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Like many other mathematicians I use mathematical software like SAGE, GAP, Polymake, and of course $\LaTeX$ extensively. When I chat with colleagues about such software tools, very often someone has an idea of how to extend an existing tool, what (non-existent) tool would be useful, or which piece of documentation should be (re)written. Due to lack of time & energy and often also programming expertise, these ideas rarely materialize.

On the other hand, every now and then I meet programmers with a strong interest in mathematics (who are often actually trained mathematicians), and who are looking for a software project to work on. However, normally they don't really know what's needed and end up doing a non-mathematical project.

This gave me the idea to ask the mathematical community to compile a wish list for mathematical software. Wishes can be very small or or something bigger. Just try to make sure that it's realistic and maybe also give an explanation why you consider your project as interesting.

And if you happen to be a programmer fulfilling one of the wishes, please leave a comment.

It would be great if you could also include an estimate on how complex your project and what the math/coding ratio is -- but this is optional.

tl;dr

  • What software tool would you like to see created?
  • What existent software tool would you like to see extended by what feature?
  • What piece of documentation is missing or should be updated/extended?

One suggestion per answer, please.

Gerry Myerson
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eins6180
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    It would be great to have software that serves up a website where mathematicians can ask other mathematicians questions that come up during their research. – Yoav Kallus Aug 07 '15 at 20:30
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    As a programmer, I really like this question, but all of the responses so far look like they would take teams of people months to complete. :( But I guess if it could be done by a single programmer in a week, it would already have been done, right? – Jashaszun Aug 07 '15 at 21:15
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    My wish would be for open-source software that can plot surfaces with realistic shading. –  Aug 08 '15 at 02:39
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    This question is way too broad. There are some subquestions that could work, but I think you really need to restrict the scope. For example "What existent software tool would you like to see extended by what feature?" would seem to allow as answer: "I would like to see feature X in Software Y (that already is present in Software Z)." I do not really believe it is useful to create a wishlist for the numerous differing software-packages in a single Q&A. –  Aug 08 '15 at 09:31
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    Likewise "What piece of documentation is missing or should be updated/extended?" solicits feedback on the documentation of a multitude of different software. Even for a single piece of software these two questions combined seem too broad, I could see one of them for one software as a very broad question (though it is not quite clear it fits here). Anyway the scope of this Q needs to be reduced drasticaly in my opinion. –  Aug 08 '15 at 09:36
  • This is protected, so I can't contribute my own answer, but here goes anyway: I'd love a software that could generate (or help the user search a database of) finite difference schemes given a few constraints such as number of dimensions, desired stencil size, desired order of accuracy, particular meshes, and location on the grid (i.e. boundary points or grid points). A similar feature would be for the user to input a PDE and for the software to offer the user a few FD schemes that might work well given that problem, its geometry, etc. – jvriesem Aug 08 '15 at 20:58
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    @quid: While I agree that my question is fairly broad, I must admit that this is by intention. The idea is to compile a list that mathematically inclinced programmers can pick a project idea from to work on. In this regard I also find it useful to have a collection of feature requests that covers more then one software tool. If this list becomes too confusing one can still split it up into several later. – eins6180 Aug 08 '15 at 21:07
  • Thanks for the clarification. How exactly do you propose to split it? –  Aug 08 '15 at 21:10
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    I'm voting to close this question as off-topic because it seems too broad for MO to be the appropriate place – Yemon Choi Aug 16 '15 at 06:33

21 Answers21

98

I think some aspects of math would be revolutionized by having a good math search engine. Recently, a question was asked on Meta.MathStackExchange about what they perceived as the greatest problems facing the site. The biggest response was that there was no search engine that indexed mathematics.

This is partly reasonable, since math is stored and documented in $\TeX$ and this can be taken as a standard. But this is also problematic, as there are multiple noncanonical ways to do things in $\TeX$. I would be remiss if I didn't say there are very many other challenging aspects of this.

As an example use case, I often have to look things up in the Gradshteyn and Ryzhik Table of Integrals and Series. It would be remarkable if there were a reasonable way to search for my expressions within the book. Even if I had to attempt multiple searches, it would almost certainly be faster. Taking it up a step, it would be great to search through TeX on the arXiv for certain expressions as well.

I think that even a relatively mediocre math search engine would be a handy start.⠀⠀⠀⠀⠀

James Smith
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davidlowryduda
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    A related discussion http://meta.mathoverflow.net/q/1564/. – Name Aug 07 '15 at 20:06
  • I've been long interested in building such a search engine but did not get time so far, nor enough interested people who are as stoked about it. But I remain interested in it! – Suvrit Aug 07 '15 at 21:46
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    You might look at http://search.mathweb.org/ – Robert Israel Aug 07 '15 at 22:21
  • @RobertIsrael: good pointer; there's also a few NSF funded academic projects building math search engines with similar scopes (afaik). – Suvrit Aug 07 '15 at 22:23
  • Springer already have a LaTeX search - http://latexsearch.com/ - but perhaps it doesn't search everything. – Gordon Royle Aug 08 '15 at 00:36
  • @GordonRoyle: In some ancient MO thread, I remember pointing out latexsearch.com --- I have never really managed to find anything useful with it though so far! – Suvrit Aug 08 '15 at 02:43
  • When you say you want to search math, do you mean finding articles involving some parts of what math/text you know like: "Hey, what was that formula for ABC with a congruence relation involving $x^2$ modulo an odd power of a prime number?" Or involving a specific formula you know like: "Where did I see $\frac{a^2+b^2}{c+d}$, in reference to XYZ?" Or a specific thing like: "What's the formula for finding PQR?" – shardulc Aug 08 '15 at 13:40
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    @shardulc ideally, all three. That is, I'd like a search engine that indexes both normal language and latex. In my example in GR, I was thinking along the lines of your second question about a specific formula, which is (as far as I know) not at all an available service. – davidlowryduda Aug 09 '15 at 01:30
  • This is actually a problem I'm facing. How best to index theorems, how to uniquely express something (say an integral) under some rules that leave the result unique, so forth. – Alec Teal Aug 10 '15 at 10:53
  • You might consider something like this (http://mathlex.org/demo) which can record the semantic meaning while still displaying LaTeX as a starting point. – Logan Aug 10 '15 at 19:46
  • There's also http://functions.wolfram.com/ – asmeurer Aug 11 '15 at 15:41
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    A review of some current math search engines have been uploaded to arXiv recently: https://arxiv.org/abs/1609.03457 – Tadashi Sep 18 '16 at 02:51
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    If one had a search engine which indexes all theorems/lemmas/... in all articles and which allows to search for (assumptions) with (conclusions). –  Jan 26 '18 at 15:28
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    There exists Approach0, which is a search engine currently only for MSE. I believe it to be some 2 years old, and its author has polished it in the meantime. I myself have been quite impressed by what it can do. Yes, it is LaTeX-aware. – Alex M. May 20 '18 at 18:54
  • This is a hard problem. To build an effective search engine you'd have to visually parse the equations from PDFs or the visual rendering of HTML/TeX, similar to handwritten equation analysis. Not sure the state of that technology. Then once you get the equations parsed, finding equivalencies based on some standard definition, which would require a formalization of mathematics such as in HoTT with Coq. Then figuring out what actually needs to be searched for from a UX perspective. – Lance May 20 '18 at 21:15
  • @lancejpollard MathPix, a OCR made for LaTeX, seems to be working very well. – Tadashi Jun 05 '18 at 23:47
53

A more modern typesetting language to replace $\TeX$. TeX is basically impossible to parse and its internals are really odd and difficult to work with, when one tries to do something advanced.

Knuth is a genius, and it was a really neat hack for the time, but, with all due respect, after 30 years of experience with computer typesetting I am sure it is possible to put together something better.

If not, at least a TeX compiler with better error messages.

Federico Poloni
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    TeX is certainly not a "neat hack". – the_fox Aug 07 '15 at 19:18
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    Please not. TeX is perfect as it is. The change of format of mathematical papers is undesirable. – juan Aug 07 '15 at 19:30
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    Tex was and is an extraordinary achievement, and is fantastic for what it is, and almost certainly superior to the alternatives, but (like any long-lived project) it has acquired a huge accumulation of cruft, much of which is rather dated. So if a dedicated team led by a clone of Knuth could write a new system, from the ground up, following the same principles as TeX and learning from what TeX’s many contributors have done in the mean time — I suspect we could end up with a significantly better system still. But there’s probably too much intertia for this to happen any time soon. – Peter LeFanu Lumsdaine Aug 07 '15 at 20:04
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    Isn't exactly this done by the developers of systems like Context, LuaTeX, ... ? – Johannes Hahn Aug 07 '15 at 20:16
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    I'll just leave a link here: http://latex-project.org/latex3.html – Andrei Smolensky Aug 07 '15 at 20:47
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    @JohannesHahn As far as I understand, Context is a macro system alternative to LaTeX, so it builds upon TeX but uses the same underlying engine and syntax. Luatex is, in my view, a very promising project: it is still TeX, but aims to support modifying all of the document properties with scripting, a bit like DOM+Javascript for HTML. – Federico Poloni Aug 07 '15 at 20:50
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    @PeterLeFanuLumsdaine TeX is certainly not fantastic for what it is. If it is superior to its alternatives, it's because there are no alternatives. A new system could still have implementations of LaTeX and TeX commands so that current LaTeX packages and old TeX/LaTeX source files remain usable. – Algernon Aug 07 '15 at 21:54
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    While a better system does exist in the platonic realm, given the bug-free monolothic design of TeX any realistic change is extremely likely to be for the worse, for there are essentially no known examples of big software that do not suffer from bugs, design-by-committee, backward- and forward-incompatibilities, poor manuals, etc. Better is enemy of good. – Boris Bukh Aug 08 '15 at 12:02
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    @juan TeX is not perfect as is. Among the desirable properties of TeX would be a format that is easier to parse. The highest rated answer at this moment is a search engine for maths, and this is precisely the type of things that are currently very difficult just because the format has lots of quirks. A different document format may make a search engine a much easier task – Andrea Ferretti Aug 11 '15 at 10:35
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    There is a lot of work being done to make LaTeX more accessible through online services such as Overleaf and ShareLaTeX (disclaimer - I'm one of the founders of Overleaf), and these are specifically looking to combine the benefits of the fantastic typesetting engine with a modern (and collaborative) interface. – John Hammersley Aug 11 '15 at 11:01
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    Many of the answers to OP's question seem to be about TeX or the related family of software. If you want to support the development of this family, or indeed just support the continued existence of versions of the software runnable on current platforms, please consider joining the people who are making that happen. Join, or donate to, a TeX Users Group (disclaimer: I am VP of TUG). – Jim Hefferon Aug 11 '15 at 13:24
  • @FedericoPoloni AFAIK modern ConTeXt heavily relies on the LuaTeX engine and (but I may be mistaken) LuaTeX and ConTeXt development in a sense are going hand in hand. – cgnieder Aug 12 '15 at 16:34
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    Yes the_fox you right and polite: a mathematician is neither a secretary nor a typographer. Yet he has to make up with each drudgery. Knuth is precise but does not understand the functional point of view. My taxes should be used to develop latex as a tool for maths and many disciplines, instead of being given to dubious publishers asking us to pay for articles written by researchers subsided by my very taxes (from an angry programmer). – Jérôme JEAN-CHARLES Aug 15 '15 at 22:09
  • TeX is good to very good as is. The problem, however, is the use of cookbook straitjacket time-varying LaTeX, and this by about 99% of users. Learning plain TeX forces you to be able to learn enough to solve TeX problems for yourself. – David Handelman Jan 26 '18 at 15:10
  • What whould you improve in LaTeX and how would you like to make it better? –  Jan 26 '18 at 15:29
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    @TKe For instance, just listing things on the top of my head (1) a more modern syntax with a proper parser (2) an easier programming interface and access to internals (Luatex partially solves this) (3) better error messages (4) a debugger (5) ability to interface with other programming languages (\write18 does not really count) (6) IEEE arithmetic. – Federico Poloni Jan 26 '18 at 16:53
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    @ FedericoPoloni Your are right an interface and well documented libraryies Here is a problem how do I define circled symbols such as union , intersection , start etc.. With a guaranty of proper scaling. I thing it is hard to produce and I could no find it in existing libraries: showing how little orthogonality there is so far. The argument often heard in favor of Tex is how useful it is which is true ONLY because we have nothing else. Writing with a quill is nice for a while only : it look as if people are deriving a pervert pleasure wallowing inside low level syntax. – Jérôme JEAN-CHARLES Jan 22 '19 at 17:35
  • I agree, that's why we have $\LaTeX$. – tomasz Aug 20 '21 at 22:41
35

I always thought it would be nice to have a real-time virtual blackboard (supporting digitizer pen), say, as an extension of Skype or similar service, where you can not only talk with a colleague but also do math together over great distances.

M.G.
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    I think there are already several services offering this -- see for instance http://www.webdistortion.com/2011/01/22/best-online-collaborative-drawing-tools/. – Federico Poloni Aug 07 '15 at 19:59
  • Thanks for your comment, but I am getting "page not found" from your link. – M.G. Aug 07 '15 at 20:03
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    @July Try with this link: http://www.webdistortion.com/2011/01/22/best-online-collaborative-drawing-tools/ – Federico Poloni Aug 07 '15 at 20:53
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    Have you tried searching for this at all? There's a TON of collaborative whiteboard solutions out there already, many of which have Skype extensions. First few results from Google: https://ziteboard.com/ http://www.groupboard.com/products/ https://syncpadapp.com/ https://idroo.com/home https://realtimeboard.com/ http://www.twiddla.com/ https://www.scribblar.com/ https://awwapp.com/ – BlueRaja Aug 07 '15 at 20:54
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    There's also hack.chat, which is text chat that allows you to input LaTeX expressions also. – user17945 Aug 08 '15 at 09:54
  • Related: Tools for long-distance collaboration. (And maybe also some other links listed in the comments here). – Martin Sleziak Aug 10 '15 at 07:49
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    @BlueRaja: I have searched, and most of them are not particularly good. Instead of naming everything google shows, it would be more helpful to point to a particular piece of software that is proved to work well. – M.G. Aug 10 '15 at 07:59
  • The tool MathCAD can be good. It is designed as a maths white board for engineers, and has the patents. Unfortunately it is now part of the much larger corporate PTC who do regular CAD tools, so it is being diverted off to supporting certified drawings. It also embeds a CAS engine. – Philip Oakley Aug 12 '15 at 15:51
32

A vastly improved support for handwritten math (e.g. via digitizer pen) incl. its conversion to typeset math would be awesome! In a long run, ideally it should be able to replace LaTeX. Just think of how much of researchers' time is spent on inputting math.

M.G.
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    One existing tool in this direction is Detexify, which can recognize handwritten symbols and tell the TeX code. This is often much faster than consulting a symbol list. Maybe the Detexify project could help build the required math recognition software? – Joonas Ilmavirta Aug 07 '15 at 20:00
  • I agree, it is a start. But I think it will take some years before we get a complete solution. Recognizing 'normal' handwriting properly is already a difficult problem (but it has improved a lot over the past few years, especially with handheld devices and 'apps' and such). And then, on top of it, one has the whole particularity of the mathematical notation. – M.G. Aug 07 '15 at 20:09
  • @JoonasIlmavirta Detexify's recognition is not so great, though the idea is. About half the time I try to use it, I give up and do a web search or consult a symbol list. – Kimball Aug 08 '15 at 03:28
  • Neural nets have made decent strides towards reliably recognizing handwritten letters. Have they been used for recognizing math symbols yet? – Emile Okada Aug 08 '15 at 08:25
  • http://www.wiris.com/ has a big step in this direction. – Mark S. Aug 09 '15 at 21:11
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    Another one: http://myscript.com/technology/technical-demonstrations/ – Stephen Aug 11 '15 at 08:41
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    I doubt that handwriting math and then digitalizing it is much faster than writing it in TeX. – Caleb Stanford Sep 24 '16 at 05:39
27

An improved version of latexdiff

A good diff software is essential for collaboratively writing articles. Latexdiff takes two tex files and outputs a new tex file with the differences highlighted (additions are underlined in blue and deletions are crossed out in red). This is very useful since it facilitates viewing the changes that coauthors have made during a round of editing, especially if some of your coauthors are not super computer-savy (e.g., they don't use diff themselves) since you can just pass them the output PDF with the marked changes.

However, my experience with using latexdiff is that the output file usually requires some manual editing before it can be compiled into a PDF, since the diff markup algorithm often messes up the latex syntax. It would be useful to have a more user-friendly latexdiff.

Yufei Zhao
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  • The online TeX Editor Sharelatex has a history feature that sounds a lot like what you are asking for. This of course helps only if all collaborators use this editor and it is only available if you pay a monthly fee or manage to recruit enough participants. – HenrikRüping Jul 25 '18 at 12:52
21

I would like software that makes the specific job of managing mathematical references easier.

When I've looked, there are BibTeX and its relatives for making sure that the whole process stays under control; many tools for managing references; tools for pulling BibTeX from MathSciNet; tools for creating BibTeX from arxiv identifiers; tools for searching these places for papers; tools for merging BibTeX files; and so on.

Using these individually simplifies individual parts of the process. But I always find that I need to:

  • copy my bibliography from my previous paper, and maybe another one,

  • merge in some more bibliography entries from my other previous paper where my coauthor managed the file, and eliminate the duplicates and inconsistencies,

  • try to figure out which papers have moved from preprints to arXiv preprints to being published, and update their entries,

  • go through new published references I'm adding, hunt them down, and copy-paste BibTeX from MathSciNet,

  • create new entries for the arXiv preprints,

  • and then some misc. extra jobs that always show up, like fixing tildes and putting capitals in braces and adding hyperlinks to bib entries that lack them because they date all the way back to my thesis.

The process is exhausting and the tools don't click together enough to make it much easier - e.g. automating pulling references from MathSciNet seems almost not worth the trouble because it involves firing up special-purpose software that's only useful for half of the new entries that I'm referencing.

user76883
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    I haven't really used it much but the postdocs at my last lab used https://www.mendeley.com/ to organize the papers they've read. You can export BibTeX from it, too. – Sumyrda - remember Monica Aug 08 '15 at 07:02
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    We mathematicians are already extremely lucky to have MathSciNet. As far as I know, in most other fields there is basically no publication database that can produce Bibtex entries of reasonable quality (names in the correct format, capitals properly escaped, consistent journal names). – Federico Poloni Aug 08 '15 at 07:07
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    There is Zotero. It is not without problems (multilingual search could be better, BibTeX export is very mediocre), but in my experience it has been the best among available software. – Boris Bukh Aug 08 '15 at 11:55
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    @FedericoPoloni We are doubly-lucky: we have not only MathSciNet, but also Zentralblatt. It has an advantage over MathSciNet that the first three search results are free (require no subscription) --- many here are in sufficiently wealthy institutions that this does not matter, but for some it is valuable. – Boris Bukh Aug 08 '15 at 17:21
  • Astronomy is also lucky has ADS. Which now covers a good bit of physics and some other sciences as well, afaik. – Kyle Aug 10 '15 at 20:52
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    DBLP now indexes most of theoretical computer science and, unlike MathSciNet, is free to use. – András Salamon Aug 15 '15 at 17:03
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    I use JabRef and get the bibtex entries from Mathscinet, Zentralblatt, or google books. If you keep a folder with the articles in pdf you can open them directly from JabRef. It still requires some discipline and I find it hard to avoid developing several "branches" of my database. – Pablo Lessa Jun 16 '17 at 13:43
  • There are several emacs packages that allow importing bibtex entries from arxiv, mathscinet, etc. without leaving the editor. Most of the other problems are solved by using Mathscinet bibtex entries (including the citation keys!) whenever possible and using a single bib database for all your writing. – pavel Jan 28 '18 at 23:33
  • @BorisBukh: the MR lookup feature of MathSciNet also gives three results and requires no subscription. It is unfortunately a bit literal, so it won't find anything if you make a typo. – R. van Dobben de Bruyn Jun 13 '18 at 13:06
20

I find it particularly cumbersome to produce good-looking mathematical illustrations. I know of several ways to make decent cartoon images in bitmap format with little hassle, but I prefer the image quality provided by vector graphics. TikZ seems to be the go-to for math-based vector graphics, but this is incredibly time consuming, even after climbing the learning curve.

I would very much like a bmp-to-tikz "converter."

Depending on the quality of the bitmap, the converter might need to iteratively suggest a vector-graphics interpretation for the user to evaluate. The user could then fine tune the TikZ code after conversion if he's extremely picky.

Dustin G. Mixon
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    Or maybe an xfig-like visual editor for tikz? – Kimball Aug 08 '15 at 15:11
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    @Kimball Inkscape has an extension to export to tikz. – Federico Poloni Aug 08 '15 at 16:53
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    Inkscape also has a feature to turn bmp to svg (e.g. by color) and then you can use the svg to tikz feature. – Sumyrda - remember Monica Aug 08 '15 at 17:36
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    Here is an example of bmp to TikZ way via potrace: http://tex.stackexchange.com/questions/172336/drawing-roman-laurel-leaves-spqr-in-tikz – percusse Aug 08 '15 at 22:17
  • Even though it costs money, for some illustrations Omnigraffle is pretty nice! There's also an MO thread on tools for mathematical illustration (for quick drawing IPE is also quite good, but not easy to get fine grained enough)... – Suvrit Aug 10 '15 at 13:23
  • Sounds rather like a beefburger to cow converter :-/ – J.J. Green Jan 26 '18 at 13:58
  • I don't really see the usefulness here. If you can make a decent cartoon images in bitmap format, you might as well make vector images (e.g. using Inkscape; if you really want, you can even convert bitmaps to vector graphics, though that won't look good). And why would you use tikz instead of plain pdf/eps/svg (except when you want tighter control over the generated image - and converting from bitmaps won't give you that). – tomasz Aug 20 '21 at 22:46
19

LaTeX support (or mode) in voice-recognition softwares for people with upper limb disabilities. Maybe via Dragon NaturallySpeaking or something new entirely. Something similar for coding as well!

user7610
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16

It would be nice to have a pdf viewer which gave the user the option to collapse and also restore individual proof sections.

Daniel Gerigk
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    I think this can be better achieved by moving away from PDFs to something like XHTML. – David Ketcheson Aug 12 '15 at 12:24
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    Philipp Kühls PhD Thesis is about algebraic surgery. However for fun he also coded his own pdf viewer with features very similar to what you are asking for. www.surgery-hotel.de – HenrikRüping Jul 25 '18 at 12:56
15

A wiki-like mathematical structure/example data base

I can imagine this better as an online service than a locally running software, but nevertheless I imagine it to be very useful.

You ever wondered (because of your research or out of curiosity) whether there is an example of a structure A (e.g. a topological space, graph, group, ...) which has the properties B, C and D, but not E? If Yes, what is an example of such a structure? If No, how can this be proven?

Wouldn't it be nice to have an online service, something like Wikipedia, where you can enter that you are browsing the database of structures A, and you specify the filters [B], [C], [D] and [not E]. You press enter and it spits out:

  • Here is an example of such a structure: ...
  • These structures are exactly the 3-dimensional compact manifolds. An example would be: ...
  • No structure can have this combination of properties. Source: ...
  • It is conjectured that no such structure exists. Source: ...
  • It is a known open problem whether there is such a structure: Read more here: ...
  • Property [B] and [C] imply [D] and [not E]. Hence you are actually looking for structures A with only [D] and [not E].
  • The database does not contain information on this combination of properties. Do you want to extend the knowledge?

"wiki-like" means that the database's knowledge can be extended/corrected by anyone $-$ like in Wikipedia. Even though this might seem like a complicated semantic search engine, I think that the strong formalization we have in mathmatics enables us to choose a strict syntax for the input.

  • We always specify the general structure we are looking for, e.g. vector space, topological space, metric space, group, ring, field, graph, function $\Bbb R\to\Bbb R$, subset of $\Bbb R$, curve (in metric space), field-automorphism, ...
  • It follows a list of properties, e.g. finite, compact, 3-dimensional, connected, Hausdorff, has inner point, metrizable, bijective, ... . Every such property can be suffixed with a [not]-operator. The listed properties are joined by conjunction.

The structures and property names are no free-form input, but chosen from a pop-up menu or by auto-completion, so that the users know what to input. The database should implement very basic reasoning, e.g.

  • If a property A implies B, and B implies C, so does A imply C.
  • If A and B contradict each other and C implies A, so C and B contradict each other too.
  • The structures can be linked, e.g. every metric space is a topological space (by its induced topology). Hence, every property which is available for topological spaces, is also available for metric spaces.

I know of several such services of varying generality: e.g. for rings, groups, graphs (here and here), polyhedra, or general counterexamples. The differences to what I am looking for can mostly be described by the following points:

  • General: I want a combined database for all/most structures. All this under a common interface.
  • Extendable: I think everyone should be allowed to add his knowledge to the database.
  • Searchable: Most of the time I know only the names (or some names or vague descriptions) of some properties of the desired structure. I do not know the structures name. Hence I want to filter by these properties. Sometimes I might be not even interested in examples, but in the relation between two properties: e.g. do they contradict each other, are they the same, does one imply the other, ...?
  • Structured: Not a loose collection of examples/counterexamples/articles, but highly interconnected and analysable data.
  • Userfriendly/Beautiful: I think mathcounterexamples and of course StackExchange is a good demonstration of these goals.

I once had an idea how this can be realized. I even asked a question on Computer Science StackExchange to see whether useful data structure for this kind of task already exist. I would love to realize such a project, but I am definitely lacking the web-developer skills, and currently also the time.

M. Winter
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    There is also Groupprops among the "things which aren't quite what you're asking for but are at least somewhat in the right direction". But the whole thing should be part of a larger "semantic web" effort: Wikimedia made a step in this direction by creating Wikidata which should, ultimately, make it possible to query Wikipedia-and-sisters in semantically useful ways (like "give me a list of French painters born between 1800 and 1835"), so maybe if this takes off, math can benefit from the infrastructure. – Gro-Tsen May 20 '18 at 18:05
12

There should be $\LaTeX$-browsers. The (relatively) new HTML 5 is great. It'd be still wonderful to have both: HTML-browsers (as today) and $\LaTeX$-browsers.

Włodzimierz Holsztyński
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    What is a LaTeX browser? – Federico Poloni Aug 09 '15 at 20:03
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    What is an HTML browser? – Włodzimierz Holsztyński Aug 10 '15 at 03:11
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    A program that takes HTML as an input and display on the screen formatted text. So, according to my definition, pdflatex + any pdf reader of your choice is a LaTeX browser. That's why I asked for your definition, not for a counter-question. – Federico Poloni Aug 10 '15 at 08:21
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    how about http://www.sharelatex.com ? "LATEX-browser" inside your "HTML browser" =) – sigrlami Aug 11 '15 at 06:24
  • When you enter url like abc.HTML in an browser then you see the html-formatted text. *** When you enter url like abc.tex in a latex browser than you see the latex-formatted file. – Włodzimierz Holsztyński Aug 11 '15 at 08:06
  • Overleaf is often used for this - it combines an online LaTeX editor with a rich-text mode to help authors new to LaTeX and those experienced users in collaboration with non-LaTeX users. You can then share your work easily via the browser. (Note - I'm one of the founders of Overleaf, and if you use it, feedback is always appreciated, thanks). – John Hammersley Aug 11 '15 at 11:06
  • Like LyX? Not a browser per se but nearly there. – Allen Aug 11 '15 at 12:24
  • This could be accomplished as a browser extension. – Rick Aug 11 '15 at 15:46
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    There's no real point in doing a whole browser for this. Just use a plugin or a dedicated website to compile LaTeX source code to an output format on the spot. – xji Aug 12 '15 at 18:36
  • This exist in the form of MathML (https://developer.mozilla.org/en-US/docs/Web/MathML) – Lanzafame Aug 14 '15 at 10:01
  • @JIXiang yeah this is too specific of a technology to have a browser dedicated to it, don't see that happening. Better math support in JavaScript components would be along these lines. Getting GitHub to support math in Markdown would be a step in this direction. https://github.com/github/markup/issues/897 – Lance May 20 '18 at 21:22
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[In short, a place we can find a more modern proof, or a proof with a different approach, than the original paper that is customary to cite.]

One of the comments mentioned moving LaTeX from PDF to an XHTML sort of environment (Stacks Project is a good example). It may be worth expanding on that as a separate answer.

One obvious advantage is that words in text mode can be searched. And it may be a step towards making LaTeX code searchable (the current top answer). Most of the non-mathematical TeX codes (bullet points, italic, include pictures) we can use markdown like here on MathOverflow.

But the real game-changer is to make it more like GitHub, where one can fork a proof in order to add missing details, or make more substantial rewriting, and "publish" it for all to see. The rest of us can vote on them. Eventually the platform should have sufficiently many of the basic theorems (with many different proofs) in all branches of mathematics that we can cite directly, instead of citing the original paper or a textbook. The citations can also be used to generate a "dependency graph", barring circular reasonings (It may not be as exhaustive in the details as in Stacks Project, or we'd lose the big pictures).

If we want to say a certain result (say, in a certain abstract theory) is important, we can just point to it and see how many important results—or results that you care about—are connected to it. It may be more fun to learn new mathematics this way, combining the best of all textbooks, old and new.

Aside from the theorems, there would be special pages that are more expository, giving historical context (say of a problem) and connecting different theorems into a coherent narrative without getting bogged down in the details of proofs.

Now, wouldn't that be the new publishing model that we have all wished for? Credit may be traced from the forking history. (Of course if we are "importing" or rewriting a proof in the literature we should give proper citations.)

liuyao
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  • You might want to check out openmathematics.org, which doens't have GitHub support for files and comments yet, but soon will have. – James Smith May 20 '18 at 18:23
  • I think the Stack Project just made their underlying system available, called Gerby https://github.com/gerby-project/gerby-project.github.io/blob/master/index.md They are now working with Jacob Lurie's (yet to unveil) project Kerodon.

    This is still a single narrative type of math writing.

     – liuyao May 23 '18 at 18:17
  • This is very interesting! Do you know anything more about Kerodon? Googling quickly, I see that he's gotten the logo sorted, but not much else is visible. – James Smith May 23 '18 at 19:24
  • Also, I see you have started your own project. Can you go into some more detail about what you hope to do with it? – James Smith May 23 '18 at 19:28
  • @JamesSmith, sorry I don't know anything about Kerodon. (I don't have the time or means to make my own project; I'm just trying out what others have set up. I used brilliant.org for a bit, but their active user base is too elementary. Now I use observablehq, which is for coding/data analysis/visualization. I think it has great potential, even as a model for some scientific publishing but perhaps not for math.) – liuyao May 24 '18 at 20:48
  • No worries about Kerodon. I'll keep a look out for it. The Observable stuff looks interesting, too. But I agree, probably not for maths. Although they list KaTeX as a dependency so I suppose you can do some maths with it. – James Smith May 24 '18 at 21:23
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What I think would be most useful to address this issue is improving the system to make requests and contributions to such software packages. I'm sure many of these systems would be happy to have more help with development.

For instance, I found that the Sage implementation of computation of zeta functions of graphs is horribly slow and I wrote a much faster implementation (using Sage), and I wanted give Sage my code so they could use it a subsequent release, but after looking at the amount of effort required to contribute code, I decided I didn't want to spend that much time on it. I was just hoping to submit my code with some comments, which an interested developer could revise to conform to standards, test and implement.

One example of something I would like implemented (say in Sage) is computation of $p$-adic integrals. E.g., given a compact open subgroup $K$ of $GL_2(\mathbb Q_p)$ and a character $\psi$ on the upper unipotent $N$, compute $\int_N 1_K(n) \psi(n) \, dn$. (Some simple cases like basic character sums might be implemented, but possibly they are only implemented mod $p$.) I once thought about trying to automate calculations for a highly computational project I had, but then decided it wasn't worth the development time for just that one project.

Kimball
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    William Stein was so put off by the Sage contribution standards, that he forked purplesage (psage) to be his own sandbox, with lower conformity. – ABCDveve Aug 08 '15 at 04:41
  • Do you refer to the Ihara zeta function ? You should at least have proposed your code to Sage, as this is rather easy. – F. C. Aug 08 '15 at 10:01
  • @F.C. Yes. Can you tell me how to do this? I had read this, which seems to require more than 10 minutes of effort. – Kimball Aug 08 '15 at 10:17
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  • get an account on trac.sagemath.org 2) open a ticket 3) attach your code to that ticket. Of course, this means that the real work is left to somebody else than you.
    • – F. C. Aug 08 '15 at 12:37
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    Code Standards are different from maintainer to maintainer, but as a professional software developer, I will tell you they absolutely exist for a reason. Relaxing code standards across the board purely for the sake of making it easier for amateur developers to make contributions would result in your software having more bugs than features. – nhgrif Aug 08 '15 at 12:49
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    @nhgrif I also believe in coding standards. I am just saying I would like the contribution system be improved to make it easier for mathematicians to contribute (which possibly includes a level of vetting by developers). Maybe already the system for this is good, but this wasn't clear from the documentation, so perhaps the documentation on this should at least be made more friendly for well-meaning but busy/lazy laymen. – Kimball Aug 08 '15 at 13:38
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    @Kimball: contributing to Sage is very easy. I highly recommend writing an email to sage-trac-account AT googlegroups DOT com as suggested on the trac page with your email id and a username of your choice! – knsam Aug 09 '15 at 18:57
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    Hi -- purple sage didn't work due to lack of time. Regarding dumping useful code on us, that we have to understand and get up to snuff for inclusion, if we had more time and bandwidth, then we would make that easy. However, we don't -- I have almost not grant support, and almost all effort is researchers in their spare time. Maybe if we someday have funding... – William Stein Aug 10 '15 at 19:37
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    From a Sage embarrassingly-non-expert: is it really true that the creator of Sage was so frustrated by contributing code to his own project that he had to make a private fork? – LSpice Aug 10 '15 at 20:37
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    @LSpice Obviously we should let William Stein comment himself, but my reading of the situation was that at the time (2010-ish), Sage had established enough momentum that it's future was assured, so he (Stein) could focus primarily on (software for) his own research area. – Gordon Royle Aug 12 '15 at 11:48