That sounds more like (philosophical) skepticism to me. ;-)

I don't know, or care, whether atheism "is a religion." In fact, I don't even know what that sentence means.

What I do know is that, like the religions, it is becoming a group identity -- an "-ism" -- as evidenced by the extremely defensive posts being made here. If it were just a collection of ideas relating to abstractions, if people didn't identify with those ideas, if people didn't see attacks on those ideas as attacks on themselves, then nobody would care enough to get angry.

Maybe that's ok. Maybe it's useful. Maybe, most atheists grew up in staunchly religious communities, and the politics of group identity, of belonging to an oppressed minority, are helpful to resist a more generally destructive culture of religious bigotry.

But for those of us who were lucky enough to grow up in a secular environment, it gets annoying. Me? I don't need to "fight back." I'm not so afraid of the concept of God that I need to destroy it. It's an abstraction. Asking whether it exists is meaningless. Do the integers exist? Mu. I like Spinoza. I'm cool with panpsychism (what makes your unfalsifiable worldview better than mine? Maybe contemplating my part in Infinity alters my outlook.). We can flirt with ideas without marrying them. Unitarian Universalists? Sometimes too New-Agey for my tastes (For me, "energy" is measured in Joules), but I think the basic idea is the right one. Jesus of Nazareth? He did say things worth hearing. The Beatitudes? The Golden Rule? I don't need to accept Old-Testament jingoism, or Paul's sexual issues, or the dogma of a politicized medieval Church, or the divinity of Christ, to recognize that they stand on their own merits (and probably predate Jesus, which is OK).

The other day, I saw a car, with two bumper stickers. One was the common "CoEXiSt" sticker. The other was a shot at Christians. They're at odds, no? Get along, I say.

Yeah... It seems reasonable to guess that the effective power of an army grows quadratically for small numbers of units, but more-or-less linearly after that -- and terrain advantages like chokepoints can shift when that transition between quadratic and linear growth happens (E.g., if three units can get through a chokepoint at a time, then the transition probably happens around three).

I suppose that if you really want an answer, you need to do some experiments, and compile some statistics! Custom maps seem like a good way do this... I'd be surprised if hardcore Starcraft players hadn't already done these kinds of studies...

Ok, let's see... In your model, a group of n units has fighting power proportional to n*sqrt(n). Good! As expected, it's slower-than-quadratic, but faster-than-linear. Sounds like what I read people measure empirically.

Specifically, say

dx/dt = -a y^q

dy/dt = -b x^q

for some a,b,q>0; for you, q=1/2. Then the quantity

D = a y^(q+1) - b x^(q+1)

is conserved.

More generally, if

dx/dt = -f(y)

dy/dt = -g(x)

then, letting F and G be antiderivatives of f and g respectively, the quantity

D = G(x) - F(y)

is conserved.

You say "employment simulation?" I say "crowdsourcing!" What's a better simulation than the real thing?

In fact, let's get rid of the whole "we might hire them" thing entirely (but don't remove that text from the website).

Best. Post.

I hadn't seen that '08 study.

Sounds like you travel in defense circles, so you've probably seen this, but I'll also point out the 2002 Millennium Challenge, where, also, technologically-superior "American" forces lost out to numbers and swarming tactics.

Indeed! There are (admittedly very simplified) models of combat that indicate that the power of a fighting force is proportional to the square of its number of members.

This is something that I stumbled across when developing simple ODE models of Starcraft combat, and later discovered is known as Lanchester's Square Law. The idea is simple: Suppose you have two opposing groups of identical combat units, with x and y members, respectively. If you assume that all units concentrate fire on the weakest enemy, then the rate at which enemy units is depleted is proportional to the number of units you have, and vice versa. In symbols,

dx/dt = -y

dy/dt = -x

It turns out that the quantity D = x^2 - y^2 is conserved by this system (to verify this, just differentiate D with respect to time, use the product rule, and substitute in from the ODEs). What this means is that the fighting power of a fighting force is proportional to its square, and when the smaller force is eliminated, the larger force will have lost as much fighting power as the smaller force had, in order to defeat it.

You can modify the equations to include constants that reflect unequal kill rates, but you will find that the equivalent conserved quantities still depend quadratically on the number of units, but only linearly on the kill rate coefficients. The conclusion to be drawn is that, given a choice between a unit that's twice as effective, and twice as many units, you should choose to have twice as many units.

All this is predicated on the accuracy of the mathematical model, of course, and that model, I freely admit, is a rather drastic simplification. However, its aesthetics are appealing, and I think it may have a grain of truth. If it does, than Rafales or Super Hornets may indeed be the better choice than F-35s.

Hmm... I do guess that's true.

Me, I'd been comparing engineering education in the US to engineering education abroad -- but that's mostly in college. The American students consistently have more practical experience, have done more projects, and have been more frequently required to invent creative solutions to problems, than many of their Indian and Chinese peers. Not because the Americans are "inherently" better -- whatever that means -- but because engineering school just works differently here.

But elementary school? I think I agree. I think it's highly variable (e.g., there are good public schools in high-property-tax areas, and private schools like Montessori Schools), but I think I agree that, even when they are good, it's only by overcoming a tradition of rote learning which still dominates -- in practice if not necessarily in theory. I am also under the impression that, until 'No Child Left Behind' emerged, elementary education had improved significantly over that of two or three generations ago. Nevertheless, yes, elementary education is definitely as much about socialization as it is about academic learning -- for both good and ill.

Finally, there is an element of tracking in education. If you were a "smart kid," if you got into honors classes, you probably were able to have a high school experience that avoided some of the rote learning that other kids were subjected to. That was my experience, at least. But, again, it doesn't happen until high school.

Overall? No. I'd say the US has been much better in this respect than many other countries. (Though "No Child Left Behind" has done its damndest to screw that up by encouraging teachers to teach-to-the-test.) However, it is like this for premeds, and that's what matters!

Why? The stakes are too high. Push up the stakes high enough, and people don't think; they memorize. Indeed, when faced with very high incentives in psychological studies, people bomb IQ tests. You can't think when something as important as a career as a doctor is on the line. (That's why classes need to be exactly as hard as necessary -- and no easier -- but also no harder!!)

It's also how biology is taught in college. "Go memorize this arbitrary chemical pathway. No, we won't talk about 'why.' Yes, you can forget it later. We all know this class is just for weeding, anyway." Partly because it's all premeds. (And partly because there's no helping the fact that, compared to physics, biology is much more about facts than principles. It's messier. Such is life.)

A lot of the algorithms that get used for formation control are designed, inherently, from a distributed point of view -- meaning, they're based just on relative distances, etc, between the different quadrotors, and could run locally on them. However, when it comes time to actually implement this stuff, it's easiest to just run everything on a PC and use a mocap system, since that's usually viewed as a sufficient proof of concept within the community. There are groups in robotics who have strapped Kinects and laser range scanners to quadrotors to do things like SLAM, so the thinking at a place like the GRASP Lab is that, since other groups are doing this perception work, they don't need to bother with it, and can focus on the part of the problem that's their niche.

It would be nice to see a setup with truly distributed sensing, but the incentives aren't really there to bother.

The part of the GRASP Lab's quadrotor work that has impressed me the most is simply the controllers they have for their quadrotors. They're not like wheeled robots in that respect; they're not even stable, passively. The lab's earlier videos (e.g., "Aggressive flight maneuvers") are still very cool. Certainly not dealing with perception parts of the problem, but that wasn't the point; the controllers were.

Of course, that's past research. What about this work? I assume it builds on those earlier controllers, but it may well be doing interesting things besides. I'd need to take a look at their new publications to see what's going on under the hood.

Yes! Which seems related, and is itself remarkable...

There is also a (developing) theory of abstraction and bisimulation. I don't know how helpful it is...

Yeah -- practically, I think it is. I mean, that's even what machine learning algorithms do, in essence: They assume a prior that assigns higher probability to lower-complexity models. The details of what Occam's Razor means then becomes a subject for debate -- there are lots of priors one could choose -- but, as an imprecise, guiding principle, it seems to do the job!

More philosophically, I'd say Occam's Razor has a dual, which is the idea that Asimov called The Relativity of Wrong. Put them together and you're looking at the tradeoff between model complexity (Occam's Razor) and model fit (Relativity of Wrong), that is precisely what the theories of learning complexity explore.

I think I agree, but I also think that the the lack of "understanding" -- the lack of "mind-sized models" -- is going to get more and more frustrating! (If unavoidable.)

It also seems that reductionalism and holism can and do complement one another. E.g., the results of simulations involving various "reductionalist" pieces can be used to refine those "reductionalist" pieces themselves. The simplest example of this would be, e.g., if there were a real-valued signal of which you had noisy measurements at different times ("local," "reductionalist" information), and then you also were able to obtain an independent measurement of its integral ("global," "holistic" information). Knowledge about the integral would improve your estimates of the signal values themselves. And all we need for this example is standard, least-squares, linear estimation.

Wait. Really? There are entire fields that do nothing but systems theory. The names shift. Cybernetics. Systems theory. Control systems. Complex networks. Cyberphysical systems. There are lots of people doing work in precisely the areas you suggest. Take a look at the NSF's "Broad Agency Announcements." There is funding.

...

I do find it a bit amazing that science works at all. In machine learning, there are notions of the complexity of learning, and one of the basic ideas is that, as the class of models you are willing to consider grows, the amount of data you need to be sure, with reasonable statistical significance, which of those models describes it, grows very rapidly -- so rapidly that it is a miracle that we have apparently learned anything at all. See "VC dimension," "Rademacher complexity," etc.

The best explanation I can come up with is that the class of physical theories the human mind can conceive is actually quite limited (or, our priors are very good), and that it is evolution, over millions of years, that has gathered the necessary data to build a brain capable of conceiving of only the right theories, and that the role of conscious experimentation is only to narrow things down within this already-restricted set.

Because if the human mind is not much more limited than we like to think, then I do not know how we know anything.