which wasn't so much about math provers, but about why category theory matters so much right now. And, as usual with Michael, how to study the `relationship between abstract knowledge and the modern world'.
Yes, we already had Jeremy Avigad, Kevin Buzzard and Johan Commelin talking about LEAN in 2022, but it was good to hear from the main creator of the Lean prover too!
(He even talked on the 7th September, a Brazilian National Holiday, and neither of us noticed it!)
Of course these last three talks say more about how I feel about the role of mathematics in the world than about how we can use provers to do more mathematics. But I think this is essential too!
I have not touched my blog much this last year. A shame. I need to write more and I need to try to correct pictures and links that stopped working. Now the programme in Bonn, on Prospects of Formal Mathematics is coming closer, so writing should help me decide to organize my time in Germany. But first things first. I need to email a few more people about our arrangements.
My Ada Lovelace heroine of 2023 is Andrée C. Ehresmann from the Université de Picardie Jules Verne
(Photo by Rabatakeu CC BY-SA 3.0, Wikimedia)
"Better late, than never," I say. Yes, Ada Lovelace Day (ALD) is in October, and it varies a little each year, a convenient leeway, for chronically late people, like myself. This year I have less of an excuse, as I had decided to honour Andrée in July already. But the year was hectic, in a good way, so I didn't get around to posting my ALD "hommage" so far. But today is the last day of the year, so here it goes, before the clock chimes, and we change back into pumpkins.
Andrée has been a role model for me in many ways. She's been a mathematician for longer than I have been alive, and I am very old indeed. She has this awesome description of her research work in her webpage. I share with her a diversified academic career (I will try to imitate the lovely graph below) and a love for category theory and its explanatory power.
I have meant to write about Andrée before. In 10/27/2016 I posted a message of Marta Bunge's in the blog "Women in Logic". Like today's post, that one was hastily written, rushing to some function or something else. But at least it was written and records my indebtedness to Marta and her friendship to me and to Andrée. So I could write lots about Andrée's work, but you can look at her pages, the "Cahiers" work for more than 40 years, and the lovely diagram above. So I'd rather leave you with Marta's words in 2016 and a picture of Marta, since I have not mourned her absence enough. The volume celebrating her work is now almost totally ready, so I will be writing more about Marta soon. She wrote:
"There is recently an article in the journal Nature about
the presence of women in science and on how to improve what seems to be
an unfair situation, which I believe. I hope that someone more
qualified than I can do this since I believe it is important for our
field. Without a shadow of a doubt I consider Andree C. Ehresmann as
someone who has done a lot for our field - not just by continuing the
work of Charles Ehresmann by publishing his Oeuvres Completes and
continuing with the journal Cahiers de Topologie et Geometrie
Differentielle (Categoriques), but by encouraging many researchers in
our field - both men and women. She certainly deserves mention as a
woman and a mathematician. I leave you with that. All the best."
The reason for this blog post is to find easily a presentation of mine that has not been recorded.
I have discussed this subject at least three times already: once in the Topos Institute internal seminar,
once with a physicist friend interested in information theory and mathematics (Simone Severini), and once in a talk for the Brazilian UD (Universal Dependencies) group. This last occasion has produced the slide deck at "Networked Mathematics: NLP tools for Better Math", May 2023.
Now Networked Mathematics is a big problem. I have talked about it in several blog posts in the Topos blog, namely:
One of our partial approaches for solving the problem is the idea of recognizing and extracting mathematical concepts from mathematical text. That means using NLP technologies for the specific domain of Mathematics.
But I have been working on other, related solutions to the problem, discussed in the presentation above. I hope to write more about these other solutions soon.
Now, because this reminded me of the Math Genealogy, I cannot resist posting this other picture. Of course there are several ways to connect a mathematician to Paul Erdős and some are nicer than others. For many years the AMS tool gave me as my shortest connection to Erdős the one via Vaughan Pratt and Frances Yao. Now it is given me the one below, which I think it is even nicer.
This year we’re having the7th Women in Logic (WiL) Workshop in Rome, associated with FSCD and CADE. Considering that we never had grants to help us defray costs of the workshop, I think getting to our seventh anniversary is a big milestone. Everyone who works for Women in Logic does it for free and because they find the cause important. You may be wondering, “what is Women in Logic?”
Women, like some other demographics groups, are very badly represented in STEM (Science, Technology, Engineering and Mathematics) subjects. While on paper we have the right to be part of our STEM communities, in reality the obstacles are very strong and sometimes even insurmountable for women participation in Academia. The infamous “glass ceiling” is still very much with us and many of our colleagues do not notice it. (ha, this is why it’s called a “glass” ceiling, right?) Even the colleagues that are notionally aware of the issues, many times do not see them, when they happen in their own lives. The sexism in STEM is systemic and entrenched, it affects both men and women. And it is not a small action like WiL that will defeat it, but WiL does help, a little, to shed some light on some of the issues.
Women in Logic (WiL) started for me in 2013, whenOrna Kupferman, who was the Program Chair of LICS that year, presented her LICS chair findings and showed this slide:
Until then, while I knew about all the prevalent sexism in society, (it’s hard to ignore the many manifestations, from murdered women to silenced ones) I thought that my community was a bit better than most. My colleagues seemed to me more helpful and more enlightened. Thus when I saw the numbers, it was a shock: how come there were no awards for women? How come there were no women on the advisory board(*)? how come only 3.7% of the submissions in 2013 were by women? It all did seem TERRIBLE!
Much more disturbing was the discovery that, despite many efforts, things were not getting better, they were even getting worse! A representative graph was provided by the AAUW (American Association of University Women) in their report entitled “Solving the Equation”, 159 pages of dismaying facts. As you can see below, while the numbers of women in Mathematics and Computer Science improved from 1960 till around 1990, from then on they dropped, being in 2013 lower than they were in 1960! Engineering, traditionally considered one of the worst areas for women, at least has increasing numbers, but from 9% in 1990 to 12% in 2013. This is a minuscule improvement, leading to, in general, the prediction that at the current rate of progress, the pay gap will not close until 2111.
So I started to think about how to do something about this sad state of affairs. Three years later, at LICS2016,Nat Shankarasked me if I wanted to say a few words about LICS and I said I’d love to, but he might not be pleased withwhat I had to say. Having talked about how bad things were in the LICS community, I decided to organize thefirst Women in Logic Workshop, which happened as an associated workshop to LICS in Reykjavik, Iceland the next year. It was easy to find several friends who wanted to do this too, so our collective was formed.
Since then the Workshop has occurred inOxford 2018 (UK),Vancouver 2019 (CA),Paris 2020 (online),Rome 2021 (online)andHaifa, 2022 (Israel). By most accounts the workshops have been very successful. The invited speakers, so far always two in each workshop, have been excellent and they showcased many different facets of logic in computing, mathematics and philosophy: from Software Engineering to Automated Reasoning to Computational Biology and Artificial Intelligence.
We have received some small grants from the Special Interest Group of the ACM for Logic (SIGLOG), the Vienna Center for Logic and Algorithms (VCLA), the Institute of Logic, Language and Computation of the University of Amsterdam (ILLC).These allowed us to help students and young researchers attend our workshop. We also had some donations to help us to organize dinners after the workshop in Oxford and Haifa, fromProf Ursula Martin(EPSRC and Wadham College) and from Microsoft Outreach, viaNikolaj Bjorner, to whom we’re also very grateful.
(Dinner at Haifa, 2022, with Alexandra Silva speaking)
Most recently we have also received important clerical support from theTopos Institute, to process these smaller grants. More substantial effort from the Topos Institute came in the shape of the help to establish aWomen in Logic website. Producing this website has been an important step to consolidate the several activities of the collective. Many thanks are due to, especially Juliet Szatko and Tim Hosgood, for all the help provided!
After the worst (we hope) of the COVID19 pandemic, we are now in the process of restructuring and rethinking how to make the collective Women in Logic more efficient. We have several social media channels, managed by several members of the original group. So we are present on the web (Women in Logic), on Facebook (Women on Logic group), on Twitter ([@WomeninLogic1](https://twitter.com/WomenInLogic1)) and on Slack (wilcrew.slack.com). We also have our blog (womeninlogic.blogspot.com) and aYouTube channel, which we haven’t, as yet, populated much. Instead we have, so far, collected already recorded talks by women in logic and posted them on our GitHub (womeninlogic.github.io/RecordedTalks/).
But much remains to be done and we want to count on all of you, women in STEM and allies, to help change the terrible state that we still find ourselves in. Because as we know is not
Nature or nurture? No, it’s bias
Researchshows that there is no inherent difference in math and science capability between girls and boys. It’s also a myth that girls aren’t interested in science: In elementary, middle, and high school, girls and boys take math and science courses inroughly equal numbers, except in engineering and AP computer science, according to the National Science Foundation.One studyfound that the apparent gender gap in mathematics is smaller in countries with greater gender equality, suggesting that gender differences in math are largely due to cultural and environmental factors, not ability.
Networked mathematics is one of my main projects at the Topos Institute (the other isDialectica categoriesand its extended family of formalisms). We have not had much luck getting the funding agencies to pay attention to it, yet. But the work is progressing, albeit slowly.
We have been working on the suggestion thatformalized mathematicswill eventually be one of the main ways of producing mathematics. It seems sensible to believe that proof assistants will become really useful tools for mathematicians, and if so, formalized math will become the usual practice, not the exception.
To discuss the issues of formalized mathematics and to prepare for the Hausdorff Institute of Mathematics trimester on“Prospects of Formal Mathematics”in 2024 we have had in the Topos Colloquium talks by several people involved with proof assistants and automated deduction.
I worked for Larry as a postdoc in the Computer Lab, Cambridge University many moons ago, which is possibly the reason I am so fascinated by formalized mathematics. His blog,Machine Logicis super interesting and I always loved the idea thatIsabellewas not conceived as a single logic, but as a generic logical framework, and it can represent reasoning in several different systems. (Together withSara KalvalaI did some small experiments inLinear Logic in Isabelle.) I still want to develop the ideas of a projectI wrote with Larry and Prof Roger Needhamon modal logics for authentication and privacy, but it is getting late.
The vast library of mathematical results in Isabelle is called AFP, the Archive of Formal Proofs, or as they say, more precisely: A vast collection of Isabelle examples and applications is available from theArchive of Formal Proofs.
August 2021
We hadKevin Buzzardtalking to us aboutWhat is the point of Lean’s maths library?Kevin’s style of online demo and talk was a huge success with the audience of applied category theorists, which helped to consolidate my ability to invite other speakers not exactly categorically minded. Kevin has this awesome sentence on his website:
We hadFlorian Rabetalking aboutMMT: A UniFormal Approach to Knowledge Representation. Florian is interested in the more general problem of implementing “a universal framework for the formal representation of knowledge”, not only mathematical knowledge, but also other generic computing systems. His system is described inhttps://uniformal.github.io/. Florian’s system MMT has been in development since 2006 and he has worked withProf. Dr. Michael Kohlhasein several versions of knowledge management atKWARC, Erlangen, Germany. One of Florian’s main goals is to obtain foundation-independent mechanized results, thus much effort is spent on languages and meta-languages. (I also worked with Florian and other colleagues on extending the Curry-Howard correspondence toGoguen’s institutionsa long time ago.)
Christoph Benzmuellertalked to us aboutLogico-pluralistic exploration of foundational theories with computers. Chris discussed his work on formalization of logic, particularly in formalizing computational metaphysics (instead of mathematics) with collaborators, includingDana Scott, Topos’ senior advisor. Personally I am very interested in Chris’ work on modal logic, also in the benchmarking of modal theorems, less so in God’s ontological existence arguments.
July 2022
Josef Urbantalked aboutCombining learning and deduction over formal math corporaand his work on combining deep learning with automated deduction. Josef, unlike many others, has an agnostic attitude to the different logical foundations and merits of different provers and implementations, so it’s interesting to discuss those kinds of issues with him.
We hadAngeliki Koutsoukou Argyrakitalking about the ProjectAlexandriain Cambridge (the one on mathematical knowledge, not theMicrosoft oneon business knowledge). Her talk was entitledThe new era of formalised mathematics and the ALEXANDRIA Project. This was very exciting because the project has a strong component in the direction we are working, that is the direction to do withmathematical language.Mathematical English poses special challenges for NLP (Natural Language Processing) tools.
So slowly, but surely we’re investigating the issues of formalization of mathematics, as well as its affordances. I hope we will continue to do so in the near future, as there are many loose ends that need tying up.
Goguen, J.A., Mossakowski, T., De Paiva, V., Rabe, F. and Schröder, L., 2007. An Institutional View on Categorical Logic.Int. J. Softw. Informatics,1(1), pp.129-152.
In the first half of the year we had an AMS Mathematical Research
Community (MRC) on Applied Category Theory, in Beaver Hollow, NY. John
Baez blogged about it in “Learn Applied Category
Theory!”
Here’s a poster advertising the whole program.
We also had a paper in the Notices of American Mathematical Society,
describing our proposal:
An MRC is a bit like a version “on steroids” of the
ACT Adjoint School, as there are many
more researchers in a research subgroup in the MRC than in a Adjoint
School group. While my ACT Adjoint School group had four students and a
TA, my MRC group had thirteen researchers. So from four to thirteen
people you have more than three times the number of people to do
interesting research with. Plenty of fun, but it can be unsettling to
think about research problems for so many people you don’t know,
especially for me (well, I haven’t had that much experience directing
other people’s research).
I was very worried to begin with, but when we got going, it was
exhilarating. Great fun to discuss serious math problems with people
interested in similar problems, with similar tastes to yours and with
plenty of ideas of their own. The program had three mentors (John, Nina
and myself), and I ended up with a group of 13, subdivided into 4
smaller groups. Things were going very well (you can check some of the
discussions in the Zulip channel [practice:Dialectica])
until two days before the meeting in upper New York state I got ill, and
ended up in the ER in the Stanford hospital. Twice.
I couldn’t go to the research meeting in Beaver Hollow, but
Jérémie Koenig,
my trusted deputy, managed to get everyone working well together into
the four groups we had established before:
The neat thing about MRCs is that they give the participants some funds
to make academic visits and I was very touched when my group decided to
come and visit me in California, using their funds. This meant that we
had another smaller meeting in Cupertino, in the middle of the Summer.
Well, it was already September and usually the weather over here is good
most of the time, but we had some bad luck, and this week was a
heatwave, the worst one this summer. But we had some fun. Now the
problem is reworking what happened in the many discussions.
A third component of the MRC program is still to come. Of course we all
hope that the math we discussed during the summer will mature nicely
into preprints and papers and programs, and these are supposed to be
presented in the
AMS Joint Mathematics Meetings
(JMM) in Boston in early January. So one organizer was chosen from each
mentor’s group and I am very happy that
Charlotte Aten
accepted to be the organizer for our group, despite being in the last
weeks of her PhD and interviewing for jobs. Then sometime in September
we were told everyone needed to write up some ideas that they hoped
could make solid papers before January 2023. I’m glad to report that
every one of our four subgroups was able to produce an abstract and I
hope the papers are in the process of being written as I write this.
Now this post was meant to be about the four subgroups above and why I
am so excited for the mathematics we’re doing. But I am a bit behind
with my processing of the knowledge we shared when people visited here
(and over Zulip!). So, while I could possibly wait for the JMM to get to
grips with more of the new math that people are producing, I think I
will try to explain, at least a little, how the themes above interact.
I started my research career working on Gödel’s Dialectica
Interpretation. On bad days I think not only have I never left the
dialectica, but even today I hardly understand the stuff, after all
these many years. Then I console myself with the thought that the stuff
is really fascinating and full of surprising consequences and
applications!
Someone else who’s awed by the Dialectica Interpretation is Jules
Hedges. His blog post
Lenses for Philosophers
shows some of the badly understood consequences and connections of the
work on categorical dialectica construction. In some ways the MRC
Dialectica categories for Computing is an attempt to clarify some of
the interrelations described in Hedges’ post.
Gödel’s Dialectica Interpretation
is a proof interpretation of
intuitionistic arithmetic into a quantifier-free theory of functionals
of finite type, called System T.
The interpretation’s original use was to show the consistency of Heyting
(or intuitionistic) arithmetic. When combined with Gödel’s double-negation
interpretation, which reduces classical arithmetic to intuitionistic
arithmetic, the Dialectica interpretation yields a reduction of the
classical theory as well.
The stated purpose of Gödel’s Dialectica paper is to present aconsistency proof of first-order Peano arithmetic PA by way of anextension of the finitary viewpoint according to which the requirementthat constructions must be of and over “concrete” objects is relaxed tobe of and over certain abstracta, namely computable functionals offinite type over the natural numbers.
Thus the Dialectica
interpretation can be considered a relaxed (or liberalized) form of
Hilbert’s program, where instead of insisting on the finitary viewpoint,
we accept ‘computable functionals of finite type’ as (sort-of)
concrete objects. To understand the interpretation itself, I recommend
A.S. Troelstra’s “Introductory Note to the Dialectica paper” in
Gödel’s Collected works, volume II,
where we can read that “Gödel presents his results as a contribution to a
liberalized version of Hilbert’s program: to justify classical systems,
in particular arithmetic, in terms of notions as intuitively clear as
possible.” From there one can graduate to more complete work of
Troelstra himself
(Metamathematical investigation of intuitionistic arithmetic and analysis
(1973)), of Avigad and Feferman
(Godel’s Functional (“Dialectica”) Interpretation)
and of
Kohlenbach
and his school.
The categorical internal versions of the Dialectica construction,
starting in de Paiva’s thesis in 1988 (out as a
technical report
in 1991), have spelled out some of the requirements and consequences of
the interpretation, especially its connection to Linear Logic. The
categorical dialectica construction was generalized to fibrations, by
Hyland
(2002),
Biering
(2008) and
Hofstra
(2011). Generalizations to dependent type theories appeared in
van Glehn
(2016) and
Moss
(2018) and then
together
(2018). More recently Trotta et al, building on this work introduced
Gödel fibrations
and
Gödel doctrines,
rekindling the
connections of the categorical dialectica construction to logic and
making sure that the three dialectica principles (Independence of
Premise, Markov Principle and Axiom of Choice) get their categorical
explanation.
The main connections of the Dialectica construction discussed in the
MRC, as anticipated above, are between ‘lenses’ and dialectica
categories, between ‘polynomials’ and dialectica categories (see Nelson
Niu’s
blog post
and Sean Moss’
talk),
between ‘games’ and dialectica categories (already hinted at in
Blass’ original paper
on games for linear logic) and between concurrency theories and
dialectica categories in the shape of
Dialectica Petri nets
Processes and their
implementation in Agda. I hope the abstracts for the extended
discussions of these four sets of connections will be available soon.
But it is clear that we’re only scratching the surface of this rich
story of interconnections. Hopefully much more will come to light in the
near future.
