Frege and his little monsters

This is NOT a serious post, in case this is not obvious. 

I don't like quantifiers, I call them Frege's little monsters.

I don't like the universal quantifier  because it brings infinity into models. Don't get me wrong, I love infinity as much as any other mathematician and I still get a twinge of pride when I explain Cantor's basic theorems (enumerability of the rationals and lack of such of the reals)  to anyone who hasn't seen them before, especially young ones. But I once attended a lecture by Phokion Kolaitis, who showed me that finite model theory is really very 'different' from model theory: nothing works the way models are supposed to work. It is not the case (as I thought before) that finite model theory is an easy (because finite and surveyable) instance of usual model theory. And one can do an awful lot without bringing infinite domains into our logic pictures.

Also universal quantification and especially vacuous quantification is horrible and very non-intuitive. The typical example of something that we feel forced to accept simply because, without it, the system is even worse.

But you have a duality, people might object. Maybe simply look at the existential quantifier and let the universal be simply its dual. Well,  this does not work for constructivists, who do not believe that 'for all' and 'there exists' are totally dual. But  also, if you chose the existential that is  not exactly dual to universal quantification, the one that has more content, the one which says which 'x' is that that makes "there exists x such that A(x)" true, well then you have a connective that has complicated binding rules, requires commuting conversions and it's not  proof-theoretically well-behaved.

Semantically it's very simple, right? If you say that there exists something such that the predicate A is true of that something, well, just show me the culprit! Call it 'a', because we have a long tradition of thinking of x's and y's as variables and thinking of a's and b's as constants and  then we know that "A(a)" is true. And that a model of this sentence should have 'a' in its domain. Seems simple and direct, but it is anything but simple.

So they are both little monsters, Frege's monstrinhos, and when we want to model them categorically not only do we need our triple adjunction, we also need the Beck-Chevelley condition (BCC), which everyone does its best not to explain. Of course the whole set-up of free and bound variables, predicates and functions, and how substitution works for them, has to be put in place and made to work, before you can add the little monsters.

Anyways I was about to post here a picture of  Frege by Renee Jorgensen Bolinger, in the style of van Gogh. Her pictures of philosophers are great and I really want to buy a few, you can see them at her site.

But she requires people to ask her for permission and I cannot be bothered, so you just need to go and look up her site!

Portuguese on my mind

Marcelo Finger and Thiago Pardo

just scored high for NLP and AI in Brazil. They now head a big, new, and shining project with IBM and FAPESP  in the new center for AI, just launched in October 2020.

The picture above is from their project NLP2 Resources to Bring NLP of Portuguese to State-of-Art description in http://c4ai.inova.usp.br/nlp2-en/. 

This is great news! I have been saying and writing for over ten years now that we need lexical and semantic open-source resources for Portuguese. 

In particular, I'm very pleased because the two blue links in the picture above, both refer to my work. 

The Universal Dependencies for Portuguese is my work with Alexandre Rademaker, Fabricio Chalub, Claudia Freitas, Livy Real, and Eckhart Bick from 2017. The second blue link SICK-BR is my work with Livy Real, Ana Rodrigues, Andressa Vieira e Silva, Beatriz Albiero, Bruna Thalenberg, Bruno Guide, Cindy Silva, Guilherme de Oliveira Lima, Igor CS Câmara, Miloš Stanojević and Rodrigo Souza. 

I have mentioned to both Marcelo and Thiago that it would be nice to get our papers on their page. These are our goals too and we have been working on them for quite a while. They are working on it!

von Neuman and understanding

So the joke goes something like this: A student asked John Von Neumann when he would start understanding things. And van Neuman replied "Young man, in mathematics you don't understand things. You just get used to them." 

 
I like this way of putting things. I can recall vividly all the ways things that I now hold dear in mathematics were once completely impenetrable. 

From the time I was first taught in high school about matrices and I kept asking myself and the teacher, but `what difference does it make if I write numbers on shapes or not?' It shouldn't make any difference how we lay out the numbers! To when I finally got the hang of Linear Algebra, after months of resisting it: the notion that a vector space could be defined by the properties it has, instead of by what it is. This was, mind-boggling and it still is, a little. 

More disturbing still was when I first learned about Natural Deduction from my first Logic teacher (Luiz Carlos Pereira) and I had the bad idea of telling him that I couldn't see the point of having axioms, sequents and natural deduction to define the "same" system.  It seemed to me that logicians hadn't gotten their formalizations sorted out, yet. Ah, the foolishness of youth.  Now I can see that some of this Bourbakianism of believing in "the most" perfect formalization of mathematical concepts is not only silly, it's bad for mathematics and for science in general. (also the most embarrassing detail is that Luiz Carlos told Prof Prawitz about my stupid remarks.) 

oh well, everyone knows I have very strong convictions, that is --translating-- that I am as stubborn as a mule. Mostly my stupidity is harmless, like, it took me forever to learn how to bike, because I was convinced that it was not possible for bikes to stay upright. or when I decided that people could not swim, as otherwise, why would anyone drown? Again it took me forever to learn how to swim, since I believed it was impossible to float.

But despite all my blunders, I still think that it's important to have your own ideas on the mathematical concepts you're taught and what they mean and don't mean. and whether they `have legs' (will go far or not). So I hope to re-ignite the more abstract side of this blog and to discuss a few more mathematics, while I can. The worst thing that can happen is that I am wrong. I have plenty of experience in this department.

ps: Check out the Wikipedia page on "The Martians", I had no idea all these guys were from Budapest!

Things I am proud of

 

In these times of pandemic, I (and I believe everyone else too) get depressed about what I have (not) been doing with my life: all the ways that I could be a socially more useful person and I am not. all the infinite hours that I spent fighting bugs in programs or bugs in my understanding of things, that I could and should have spent fighting the bad guys in the actual world.

So as I way to cheer myself up I thought I'd write a bit about stuff I have been doing that I think is cool, that, to put it the Marie-Kondo way, gives me joy. Funnily enough, these things are hard to put in resumes or curriculum vitae. But since this is "candy for the soul" and too much candy does make you sick, I think I will do it in small doses, in several posts, with a decent amount of time between them.

Because I was looking for something else, I found the message from 2002 when Bob Rosebrugh invited me to be an editor of TAC (Theory and Applications of Categories). TAC was one of the cleverest things that category theorists did very early on (together with managing to keep a mailing list going). We've had one of the first open-source journals in Mathematics,  since 1995. And I was lucky enough to be invited to its Editorial Board in 2002. Maybe it was the allure of "industrial mathematics", I was in Xerox PARC then. Who knows? 

When I joined  TAC's Editorial Board there was only one woman there, Susan Niefield. She had been the only woman there since 1995.  Now there are six of us, better. TAC has published around 770 articles in its entire career, so far. More recently, having been active in NLP where a single conference, EMNLP 2020 has had the following data:

After receiving 3677 submissions, 3359 of these went through review, of which 754 were accepted to EMNLP and 520 were accepted to Findings of EMNLP. This gives an acceptance rate of 22.4% for EMNLP and a further 15.5% for Findings.

I found out the extent to which the numbers can be different. very different indeed. 

Calling a `citation' the minimum unit of measurement of productivity in Academia is very misleading too. Everyone knows this! But as we are always reminded (e.g. Dunne's short summary) people measure what they can, or "You get what you measure".  But more than individual researchers' gaming the system and/or groups of scientists or publishers ganging up in 'citation farms' (which Dunne discusses), there are also the societal prejudices and old structures conspiring against women, black or brown researchers, gender non-conforming researchers, researchers not from the Global North, etc that change the landscape of academic fields. And keeping working at pointing out these things, in the long run, can be extremely tiring. Deadlines always coincide (Murphy's Law), disease and small (and big disasters) always occur, and constructing things (even simple, small ones like a workshop) is always much more work and time spent than you could possibly estimate.

So, to begin with, a list of things I'm proud of, and I might (or not) discuss these in future blog posts, as time permits:

1. Editorial Boards of TAC, Logical Methods in Computer Science, Logica Universalis, Compositionality.

2. Industry Advisory Board of the Masters in NLP program of UC Santa Cruz.

3. Scientific Advisory Board of the Institute of Logic, Language and Computation (ILLC), University of Amsterdam.

4. Council of the Division of Logic, Methodology and Philosophy of Science and Technology of the International Union of History and Philosophy of Science and Technology, 2020-2023.

5. Ambassador for Logic, Vienna Centre for Logic and Algorithms. 

Special mention to the ``Encontro Brasileiro de Mulheres Matematicas" at IMPA in 2019, where I talked about how Applied Category Theory is the way I want to connect algebra, programming, and logic, but especially why I think we need to pay attention to gender gaps in maths, computer science, and logic.

Counting Intuitions

 This post is an exercise in thinking about vague things.

I think everyone can agree that there are infinity more ways for things to be bad, to not work, as there are for them to work. For things to work, you need a big conjunction of things, you need to be healthy, you need a decent house,  you need good food, you need friends, you need amusements, you need purpose, you need things to be good for your friends, etc...you know the list just keeps growing. While for things to not work, only one of them must be missing. So clearly it's much easier for things to not work, then for them to work, regardless of whichever priority order you put on your personal list.

Even people like me, who do not like probabilities and that have difficulties with them can see that the bad scenarios are much more probable in the big scheme of things than the good ones.

But there is some help to be had. By and large it seems that verifying that something is true is much easier than discovering when something is true. This (possibly very large) gap between the easiness of checking a given answer versus the hardness of coming up with a possible answer is one of the mathematicians most used tools. Note that we do not have a proof for it, it is just obvious for, say, equations of second degree or systems of linear equations. But we extrapolate, we reckon it might happen in all branches of mathematics.

Another helping tool is symmetry: we believe it is everywhere and we love it. It does help to half the work in many situations and the Universe does seem to have a penchant for it. or maybe it's just us, humans, that see it everywhere, when it's only there in some very special places. I don't know of any attempt to measure the amount of symmetry of the Universe, but I know that mathematicians, if they can,  will make things symmetric: symmetric things are prettier.

And a third helping tool is `adversarial thinking', in whichever way you may want to think about it. So this might be games, where there are proponents and opponents and they battle their wits over the truth or falsity of propositions (the mathematician thinking about it might play both sides and hence, perhaps see more clearly the weaknesses of arguments of the other side). Or it might be adversarial training in machine learning, which I don't really enough about to pass judgement on.

In any case, these generic tools are about trying to make the problem easier, about simplifying problems by trying to see what would happen, if they were indeed simpler and easier. 

But of course we know that many times things that are simpler, that look intuitive and clear are just plainly  wrong. The Sun does not move around the Earth; heavier things do not fall faster; things that do seem to stop if not subject to forces non-stop,  actually would keep going non-stop if the attrition of other forces did not stop them. Similarly, lovely graphs in Geometry prove wrong things, because you cannot trust graphs (Escher pictures anyone?)

So one of main points of the apprenticeship in Mathematics is learning to distrust your intuitions. A bit like philosophers who start asking "why" about any and everything, mathematicians have to learn to read the books of they favorite authors doubting every word and trying to prove or disprove every sentence. Being a mathematician is about verifying always; trusting only in special occasions, if at all.

Partiality Insights

 No, I don't mean political nepotism nor do I mean favoritism within families. 

By partiality, I just mean the prosaic fact that, sometimes, functions are not defined everywhere. 

From humdrum 'step functions' 

to the beautiful poetry of '1/x':

Anyone, mathematician or not, can see the beautiful poetry of missing a zero, but gaining two infinities!

However, Category Theory has only total functions and we need to deal, with grace, as point out Cockett and Garner, with partial functions. What can we do?

It turns out that several partial solutions are available and here are some of the ones I know about:

1. We can use some 3-valued logic, where the third truth-value is some sort of undefined (and there are a few extra choices to be made here);

2. We can use the exceptions monad T(A)= A+1, where A stand for the normal values and 1 is the error of type A;

3. We can talk, like Fourman and Scott do, of "existentials that are uniquely defined";

4. We can try to choose between de Paola and Heller's 'dominical categories', Rosolini's 'P-categories' or Cockett's restriction categories.

Now, if we were to do dialectica constructions, paying attention to partiality, which of these alternatives would be easiest for us? Are there other constructions that are better?

Logic: a quote or two

Confirming once again that nothing is black or white, but an infinitude of greyness, TU - Wien is celebrating World Logic Day 2021. As they wrote:

To enhance public understanding of logic and its implications for science, technology and innovation, in 2019 UNESCO proclaimed the 14 of January the"World Logic Day". The date was selected in honour of Alfred Tarski (born on January 14th) and Kurt Gödel (who died on this date).  We, as the Vienna Center for Logic and Algorithms (VCLA at TU Wien), would like to celebrate World Logic Day 2021.

[...]

We would be honoured if you would be an Ambassador and Supporter of World Logic Day 2021. In the affirmative case, please send us a quote about logic not exceeding 50 words. 

I do want to be an Ambassador and Supporter of Logic, not of Logic Day, so I have been thinking about it on and off for some days. 

I'm not good with epigrams and such-like. I wish I could make slogans like some of my friends. Maybe I should crowdsource this task on Twitter. 

But I do feel that one of the best things ever said about logic is the cartoon from the New Yorker above. This cartoon used to hang from Martin Hyland's door in the old DPMMS building in Mill Lane, when I was doing my phd.