Monty Haul

(Warning: this one involves probability, so if you don’t understand probability you may not follow it. I’ll have to do one on probability at some point.)

It’s called the Monty Haul Problem, but the heck with that. Let’s tell a story.

A travelling salesman stops for the night at a farm, and the farmer informs him he’ll be sleeping in the barn. “There are three doors in the barn,” says the farmer. “Behind one is the comfiest bed this side of the Ozarks, but behind the other two are heaps of dung. Now, I’ll let you pick a door, and then I’ll tell you one of the doors you didn’t pick has dung behind it, and if you want to switch to another door, you just say the word. But once you’ve decided on a door, that’s where you’re sleeping.”

The salesman picks door number one. “Well, door number three has dung,” says the farmer with a grin. “Now, you sure you want door number one.

What should the salesman say here?

Before you break something raising your hand, I’ll tell you: he should switch to door number two, and not because I know that that’s the one with the comfy bed. I don’t know; I’m not the farmer. We’ll ask him in a bit.

Turns out, if someone gives you a choice between two hidden bad options and one hidden good one, then takes away one of the bad ones and asks if you want to switch, you should. It’s not a guarantee, but it’s a question of odds. And no one believes this.

Let me try to explain.

You have a 1/3 chance of picking correctly, right? If you get to pick and then let it ride, you still have a 1/3 chance of being right, right? Wrong.

Let’s examine: by removing one bad option, the farmer has made it so that if you switch, the only way you lose is if you picked right in the first place, right? If you pick right, then switch away, you’re guaranteed of a loss here and winding up sleeping in dung, but if you pick wrong, because there’s now only one other option, the right one, you will be sleeping pretty.

So accepting that the farmer has changed the rules of the game from “randomly pick the correct door” to “randomly pick the incorrect door, then switch to the guaranteed correct door,” you can now hopefully see that in the first choice, your chances of picking the incorrect door are 2/3. Now I will take 2/3 over 1/3 any day of the week.

But let’s frame it another way, if you’re still having trouble.

Suppose, instead of one salesman, we have three, one for each door. None of them know anything more than the first, and they all get the same spiel. And since I’ve asked the farmer, the bed is behind door number one. As we’ve seen, salesman 1 picks door one, but then because he was paying attention, he changes to door two. Dung for him. This isn’t shaping up well.

Salesman 2 picks door two. The farmer shows that door three is dung as before. Salesman 2 changes to door one. Winner winner chicken dinner.

Salesman 3 picks door three. The farmer shows that door two is dung. Salesman 3 changes to door one. Also sleeping pretty (not together; these are on successive nights).

Two of the three salesmen won by changing; one lost. If they’d all stayed put on their choices, two would have lost and one would have won. If you take this and extrapolate out, we get that 2/3 : 1/3 division again. I could do a random distribution of beds, picks, and removals (bearing in mind that the removal must be a dung door) and get the same breakdown, give or take a few decimal places.

Counterintuitive, ain’t it? Don’t feel bad; some terribly smart people have agreed with you.

Weight, Gravity, and Phantom Phorces

Everyone who talks about zero gravity is wrong. Just because you don't feel something doesn't mean it's not there. If you jump out of a plane, you won't feel gravity; it'll feel like a strong wind is pushing you up, if anything. But guess what, Jackson; whether the ground hits you or you hit the ground, the two of you are going to collide and it'll be gravity, the force you couldn't feel, making it happen.

It is technically correct to say that one is experiencing zero gees, in the same way that g-force can be used to talk about acceleration, but that’s actually inertia acting on your body rather than an increase or decrease in gravity. But even in a case where one isn’t experiencing gravity, it’s still present, but from your relative perspective it doesn’t act on you any differently than your surroundings.

We’ve all had a moment of low-g, either when driving over a hill too fast or in an elevator or even on a roller coaster or some other ride designed to cause its riders to experience various g-force effects. It’s that momentary drop when your stomach turns and you feel like you’re rising out of your seat. Actually, that was negative g-force; your body was still going up, but your context began moving down, so you briefly weighed less than nothing.

How is that possible? You weigh what you weigh, right? When you step on a scale, the scale says you weigh some number of pounds (or kilograms or quatloos or whatever) and that’s that. That’s how heavy you are. Sure, on other planets you weigh a different amount, but those are other planets.

They are other planets, and the reason you weigh a different amount on those planets is because their gravitational pull is greater or less. But gravity isn’t weight. Gravity is just a force like any other, and there are plenty of others, like inertia or jetpacks or being punched by Drederick Tatum, all of which could cause you to “weigh” something.

Weight is merely the effect of a force acting on a mass. Yes, if you jump onto a scale it will register your "weight" as being higher when you land than after the force has been dissipated by friction and the mass of the planet, but you can cause a scale to register "weight" upside down by pushing against it; stick a bathroom scale to your ceiling (don't really do this) and then tie yourself to a pulley and hoist yourself into the air upside down until you can put your feet on the scale and push up with them, and you'll have weight according to the scale. In fact, you'll have weight equal to the force with which you are pulling up minus the force with which gravity pulls you down.

In fact, you can do conversions from pounds to Newtons because "pounds" are just a measure of force, which is why you can take a scale (a hanging scale this time), tie one end to a stationary object and the other to a rope, and measure the force of of a pull on that rope. Surely you've done this in the grocery store: you pull down on the hanging scale a little and make the arrow move even though you're not actually "weighing" anything. Scales are scales, whether you use them to measure weight or call them force gauges and measure force.

Weight is basically a construct which is convenient; from weight and the specific force of gravity of the place where the weight was measured, you can determine mass by conversion. That’s how most scales work; they’re telling you your mass based on the fact that you’re close enough to sea level to be within the margin for error, so if the scale is calibrated with that in mind, it can convert to mass from weight.

In fact, since gravity is usually considered as acceleration (9.8 m/s^2), one could call “weight” the force which causes that acceleration. But that would imply that weight is gravity, since gravity is a force which causes the acceleration caused by gravity. Are you confused yet?

Actually, “weight” is just a measurement of force. If you can use a scale to measure any kind of force, all that you’re doing when you measure “weight” is measuring the force with which the object on the scale is being pushed against the scale. In most familiar cases, that force is gravity. But it doesn’t have to be, and it doesn’t have to be such an artificial situation as strapping yourself to the ceiling with ropes.

Remember 2001: A Space Odyssey? If you don’t, you probably still know about the scene where the astronaut is jogging around a circle, only he’s jogging on the inside of a circle. If he stopped, he could stand inside that tube without difficulty, and if he put a scale on the ground he could weigh himself, even though we know he’s in outer space where gravity is, if not 0, at least small enough that it wouldn’t make him weight enough to register on a scale. So he has weight; that’s what’s keeping him standing rather than just floating around. But it’s not gravity.

What’s giving him weight is the centrifugal effect (not a force, which I’ll discuss elsewhere, but for the purposes of this discussion it’s acting as one). A force is pressing him against the scale. Thus, weight would, in this case, be a measurement of that force, rather than gravity.

Similarly, if you were to stand with your head facing the nose-cone of a rocket as it was launching, a scale beneath you would register a weight which was more than you typically weigh. That’s “feeling gees.”

The crazy thing is that (ready for the brainfuck) if you were to spin up a tube to a constant rotational speed, then get inside it and have it dropped from an airplane while still spinning that speed, you’d feel the same weight you would in space. If you’re in a situation where you would otherwise be “weightless” and you do something which would give you “weight,” you’ll experience it the same way you would have had you actually been in a situation with no gravity.

Astronauts are living that particular brainfuck constantly; in orbit, you’re not outside of gravity at all, you’re just in constant free-fall. It’s like The Hitchhiker’s Guide to the Galaxy says: "There is an art to flying, or rather a knack. Its knack lies in learning to throw yourself at the ground and miss. ... Clearly, it is this second part, the missing, that presents the difficulties." That’s what orbit is, actually; if you accelerate just enough so that as you fall toward the planet you’re always just missing it, you’ll be in orbit. A strange definition, but oddly an accurate one.

But if gravity is a force that attracts, and a rocket is a force which propels, how can the two of them wind up with the same results, relatively-speaking (to say nothing of the more bizarre case of rotation, which isn’t necessarily a force at all). It all comes back to Newton’s Laws of Motion, specifically #3. In the case of the force pushing against you, the scale is being pushed against you, and you are pushing back against it. In the case of gravity, it’s pulling you against the scale, which means you’re pushing against the scale.

Relativity means that, if the thing which gravity would pull you against is being pulled with the same force by gravity, then with respect to it, you aren’t pushing it at all. So in a room sitting on the ground where the ground is keeping gravity from accelerating the floor toward the center of the Earth, you have weight, but if you drop that room out of a plane, you and it will both be accelerating under the same force and you’ll have no weight in that context.

Okay, so that makes sense for gravity. But why isn’t it the same way with the rocket? You’re being pushed along with the rocket, right? Why isn’t it the same for the spinning tube? You’re part of that tube, so why do you have weight?

The answer is inertia, and to talk about that, we have to talk about another phantom phorce, in this case centrifugal. Because if weight is just a way of measuring force, what is it measuring when you’re spinning with no force being applied? There must be some kind of force which is pushing you against the wall of the tube, right?

The Magical Idealized Land of Physics

If you stick around in physics long enough, you'll realize that most of what physicists say doesn't work in the real world.  Or rather, most of it isn't exactly right, though "exactly" can mean anything from, "to within a micron or so," to, "unless we're talking about someplace with air."  Even the things which are good enough for everyday use in the real world aren't quite right because to make them account for complications would make them unusably complicated.

So if you're never going to fly a spaceship while trying to boil an egg back on Earth, you can usually get by with The Magical Idealized Land of Physics™ or The MILP as I shall henceforth refer to it.

In The MILP, anything that we don't care about doesn't happen.  Unless we're talking about Relativity (and if we are, we're probably ignoring other things) The MILP doesn't experience relativistic effects; no time dillation, a fixed point of reference (usually the observer of whatever it is we're talking about), no General Relativity at all.  Unless we're talking about friction, there's usually no friction, and if there is, it's only the type of friction we care about.  Cosmic constants stay constant even if we're not at sea level (believe it or not, gravity affects you slightly less on Mt. Everest than it does at the bottom of the Mariana Trench).

So, unless a physicist says otherwise, they're usually talking about The MILP.

And this is a good thing.  Because without The MILP, I couldn't talk to you about physics because I wouldn't understand it, or I would have invented The MILP for myself just so I could reduce the complications down to a point where I could understand things.  The MILP makes it possible to talk about the real world without dealing with all the crap the real world wants to throw at you.  All scientists use some kind of MIL, be it the MILB for biology or the MILC for chemistry, etc.

This has implications to philosophy too: it turns out that the only way to represent a thing completely is with the thing itself.  And if all you can do to explain the world is to point and say, "There's my explanation," any time anyone asks you for a reason, that's not terribly helpful.

So embrace The MILP™ because it's the only way to talk about anything.

Inertia and Newton

Sir Isaac Newton was wrong about pretty much everything about which he was right.  His calculus was brilliant, but his crazy notation for it set English mathematics back decades if not centuries.  His theory of gravity is sound enough, but only as an abstraction.  And his Laws of Motion were revised by Euler and then made quaint by relativity.  And yet we keep talking about him.  Why?  Because, like most things, unless you're a purist, a rough abstraction which accurately describes the situations in which you find yourself is good enough.  Hell, even purists have to deal with differential equations, which are one hand-wave away from being well-informed guesses.

I don't have much patience for the arcane if it can't be made plain, and even Newton can be arcane, so let's see if we can't demystify him slightly.

Via Wikipedia:

Law I: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.

Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.

Law III: To every action there is always an equal and opposite reaction: or the forces of two bodies on each other are always equal and are directed in opposite directions.

Yeah.

In the first law, everyone says, 'A body at rest tends to remain at rest, and a body in motion tends to remain in motion.'  Which gives the impression that he meant 'tends to' (which he never actually said) to mean that an object wants to remain in some way.  Objects don't want.  The universe doesn't want.  "Tends" in this case simply means "has a tendency to," which means "will, all things being equal."

So, rather, let us say that an object's state of motion, either at rest or moving in a straight line, doesn't change without something making that change.  If something is moving, it doesn't slow down or speed up without something making it slow down or speed up, and if something is still, it doesn't start moving without something making it move.  Seriously, it took all of human history up to Newton to write that down.  People are idiots.  It seems screamingly obvious when you look at it; things don't change unless something changes them.

But it holds a key idea: inertia. The thing is, what is it that makes objects stay in their state of motion?  It's not a desire, either on the part of the object or the universe.  It's a property of mass, an energy that mass has.  Crazily enough, no matter how fast an object is going, its inertia never changes; therefore an object always requires the same amount of force to accelerate the same amount.  But we'll get to that in a minute.

People also use "inertia" to mean "momentum," which it isn't.  An object in motion has momentum, which increases proportional to both mass and speed.  A car with half the mass of another car travelling at twice the speed will have the same momentum and would exert the same force when it hit a wall.  But it would take 1/2 the force to accelerate the car with half the mass, no matter how fast or slow the two were going.

Which leads directly to our second law, which is typically expressed as a math equation, F=ma.  Or rather, it should be a = F/m, as if we were saying that "alteration is proportional to the Force motive," which is what Newton said.  But what it really means is that acceleration is proportional to force as it acts on mass.

What does it really mean?  It means that if you push something hard, you'll accelerate it faster than if you push it more softly.

When I say it like that, again, you're thinking, "What the hell, rest of history leading up to Isaac Newton?"  But it's an important thing to know; if it weren't true we'd be like really bad golfers who, when they hit the ball, have no idea how far it will go.  There would be no correlation between the amount of force applied to an object and the change in its motion.

Newton also adds the part that most people take for granted or don't understand; if you push an object in a direction, it'll accelerate in that direction.  Again, no big news there, but that means that, in our crazy world of vectors (that's a quantity which has a direction) rather than scalars (which is just a quantity) we don't have to worry about throwing a ball and having it go off at a right angle to the direction we threw it (unless we're really bad at throwing balls, but that's not a violation of the basic laws of physics).

The third law is the trickiest and the easiest to wave your hands at and explain away, and yet it gets quoted a lot, particularly in places where it doesn't belong.  If you're using Newton's Third Law of Motion to describe love, or politics, or really anything but Newtonian physics, you're doing it wrong.  In Newtonian physics, however, for every action there is an equal and oposite reaction.

Why?  That's pretty silly if you think about it; that means that if I push against a wall, the wall pushes against me.  if I throw a ball, the ball throws me.  In Soviet Russia, car drives you.  What, Newton was doing Yakov Smirnov?

Let's break it down here; no need to consider any force but pushing, because everything follows from that.  So if you apply force to an object, the object applies force back.  It's almost relativistic.  Consider; if you push against a wall, does it feel any different than having the wall push against you?  It does because your muscles are exerting that force, so you feel them working.  But supposing you have to hold something up.  Suppose the wall was about to fall on you.  The only way you'd be able to keep the wall in place is to push back with the same force as it's pushing against you (gravity is a force; the "Force of Gravity" isn't about the Jedi).

That push-back comes from inertia.  Yes, if you push harder, the push-back is equal (and opposite, which means that it won't push sideways because vectors complicate things).  But the acceleration caused by that force will be dictated by inertia.  So, in The MILP™, if you push against an object with exactly the same mass as yourself, both you and it will move away from each other with equal acceleration and wind up traveling at the same speed in opposite directions (let's leave relativistic motion out of this for the moment).  If you push an object half your mass, it will move away at twice the acceleration as you and wind up traveling at twice the speed you do (but, in that perfect ideal, you'll still wind up traveling at the same speed as in the first example.  Why?  Because while the forces were equal, the inertia wasn't, which means that the same force acting on half the mass resulted in twice the acceleration.

That's the big takeaway from Newton, actually: objects have inertia.  You could, I suppose, phrase all three laws as just that.  But then you'd have to define "inertia' and you'd wind up saying, "Well, inertia means that objects at rest tend to stay at rest..." and you'd wind up defining inertia by Newton's Laws of Motion.