Mathematical Information II

Introduction

This is the second post about Mathematical Information inspired by talks the AMS meeting in Seattle in January, 2016. The first post was Mathematical Information I. That post covered, among other things, types of explanations.

In this post as in the previous one, footnotes link to talks at Seattle that inspired me to write about a topic. The speakers may not agree with what I say.

The internet

Math sources on the internet

Publishing math on the internet

  • Publishing on the internet is instantaneous, in the sense that once it is written (which of course may take a long time), it can be made available on the internet immediately.
  • Publishing online is also cheap. It requires only a modest computer, an editor and LaTeX or MathJax, all of which are either free, one-time purchases, or available from your university. (These days all these items are required for publishing a math book on paper or submitting an article to a paper journal as well as for publishing on the internet.)
  • Publishing online has the advantage that taking up more space does not cost more. I believe this is widely underappreciated. You can add comments explaining how you think about some type of math object, or about false starts that you had to abandon, and so on. If you want to refer to a diagram that occurs in another place in the paper, you can simply include a copy in the current place. (It took me much too long to realize that I could do things like that in abstractmath.org.)

Online journals

Many new online journals have appeared in the last few years. Some of them are deliberately intended as a way to avois putting papers behind a paywall. But aside from that, online journals speed up publication and reduce costs (not necessarily to zero if the journal is refereed).

A special type of online journal is the overlay journalG. A paper published there is posted on ArXiv; the journal merely links to it. This provides a way of refereeing articles that appear on ArXiv. It seems to me that such journals could include articles that already appear on ArXiv if the referees deem them suitable.

Types of mathematical communication

I wrote about some types of math communication in Mathematical Information I.

The paper Varieties of Mathematical Prose, by Atish Bagchi and me, describes other forms of communicating math not described here.

What mathematicians would like to know

Has this statement been proved?G

  • The internet has already made it easier to answer this query: Post it on MathOverflow or Math Stack Exchange.
  • It should be a long-term goal of the math community to construct a database of what is known. This would be a difficult, long-term project. I discussed it in my article The Mathematical Depository: A Proposal, which concentrated on how the depository should work as a system. Constructing it would require machine reading and understanding of mathematical prose, which is difficult and not something I know much about (the article gives some references).
  • An approach that would be completely different from the depository might be through a database of proved theorems that anyone could contribute to, like a wiki, but with editing to maintain consistency, avoid repetition, etc.

Known information about a conjecture

This information could include partial results.G An example would be Falting’s Theorem, which implies a partial result for Fermat’s Last Theorem: there is only a finite number of solutions of $x^n+y^n=z^n$ for integers $x, y, z, n$, $n\gt2$. That theorem became widely known, but many partial results never even get published.

Strategies for proofs

Strategies that are useful in a particular field.

The website Tricki is developing a list of such strategies.

It appears that Tricki should be referred to as “The Tricki”, like The Hague and The Bronx.

Note that there are strategies that essentially work just once, to prove some important theorem. For example, Craig’s Trick, to prove that a recursively enumerable theory is recursive. But of course, who can say that it will never be useful for some other theorem? I can’t think of how, though.

Strategies that don’t work, and whyG

The article How to discover for yourself the solution of the cubic, by Timothy Gowers, leads you down the garden path of trying to “complete the cubic” by copying the way you solve a quadratic, and then showing conclusively that that can’t possibly work.

Instructors should point out situations like that in class when they are relevant. A database of Methods That Work Here But Not There would be helpful, too. And, most important of all, if you run into a method that doesn’t work when you are trying to prove a theorem, when you do prove it, mention the failed method in your paper! (Remember: space is now free.)

Examples and Counterexample

I discovered these examples in twenty minutes on the internet.

Discussions

“Mathematical discussion is very useful and virtually unpublishable.”G But in the internet age they can take place online, and they do, in discussion lists for particular branches of math. That is not the same thing as discussing in person, but it is still useful.

PolymathG

Polymath sessions are organized attempts to use a kind of crowdsourcing to study (and hopefully prove) a conjecture. The Polymath blog and the Polymath wiki provide information about ongoing efforts.

Videos

  • Videos that teach math are used all over the world now, after the spectacular success of Khan Academy.
  • Some math meetings produce videos of invited talks and make them available on You Tube. It would be wonderful if a systematic effort could be made to increase the number of such videos. I suppose part of the problem is that it requires an operator to operate the equipment. It is not impossible that filming an academic lecture could be automated, but I don’t know if anyone is doing this. It ought to be possible. After all, some computer games follow the motions of the player(s).
  • There are some documentaries explaining research-level math to the general public, but I don’t know much about them. Documentaries about other sciences seem much more common.

References

The talks in Seattle

  • List of all the talks.
  • W. Timothy Gowers, How should mathe­matical knowledge be organized? Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.
  • Mathematical discussions, links to pages by Timothy Gowers. “Often [these pages] contain ideas that I have come across in one way or another and wish I had been told as an undergraduate.”
  • Colloquium notes

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One thought on “Mathematical Information II”

  1. Another well-known book of counterexamples is “Counterexamples in Topology” by Steen & Seebach Jr, also published by Dover.

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