I don't want to involve myself in an endless topic of debate by discussing the treatment of slaves, towards whom we Romans are exceptionally arrogant, harsh, and insulting. But the essence of the advice I'd like to give is this: treat your inferiors in the way in which you would like to be treated by your own superiors. And whenever it strikes you how much power you have over your slave, let it also strike you that your own master has just as much power over you. "I haven't got a master," you say. You're young yet; there's always the chance that you'll have one.

--Seneca, Letter XLVII

1) Can aspects of grooming as opposed to selecting/testing be steelmanned, are there corner cases when it could be better?

How about selecting someone to groom? There was a line of Roman Emperors--Nerva, Trajan, Hadrian, Anoninus Pius, and Marcus Aurelius--remarkable in that the first four had no children and decided to select someone of ability, formally adopt him, and groom him as a successor. These are known as the Five Good Emperors and their rule is considered to be the height of the Roman Empire.

It would be condescending to talk in short bursts of wisdom.

It would be condescending for the master too, to talk in short bursts of wisdom to his disciples, as long as he was alive. The issue is rather that once he dies, and the top level disciples gradually elevate the memory of the master into a quasi-deity, pass on the thoughts verbally for generations, and by the time they get around to writing it down the memory of the master is seen as such a big guy / deity and more or less gets worshipped so it becomes almost inconceivable to write it in anything but a condescending tone. But it does not really follow the masters were just as condescending IRL.

You can see this today. The Dalai Lama is really an easy guy, he does not really care how people should behave to him, he is just friendly and direct with everybody, but there is an "establishment" around him that really pushes visitors into high-respect mode. I had this experience with a lower lama, of a different school, I was anxious about getting etiquette right, hands together, bowing etc. then he just walked up to me, shook my hand in a western style, did not let it go but just dragged me halfway accross the room while patting me on the back and shaking with laughter at my surprise, it was simply his joke, his way of breaking the all too ceremonious mood. He was a totally non-condescending, direct, easy-going guy, who would engage everybody on an equal level, but a lot of retainers and helpers around him really put him and his boss (he was something of a top level helper of an even bigger guy too) on a pedestal.

This puts a big constraint on the kind of physics you can have in a simulation. You need this property: suppose some physical system starts in state x. The system evolves over time to a new state y which is now observed to accuracy ε. As the simulation only needs to display the system to accuracy ε the implementor doesn't want to have to compute x to arbitrary precision. They'd like only have to compute x to some limited degree of accuracy. In other words, demanding y to some limited degree of accuracy should only require computing x to a limited degree of accuracy.

Let's spell this out. Write y as a function of x, y = f(x). We want that for all ε there is a δ such that for all x-δ<y<x+δ, |f(y)-f(x)|<ε. This is just a restatement in mathematical notation of what I said in English. But do you recognise it?

One problem is that the function f(x) is seldom known exactly. In physics, we usually have a differential equation that f is known to satisfy. Actually computing f is another problem entirely. Only in rare cases is the exact solution known. In general, these equations are solved numerically. For a system that evolves in time, you'll pick an increment. You take the initial data at t_0 and use it to approximate the solution at t_1, then use that to approximate the solution at t_2, and so on until you go as far out as you need. At each step you introduce an error and a big part of numerical analysis is figuring out what happens to this error when you take a large number of steps.

It's a feature of chaotic systems that this error grows exponentially. Even a floating point error in the last digit has the potential to rapidly grow and come to dominate the calculation. In the words of Edward Lorenz:

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Isn't that true by definition? SCOTUS is the final authority on what is constitutional, no?

It seems cleanest to see Constitutional as a two-place word, and to point out that the government's written policy is to accept CurrentSCOTUS!Constitutional as the binding word. (The SCOTUS can overrule a previous version of itself, for example, which means it's not quite final.) It's popular to describe SCOTUS decisions as "morally wrong," but more relevantly, it seems that they could make decisions that are "logically wrong" and thus aren't Constitutional in some other important sense.

There's also commentary here and there about what the Constitutional duties of the non-SCOTUS arms of the government are; the President does have, as part of his oath of office, defending the Constitution, which presumably could require him to stop an insane SCOTUS out to wreck everything, but mostly people discuss in context of presidents signing laws they believe to be unconstitutional.

The debate over what is right is different from the debate over what is legal. Laws are generally written in an attempt to reflect what we believe is right. If a conflict should later appear, then the appropriate course of action is to change the law. It's a very dangerous precedent for a government to openly flaunt laws on the grounds that it's "right" to do so.

This is a great point. Other than fairly easy geometric and time symmetries, do you have any advice or know of any resources which might be helpful towards finding these symmetries?

Here's what I do know: Sometimes you can recognize these symmetries by analyzing a model differential equation. Here's a book on the subject that I haven't read, but might read in the future. My PhD advisor tells me I already know one reliable way to find these symmetries (e.g., like how to find the change of variables used here), so reading this would be a poor use of time in his view. This approach also requires knowing a fair bit more about a phenomena than just which variables it depends on.

PCA and other dimensionality reduction techniques are great, but there's another very useful technique that most people (even statisticians) are unaware of: dimensional analysis, and in particular, the Buckingham pi theorem. For some reason, this technique is used primarily by engineers in fluid dynamics and heat transfer despite its broad applicability. This is the technique that allows scale models like wind tunnels to work, but it's more useful than just allowing for scaling. I find it very useful to reduce the number of variables when developing models and conducting experiments.

Dimensional analysis recognizes a few basic axioms about models with dimensions and sees what they imply. You can use these to construct new variables from the old variables. The model is usually complete in a smaller number of these new variables. The technique does not tell you which variables are "correct", just how many independent ones are needed. Identifying "correct" variables requires data, domain knowledge, or both. (And sometimes, there's no clear "best" variable; multiple work equivalently well.)

Dimensional analysis does not help with categorical variables, or numbers which are already dimensionless (though by luck, sometimes combinations of dimensionless variables are actually what's "correct"). This is the main restriction that applies. And you can expect at best a reduction in the number of variables of about 3. Dimensional analysis is most useful for physical problems with maybe 3 to 10 variables.

The basic idea is this: Dimensions are some sort of metadata which can tell you something about the structure of the problem. You can always rewrite a dimensional equation, for example, to be dimensionless on both sides. You should notice that some terms become constants when this is done, and that simplifies the equation.

Here's a physical example: Let's say you want to measure the drag on a sphere (units: N). You know this depends on the air speed (units: m/s), viscosity (units: m^2/s), air density (units: kg/m^3), and the diameter of the sphere (units: m). So, you have 5 variables in total. Let's say you want to do a factorial design with 4 levels in each variable, with no replications. You'd have to do 4^4 = 256 experiments. This is clearly too complicated.

What fluid dynamicists have recognized is that you can rewrite the relationship in terms of different variables, and nothing is missing. The Buckingham pi theorem mentioned previously says that we only need 2 dimensionless variables given our 5 dimensional variables. So, instead of the drag force, you use the drag coefficient, and instead of the speed, viscosity, etc., you use the Reynolds number. Now, you only need to do 4 experiments to get the same level of representation.

As it turns out, you can use techniques like PCA on top of dimensional analysis to determine that certain dimensionless parameters are unimportant (there are other ways too). This further simplifies models.

There's a lot more on this topic than what I have covered and mentioned here. I would recommend reading the book Dimensional analysis and the theory of models for more details and the proof of the pi theorem.

(Another advantage of dimensional analysis: If you discover a useful dimensionless variable, you can get it named after yourself.)

This is part of what makes consciousness a hard problem. Since consciousness is responsible for our perception of the world, it's very hard to take an outside view and define it in terms of other concepts.

Really? What's the quale of a number? I think we can investigate consciousness scientifically precisely because science is one of our very few investigation methods that doesn't amount to introspecting on intuitions and qualia. It keeps working, where previous introspective philosophy kept failing.