Henry D’Angelo

P.S. this reminds me of how we discussed the definition of a game. You suggested something like: a set of rules with a goal attached. I said something like: a set of constraints that an agent adopts to further their own goals. I believe when it comes to games, just as it does with signs, you need to highlight the thing "outside" the game (the "concept"), not just the game itself (the "object" or sign). Ultimately, I think what matters most is the "bridge" -- from within the game itself, to the player who is "outside" the game.

This is beautiful. I will respond to one specific thought --- the claimed target of math being omnipotence.

To set the conceptual stage for my response, let me first appeal briefly to some notes.

Question: How do you quantify randomness or "patternness"? One of my favorite proposals is the concept of Kolmogorov complexity, which quantifies the randomness of strings relative to a computer.

Informally, the Kolmogorov complexity K(x) for a string x is equal to the length of the shortest computer program that can output x. So, what does this mean?

Consider the ζ (which is just “10” repeated 50 times):

ζ: 1010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010

This string is 100 characters long. But its Kolmogorov complexity, K(ζ), seems to be less than 100. Why? Because you could (in a given language) program ζ in less than 100 characters. I essentially just did; in pseudocode, it would be something like:

print(“10” * 50)

This program, despite being shorter than ζ itself, would output the string correctly. So you get the idea: the Kolmogorov complexity of an object in effect reflects how long it takes to be described (though the specific measure varies across different languages and computers, the idea is consistent across).

The basic thing to note is this: there is a sense in which ζ seems not to be random; intuitively, it is the fact that it has a pattern. And it is this pattern—repeating “10”—that enables us to describe ζ in a compressed way (that is, using a program that takes less characters than the string itself). Kolmogorov complexity is meant to formalize this intuition — that fundamentally, patterns enable compressibility, and randomness is incompressibility.

So, Kolmogorov complexity (or algorithmic information) captures the intuition of patterns being dual to randomness, which is neat. That said, it's not generally computable, and there are some technical things needed to make it work formally (e.g. the idea of prefix-free Kolmogorov complexity).

Now, how does algorithmic information relate to the question of math's potential for omnipotence?

Informally, I generally think of math as follows: "the methods of compression, universal across effective compressers, in the limit of data."

So, what is an "effective compresser"? Roughly, imagine there is an algorithm that receives datapoints (in the "real world", think sense-experience), and then performs some kind of "operation" to encode them. There is time to try different schemes. Loose example: a scientist who is reasonably intelligent. The scientist observes the world. He seeks a "theory" to understand these datapoints, and he has his lifetime to do so. His basic goal is to find "programs" for compression that allow him to parsimoniously "recreate" these datapoints, which in turn, helps him to predict them (assuming his methods of compression identify true "hidden causes", and ignoring the problem of induction).

Eventually the scientist will arrive on certain programs for compression. Maybe he'll find numbers. In fact, with enough time, he is sure to discover "something like" numbers. So perhaps there are indeed methods of compression that are universal or "inevitable" for effective compressers with enough datapoints. This is what I imagine math aspires to.

For example, maybe addition is one of those universal methods of compression. What I'm saying here is that if you're trying to compress datapoints, and you have enough datapoints, eventually "something like" addition (up to isomorphism?) will be strictly useful to encode. Maybe. And, so any compresser that's good enough and has datapoints, will do so. That is to say, perhaps any scientist that is sufficiently advanced and has enough time / data, will eventually discover addition.

Quantity, length, etc. these things strike me as being potential "universals" in this above sense.

But then...

Are there such universals? Does this concept even make sense?

A problem is that Kolmogorov complexity is computer/language dependent. You can force it to be a universal computer, sure. But the reality is that relative to any computer, even a universal one, the vast majority of strings are random (I can prove this fact quite easily, if you'd like -- the basic idea is that if you define "not random" as "described with a short program", you run into the problem that there are many more long strings than short strings, so short inputs can cover only a small part of the terrain) then there is always a computer to which those "random" strings aren't random.

The extent to which patterns or randomness are necessarily observer dependent is a genuine facet of math being limited. Even in the formal world, compression is limited.

Further yet, there is a deeper point, which perhaps is related to what you are getting at. The point is this: it is not just the question of whether "datapoints" (left undefined) are patterned (compressible) or not, BUT the fact that it is a kind of fluid act or "process" of actively receiving datapoints, being affected by them, making sense of them, etc. that is itself a totally different thing. You can never "complete" this task.

You don't get anything whatsoever about firsthand experience from "universals"; in fact, firsthand experience might be described as "that which is not universal".

A scientist who observes the world and discovers some beautiful pattern or order; this is a beautiful thing. Is there any way to "compress" this experience? Maybe not. Some things are defined by their limitations. Can anything be defined from the outside? For that matter, can anything be defined from the inside?

Fixing things in formal terms delimits those things, but delimit the world it does not.

I am still in synthesis mode, but these are the sparks that were struck for me. Hopefully some are enjoyable.

You know, I imagine you are aware that the "gather brilliant people, then get out of the way" philosophy is something I believe deeply in.

I've long worshipped the ideal of the Institute for Advanced Study, which is exactly as you describe: "gather smart people and give them absolute security in terms of salary, legalization, career, etc., while imposing no requirements." I loved reading the Usefulness of Useless Knowledge by Abraham Flexner, and I would still recommend it to anyone. Freedom, curiosity, and doing something "for its own sake"; these are the ideals I devote my life to.

That said, as I grow older I think it is more complicated. Why? I'll name drop some more and give a Feynman quote:

When I was at Princeton in the 1940s I could see what happened to those great minds at the Institute for Advanced Study, who had been specially selected for their tremendous brains and were now given this opportunity to sit in this lovely house by the woods there, with no classes to teach, with no obligations whatsoever. These poor bastards could now sit and think clearly all by themselves, OK? So they don't get any ideas for a while: They have every opportunity to do something, and they're not getting any ideas. I believe that in a situation like this a kind of guilt or depression worms inside of you, and you begin to worry about not getting any ideas. And nothing happens. Still no ideas come. Nothing happens because there's not enough real activity and challenge: You're not in contact with the experimental guys. You don't have to think how to answer questions from the students. Nothing!

— Richard Feynman

I think this quote isn't completely wrong. Broadly, I think it is healthy to have some friction. People get ideas from the most inexplicable places. The feedbacks of human life can keep you grounded. And, when ideas don't come, having some practical, well-defined task can keep you from falling into a rut. I mean, when I have logistical tasks to do, I often feel better afterwards, even if it's not what I am really passionate about. It's often just healthy.

Again, I say all this even as someone who worships the freedom ideal. I think there is an antinomy within freedom. For one thing, the more free you are, the more responsible you are. And, when you do have a constraint, transcending it is freedom. What I mean here is this: sometimes it's easier to have freedom defined out of something else, than freedom qua freedom.

This train of thought is one reason why I want to start a company. Broadly, the financial constraint of a company is a good source of friction; this bolsters the value of freedom.

What I imagine: give researchers freedom, with the exception of the friction from having to at times work on a very practical goal (providing a service people value).

I think with a gym for the mind, this "practical goal" is complementary to the more basic goal of understanding how the human mind works.

Whenever basic research is going slow, you have "guppy" problems to help you feel productive (which I think is important for people).

I also think with a company, in exchange for working on these "dirtier" problems, researchers would have capital to aid their ideas. E.g., in the long run I think our "gym for the mind" might want to buy, say, functional MRI machines.

But of course, researchers should be given time to focus on what they think is interesting, and in doing so, should not be policed in any capacity. (I wonder if that's easier in a private company than in a public-facing university that receives tax dollars?)

Anyways, that's a high level overview of my position. Roughly, the ideal of absolute freedom is great, but it could also be true that the journey (moving towards absolute freedom) may be better than the destination (being "at" absolute freedom).

Still, I am open for conversation on this point. I think it varies per person what the right level is. I myself want a lot of freedom!

It's remarkable how similar the question of how to create a clock is, to the question of how to do science at all! The question is, what phenomena can we bundle together to form some invariant?

Goodness me, a banger. I'll hopefully come back to this as I integrate into my thinking, but here is the high level point I connect on-- oftentimes, "ground truth" is actually the stacking of relations on top of each other. There is a bit of a "Kantian turn" one must do in thinking -- instead of trying to capture the thing-in-itself, one has to find the right phenomena to, in some way, restrict or lasso the thing-in-itself.

"How can we be sure that what survives is the truth?" That's the question of this entire enterprise, my friend.

Gyms give clear feedback. You lift more and more weight. You see yourself in the mirror with a nice pump. Your body simply feels better after a workout.

What is the feedback on twitter? What would it even mean to be a good twitter user? People just like being told they're good, or that they need to improve, IF, that is, they trust the mechanism for generating this feedback.

I think that good feedback for the brain is harder -- it's the whole problem of education.

Ultimately I think the best starting point is the people. If the Aeneas community attracts people who want to learn, who are intellectually honest, who are fundamentally curious, etc., then we're already far along. The acclaim of your peers means a lot, if you acclaim your peers.

I think, virtually always, the key thing is: effective feedback. In math, you have logical rigor; in science, predictions are tested by experiment; in the gym, you witness how much weight you can lift; in a game, you win or lose. These pursuits are all so wonderful because they are honest, i.e., they give effective feedback. But then, the problem is that so many ideas aren't in that stage to go "into the arena" -- being at the stage where feedback makes sense is non-trivial. What's the solution? Well, now you get into the idea you and I love most: communication is perhaps the most important feedback loop there is. Trying to explain something well--putting it in simple, intuitive and precise language--leads to better understanding for all. Why? Well, having the "universals characterization" of an idea--again, the idea put into its simplest, most intuitive, most precise form--in fact amounts to understanding. Understanding an idea just is having access to the idea's simplest form. So trying to communicate effectively, as it turns out, gives the perfect "game" or feedback to aid understanding. Put another way: communication is the game for which winning is understanding.