Ideas for math tools
Ideas for math tools
Mathematicians have some great software available. Even domain-specific systems like Macaulay2, a programming language for commutative algebra and algebraic geometry. And then there are some amazing projects like the OEIS that, in addition to maintaining a huge database of integer sequences, has a search engine for integer sequences and a database of “facts” and relationships between sequences.
But I think there’s a lot more cool stuff, on a smaller and more tractable scale, that could be done today if someone put in the time and energy. Here are some random ideas in that vein. Apologies if it’s been done, I’d love to hear from you with references.
An open problem tracker. Today open problems are tracked entirely manually, with conjectures scattered through papers and no clear indication of whether a problem has been solved. OEIS’s “under the fold” submissions with facts about particular integer sequences is probably the closest thing I’ve seen to a structured database of open conjectures and results, and it’s not all that structured. It would be cool to have a stackexchange-type website where each entry is a problem statment and it can be solved when a paper is published that claims to solve it, and with sufficient time and corroboration. Imagine it also has crossreferences to variants, generalizations, specializations, and updates on progress and retracted attempts.
A (good) search engine for math.
This one has been tried.
SearchOnMath searches a restricted portion of the web
like MathOverflow and Wikipedia, and includes formula support,
as does this page.
However, nothing seems to properly search even arXiv papers,
not even arXiv’s search engine.
When I try to search for the term qLTC conjecture,
a fairly distinctive term which appears in this recent paper,
and arXiv returns a single hit for this paper
which does not contain the term at all.
Nobody seems to have managed to handle
even basic keyword searching in a corpus of PDFs,
except for Google, of course.
I see a space for a search engine whose index
includes all major preprint servers,
plus (configurably) MathOverflow, Wikipedia, math blogs, etc.,
and indexes PDFs properly,
and has decent formula support.
A math-specific blogging platform. Imagine a Wordpress.com clone (hosted blogging platform) with native MathJAX support, conversion to/from LaTeX, import/export to/from other blogging platforms, special features for the math community like tooling for collaborative problem solving (a la Polymath projects), tools for converting blog posts into books, citation management/tracking, a nice front page.
A PDF reader that defines things you click on. I got this idea from this tweet where a neural network attempts to explain what highlighted code is doing in English. Imagine being able to click on a symbol in a math paper, and based both on the paper’s earlier text and the general background of mathematical discourse, it could say something like, “N (contextually usually an integer), is defined in Sec 3.2 as 2^B, where B is the dimension of a blah blah blah.” It could the placement of a symbol in a formula, and even arXiv classification labels, to specialize its “contextually” phrases to what’s commonly used in the subfield. If the symbol is “epsilon” and it’s in an inequality in a paper that uses words like “tight” and “bound”, it might say , “epsilon, often a small positive constant in analysis, …” vs if it shows up in a PL-type computer science paper with words like “string” and “alphabet” it woudl say “epsilon, often the empty string in compilers, …”). People sometimes annotate their papers with colors or even have online versions of their papers with tooltips that hover with reminders of the definition of each symbol. This would make that automatic, and retroactively work on papers without such fancy metadata. It’s probably not a game changer, but it would be a nice aid to reading a symbol-heavy or context-heavy paper. Deep learning should be up to this task by now. Also a risk that having this widespread would just enable the poor writers to write even MORE symbol heavy, inscrutable papers.
Dedicated tool for counterexample search. Most mathematicians I talk to who use computers for their research tell me they use it mainly as a quick counterexample search when they have a testable conjecture. Sometimes they use dedicated tooling specific to their field (e.g., Macaulay2), but most of the time they hack together something in a general purpose language like Mathematica or Sage or just vanilla Python. It seems like there would be an appetite for a tool more specifically suited to the problem of searching for counterexamples. Either by having some means to accelerate the search (e.g., making it faster, and using unbounded arithmetic by default), or parallelize the search, and having some means to record the results in a verifiable or reproducible manner.