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Open Source Math
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.'""
Thanks for the article, now some crazed company is going to try to copyright math.
about the money.
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 the compiler. And if you hand compile the compiler, how can you trust the CPU? 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
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.
Do you even lift?
These aren't the 'roids you're looking for.
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 - I might have to download it and see if the rusty old brain cells can figure out how to play with it ;)
One swallow does not a fellatrix make
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.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
TWW
"Encyclopedia" is to "Wikipedia" what "Library" is to "Some people at a bus stop"
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.
IMHO, it would be an important contribution to establish an open proof exchange format that make it possible to... exchange proofs between different tools: theorem prover, proof checker, etc. Possibly this format would have a translator to a human-readable format (e.g. based on TeX) that would also make it possible for humans to review the proof process.
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.
SJW n. One who posts facts.
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.
not entirely on-topic, but i figured the slashdot community might be interested in this tool.
OpenModelica
a very nice modelling package that can help you with practical mathematics issues like mathematica might.
cheers.
Peter
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.
Ruby Neural Evolution of Augmenting Topologies
I would think that hardware errors would be an even worse problem, like the old Pentium bug, since they are so insidious.
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: 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: 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
Bastard Operator From 193.219.28.162
We may also mention Coq, a proof assistant wich is available under LGPL and runs on OCaml (which in turn is also open sourced and available on Linux).
/. (article about machine assisted proofs).
This is a tool that can help mathematician prove their theorems.
It was notably being used in the proof of the four color theorem, as mentioned on
"Sufficiently advanced satire is indistinguishable from reality." - [Tips: 1DrYakQDKCQ6y52z6QbnkxHXAocMZJE61o ]
Look around you. Look around you!
That's how I learned maths in high school.
I am defenseless. Use your button. Mod me down with all of your hatred.
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.
"She's a scientist and a lesbian. She's not going to let it slide." Orphan Black
... 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) and you will be asked for... money. Well that's their interpretation of "... In mathematics information is passed on free of charge..." cheers
http://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
I'm too lazy to see if that's the IEEE 754 result or not (but I suspect it is). But three things in Python's defense:
Dewey, what part of this looks like authorities should be involved?
Sage( http://www.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/ ) 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/ .
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.
Python calculated exactly what its documentation says it will do: ((1 minus the IEEE-754 double closest to 1/100) rounded to the nearest IEEE-754 double). It's not Python's fault if you don't know the basics of floating-point arithmetic. Mathematicians who use or write numerical software do.
I recommend reading What Every Computer Scientist Should Know About Floating-Point Arithmetic.
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, 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. 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.
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. But the algorithm IS the proof. If the theorem is true, running the program produces YES or TRUE or whatever and if it's false the algorithm produces NO or FALSE or something similar.
If we don't know that the algorithm works (it is proven to do so) and that the software also works (THAT is furthermore proven to do so) we have no way of knowing whether that YES or NO our program produced actually proves the theorem. Now assume that the program found a counterexample somewhere between case #50.000 and case #55.000. What if it was a precision error? If we don't have access to whatever it was that could have produced such an error - if we don't have ALL the source and ALL other information about the software, the results are useless.
Much of the guts of Mathematica -- the high level functions -- are available for perusal at http://functions.wolfram.com/ Much of the reason why Mathematica works so well is due to careful consideration of branch cuts.
For example, what is Log[a b]? If you look in almost any math book, you'll find Log[a b] = Log[a] + Log[b]. That works fine most of the time, but tends to fail in devastating ways when combined with other functions. The open source programs tend to follow this "good enough" approach.
Log[a*b] == Log[a] + Log[b] + 2*I*Pi*Floor[(Pi-Arg[a]-Arg[b])/(2*Pi)]
is the very careful way to handle the log function.
http://functions.wolfram.com/ElementaryFunctions/Log/16/04/0005/
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:
Or try a few titles which might be a little more familiar to Slashdotters:
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
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.
I've applied for a patent for the carefully aimed use of airborne chairs for intimidation of employees, particularly when they are threatening to resign to go to work for a competitor.
Would these be the same kind of good-for-nothing, lazy, worthless asses who brought us Special Relativity while working in a lowly position in the Patent Office in Bern? You know, the kind who got together with friends to peruse and discuss the latest freely available scientific texts, the same texts that led him to revolutionise science more than anyone since Newton?
The books in the Princeton Library are free, thanks to the generousity of far-seeing individuals who realised that their money was better spent on a library than a new yacht. They, at least, saw the benefit of sharing knowledge with everyone, regardless of their means. I can only hope that, somewhere in that misanthropic little husk you call a heart, you will some day find room for a similar spirit of openness and sharing.
Crumb's Corollary: Never bring a knife to a bun fight.
The issue is not whether software companies should make their source code open - the real issue is should mathematicians accept proprietary applications as proof of theorums?
As pointed out in the editorial, software developers make mistakes, and this is true regardless of whether that developer is a proprietary software vendor, or a free/open source software project. There is one key difference however, the validity of any given proof can be determined independently when using free/open source code by the very nature of the product (availability of source code). There is no validation for proprietary software beyond the assurances of the company involved.
When mathematic theory becomes applied mathematics (such as the creation of buildings, bridges, aircraft, or thermonuclear devices), which proof would you prefer to hang your life upon - Microsoft's guarantee, or independent verification and peer review? This becomes ever more critical as we create more complex systems that can not be easily verified by hand, yet rushed into applied use by the expediency/efficiencies they deem to provide.
Lodragan Draoidh
The more you explain it, the more I don't understand it. - Mark Twain
It's called Math You Can't Use - by Ben Klemens. Makes a bunch of great points in favor of open source, too.
Luckily for you, you haven't the foggiest idea how mathematics is developed. As it happens, I am a trained mathematician, and I work in finance, so I think I'm relatively qualified to comment. Your naive analysis makes a great deal of sense when you're dealing with procedures that are simple enough for a few smart people to come up with, but modern mathematics has become so complex that we only advance by standing on each others shoulders.
At my firm we use a lot of "proprietary mathematics", but guess what: none of it is stuff that wasn't discovered by some academic, we just combine their results in innovative ways. We depend completely — and it's not just us, it's everyone in the industry — on the results of academics.
Firms like Renaissance hire a bunch of PhDs who then apply their years of research in academia to generate the returns they do. Derivatives pricing in particular is so complex that it's not that the math isn't freely available for anyone to peruse, it's that it takes years of study to properly understand how it even works. So when a firm hires one of these guys, they aren't hiring them hoping that they'll come up with the next Black-Scholes-Merton pricing theory on company time. They're hoping that they understand the theory that already exists well enough to be able to apply it to real-world data and tweak it as necessary. That's what all those proprietary processes really are: not new, exciting theory, but old theory, properly combined and tested and written up into algorithmic trading or security selection algorithms.
Now, cryptography is not my thing, that's not the field I studied or what I have an interest in, but it may be that the NSA has a bunch of tech that no one else has. However, the NSA is not a private firm, and they'll be funded no matter what happens, so they can afford to employ thousands of brilliant cryptography researchers (in the same way a university does) and sit on the results. A private company can't do that. You can point to them as an example of how "it makes sense to keep results closed" if you want, but understand: they are the exception, not the rule. No one else works the way they do.
Universities, many of the best of which are private, depend on their professors publishing new and innovative stuff nearly non-stop. It's not altruistic; they make their money on tuition and they can justify charging 40k a year for an education only if they employ the best and the brightest. The measure of the best and the brightest is in their scholarly output. It's no coincidence that Harvard, Princeton, Stanford, MIT, and Yale have so many Nobel Prize and Fields Medal winners in their employ. Reputation is what makes or breaks a school; without it, they have no income, because the best students want to study at a university with pedigree.
So while your (cynical) view of the world may "feel" right if you don't know what you're talking about, thankfully for all of us, open mathematical and scientific development will always trump private proprietary systems. That may not be true of software, I don't know. But when it comes to research, academia wins.
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/ 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.
Mathomatic 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
Many thanks for writing it, George Gesslein II.
Trusted Computing FAQ | Free Dawit Isaak!
cripple itself with IEEE-754 standards needlessly
The IEEE 754 standard is very well designed and ensures floating-point arithmetic to be accurate, efficient, and compatible across platforms.
But it does mean that when there is a precise answer and it is calculatable (I think the former demands the latter), that I do minimally want Python to store it fully in memory and to not print out questionable and/or incorrect answers.
So you'd be happy if Python generates an exact answer when you ask it to compute 1 + 1e-100000 and it allocates several kilobytes of memory to represent the result? Would you still be happy with this behavior if you are doing a numerical computation involving a million numbers? Exact rational arithmetic is, in general, much slower than floating-point arithmetic, and the cost grows the more operations you perform.