Stating obvious consequences

Sometimes authors don’t state the far-reaching consequences of their work. This could be humility or could be that they perceive it as obvious.

Such a norm seems to be quite prevalent in maths. I think there’s also a lot of signalling going on not stating the obvious: I guess that’s one way to project smartness.

I recall the following conversation from my undergraduate lecture:

Me: I really like this result in algebraic number theory showing that triplets of real numbers can not be turned into a field.

Friend: But why would you even want to turn triplets into a field!?

Me: You remember you told me that complex analysis is your favourite topic at the moment? Come to think of it, most of it is only possible due to the fact that we turned pairs of real numbers into a field. Perhaps if we could do the same with triples we’d develop another fascinating subject. But we won’t as it’s not possible.

Friend: I never made this connection, I guess such investigation makes sense now.

However the above motivation behind investigating whether triplets could be turned into a field wasn’t stated in the course. It wasn’t that the lecturer didn’t have the time. It’s just the way things are done in many courses at the moment: we are presented with results often without their significance explicitly stated.

So I am going to state significance behind things that are of interest to me explicitly right here, right now.

Cox’s theorem

Cox’s theorem says: there’s one, objective, reality: out there to be discovered. Your job is to approximate such reality with the information that you have. Since you can only have limited amount of information, you need a theory to deal with uncertainty. Turns out probability theory is a unique way to deal with uncertainty.

Roughly speaking, if you want to adhere to some commonsense assumptions of reasoning, then you should assign a probability to every statement that you believe, as a consequence. And then update such number according to the laws of probability theory when you encounter new information. The higher probability you assign, the more confident you are. Of course, in practice you should only strive to approximate this framework, as the mother nature didn’t equip our human brains with such calculators.

Why is this highly significant? Philosophers argued for centuries about different epistemologies, developed and spread retarded memes as a result.

Such as: you can never be sure of anything or every belief is valid. Such ideas blatantly violate the conclusion of the Cox’s theorem: the certainty you assign to beliefs does not have only 3 values: (true, false, I don’t know), it has an entire uncoutable range of them. I wrote more about it in Plausibility = Just Double | DoNotKnow. And it’s all right there, in the boring equations. Work of a dozen of philosophers undermined by one physicist, R. T. Cox.

Another thing about Cox’s theorem is that it is single-valued. Turns out how human feel about a statement: whether it’s offensive or noble doesn’t affect its veracity. Again, this is all just a consequence of the proof. So taking an offence doesn’t disprove a statement and people identifying with social justice movement ought to start taking this into account.

Work by Daniel Kahneman

More and more people I speak to read “Thinking, Fast and Slow”. And then said “meh” and went back to living their lives the way they lived before. I hear the following sentiments:

Kahneman is telling me that people, en-masse, are irrational? Who would’ve thought!

Daniel’s mighty thesis that people are, in fact, irrational is hard to dispute. But there’s more to it than just that. Together with Tversky he identified some areas where people are irrational and guided us how to avoid such common pitfalls. And, perhaps, some readers take notice and adjust.

However, what really should be happening is the following:

Oh my god, everything I’ve ever believed in, ever wanted to achieve and ever argued for could well be deeply mistaken. Time and time again Daniel’s book highlighted that our intuitions are often wrong. This wasted millions of man-hours spent on achieving stuff inefficiently by people who didn’t really want to achieve it had they thought about it more logically. I need some new, solid knowledge on how to be sure that I avoid similar traps.

Turns out Cox’s theorem and Daniel’s work go hand in hand together. For once you believe that you need a systematic way to arrive at epistemic truths without System 1 interfering, Cox’s theorem fits the bill.

People who were thinking about the theory behind Artificial Intelligence also ran into this problem. How to teach a computer to reason in uncertainty and how to make it value what we, humans, value?

The technical details of all of this haven’t yet been worked out, but a first-order approximation framework began to emerge. And people in charge of such theoretical AI designs needed to unite some work in epistemology, decision-theory, game-theory, ethics and more for their answers.

However, as a nice side effect such people stumbled upon ways how to make us, humans more strategic: we should just strive to approximate such idealised framework, especially in the areas where the effect of System 1 is particularly strong. And, as it has been discovered, Kahneman’s work was far from exhaustive. “People are crazy, the world is mad,” - is the sentiment from the AI people.

This is how lesswrong was born in 2009 which provided some of the answers to:

OMG my brain is a massive liar, how do I mitigate the damage?

You’re welcome, crazy people.