Excellent post, I like the breakdown. So let's take a simple example for clarification:
"Why do race cars sound high-pitched when they are coming toward you and low pitched when they're heading away?"
Level 0: "That's because of the Doppler effect."
Level 1: "The frequency is f = (v + vr)/(v+vs) * f0 . Make sure you get the signs right."
Level 2: "Sound waves have a wavelength, but the wavelength is shortened when the car is coming toward you, because the car 'catches up' with its own waves, and lengthened when it's heading away. Since the waves travel at a constant speed in the medium, a short wavelength implies a high frequency, and frequency is what we hear as pitch. Yes, I know the equation. The phenomenon also happens for light, although the equation is different."
Level 3: "No, I don't remember the equation. Here, gimme a minute and I'll derive it."
Well done. Coincidentally, I think my first frustration regarding (different meanings of) understanding was about that exact issue, back in high school. When my friends and I had learned about the Doppler effect, we were all at Level 1, maybe with minor inroads into Level 2. The problem? I classified that as "not understanding", while my friends called it "complete understanding". So we had conversations like this:
me: Okay, I don't really get this Doppler effect. I mean, why should coming closer make the pitch higher?
them: Because the equation says...
me: Well, I know the equation, but I don't get it at the gut level. What is it about moving closer that shortens the waves like that?
them: Because it like, compresses the waves.
me: But why should the sound wave act like some stiff rod connecting me to the car? Why do you get to apply that kind of reasoning to sound waves?
them: Because that's what the equations say!
me: Argh!
Btw, another important thing about Level 3 is that you could change the question to "... low-pitched when they are coming toward you ..." and vice versa, and the Level 3 rationalist would still derive the same result, except to add that, "wait, I'm confused -- are you sure you're reporting that right?"
There could be an interesting parallel here with "levels of misunderstanding". Perhaps it could help explain how difficult it is for a homeopathist or creationist to change their mind once they've hit the "engtangled truths" or "truly part of you" stages.
That's probably closer to the truth than one might think. Once a belief system moves beyond rote memorization of its basic principles and becomes associated with other domains, non-rational beliefs can get very heavily embedded with outside belief networks. The feedback loop that can be created by having just a few anecdotal connections to an already established system would be severe.
The key factor is that, for people who are not strict rationalists already, the "correlation=causation" attitude is quite strong, so any neuronal links I make from new information to outside branches of knowledge can freely flow right back the way they came. Where the rationalist would have to find additional evidence to ingrain a belief, the fundamentalist is free to draw from his outside branches of knowledge to find reverse reinforcement to support the belief he's trying to learn.
Of course, we all do this to a certain extent, bootstrapping our new, tenuous beliefs by looking for associations we can make to older, more familiar territory. But fundamentalists can get through the neuronal rut-treading faster than rationalists, allowing a belief system to become ingrained that much faster...
Dilution of good content is subtraction, if not as bad as the addition of bad content. I really do have no desire to see a bare "LOL", and will continue to vote accordingly.
Just remembered something which either is a generally useful test for Level 3, or deserves a distinct level of its own: having the ability to guide someone through constructing the knowledge in the first place.
I've just spent the afternoon tutoring a friend of mine's kid, eighth-grader age, who's math-averse. This is a particularly good test, I find, of how well you understand a given bit of material. (And sadly, it seems that his teacher has no particular knack for explaining.)
To someone who already groks math a bit, something like factoring a sum of integers raised to some power is "intuitively obvious". There is no need to spell out the component intuitions of algebra, so you can just say "find the common factor, divide each term in the original sum by that, and you're done".
When you find yourself explaining that to a math-challenged kid you realize that "find the common factor" isn't an obvious, one-step operation, you have to slow way down and break it up: there are usually many different common factors but when the teachers says "the" what they mean is the greatest one, for instance, and finding a common factor might involve decomposing...
Hm; offhand, I would think level 2 should be split up. There's the level where you can see the analogies to other areas, but can't formalize them - you can use them to reason analogically, but can't be quite sure that what you're doing make sense. Then there's the level where you actually, well, understand the connections to other areas. Does this distinction still make sense outside of mathematics?
Where in this system would you place a thorough and accurate, but superficial model that described the phenomenon? If I've made a lot of observations, collected a lot of data, and fit very good curves to it, I can do a pretty good job of predicting what's going to happen--probably better than you, in a lot of cases, if you're constrained by model that reflects a true understanding of what's going on inside.
If we're trying to predict where a baseball will land, I'm going to do better with my practiced curve-fitting than you are with your deep understanding ...
Can we erase the relevant portion of the OPs memory, and see if he can re-derive these classifications?
What about fuzzy analogical understanding? This is understanding some process or even only with reference to an inexact metaphor: people who think of electricity as a fluid flowing through conductors, think of anatomy as if the body was a society and each part and industry or job, think of animal behavior in terms of the equivalent or similar behavior in humans, etc. This is extremely common in my experience.
I suppose it belongs somewhere between levels 1 and 2.
I'm putting in a kind word for "guessing the teacher's password". Sometimes it's a useful preliminary to getting better understanding. In my case, especially if the "teacher" is stuff rather than a person, blundering around for a while gives me enough raw material to develop conscious theories.
When reading or watching Feynman, I really see the level-III understanding; it's such a pleasure to see him go past all the formulae and really understand what's going on.
The thing with physics is that it's like a recursive level-III, where today's level-III is like tomorrow's level-II or even I.
The level analogy suggests that someone with level 2 understand would usually understand a subject better then a person with level 1 understanding.
Not all caluclations are equal and there will be people with level 1 understanding who can calculate and predict better then some people with level 2 or level 3 understanding.
I've been talking to a lot of recruiters as I interview for web developer positions. The recruiters don't code, but they are very familiar with all the technologies and buzz words. That made me think of this article. They use the words and often do so in a sensible way, but they don't have a clue as to why their sentences make sense.
Example: "Yes, we use Angular on the front end because it provides the tools to solve the problems we have." That's a sensible answer, but the recruiter doesn't know:
Warning, aiming high too frequently while young may be hazardous to your grades and hence instrumentally irrational.
It seems to me that stage 3 just means that you use correct scientific methods to learn & expand your knowledge (or am I missing something ?). If that is correct, wouldn't that mean you could essentially recreate the entire body of human knowledge given enough time & persistence ?
The only knowledge that seems absolutely essential to me then is the scientific method itself. Given my human psychology I'm reasonable certain that without that knowledge I would dream up an entire pantheon of gods to explain away everything and just stop there.
You are quite right. I thought about mentioning the Cox-Jaynes road to Bayes' theorem in my post, but decided that someone trying to reconstruct Bayes' theorem would be more likely to get to it by muddling through intuitively via conditional probability.
Related to: Truly Part of You, A Technical Explanation of Technical Explanation
Partly because of LessWrong discussions about what really counts as understanding (some typical examples), I came up with a scheme to classify different levels of understanding so that posters can be more precise about what they mean when they claim to understand -- or fail to understand -- a particular phenomenon or domain.
Each level has a description so that you know if you meet it, and tells you what to watch out for when you're at or close to that level. I have taken the liberty of naming them after the LW articles that describe what such a level is like.
Summary: You have no understanding, because you don't see how any outcome is more or less likely than any other.
Description: This level is only included for comparison -- to show something that is not understanding. At this point, you have, a best, labels that other people use when describing the phenomenon. Maybe you can even generate the appearance of understanding on the topic. However, you actually have a maximum entropy probability distribution. In other words, nothing would surprise you, no event is more or less likely to happen, and everything is consistent with what you "know" about it. No rationalist should count this as an understanding, though it may involve knowledge of the labels that a domain uses.
Things to watch out for: Scientific-sounding terms in your vocabulary that don't correspond to an actual predictive model; your inability to say what you expect to see, and what you would be surprised by.
Summary: You can successfully predict the phenomenon, but see it as an independent, compartmentalized domain.
Description: This is where you can predict the phenomenon, using a generative model that tells you what to expect. You are capable of being surprised, as certain observations are assigned low probability. It may even be tremendously complicated, but it works.
Though low on the hierarchy, it's actually a big accomplishment in itself. However, when you are at this stage, you see its dynamics as being unrelated to anything else, belonging to its own domain, following its own rules. While it might have parallels to things you do understand, you see no reason why the parallel must hold, and therefore can't reason about how extensive that relationship is.
Things to watch out for: Going from "It just works, I don't know what it means" to "it doesn't mean anything!" Also, becoming proud of your ignorance of its relationship to the rest of the world.
Summary: Your accurate model in this domain has deep connections to the rest of your models (whether inferential or causal); inferences can flow between the two.
Description: At this stage, your model of the phenomenon is also deeply connected to your model of everything else. Instead of the phenomenon being something with its own set of rules, you see how its dynamics interface with the dynamics of everything else in your understanding. You can derive parameters in this domain from your knowledge in another domain; you can explain how they are related.
Note the regression here: you meet this stage when your model for the new phenomenon connects to your model for "everything else". So what about the first "everything else" you understood (which could be called your "primitively understood" part of reality)? This would be the instinctive model of the world that you are born with: the "folk physics", "folk psychology", etc. Its existence is revealed in such experiments as when babies are confused by rolling balls that suddenly violate the laws of physics.
This "Level 2" understanding therefore ultimately connects everything back to your direct, raw experiences ("qualia") of the world, but, importantly, is not subordinate to them – optical illusions shouldn't override the stronger evidence that proves to you it's an illusion.
Things to watch out for: Assuming that similar behavior in different domains ("surface analogies") is enough to explain their relationship. Also, using one intersection between multiple domains as a reason to immediately collapse them together.
Summary: Your models are such that you would re-discover them, for the right reasons, even they were deleted from your memory.
Description: At this stage, not only do you have good, well-connected models of reality, but they are so well-grounded, that they "regenerate" when "damaged". That is, you weren't merely fed these wonderful models outright by some other Really Smart Being (though initially you might have been), but rather, you also consistently use a reliable method for gaining knowledge, and this method would eventually stumble upon the same model you have now, no matter how much knowledge is stripped away from it.
This capability arises because your high understanding makes much of your knowledge redundant: knowing something in one domain has implications in quite distant domains, leading you to recognize what was lost – and your reliable methods of inference tell you what, if anything, you need to do to recover it.
This stage should be the goal of all rationalists.
Things to watch out for: Hindsight bias: you may think you would have made the same inferences at a previous epistemic state, but that might just be due to already knowing the answers. Also, if you're really at this stage, you should have what amounts to a "fountain of knowledge" – are you learning all you can from it?
In conclusion: In trying to enhance your own, or someone else's, understanding of a topic, I recommend identifying which level you both are at to see if you have something to learn from each other, or are simply using different standards.