This morning, Deyan Ginev announced on the LaTeXML mailing list that the first alpha version of LaTeXML with LaTeX to EPUB support is now available. This is a very good news for people willing to encourage researchers to move from offline formats to more modern Web formats. Although, some people
had already been successful to combine LaTeX-to-XHTML converters
and XHTML-to-EPUB converters, this is the first tool that I'm aware of that can do the direct LaTeX to EPUB3 (XHTML+MathML) conversion. I already mentioned a couple of Gecko-based EPUB tools in my previous blog post, so let's have a look at three of them. Feel free to mention more Gecko-based EPUB tools in the comments, I'm particularly interested to hear about FirefoxOS applications that would be similar
to Apple's iBooks.

I have updated the LaTeXML samples based on Boris Zbarsky's thesis that we demonstrated at the Innovation Fairs in Santa Clara & Brussels. This shows how to generate the traditional PDF version, the Web version, the Web version with MathJax fallback and now the EPUB version! Here are some screenshots using the Firefox extension Lucifox:

Boris' Thesis in Lucifox ; page 2

Boris' Thesis in Lucifox ; page 4

I have intentionally not shown the diagram that are incorrectly converted by LaTeXML due to missing Xy-pic support (this is still in development). However,
Gecko supports mixing SVG and MathML via the foreignObject element so this would not be a problem for Gecko-based EPUB readers. Here are some screenshots of an ebook about
regular polygon that can be constructed with compass and straightedge that I have created with the help of itex2MML. They are viewed in EPUBReader which is another Firefox extension:

EPUBReader, Constructible Numbers

EPUBReader, Cyclic Galois Extension

Lucifox and EPUBReader have a big drawback: they do not support EPUB pages with the "scripted" property. This means that you can not use Javascript to create dynamic ebooks with live samples or interactive exercices... but this is one of the reason to use Web formats! Fortunately, there is a XUL application called AZARDI that supports this feature. I have created another ebook that shows an interactive
course on matrices. Click on the image to see the video on YouTube:

update 2013-10-15: since I got feedback, I have to say that my funding plan is independent of my work at MathJax ; I'm not a MathJax employee but I have an independent contractor status. Actually, I already used my business to fund an intern for Gecko MathML developments during Summer 2011 :-)

Retrospect

Since last April, I have been allowed by the MathJax Consortium to dedicate a small amount of my time to do MathML development in browsers, until possibly more serious involvement later. At the same time, we mentioned this plan to Google developers but unfortunately they just decided to drop the WebKit MathML code from Blink, making external contributions hard and unwelcome...

Hence I have focused mainly on Gecko and WebKit: You can find the MathML bugs that have been closed during that period on bugzilla.mozilla.org and bugs.webkit.org. For Gecko, this has allowed me to finish some of the work I started as a volunteer before I was involved full-time in MathJax as well as to continue to mentor MathML contributors. Regarding WebKit, I added a few new basic features like MathML lengths, <mspace> or <mmultiscripts> while I was getting familiar with the MathML code and WebKit organization/community. I also started to work
on <semantics> and <maction>.
More importantly, I worked with Martin Robinson to address the design concerns of Google developers and a patch to fix these issues finally landed early this week.

However, my progress has been slow so as I mentioned in my previous blog post, I am planning to find
a way to fund MathML developments...

Why funding MathML?

Note: I am assuming that the readers of this blog know why MathML is important and are aware of the benefits it can bring
to the Web community. If not, please check
Peter Krautzberger's Interview by Fidus Writer or the MozSummit MathML slides for a quick introduction.
Here my point is to explain why we need more than volunteer-driven
development for MathML.

First the obvious thing: Volunteer time is limited so if
we really want to see serious progress in MathML support we need to give a
boost to MathML developments. e-book publishers/readers, researchers/educators who are stuck outside the Web in a LaTeX-to-PDF world, developers/users of accessibility tools or the MathML community in general want good math support in browsers now and not to wait again for 15 more years until all layout engines catch up with
Gecko or that the old Gecko bugs are fixed.

There are classical misunderstandings from people thinking that non-native
MathML solutions and other polyfills are the future or that math on the Web could be implemented
via PNG/SVG images or Web Components.
Just open a math book and you will see
that e.g. inline equations must be correctly aligned with the text or
participate
in line wrapping. Moreover we are considering math on the Web not math on paper,
so we want it to be compatible with HTML, SVG, CSS, Javascript,
Unicode, Bidi etc and also something that is fast and responsive. Technically,
this means that a clean solution must be in the core rendering engine,
spread over several parts of the code and must have strong interaction with the
various components like the HTML5 parser, the layout tree,
the graphic and font libraries, the DOM module, the style tree and so forth.
I do not see any volunteer-driven
Blink/Gecko/WebKit feature off the top of my head that has this
characteristic and actually even SVG or any other kind of language for
graphics have less interaction with HTML than MathML has.

The consequence of this is that it is extremely difficult for volunteers
to get involved in native MathML
and to do quick progress because they have to understand
how the various components of the Blink/Gecko/WebKit code work and be sure to do
things correctly. Good mathematical rendering is already something hard by
itself, so that is even more complicated when you are not writing an isolated
rendering engine for math on which you can have full control.
Also, working at the Blink/Gecko/WebKit level requires technical skills above the average
so finding volunteers who can work with the high-minded engineers of
the big browser companies is not something easy.
For instance, among the enthusiastic people coming to me and
willing to help MathML in Gecko, many got stuck when e.g. they tried to build
the Firefox source or do something more advanced and I never heard back from
them.
In the other direction, Blink/Gecko/WebKit paid developers are generally
not familiar with
MathML and do not have time to learn more about it
and thus can not always provide a relevant review of the code, or they may
break something while trying to modify code they do not entirely understand.
Moreover,
both the volunteers and paid staff have only a small amount of time to do
MathML stuff while the other parts of the engine evolve very quickly,
so it's sometimes hard to keep everything in sync.
Finally,
the core layout engines have strong security requirements that are difficult to satisfy in a volunteer-driven situation...

Beyond volunteer-driven MathML developments

At that point, there are several options. First the lazy one: Give up
with native math rendering, only focus on features that have impact on the
widest Web audience (i.e. those that would allow browser vendors to get more market share and thus increase their profit), thank the math people for creating the Web and kindly ask them to use
whatever hacks they can imagine to display equations on the Web. Of course as a
Mozillian, I think people
must decide the Web they want and thus exclude this option.

Next there is the ingenuous option: Expect that browser companies
will understand the importance of math-on-the-Web and start investing
seriously in MathML support. However, Netscape and Microsoft
rejected the <MATH> tag from the 1995 HTML 3.0 draft and the browser
companies have kept repeating they would only rely on volunteer contributions
to move MathML forward, despite the repeated requests from MathML folks and other scientific communities. So that option is excluded too, at least in the short
to medium term.

So it remains the ambitious option: Math people and other interested parties
must get together and try to fund native MathML developments. Despite the effort
of my manager at MathJax to convince partners and raise funds, my situation has
not changed much since April and it is not clear when/if the MathJax Consortium
can take the lead in native MathML developments. Given my expertise
in Gecko, WebKit and MathML, I feel the duty to do something.
Hence I wish to reorganize
my work time: Decrease my involvement in MathJax core in order to increase
my involvement in Gecko/WebKit developments. But I need the help of the
community for that purpose. If you run a business with interest for math-on-the-Web
and are willing to fund my work, then feel free to contact me directly by
mail for further discussion. In the short term, I want
to experiment with
Crowd Funding as
discussed in the next section. If this is successful we can think
of a better organization for MathML developments in the long term.

Crowd Funding

Wikipedia defines
Crowd funding as
"the collective effort of individuals who network and pool their money, usually
via the Internet, to support efforts initiated by other people or organizations". There are several Crowd Funding platforms with similar rule/interface.
I am considering Catincan which is specialized in Open Source Crowd Funding, can be used by any backer/developer around the world, can rely on Bugzilla to track the bug status and
seems to have good process to collect the
fund from backers and to pay developers.
You can easily login to the Catincan Website if
you have a GitHub, Facebook or Google account (apparently
Persona is not supported yet...). Finally, it seems to have a communication interface between backers and
developers, so that everybody can follow the development on the funded
features.

One distinctive feature of catincan is that only well-established Open
Source projects can be funded and only developers from these projects can
propose and work on the new features ; so that backers can trust that the
features will be implemented. Of course, I have been working on Gecko, WebKit and
MathML projects
so I hope people believe I sincerely want to improve
MathML support in browsers and that I have the skills to do so ;-)

As said in my previous blog post, it is not clear at all (at least to me)
whether Crowd Funding can be a reliable method, but it is worth trying. There are
many individuals and small businesses showing interest in MathML, without
the technical knowledge or appropriate staff to improve MathML in browsers. So if each
one fund a small amount of money, perhaps we can get something.

One constraint is that each feature has 60 days to reach the
funding goal. I do not have any idea about how many people are willing
to contribute to MathML and how much money they can give.
The statistical sample of projects currently funded is too small to extract relevant
information. However, I essentially see two options:
Either propose small features
and split the big ones in small steps, so that each catincat submission
will need less work/money and improvements will be progressive with
regular feedback to backers ;
or propose larger features so they look more attractive and exciting to people
and will require less frequent submissions to catincat.
At the beginning, I plan to start with the former and if the crowd funding is
successful perhaps try the latter.

Status in Open Source Layout Engines

Note: Obviously, Open Source Crowd Funding does not apply
to Internet Explorer, which is the one main rendering engine not mentioned below. Although
Microsoft has done a great job on MathML for Microsoft Word, they did not
give any public statement about MathML in Internet Explorer and all the bug
reports for MathML have been resolved "by design" so far. If you are interested
in MathML rendering and accessibility in Internet Explorer, please check
Design Science blog for the latest updates
and tools.

Blink

Note: I am actually focusing on the history of Chromium here but of course there are other Blink-based browsers. Note that programs like QtWebEngine (formerly WebKit-based) or Opera (formerly Presto-based) lost the opportunity to get MathML support when they switched to Blink.

Alex Milowski and François Sausset's first MathML implementation did not
pass Google's security review. Dave Barton fixed many issues in that implementation and as far as I know, there were not any known security vulnerabilities when Dave submitted his last version. MathML was enabled in Chrome 24 but Chrome developers had some concerns about the design of the MathML implementation in WebKit, which indeed violated some assumptions of WebKit layout code. So MathML was disabled in Chrome 25 and as said in the introduction, the source code was entirely removed when they forked.

Currently, the Chromium Dashboard indicates that MathML is shipped in Firefox/Safari, has positive feedback from developers and is an established standard ; but the Chromium status remains "No active development".
If I understand correctly,
Google's official position is that
they do not plan to invest in MathML development but will accept external
contributions and may re-enable MathML when it is ready
(for some sense of "ready" to be defined).
Given the MathML story in
Chrome, it seems really unlikely that any volunteer will magically show up and be willing to
submit a MathML patch. Incidentally, note the
interesting work
of the ChromeVox team regarding MathML accessibility:
Their recent video
provides a good overview of what they achieve (where Volker Sorge politely regrets
that "MathML is not implemented in all browsers").

Although Google's design concerns have now been addressed in WebKit, one
most serious remark from one Google engineer is that the WebKit MathML implementation is
of too low quality to be shipped so they just prefer to have no MathML
at all. As a consequence, the best short term strategy seems to be improving
WebKit MathML support and, once it is good enough, to submit a patch to
Google. The immediate corollary is that if you wish to see MathML in Chrome
or other Blink-based browsers you should
help WebKit MathML development. See the next section for
more details.

chromatic

Actually, I tried to import MathML into Blink one day this summer. However,
there were divergences between the WebKit and Blink code bases that made that
a bit difficult. I do not plan to try again anytime soon, but if someone is
interested, I have published my script and patch on GitHub. Note there may be even more divergences now and the patch is
certainly bit-rotten. I also thought about creating/maintaining a "Chromatic"
browser (Chrome + mathematics) that would be a temporary fork to let Blink
users benefit from native MathML until it is integrated back in Blink. But
at the moment, that would probably be too much effort for one person and
I would prefer to focus on WebKit/Gecko developments for now.

WebKit

The situation in WebKit is much better. As said before, Google's concerns
are now addressed and MathML will be enabled again in all WebKit releases
soon.
Martin Robinson is interested in helping the MathML developments in
WebKit and his knowledge of fonts will be important to improve operator
stretching, which is one of the biggest issue right now.
One new volunteer contributor, Gurpreet Kaur, also started to
do some work on WebKit like support for the *scriptshifts
attributes or for the <menclose> element. Last but
not least, a couple of Apple/WebKit developers reviewed and accepted
patches and even helped to fix a few bugs, which made possible to move
development forward.

When he was still working on WebKit, Dave Barton opened bug 99623 to track the top priorities. When the bugs below and their related dependencies are fixed, I think the rendering in WebKit will be good enough to be usable for advanced math notations and WebKit will pass the MathML Acid 1 test.

Bug 44208:
For example, in expression like
$\mathrm{sin}\left(x\right)$,
the "x" should be in italic but not the "sin". This is actually slightly
more complicated: It says when the default mathvariant
value must be normal/italic.
mathvariant is more like
the
text-transform CSS property in the sense that it remaps
some characters to the corresponding mathematical characters (italic, bold, fraktur,
double-struck...) for example
$\mathfrak{A}$ (mathvariant=fraktur A)
should render exactly the same as $\U0001d504$ (U+1D504).
By the way, there is the related bug 24230 on Windows, that prevents to use plane 1 characters.
The best solution will probably be to
implement mathvariant correctly. See also Gecko's ongoing work by James Kitchener below.

Bug 99618: Implement <mmultiscripts>, that allows expressions like
${}_{6}{}^{14}\mathrm{C}$ or $R_{i}{}_{j}{}_{;}{}^{j}=\frac{1}{2}S_{;}{}_{i}$. As said in the introduction, this is fixed in WebKit Nightly.

Bug 99614: Support for stretchy operators like in
${\left(\frac{\overline{{z}_{1}+{z}_{2}}}{3}\right)}^{4}$. Currently,
WebKit can only stretch operators vertically using a few Unicode constructions
like ⎛ (U+239B) + ⎜ (U+239C) + ⎝ (U+239D) for the left parenthesis.
Essentially only similar delimiters like brackets, braces etc are supported.
For small
sizes like $(\text{}$ or for large operators like
$\sum {n}^{2}$ it is necessary to use non-unicode glyphs in various math fonts, but this
is not possible in WebKit MathML yet. All of this will require a fair amount of
work: implementing horizontal stretching, font-specific stuff,
largeop/symmetric properties etc

Bug 99620:
Implement the operator dictionary. Currently, WebKit treats all the operators the same way, so for
example it will use the same 0.2em spacing before and after parenthesis, equal sign or invisible
operators in e.g.
$f\left(x\right)={x}^{2}$. Instead it should use the information provided by the MathML operator dictionary. This dictionary also specifies whether operators are stretchy, symmetric or
largeop and thus is related to the previous point.

Bug 119038: Use the code for vertical stretchy operators to draw the radical symbols
in roots like $\sqrt{\frac{2}{3}}$. Currently,
WebKit uses graphic primitives which do not give a really good rendering.

Bug 115610: Implement <mspace> which is used by many MathML generators
to do some spacing in mathematical formulas. As said in the introduction, this is fixed in WebKit Nightly.

In order to pass the Mozilla MathML torture tests, at least displaystyle and scriptlevel must be implemented too, probably as internal CSS properties. This should also allow to pass
Joe Java's MathML test, although that one relies on the infamous <mfenced>
that duplicates the stretchy operator features and is implemented inconsistently
in rendering engines. I think passing the MathML Acid 2 test will require slightly more effort,
but I expect this goal to be achievable if I have more time to work on WebKit:

Bug 120164: Implement negative spacing for <mspace> (I have an experimental patch).

Bug 85730: Implement <mpadded>, which is also used by MathML generators to do some tweaking of formulas. I have only done some experiments, that would be a generalization of <mspace>

Bug 85733: Implement the href attribute ; well I guess the name is explicit enough to understand what it is used for! I only have some experimental patch here too. That would be mimicing what is done in SVG or HTML.

Bug 120059 and
bug 100626: Implement <maction> (at least partially) and <semantics>,
which have been asked by long-time MathML users Jacques Distler and Michael Kohlhase. I have patches ready for that and this could be fixed relatively soon, I just need to find time to finish the work.

In general passing the MathML Acid 2 test is not too hard, you merely only need to implement those few MathML elements whose exact rendering is clearly defined by the MathML specification. Passing the MathML Acid 3 test is not expected in the medium term. However, the score will
naturally increase while we improve WebKit MathML implementation. The priority
is to implement what is currently known to be important to users.
To give examples of bugs not previously mentioned: Implementing menclose or fixing various DOM issues like bugs 57695, 57696 or 107392.

More advanced features like those mentioned in the next section for Gecko
are probably worth considering later (Open type MATH, linebreaking,
mlabeledtr...). It is worth noting that Apple has already
done some work on accessibility (with MathML being readable by VoiceOver
in iOS7), authoring and EPUB (MathML is enabled in WebKit-based ebook
readers
and ibooks-author has
an integrated LaTeX-to-MathML converter).

Gecko

In general I think I have a good relationship with the Mozilla community and most people have respect for the work that has been done by volunteers for almost 15 years now. The situation has greatly improved since I joined the project, at that time some people claimed the
Mozilla MathML project was dead after Roger Sidge's departure.
One important point is that Karl Tomlinson has worked
on repairing the MathML support when Roger Sidge left the project. Hence
there is at least one Mozilla employee with good knowledge of MathML who
can review the volunteer patches. Another key ingredient is the work that has recently been made by Mozilla to increase engagement of the volunteer
community like good documentation on MDN, the #Introduction channel, Josh Matthews' mentored bugs and of course programs like GSOC. However, as said
above, it is one thing to attract enthusiastic contributors and another thing
to get long-term contributors who can work independently on more advanced features. So
let's go back to my latest Roadmap for the Mozilla MathML Project and see what has been accomplished for one year:

Font support: Dmitry Shachnev created a Debian package for the
MathJax fonts and Mike Hommey added MathJax and Asana fonts in the list
of suggested packages for Iceweasel. The STIX fonts have also been
updated in Fedora and are installed by default on
Mac OS X Lion (10.7). For Linux distributions, it would be helpful
to implement Auto Installation Support. The bug to
add mathematical fonts to Android has been assigned in June but no more progress has happened so far.
Henri Sinoven opened a bug for FirefoxOS but there has not been any progress there either.
I had some patches to restore the "missing MathML fonts" warning (using an information bar) but it was refused by Firefox reviewers. However, the code to detect missing MathML font could still be used for the similar bug 648548, which also seems inactive since January. There are still some issues on the MathJax side that prevent to integrate Web fonts for the native MathML output mode. So at the moment the solution is
still to inform visitors about MathML fonts or to add MathML Web fonts to your Web site. Khaled Hosny (font and LaTeX expert) recently updated my patches to prepare the support for Open Type fonts and he offered to help on that feature.
After James Kitchener's work on mathvariant, we realized that we will
probably need to provide Arabic mathematical fonts too.

Spacing: Xuan Hu continued to work on <mpadded> improvements and I think his patch is close to be accepted. Quentin Headen has done some progress on <mtable> before focusing on his InstantBird GSOC project. He is still far from being able to work on
mtable@rowspacing/columnspacing but a work around for that has been added
to MathJax. I fixed the negative space regression
which was missing to pass the MathML Acid 2 test and is used in MathJax. Again, Khaled Hosny is willing to help to use the spacing of the Open Type MATH, but that will still be a lot of work.

<mlabeledtr>: A work around for native MathML has been added in MathJax.

Linebreaking: No progress except that I have worked on fixing a bug with intrinsic width computation. The unrelated printing issues mentioned in the blog post have been fixed, though.

Operator Stretching: No progress. I tried to analyze the regression more carefully, but nothing is ready yet.

Tabular elements: As said above, Quentin Headen has worked a bit on cleaning up <mtable> but not much improvements on that feature so far.

Token elements: My patch for <ms> landed and I have done significant progress on the bad measurement of intrinsic width for token elements (however, the fix only seems to work on Linux right now). James Kitchener has taken over my work on improving our mathvariant support and doing related refactoring of the code. I am confident that he will be able to have something ready soon. The primes in exponents should render correctly with MathJax fonts but for other math fonts we will have to do some glyph substitutions.

Dynamic MathML: No progress here but there are not so many bugs regarding Javascript+MathML, so that should not be too serious.

Documentation: It is now possible to use MathML in code sample or
directly in the source code. The MathML project pages have been entirely migrated to MDN. Also, Florian Scholz has recently been hired by Mozilla as
a documentation writer (congrats!) and will in particular continue the work he started as a volunteer to document MathML on MDN.

I apologize to volunteers who worked on bugs that are not mentioned above or who are doing documentation or testing that do not appear here. For a complete list of activity since September 2012, Bugzilla is your friend. There are two ways to consider the progress above.
If you see the glass half full, then you see that several people have continued
the work on various MathML issues, they have made some progress and we now pass
the MathML Acid 2 test. If you see
the glass half empty, then you see that most issues have not been addressed
yet and in particular those that are blocking the native MathML to be enabled
in MathJax: bug 687807, bug 415413, the math font issues discussed in the first point, and perhaps linebreaking too. That is why I believe we should go beyond volunteer-driven MathML
developments.

Most of the bugs mentioned above are tested by the MathML Acid 3 tests and we will win a few points when they are fixed. Again, passing MathML Acid 3 test is not a goal by itself so let's consider what are the big remaining areas it contains:

Improving Tabular Elements and Operator Stretching, which are obviously important and used a lot in e.g. MathJax.

Linebreaking, which as I said is likely to become fundamental with small screens and ebooks.

Elementary Mathematics (you know addition, subtraction, multiplication, and division that kids learn), which I suspect will be important for educational tools and ebooks.

Alignment: This is the one part of MathML that I am not entirely sure is relevant to work on in the short term. I understand it is useful for advanced layout but most MathML tools currently just rely on tables to do that job and as far as I know the only important engine that implements that is MathPlayer.

Finally there are other features outside the MathML rendering engines that
I also find important but for which I have less expertise:

Transferring MathML that is implementing copy/cut/drag and paste. Currently, we can do that by treating MathML as normal HTML5 code or by using the "show MathML source" feature and copying the source code. However, it would be best to implement a standard way to communicate with other MathML applications like Microsoft Word, Mathematica, Mapple, Windows' Handwriting panel etc I wrote
some work-in-progress patches last year.

Authoring MathML: Essentially implementing things like deletion, insertion etc maybe simple MathML token creation ; in Gecko's core editor, which is used by BlueGriffon, KompoZer, SeaMonkey, Thunderbird or even MDN. Other things like integrating Javascript parsers (e.g. ASCIIMath) or equation panels with buttons like are probably better done at the higher JS/HTML/XUL level. Daniel Glazman already wrote math input panels for
BlueGriffon and
Thunderbird.

MathML Accessibility: This is one important application of MathML for which there is strong demand and where Mozilla is behind the competitors. James Teh started some experimental work on his NVDA tool before the summit.

EPUB reader for FirefoxOS (and other mobile platforms): During the
"Co-creating Action Plans" session, the Mozilla Taipei people were thinking
about missing features for FirefoxOS and this idea about EPUB reader was my modest contribution.
There are a few EPUB readers relying on Gecko and it would be good to check if they work in
FirefoxOS and if they could be integrated by default, just like
Apple has iBooks. BTW, there is a version of BlueGriffon that can edit EPUB books.

Conclusion

I hope I have convinced some of the readers about the need to fund MathMLin browsers. There is a lot of MathML work to do on Gecko and WebKit but both projects have volunteers and core engineers who are willing to help. There are also several individuals / companies relying on MathML support in rendering engines for their projects and could support the MathML developments in some way. I am willing to put more of my time on Gecko and WebKit developments, but I need financial help for that purpose. I'm proposing catincan Crowd Funding in
the short term so that anyone can contribute at the appropriate level, but other alternatives to fund the MathML development can be found like asking Peter Krautzberger about native MathML funding in MathJax,
discussing with Igalia about funding Martin Robinson to work more on WebKit
MathML or contacting me directly to establish some kind of part-time
consulting agreement.

Please leave a comment on this blog or send me a private mail, if you
agree that funding MathML in browsers is important, if you like the crowd funding idea and plan to contribute ; or if you have any opinions about alternative funding options. Also, please tell me what seem to be the priority for you and
your projects among what I have mentioned above
(layout engines, features etc) or among others that I may have forgotten. Of course,
any other constructive comment to help MathML support in browsers is welcome. I plan to submit features on catincan soon, once I have more feedback on
what people are interested in. Thank you!

I'm back from a great Mozilla Summit 2013 and I'd just like to write
a quick blog post about the MathML booths at the Innovation Fairs.
I did not have the opportunity to talk with the MathML people who
ran the booth at Santa Clara yet. However, everything went pretty
well at Brussels, modulo of course some demos failing when done in
live... If you are interested,
the slides and other resources are available on my GitHub page.

Many Mozillians did not know about MathML or that it
had been available in Gecko since the early days of the Mozilla project.
Many people who use math (or just knowing someone who does)
were curious about that feature and excited about the MathML potentials.
I appreciated to get this positive feedback from Mozillians willing
to use math on the Web and related media, instead of the scorn or hatred
I sometimes see by misinformed people. I expect to provide more
updates on LaTeXML, MediaWiki Math and MathJax when their next versions
are released. The Gecko MathML support improves slowly but there has
been interesting work by James Kitchener recently that I'd like to
mention too.

Let's do an estimation
à la Fermi:
only a few volunteers have been contributing
regularly and simultaneously to MathML in Gecko while most
Mozilla-funded Gecko projects have certainly development teams that are
3 times as large. Let's be optimistic and assume that these
volunteers have been able to dedicate a mean of 1 work day
per week, compared to 5 for full-time staff.
Given that the Mozilla MathML project will celebrate its 15 years
next May, that means that the volunteer work
transposed in terms of paid-staff time is only
$\le \frac{15}{3\cdot 5}=1$
year. To be honest, I'm disregarding here the great work made by the
Mozilla NZ team around 2007 to repair MathML after the Cairo migration.
But still, what we have achieved in quality and
completeness with such limited resources and time is really impressive.

As someone told me at the MathML booth, it's really frustating
that something that is so important for the small portion of
math-educated people
is ignored because it is useless for the vast majority of people. This
is not entirely true, since even elementary mathematics taught at school
like the one of
this blog post are not easily expressed with standard HTML
and even less in a way accessible to people with visual
disabilities. However, it summarizes well the feeling MathML folks had
when they tried to convince
Google to
accept the volunteer work on MathML, despite its low quality.

As explained at the Summit Sessions, Mozilla's mission is different and
the goal is to give people the right to control the Web they want.
The MathML project is perhaps one of the oldest and successful
volunteer-driven Mozilla project that is still active and
demonstrates concretely the idea of the Mozilla's mission with e.g. the
work of Roger Sidge who started to write the MathML
implementation when Netscape opened its source code or the one of
Florian Scholz who made MDN one of the most complete Web resource for
MathML.

Mozilla Corporation has kept saying they don't want to invest in MathML developments and the focus right now is clearly on other features like FirefoxOS. Even projects that have a larger audience than the MathML support like the mail client or the editor are not in the priorities so someone else definitely need to step in for MathML. I've tried various methods, with more or less success, to boost the MathML developments like mentoring a GSoC project, funding a summer internship or relying on mentored bugs. I'm now considering crowd funding to help the MathML developments in Gecko (and WebKit). I don't want to do another Fermi estimation now but
at first that looks like a very unreliable method. The only revenue generated by the MathML project so far are the
$2\frac{\lfloor 100\cdot \pi \rfloor}{100}=2\cdot 3.14=6.28$ dollars to the Mozilla Fundation via contributions to
my MathML-fonts add-on, so it's hard to get an idea of how much people would contribute
to the Gecko implementaton.
However, that makes sense since the only people who showed interest
in native MathML support so far are individuals or small businesses
(e.g. working on EPUB or accessibility) and I think it's worth
trying it anyway. That's definitely
something I'll consider after MathJax 2.3 is
released...

Last November, I tried to provide
some details
of the proof given in chapter 7,
regarding the fact that the continuum hypothesis implies the
existence of a Ramsey ultrafilter.
Peter Krautzberger
pointed out that the proof could
probably work assuming only Martin’s Axiom.
This was indeed proved by Booth in 1970 and the missing argument is actually
given in exercise 16.16. For completeness, I copy the details on this blog
post.

Remember that the proof involves contructing a sequence
${(X_{\alpha})}_{{\alpha<2^{{\aleph_{0}}}}}$ of infinite subsets of $\omega$.
The induction hypothesis is that at step $\alpha<2^{{\aleph_{0}}}$,
for all $\beta_{1},\beta_{2}<\alpha$ we have $\beta_{1}<\beta_{2}\implies X_{{\beta_{2}}}\setminus X_{{\beta_{1}}}$ is finite.
It is then easy to show the result for the successor step,
since the construction satisfies
$X_{{\alpha+1}}\subseteq X_{\alpha}$. However at limit step, to ensure that
$X_{\alpha}\setminus X_{\beta}$ is finite for all $\beta<\alpha$, the proof
relies on the continunum hypothesis. This is the only place where it is used.

Assume instead Martin’s Axiom and consider a limit step
$\alpha<2^{{\aleph_{0}}}$. Define the forcing notion
$P_{\alpha}=\{(s,F):s\in{[\omega]}^{{<\omega}},F\in{[\alpha]}^{{<\omega}}\}$
and $(s^{{\prime}},F^{{\prime}})\leq(s,F)$ iff
$s\subseteq s^{{\prime}}$, $F\subseteq F^{{\prime}}$ and $s^{{\prime}}\setminus s\subseteq X_{\beta}$
for all $\beta\in F$.
It is clear that the relation is reflexive and antisymmetric. The transitivity
is almost obvious, just note that if $(s_{3},F_{3})\leq(s_{2},F_{2})$ and
$(s_{2},F_{2})\leq(s_{1},F_{1})$ then
for all $\beta\in F_{1}\subseteq F_{2}$ we have
$s_{3}\setminus s_{1}\subseteq s_{3}\setminus s_{2}\cup s_{2}\setminus s_{1}%
\subseteq X_{\beta}$.

The forcing notion satisfies ccc or even property (K):
since ${[\omega]}^{{<\omega}}$ is countable,
for any uncountable subset $W$ there is $t\in{[\omega]}^{{<\omega}}$ such
that $Z=\{(s,F)\in W:s=t\}$ is uncountable. Then any
$(t,F_{1}),(t,F_{2})\in Z$ have a common refinement $(t,F_{1}\cup F_{2})$.

For all $n<\omega$, define $D_{n}=\{(s,F):|s|\geq n\}$.
Let $\beta_{1}>\beta_{2}>...>\beta_{k}$ the elements of $F$.
We show by induction on $1\leq m\leq k$ that $\bigcap_{{i=1}}^{m}X_{{\beta_{i}}}$
is infinite. This is true for $m=1$ by assumption. If it is true for
$m-1$ then

The left hand side is infinite by induction hypothesis. The second term
of the right hand side is included in $X_{{\beta_{1}}}\setminus X_{{\beta_{m}}}$ and thus
is finite. Hence the first term is infinite and the result is true for $m$.
Finally, for $m=k$, we get that
$\bigcap_{{\beta\in F}}X_{\beta}$ is infinite. Pick $x_{1},x_{2},...,x_{n}$ distinct
elements from that set and define $(s^{{\prime}},F^{{\prime}})=(s\cup\{x_{1},...x_{n}\},F)$.
We have $(s^{{\prime}},F^{{\prime}})\in D_{n}$, $s\subseteq s^{{\prime}}$, $F\subseteq F^{{\prime}}$ and
for all $\beta\in F$,
$s^{{\prime}}\setminus s=\{x_{1},...x_{n}\}\subseteq X_{\beta}$. This shows that
$D_{n}$ is dense. For each $\beta<\alpha$,
the set $E_{\beta}=\{(s,F):\beta\in F\}$ is also dense:
for any $(s,F)$ consider $(s^{{\prime}},F^{{\prime}})=(s,F\cup\{\beta\})$.

By Martin’s Axiom there is a generic filter $G$ for the family
$\{D_{n}:n<\omega\}\cup\{E_{\beta}:\beta<\alpha\}$ of
size $|\alpha|<2^{{\aleph_{0}}}$.
Let $X_{\alpha}=\{n<\omega:\exists(s,F)\in G,n\in s\}$.
For all $n<\omega$, there is $(s,F)\in G\cap D_{n}$ and so
$|X_{\alpha}|\geq|s|\geq n$. Hence $X_{\alpha}$ is infinite. Let $\beta<\alpha$
and $(s_{1},F_{1})\in G\cap E_{\beta}$.
For any $x\in X_{\alpha}$, there is $(s_{2},F_{2})\in G$ such that $x\in s_{2}$.
Hence there is $(s_{3},F_{3})\in G$ a refinement of $(s_{1},F_{2}),(s_{2},F_{2})$.
We have $x\in s_{2}\subseteq s_{3}$ and $s_{3}\subseteq s_{1}\subseteq X_{\beta}$.
Hence $X_{\alpha}\setminus X_{\beta}\subseteq s_{1}$ is finite and the induction
hypothesis is true at step $\alpha$.