemented. Ergo, it is not a proof.

      Computing, in mathematics, is a source of fresh problems and a vehicle to explore and gain insight about mathematical structures. The AMS is far more concerned about good exploratory algorithms getting swept up by Wolfram Inc., and Mathworks, and the like, and never being seen by mathematicians again.

      Regarding which language is approriate for mathematics, the answer is whichever clearly expresses the idea you're trying to write. Lexical scoping is familiar to us. I know I prefer it, since it lessens my cognitive load. I prefer dynamically typed languages. I need the ability to construct anonymous functions efficiently. And I would prefer automatic memoization. Development time is always an issue. Most languages don't come with extensive mathematical algorithm libraries. So you'll either have to write them yourself (time consuming; boring, unless you're into that stuff) or find some. I've used Perl, Ruby, Scheme, and C.
      • Re: (Score:3, Interesting)

        I fear you and/or the AMS are giving too much credit to the big names in mathematical software. Sure, they have some bright people and they do some useful research in their own right, but they're still only human. They make mistakes, their software has bugs, and they don't know lots of deep secrets that the rest of academia don't. In fact, the development practices at certain high profile mathematical software companies leave a lot to be desired; they tend to hire PhD types, who know a lot about mathematics

        • Re: (Score:2, Interesting)

          by poopdeville (841677)
          I fear you and/or the AMS are giving too much credit to the big names in mathematical software.

          I can see why you might think that, but my point had little to do with commercial software houses. My main point was that computer-assisted "proofs" are not proofs in the mathematical sense. They're "results" that rest "scientifically" on the software and hardware and real world. It really doesn't matter whether I use my implementation of Newton's Method or Mathematica's. Neither should be trusted in a proof.

          I
          • Fair enough.

            The thing that always gets me about the concept of a mathematical proof is that it seems to be turtles all the way down. Sure, if you're proving, say, a simple result in group theory derived trivially from axioms, then the proof can be quite convincing. But recent proofs for some famous results run to many pages, and as experience shows, a "proof" can turn out to be completely undermined by a simple flaw in the logic that the person presenting the proof missed. Even if you break the steps down

            • Re: (Score:3, Interesting)

              by eh2o (471262)
              One thing that I find interesting about mathematical proofs is that they keep getting smaller and easier to explain. If you consult text books of just a few decades ago, they are significantly more verbose than today's equivalent. For example, "Linear Algebra" by Friedberg, Insel and Spense (a standard text), 1979 (fist edition), is almost twice the number of pages as "Linear Algebra Done Right" (Axler, 1996), a book that covers the same material.

              Furthermore the advent of computers has made the illustrati
              • Re: (Score:3, Interesting)

                It seems to me that you're confusing lack of verbosity with changes of mathematical points of view. It's true that definitions get refined over time, but a changed point of view does not always equate with improvement. Every generation works on slightly different mathematical questions, and changes of viewpoint often merely reflect the problems and interests of the previous generation. There are plenty of "dead" fields that nobody works in anymore.

                Try reading books from the early 20th century [uni-bielefeld.de], and ask you

      • by rucs_hack (784150)
        The AMS isn't worried about the correctness of these "proofs." They aren't proofs. It is logically possible for one of these programs to return the wrong answer, even if the program is correctly implemented. Ergo, it is not a proof.

        I might be wrong, but it occurs to me that a program which 'proves' a mathematical hypothesis can only, on inspection, be shown to be a proof of the program itself, not the initial hypothesis.

        The problem with software is that it can be made to do anything. Want to model colliding
        • Re: (Score:3, Insightful)

          by Goaway (82658)
          You are confusing maths and physics. Mathematicians do not care about galaxies, nor the "real world" at all. Their proofs and theorems live entirely in the world of abstract logic.
      • Re: (Score:3, Informative)

        by khallow (566160)

        The AMS isn't worried about the correctness of these "proofs." They aren't proofs. It is logically possible for one of these programs to return the wrong answer, even if the program is correctly implemented. Ergo, it is not a proof.
        That is incorrect. Even with normal proofs, it is possible to return the wrong answer, and frankly computer proofs scale better than strictly human-based ones do. So I think it's quite reason to call them "proofs".
    • by Gadzinka (256729) <rrw@hell.pl> on Sunday November 18, 2007 @01:17PM (#21398547) Journal

      Python seems to deftly marry precision with looseness. When code is laid out in Python I find it is easier to see what it's trying to do than other languages. It's aesthetic qualities aside, it supports a number of features out of the box which I imagine would be ideal of mathematicians. To list a few, it's treating of lists and tuples as first class objects, support for large integers, complex numbers, it's ability to integrate with C for high-performance work.

      I often think of Python as "basic done right" and it's ideal for mathematicians (or anybody) who don't want to think about programming but the problem at hand.
      I could also recommend Ruby for the job. It has all the features you recommend, and more. If you could forget for a moment about the monstrosity that is Rails (I don't know, lobotomy might do the trick), the language in itself is quite beautiful.

      There is one special feature of Ruby, that I miss in every single programming language I used since: iterator methods. Any time I want to iterate over elements of an array or hash I just do:

      myhash.each_pair do |key,val|
        puts "#{key}: #{val}"
      end
      That's it, instant "anonymous function" given as a parameter in estetically pleasing syntax. In fact, "for" loop in Ruby is just obfuscated way of calling method #each on an object. But the madness doesn't stop here:

      File::open("somefile.txt") do |fh|
        fh.each do |line|
            puts line
        end
      end
      It's a pity that so many people disregard Ruby as a "platform for Rails". It is a feature complete countepart to Python, and as my company high volume systems can attest, can handle anything other languages can handle.

      Robert
    • by DrYak (748999) on Sunday November 18, 2007 @01:30PM (#21398637) Homepage
      We may also mention Coq [wikipedia.org], a proof assistant wich is available under LGPL and runs on OCaml (which in turn is also open sourced and available on Linux).

      This is a tool that can help mathematician prove their theorems.
      It was notably being used in the proof of the four color theorem [wikipedia.org], as mentioned on /. [slashdot.org] (article about machine assisted proofs).
    • Re: (Score:3, Insightful)

      by Dare nMc (468959)

      You want to verify that the whole software stack is operating correctly before you can trust the result. Having the source-code is a pre-requisite to this exercise.

      I disagree, it is certainly possible to prove to a reasonable certainty what a black box is doing. It may be easier, or more though to prove looking into the box.
      As you say, for all practicality no one is going to be able to confirm the entire software stack, by looking at the code for any proof. unless your running the final step on a basic st

    • by Bert64 (520050)
      Well, you're only really responsible for the correctness of your own code.
      As to the compiler and CPU, so long as you use a combination that have been verified as correct by other mathematicians you should be fine.
    • Re: (Score:3, Funny)

      by ozbird (127571)
      The are all manor of subtle mistakes... Irony?
  • by larry bagina (561269) on Sunday November 18, 2007 @12:20PM (#21398069) Journal
    The article (which is actually a PDF, thanks for the warning) uses proprietary fonts (LucidaBright). While it was typeset with TeX (open), only the PDF (closed and uneditable) is provided.
    • by Main Gauche (881147) on Sunday November 18, 2007 @12:48PM (#21398289)

      "While it was typeset with TeX (open), only the PDF (closed and uneditable) is provided."

      Indeed. Now we are left wondering whether the TeX code is buggy. Like maybe an extra character accidentally slipped into the file.

      therefore mathematics software should %not
      be open source!

      Now we'll never know.

    • PDF rant. (Score:5, Insightful)

      by serviscope_minor (664417) on Sunday November 18, 2007 @12:54PM (#21398347) Journal
      Why does this keep coming up on ./? What is wrong with PDF? It's undeitable, sure, that's kind of the point. However, the spec is accessible, and there are plenty of open readers, e.g. xpdf and ghostscript.

      Really, what is wrong with PDFs and why should they require a warning?

      By the way, all scientific papers are disseminated by PDF.
      • by Tango42 (662363)
        "By the way, all scientific papers are disseminated by PDF."

        Actually, most scientific papers I see are disseminated as PostScript (often with a PDF option for people without ghostscript or similar installed - basically, non-academics).
        • Actually, most scientific papers I see are disseminated as PostScript (often with a PDF option for people without ghostscript or similar installed - basically, non-academics).

          Not in my experience. PS is opten an option, but not always. LNCS (Springer?) for instance only offer as PDF. I think Elsevier and the IEEE are like that as well.
        • by saforrest (184929)
          Actually, most scientific papers I see are disseminated as PostScript (often with a PDF option for people without ghostscript or similar installed - basically, non-academics).

          Perhaps it depends on the field. In my experience, in computer science all recent papers are provided either as PDF alone or PDF + PostScript, and in my (very limited) experience with refereed publications, PDF is the accepted standard.

          PDF has a lot of advantages over PostScript, the most obvious of which is internal hyperlinks.
          • by Tango42 (662363)
            It may well depend on the field - my experience is with Maths papers. Also, I'm thinking of pre-prints rather than papers from journals - journals are more commonly PDF, now I think about it. But my point stands - PDF is far from universal.
            • by lahvak (69490)
              I think lot of preprints used to be postscript because people simply ran TeX and dvips. With pdftex becoming more popular, I expect that is probably soon going to change.
      • I would like a warning because I usually don't click on links to PDFs unless I really need the info. Not because it's proprietary or whatever, they just take a long time to load, and if it's a big one, my browser hangs while it's rendering.
        • Re: (Score:3, Informative)

          I would like a warning because I usually don't click on links to PDFs unless I really need the info. Not because it's proprietary or whatever, they just take a long time to load, and if it's a big one, my browser hangs while it's rendering.

          Then get a better PDF reader. Even on a very slow computer, xpdf or ghostview have subsecond load times. If you use mozilla related browsers, then plugger will let you "embed" decent PDF readers. In fact if you install mozplugger under Ubuntu, it uses evince by default.
          • PDFs sucks in the default reader, and it often requires external shitty setup. This makes the format suck (on the web) for many/most people. Thus, it is courteous to give a warning. Whether a warning is unnecessary for YOU doesn't matter - it's courteous because the format is annoying for a large enough fraction to matter.

            For me, I find it particularly annoying because the default Adobe PDF plugin on Windows sometimes crash my browser. I think that's true for many others, too, though I don't know that

            • PDFs sucks in the default reader, and it often requires external shitty setup. This makes the format suck (on the web) for many/most people. Thus, it is courteous to give a warning. Whether a warning is unnecessary for YOU doesn't matter - it's courteous because the format is annoying for a large enough fraction to matter.

              For me, I find it particularly annoying because the default Adobe PDF plugin on Windows sometimes crash my browser. I think that's true for many others, too, though I don't know that for s
      • Re:PDF rant. (Score:4, Interesting)

        by izomiac (815208) on Sunday November 18, 2007 @03:00PM (#21399305) Homepage
        The main reason they should need a warning is because they aren't webpages. Either they get loaded through a browser plugin or they must be downloaded. In the former case, most browser plugins are slow to load, and nearly impossible to stop from loading, so a warning is nice. In the latter case they take a bit of effort to open and often people are too lazy (a warning isn't critical though). In both cases they are more inconvenient to use than HTML or text, so that's why I personally don't care for them. (IMHO, for online documents: html >= txt > rtf > pdf > jpg >> doc)
    • by StormReaver (59959) on Sunday November 18, 2007 @01:01PM (#21398401)
      "While it was typeset with TeX (open), only the PDF (closed and uneditable) is provided."

      PDF is neither closed nor uneditable. Adobe publishes the complete PDF format for anyone to use free of charge. It may not be FSF Free (since Adobe requires that implementers adhere to certain rules that violate the principle of Free), but it's definitely not closed. Also, KWord will import it for further editing, text and images, so it's not uneditable (even if it's not ideal).

      I agree with your main point, but let's cut PDF some slack.
    • Re: (Score:3, Informative)

      by 1u3hr (530656)
      The article (which is actually a PDF, thanks for the warning) uses proprietary fonts (LucidaBright). While it was typeset with TeX (open), only the PDF (closed and uneditable) is provided.

      I think (hope) you're joking, but several people who responded seem to be taking this at face value. It's wrong in several ways. PDF is an open format, and if you look at the file info, you see that this particular PDF was generated with Ghostscript. And it's quite simple to edit PDFs. Not as easy as, say HTML, but much

    • seriously, wtf? (Score:5, Informative)

      by tetromino (807969) on Sunday November 18, 2007 @02:13PM (#21398929)

      The article (which is actually a PDF, thanks for the warning) uses proprietary fonts (LucidaBright). While it was typeset with TeX (open), only the PDF (closed and uneditable) is provided.
      Oh, where to begin...
      1. The only reason you would need a "PDF warning" is that you use an operating system with poor support for the format (i.e. Windows). Switching to a real OS, among other benefits, will make reading math papers (which are almost always in PDF format) a pleasure.
      2. PDF is an open standard [adobe.com], which has been implemented by many different parties: Adobe and Apple have closed-source implementations; freedesktop.org's poppler and cairo libraries are Free software.
      3. The fontface chosen by AMS is orthogonal to the content of the paper - you can easily copy-paste the text and use Computer Modern, Dejavu, Liberation or any other open-source font of your choice. Why would a proprietary font embedded in a PDF file bother you any more than the proprietary fontface of a book?
      4. First of all, PDF is editable [petricek.net]. And second, why would you want to edit this particular document? Remember, it's copyrighted by AMS - if you can't prove fair use, you do not have the right to distribute a modified version.
  • The article is a very well argued opinion piece, and is correct in that only open-source software should ever be used in a proof.

    It is fundamental to mathematics that other mathematicians in the same field can check a proof, and the use of closed source software makes that logically impossible, for without access to the source of the application, it is not possible to guarantee that any particular operation has been implemented correctly.

    He's also plugging his own open source project, SAGE [sagemath.org] - I might have to

    • by tcgroat (666085)
      If software is used in a formal mathmetical proof, then the software itself must be subjected to rigorous mathematical proof. Every step must be justified based on accepted postulates and previously proven theorems, or else the work isn't rigorous and doesn't qualify as mathemetically "proven". As I repeatedly tell my daughter about her alegbra, you must show your work: it isn't just coming up with the "right answer", it's about how you know it's the right answer. Opaque software isn't mathematic proof, it'
    • by s20451 (410424) on Sunday November 18, 2007 @01:21PM (#21398571) Journal
      Well, don't get your panties in a big bunch over this. Humans make mistakes in proofs all the time, many of which are not caught before publication (and many not even for some time afterward).

      Also, although it's not in the field of theorem-proving, the mathematical package I use the most -- MATLAB -- is a million times better than the open source equivalent, Octave. I'm not going to use Octave simply because I can inspect the code, because who does that? An error in a software proof would be pretty obvious if it were checked with another independently written piece of software. With MATLAB, I can write my own alternative algorithm using C if I need to, though with significantly more effort and annoyance.

      Furthermore, mathematicians are smart people who are fully aware of the implications of their assumptions, probably moreso than any other group of people I have encountered. Reading the set of comments accompanying this article, saying what mathematicians should and should not consider a proof, is like watching monkeys trying to use a can opener.
      • I'm not 100% sure, but I'm pretty sure that the source for many of MATLAB's functions (albeit copywrighted) is available for inspection.
  • by davidwr (791652) on Sunday November 18, 2007 @12:23PM (#21398107) Homepage Journal
    Algorithms cannot be protected by copyright, only by patents and trade secrets. If the algorithm is a trade secret, it has no place in a mathematical proof because it cannot be shared with the world and verified or refuted by anyone interested in doing so.

    If the algorithm is part of a patented device or piece of software, its use in a mathematical proof is not subject to the patent on the grounds that pure math cannot be patented.

    If journals and academic societies refused to publish proofs based on trade secrets and insisted on a covenant not to enforce the patent against researchers doing purely mathematical research or those who publish the research, the problem would mostly go away. An alternative to the covenant is congressional action or a court ruling that says with absolute clarity that mathematical research is exempt from math-related patents directly related to the research.

    --

    Personally, I'm against all such patents but I'm not holding out hope that Congress or the Courts will agree with me.
  • Not Proven (Score:4, Insightful)

    by nagora (177841) on Sunday November 18, 2007 @12:29PM (#21398141)
    If a "proof" is published with some steps or information excluded then it's not a proof, it's just an assertion.

    TWW

    • ...and I don't think journals should accept papers which don't include proofs verifiable only with closed-source software.
    • Re:Not Proven (Score:5, Informative)

      by ciaohound (118419) on Sunday November 18, 2007 @12:43PM (#21398251)
      As a high school math teacher, I am familiar with some of the details of Thomas Hales' proof of Kepler's "Cannonball" Conjecture, concerning the most efficient way to stack spheres. When he first published his proof in 1996, he included the source code for the programs that were used to do the calculations for the thousands of possible sphere configurations. I think most of the code was actually written by his graduate assistant. At first that struck me as cheating -- "... and then this program runs. Q.E.D." -- but then I realized that if anyone else was to verify his results, they would need the programs. There are just too many calculations to perform without software, which is why the conjecture went unproven for four hundred years. But without the source code, it would smack of charlatanism.
  • by calebt3 (1098475) on Sunday November 18, 2007 @12:35PM (#21398185)

    Increasingly, proprietary software and the algorithms used are an essential part of mathematical proofs
    Like Excel's 65,535-equals-100,000 formula?
  • by MacTO (1161105) on Sunday November 18, 2007 @12:38PM (#21398209)
    This problem goes beyond mathematics, and reaches into many of the sciences. Mathematicians and scientists often place undue trust in complex software systems, simply as a matter of getting the work done faster rather than producing higher quality research. Sometimes it is a case of handling large volumes of data, in which case human intelligence and discretion is a bottleneck. Sometimes it is a matter of finding numerical solutions where analytic ones are difficult (if not impossible) to find at present. And, in the case of mathematics, I'm guessing that they are using it as a shortcut for those difficult analytic solutions.

    Then again, I must really ask if the mathematician in question understands what they are doing if they are using software as a shortcut for difficult analytic solutions. After all, if they don't understand the algorithms well enough to do the work themselves, who is going to say that they understand the limitations of the rules that they are asking the computer to apply.
    • Re: (Score:3, Insightful)

      by jhfry (829244)
      I thought the same thing... shouldn't mathematic proofs be independent of outside influence, shouldn't they stand on their own and make as few assumptions as possible. I figured that a proof, properly done, would be a large step by step solution to the problem.

      Then I realized that many proofs aren't concerned with single-input single-output situations, but instead may require thousands of iterations based upon large sets of inputs. You can't do that by hand.

      I am certain, that because computers/software ar
    • Re: (Score:2, Interesting)

      by mathcam (937122)

      And, in the case of mathematics, I'm guessing that they are using it as a shortcut for those difficult analytic solutions.

      This is certainly one application, but the use of computers in the more "pure" aspects of mathematics is nothing to sneeze at either. Programs like GAP for group theory, PARI for number theory, and Macaulay for commutative algebra and algebraic geometry play a significant role in the development of their respective subjects. For example, there's very little you can say about the Monster group [wikipedia.org] without the aid of computer calculations -- it's not that researchers don't understand the algorithms involved, i

  • ...is this saying the American Mathematical Society is accepting proprietary software used in proofs?

    Seems the only problem here is one of the position of the AMS regarding what is acceptable.
  • I don't understand: don't these automatic theorem provers provide the steps they took to prove the theorem? As long as those steps are provided and can be verified, I don't see why we care how the proof was obtained. We don't always know how proofs obtained by humans were obtained either; they don't tell us what they had for breakfast that day or what inspired them.

    There's probably not much insight that can be obtained by the source code of the theorem prover, you can always just assume that it was brute fo
    • Re: (Score:3, Informative)

      by flajann (658201)
      The advantage of having the source code is that, in a lengthy proof that involves thousands of steps that may be hard to follow, one may have an easier go at proving that the software did the steps correctly. At least, if a bug were found that would save you many hours over sweating over the actual proof!!!

    • Re: (Score:2, Insightful)

      by popmaker (570147)
      Suppose the only known way to prove something is to check all cases. Suppose the cases number about 1.000.000. It would take any human practically an eternity to check them all. Suppose checking each case is somewhat trivial, even though each case takes some considerable time and the results of each case is not generalizable to the other ones (so that we absolutlely HAVE to check the all). The mathematician working on the problem creates an algorithm for this work. It can be proven that the algorithm works.
  • Open Formats (Score:3, Insightful)

    by iamacat (583406) on Sunday November 18, 2007 @12:55PM (#21398357)
    Proprietary math software is not a problem as long as the end result can be exported into a fully documented format and can be then verified by open software, including human mathematicians.
  • openmodelica.... (Score:3, Interesting)

    by dohmp (13306) <dohmp@y[ ]o.com ['aho' in gap]> on Sunday November 18, 2007 @01:02PM (#21398409)
    not entirely on-topic, but i figured the slashdot community might be interested in this tool.

    OpenModelica [ida.liu.se]

    a very nice modelling package that can help you with practical mathematics issues like mathematica might.

    cheers.

    Peter
  • by Ardeaem (625311) on Sunday November 18, 2007 @01:09PM (#21398459)
    There are some programs which can aid proofs that are closed source. This doesn't HAVE to mean that steps of the proof are omitted. Take, for example, Mathematica for the Web [calc101.com]. It can spit out a result, including all the steps (try a derivative). Or check out a sample Otter proof [anl.gov]. Mathematica is closed source, Otter is open source. However, even if both of these were closed source, all the steps would be laid bare for all to see.

    In other cases, like the proof of the four color theorem, it seems like the source code is important to see, but not essential. Pseudocode should suffice. Providing pseudocode is akin to saying things like "Simplifying expression (1) yields..."; we don't have to provide EVERY step, but with pseudocode you have enough to determine whether the algorithm itself will work. Checking the source code beyond that is akin to checking someone's algebra.

    Just because we don't know how the program arrived at the steps it did doesn't mean that we shouldn't use it; we can usually check the steps. After all, the human brain has been a closed-source proof machine for thousands of years, and no one has complained about that :) Just require pseudocode in computer aided proofs, and it should be sufficient.

    • Re: (Score:3, Insightful)

      by mopslik (688435)

      In other cases, like the proof of the four color theorem, it seems like the source code is important to see, but not essential. Pseudocode should suffice. Providing pseudocode is akin to saying things like "Simplifying expression (1) yields..."; we don't have to provide EVERY step, but with pseudocode you have enough to determine whether the algorithm itself will work. Checking the source code beyond that is akin to checking someone's algebra.

      Perhaps I'm being too pessimistic, but shouldn't the source code

      • by Ardeaem (625311)
        But with the pseudocode, you can write your own program in whatever language you like to verify the results. In my opinion, proving something doesn't obligate you to show every single step. We all omit things in proofs, especially steps which can be verified easily by others, like algebraic simplification, etc. The pseudocode is the minimum acceptable transparency in a computer-aided proof.
    • by HiThere (15173)
      I've encountered too many programs where the source code doesn't match the documentation. For some of your arguments, that's not fatal. If the entire proofs are made explicit, then you can argue that it's like not being able to peer inside the skull of the mathematician. In the cases, however, where you are depending on the results of computational steps (as in the four color proof), those steps need to be made open and explicit.

      Pseudocode is not sufficient. You don't know that it actually reflects the
  • If you are using a software tool/package, then it must have been subject to mathematically rigourous tests to demonstrate it's own correctness. If not, then the foundation of any proofs that use it must be in doubt.

    So, if you use a closed product, how can that have been proved corect (independently of the supplier, of course) without recourse to the source code?

  • by LM741N (258038) on Sunday November 18, 2007 @01:15PM (#21398525)
    I would think that hardware errors would be an even worse problem, like the old Pentium bug, since they are so insidious.
  • by fermion (181285) on Sunday November 18, 2007 @01:37PM (#21398699) Homepage Journal
    This seems to fall under the realm of researchers using tools they do not understand. Black box science does not work. As has been mentioned, the results cannot be shown to be valid.

    A recall a few recent incidents in which papers had to be retracted because the machine did not do what the researchers thought it did. I have personal experience in which the spectroscopy generated by the computer did not reflect reality. If the researcher does not know how to use a tool, then he or she does not know when that tool is being misused.

    I am not sure something like mathematica is the issue. Wolfram seems to use standard standard well known algorithm. Almost every academic institution has a license, so, given the data, any number of people can rerun the analysis. Likewise the algorithms can be tested with simpler data sets to understand how they work and breakdown. I would be more worried about homegrown software.

  • ... try to read a paper from their journal (JAMS http://www.ams.org/jams/2003-16-03/S0894-0347-03-00422-3/S0894-0347-03-00422-3.pdf [ams.org]) and you will be asked for... money. Well that's their interpretation of "... In mathematics information is passed on free of charge..." cheers
    • Re: (Score:3, Informative)

      by William Stein (259724)
      The AMS did not write that article. I wrote the article as an opinion piece and the AMS published it. They do not necessarily agree with the points made in the article.

      By the way, the article is not about formal automated proofs. It is about what is now standard procedure in mathematical research, namely proofs that look like this:

      [Formal mathematical argument] ... and (using [Mathematica|Magma|...]) we deduce that [...].

      It's incredibly common right now when reading published mathematical papers to see
  • by ClarkEvans (102211) on Sunday November 18, 2007 @01:43PM (#21398745) Homepage
    http://metamath.org/ [metamath.org] has been around for 15 years or so; it has a very nice text-based proof expression, a huge library of existing proofs and a graphical visualization tool
  • Sage (Score:3, Informative)

    by Anonymous Coward on Sunday November 18, 2007 @02:00PM (#21398859)
    Sage( http://www.sagemath.org/ [sagemath.org] ) is currently the most full=featured open-source computer algebra system. It is being developed by the two authors of the AMS opinion piece (and many others including myself). Our goal is to provide a free, viable, open-source alternative to Mathematica, Maple, MATLAB, and Magma. Some nice features of Sage include:

    * It uses Python as its programming language so that you can use any existing Python modules with your Sage programs.
    * Sage also includes Cython ( http://www.cython.org/ [cython.org] ) which is based on Pyrex and allows one to easily compile Python code down to C for speed.
    * Sage's notebook interface with also interface with pretty much every existing computer algebra system, open-source or not.
    * Sage includes Maxima, GAP, Scipy, Numpy, and many other open source math packages.
    * A very active developer community. If there is something that you need Sage to do, chances are that there will be a number of developers willing to help you out.

    For some screenshots, see http://www.sagemath.org/screen_shots/ [sagemath.org] .

    One of the things that Sage needs most now is more users. So, if you have an interest in open source math software, definitely check out Sage.
  • Mathematics goes in an out of the phases of being secretive and open.

    Pythagoreans were very secretive. So were statisticians in the 19th century. I am pretty sure investment bankers do a great deal of math that they don't want anyone to ever see because it gives them an edge in the market.

    It's sort of like gun powder. When first discovered, the secret is tightly controlled because it would gives advantage over the competition. Then the competition realizes that it is being consistently beaten and tr

  • by mathgenius (526070) <simon@@@arrowtheory...com> on Sunday November 18, 2007 @03:16PM (#21399463) Homepage
    These guys are advocating setting up a peer-review process for open source software in machine learning. The idea is that this would encourage researchers to spend more time on the software component of the publication, and perhaps produce something that others can use aswell.
    The article is in the Journal of Machine Learning Research. [mit.edu]
  • by ndru82 (1043778) on Sunday November 18, 2007 @08:35PM (#21402001)
    It's called Math You Can't Use - by Ben Klemens. Makes a bunch of great points in favor of open source, too.
  • OpenAxiom (Score:3, Insightful)

    by andhow (1128023) on Monday November 19, 2007 @12:18AM (#21403421)
    On the subject of open source math, Axiom is an interesting 30 years-and-running project started (I believe) at IBM research that recently became open source. A new branch of the project started recently: http://www.open-axiom.org/ [open-axiom.org] and has several people actively working on it. It differs perhaps most significantly from Maple and Mathematica in its use of strong static type checking. This allows its library creation language to be compiled into C, which gets compiled and loaded back into the interactive top-level. Altogether, a very neat system and a gigantic resource.
  • by vinsci (537958) on Monday November 19, 2007 @03:23AM (#21404559) Journal

    Mathomatic [mathomatic.org] is a quick and handy package to use for your more everyday math problems. It celebrates 20 years since its first release this year and is still updated with its latest release, version 12.8.0, just three days ago.

    For programmers, its 'code' command converts your math problem to source code in your choice of several programming languages.

    For more info see http://www.mathomatic.org/math/adv.html [mathomatic.org]

    Many thanks for writing it, George Gesslein II [mathomatic.org].

Never trust a computer you can't repair yourself.

Working...