Gödel for Dummies

Gödel’s theorems say something important about the limits of mathematical proof.

Proofs in mathematics are (among other things) arguments. A typical mathematical argument may not be “inside” the universe it’s saying something about. The Pythagorean theorem is a statement about the geometry of triangles, but it’s hard to make a proof of it using nothing but points and lines in a plane! For instance, Euclid’s own proof (per Wikipedia) starts like this:

  1. Let ACB be a right-angled triangle with right angle CAB.
  2. On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.[13]
  3. From A, draw a line parallel to BD and CE. It will perpendicularly intersect BC and DE at K and L, respectively…

The proof uses words, symbols and other tools that aren’t points and lines in a plane. Euclid’s proof thus is about geometry but isn’t inside geometry.

How does this apply to logic? Logic is (among other things) the mathematics of arguments. It’s reasonable to wonder if proofs about logic fit inside logic! More precisely, given a particular system of logic S, we can ask what S can and can’t say about itself.

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What is intelligence?

Intelligence as Awareness and Response

The most primordial kind of intelligence is awareness of relevant features in the environment, coupled with responses to relevant information. This environmental awareness-response type of intelligence only makes sense in the light of goals (“relevant” to what?) – from a single-celled organism responding to the presence of food by consuming it, to a human noticing that a plant is dry and watering it.

Evolution itself has acquired a great deal of intelligence; DNA is the transmissible record of the information evolution has acquired about the environment, from the perspective of billions of organisms with future existence as a “goal.” Simple organisms are still very viable, but the computational process of evolution has revealed that increasingly complex organisms that extract a great deal of information (and energy) from their surrounding systems are also extremely viable, especially over short time frames. Organisms have evolved increasingly complex neural systems and senses that reach into new domains of relevant information. Humans have created instruments that do the same.

Games vary in the amount of “luck” that is available. A solved game presents no opportunities for randomness, no luck – but even very complex games present different amounts of luck depending on the level of play. One measure of luck available in a game is the distance from the best player to the ideal player; as chess becomes computationally solved, its skill component overtakes its luck component. Games, like awareness-response intelligence, only make sense in the context of goals. Awareness-response intelligence extracts as much information as it can about the world relevant to its goals so that “luck” is as small a factor as possible. Sources of apparent randomness must either be controlled, studied until predictable, or, if these are not possible, responded to with optimal probabilistic strategies.

The “intelligence” apparently contained in complex economies (the “invisible hand” of the market, ecosystems) is of the awareness-response type.

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What is statistics?

I put the call out on twitter for ideas for my first post, and Gabe asked this:

https://twitter.com/GabrielDuquette/status/488858307885424640

I suppose I did ask for statistics questions. This one is a bit tough to answer because, like I hinted at on twitter, a wide variety of things get called statistics by the people doing them, statisticians do an even wider variety of things, and to muddy the waters even more, lots of things that typically get categorized as statistics often are also categorized as other things like machine learning or computer science. I suppose I should blame the computer people.

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Rationality Is Not A Martial Art

From the beginning, Yudkowsky’s sequences follow a running metaphor of rationality as martial art. This implies a bunch of shitty stuff. I’m going to describe why it’s shitty, and then propose an alternative metaphor that I think is somewhat less shitty.

  • Martial arts are individual pursuits. They are typically practiced in a social context, yes, students practicing together, masters passing their wisdom on to students. But they’re mostly not about fighting together, just training. Rationalists, like other humans, need to work together to complete large projects.
  • Martial arts are personal. They are specifically about what a human mind can do with a human body. Rationalists are encouraged to make and use tools.
  • Martial arts are only good for one thing: physical conflict with other humans. Rationality is broadly applicable, in almost any context or for any purpose.
  • It’s straightforward to identify skilled martial artists by holding fighting tournaments. What sort of tournament do you hold to test rationalists? Assessing rationality in humans is Hard.
  • Martial arts are competitive. They are about becoming the best fighter (comparative) and not about becoming the true fighter (absolute), whatever that would even mean.
  • In martial arts, your opponents are always human.

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Ice Under the Bridge

The BBC is renowned worldwide for the high quality of its journalism. Take this article from 2005:

A thirsty thief is being blamed for downing a bottle of water, valued at £42,500, at a literary festival.

The two-litre clear plastic bottle containing melted ice from the Antarctic was devised to highlight global warming by artist Wayne Hill…

Its value was worked out by the artist from the damage worldwide of the entire ice sheet melting – he estimates between £6 trillion and £9 trillion – and the relative amount of damage from two litres of water.

Hmm. Something seems off about those numbers. Let’s check them!

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A Very Funny Joke

A long time ago, I decided that if I were ever to get a tattoo, I would get a tattoo of an irregular black splotch.

“Say,” people would ask, indicating my tattoo, “what does that represent?”

“What do you mean, what does it represent?”

“You know, what’s it for, what does it symbolize?”

“This? It doesn’t mean anything.”

“Why’d you get a tattoo of a meaningless black splotch, then?”

“Oh, this isn’t a tattoo. It’s cancer.”

Contrarianism Explained, or Postmodernism: the Drinking Game

(Slightly edited from something I wrote over 20 years ago.)

POSTMODERNISM: THE DRINKING GAME
RULE 1: If anyone, at any time, for any reason, believes in, supports, or likes a person, place, or idea, it’s only because they haven’t uncovered the fundamental contradictions underlying that belief; you are allowed to laugh at them because they are Less Jaded than you.

RULE 1a: If everyone disbelieves in, attacks, or dislikes a person, place, or idea, it’s only because they haven’t uncovered the fundamental contradictions underlying that disbelief; you may support that person, place, or idea, and you are allowed to laugh at the other players because they are Less Perceptive than you.

RULE 2: Never explain the rules.