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.