Friday, December 13, 2013

The Spherical Cows of Economics

Some conservative oppose all regulations by appealing to the free market. They say the free market is the best way of distributing goods, and any regulation makes it less free. Therefore regulations make the market less efficient, which is bad.

However, the free market is only efficient under certain conditions. For example, everyone involved has to have perfect information about the costs and benefits of their decisions. Another is that there must be no externalities, that is, all the costs and benefits of a decision must be borne by the people making the decision, not by third party bystanders.

The thing is, these conditions rarely, if ever, actually hold. They're the spherical cows of economics. And the free market is inefficient to the extent that these conditions are false.

But government regulations can help improve these conditions, and thus make the market more efficient. For example, by requiring drug manufacturers to disclose their drugs' side effects, they can help lessen information imbalance, allowing people to make more rational decisions. Further, they can use taxes and subsidies to internalize the costs and benefits of externalities.

So regulations do not make the free market less free. In fact, the free market needs regulations to be free.

Friday, December 6, 2013

De Morgan's Law and Duality

Before, I gave the challenge of replicating the boolean OR function using only NANDs (and replicating AND using only NORs). I'm going to show you the solution, using truth tables. A truth table is a table where you list out every possible combination of values of the inputs, and the corresponding value of the output. Here's a truth table showing the AND, OR, NAND and NOR functions.
XYX∧YX∨YX NAND YX NOR Y
FFFFTT
FTFTTF
TFFTTF
TTTTFF

You can show that two functions are equivalent by showing that they have the same output as each other for every possible combination of inputs. Here's a simple example to show that ¬X = X NAND X.

X¬XX NAND X
FTT
TFF

If you're not sure what one of the outputs should be, plug in the input values and evaluate it. F NAND F = T, and T NAND T = F, which you can see from the first truth table.

So, here's the solution to the first part of the challenge: X∨Y = (X NAND X) NAND (Y NAND). Here's the truth table.

XYX NAND XY NAND Y(X NAND X) NAND (Y NAND Y)
FFF NAND F = TF NAND F = TT NAND T = F
FTF NAND F = TT NAND T = FT NAND F = T
TFT NAND T = FF NAND F = TF NAND T = T
TTT NAND T = FT NAND T = FF NAND F = T

Try to figure out the second part of the challenge (That is replicate the AND function using only NOR) using a truth table now. I'll wait.

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The solution to the second part is X∧Y = (X NOR X) NOR (Y NOR Y). Interestingly, it's exactly the same as the first part, except with ∨ replaced by ∧ and NAND replaced by NOR. I'll talk about that more in a bit, but first notice that since (X NAND X) = ¬X, the solution to the first part can be rewritten as X∨Y = ¬X NAND ¬Y. Also, since (X NAND Y) = ¬(X∧Y), it can be further changed to X∨Y = ¬(¬X∧¬Y). The same way, the solution to the second part can be rewritten as X∧Y = ¬(¬X∨¬Y).

This is an example of De Morgan's Law, which says that ¬(X∧Y) = ¬X∨¬Y and ¬(X∨Y) = ¬X∧¬Y. De Morgan's Law is a very important thing to remember when doing Boolean algebra. It's a useful way of simplifying expressions, and it's vital for programmers to know.

De Morgan's Law also ties back into the other point I made. Notice that the two forms of De Morgan's Law are the same, except with ∧ and ∨ swapped? In fact, you can take any boolean expression, swap ∧ and ∨, and swap T and F, and the expression will mean the same thing.* For example, X∧T=X, swapped X∨F=X. Also, X∧F=F, swapped X∨T=T. This property is called duality.

Why does this happen? Keep in mind that the labels we use are arbitrary. It doesn't matter if we use T and F or 1 and 0, or if we use ∧ and ∨ or AND and OR. What matters is the relationships that hold between the symbols. And the way we've defined them, the relationships between T and ∧ are exactly the same as the relationships between F and ∨. That is, X AND Y is true if and only if both X and Y are true. X OR Y is false if and only if both X and Y are false. Those say exactly the same things, just with the names changed.

*If the expression contains XOR, or other functions besides ∧, ∨ and ¬, those need to be rewritten to use only ∧, ∨ and ¬. Or they can be swapped with their own dual, for example, the dual of XOR is XNOR. ¬ is it's own dual.

Monday, December 2, 2013

What Goals Should You Have?

Last time, I ended with the question, "What goals should we have?". Before, I said that "should" only makes sense in reference to goals, so how can this question be answered?

Obviously, using a goal to justify itself is circular reasoning. But you could justify a goal using other goals. If a goal helps you achieve your other goals, you should have it, in the same way you should do anything else that furthers your goals. Conversely, if a goal hinders your other goals, you shouldn't have it, in the same way you shouldn't do anything else that hinders your goals.

But then, what about the other goals? How do you determine whether or not you should have them? In the same way, referring to each other. This will form an infinite regress of self-reference, but that's not necessarily insurmountable. It could probably be represented similarly to Google's PageRank algorithm, which determines the importance of a website based on the number and importance of websites that link to it.

But doesn't that end up being just as circular as before? Well... Yes. And given two or more sets of goals which support each other equally well, and the unlimited ability to modify your goals, I don't know how you could determine which set of goals you "should" adopt. For that matter, I don't know what "should" means in that context.

But as it happens, I don't think we do have the unlimited ability to modify our goals. I think our goals are at least partially constrained by our biological nature. We need to eat. You can deliberately refrain from doing so, but I think that's more acting on a conflicting goal rather than not having the goal to eat.

And if that is the case, then the goals you can't change give you a starting point to base the goals you can change around.

Thursday, November 21, 2013

Boolean Algebra

You may have heard that computers only use the numbers 0 and 1. It's true. Everything computers do is done using electronic switches, and those switches can only be either on or off. (That's not strictly necessary. It's possible to make an electronic switch that has in-between states, but on/off is the simplest and easiest way to make a switch.) But how can computers calculate larger numbers or do all the cool things computers do, if they can only use 0 and 1? Well, one of the fundamental ideas behind how computers work is Boolean algebra.

Boolean algebra is a type of math that only uses two values. These two values are usually called true and false, or 1 and 0. There are three basic operations of Boolean algebra.

NOT X, also written as ¬X is simply the opposite of X. If X is true, ¬X is false. If X is false ¬X is true. Unsurprisingly, ¬¬X=X.

X AND Y, also written as X∧Y is true if both X and Y are true. Otherwise, it's false. The AND operation is also sometimes written as multiplication, because it works the same way. 0*0=0, 0*1=0 and 1*1=1. And just as with multiplication X*0=0 and X*1=X, no matter what X is, X∧F=F and X∧T=X, no matter what X is.

X OR Y, also written as X∨Y is true if X is true, Y is true or both are true. The OR function is sometimes written as addition. Similar to normal addition, 0+0=0, 0+1=1 and 1+1=... Well, there's no 2, so 1+1=1. This has the properties that X∨F=X, just like X+0=X, and X∨T=T, unlike normal addition.

You can put these operations together to get more complicated operations. For example X XOR Y, also written as X⊕Y, is true if X is true, or Y is true, but not both. It can be built out of the other operations like this (X∨Y)∧¬(X∧Y). Similarly, NAND, NOR and XNOR are made by putting a NOT in front of an AND, OR and XOR, respectively.

Although I said there are three basic operations of Boolean algebra, there really only needs to be one. You can make every single Boolean operation out of just NAND or just NOR. For example, ¬X = X NAND X. X∧Y = ¬(X NAND Y) = (X NAND Y) NAND (X NAND Y).

Here's a challenge for you: How can you make the OR operation, using or NAND, or complementarily, how can you make AND using only NOR?

Wednesday, November 13, 2013

The Fundamental Question of Morality

The more I think about my last post, the more I think that "What should we do?" is the most fundamental question of morality. Should you kill one person to save five? Should you ever lie under any circumstance? It seems all moral dilemmas boil down to questions of action and decision.

But this alone doesn't really clarify matters much. It just pushes the ambiguity onto the word "should". What does it mean to say that you "should" do something?

Well, I came up with a partial answer here. As far as I can tell, for a "should" question to make sense, a goal is required. And I think in practice, whenever you make a decision, you do so for reasons, which can be described as goals.

But morality isn't just about how to achieve your goals. I think most people would say it's immoral for a sociopath to kill, even if that's his goal. In fact, I think most people would say it's immoral for someone to have such a goal.

So, maybe the most fundamental question of morality isn't "What should we do?" but rather, "What goals should we have?". But I just said "should" requires goals. How can you say what goals you should have without referring to goals? Is it even possible? If not, how can you answer the question?

Saturday, November 9, 2013

Wait, What's This Morality Stuff Again?

It turns out that building meta-ethics from the ground up is somewhat difficult. So I'm going to take a step back, and explicitly describe some fundamental ideas about what morality is. Hopefully this will help point in the right direction to look.

Morality is about answering the question, "What should we do?"

This implies that morality is about willful actions. Morality doesn't apply to things that can't act, and can't make choices.

But morality isn't just about how to achieve your goals. It has more to do with what your goals should be.

Who is left unspecified. Usually, that's interpreted to mean that the answer should be universal, that it should apply to everyone in the same way. I'm not so sure about that. Maybe it just means the question needs more detail to have an answer.

However, morality does seem to be concerned with controlling others' actions. It's frequently used to justify doing something "bad" to someone else if they did something "bad" to you. But then again maybe that's a misapplication...

So, that's not a lot, but it's something. Maybe it will be helpful.

Monday, October 28, 2013

Viewer's Guide to the ISS

Did you know the International Space Station is easily visible to the naked eye? In fact, at it's brightest, it can outshine Venus! The problem is that it moves very fast (17,130 mph) so you need to know when to look, or you'll miss it.

So, how do you know when the ISS will be overhead? One great resource is Heavens-Above. It will tell you exactly when the ISS will be visible, how long it will be visible for, and where in the sky it will be. It also has information on lots of other satellites and other things of astronomical interest.

But sometimes it will be months before the ISS passes over your location at night. You'll need some kind of reminder after that long. That's the purpose of Spot The Station. Sign up for it, and it will send you an email 12 hours before the next sighting.

There are also apps for mobile devices that are very useful. A good one I just got is ISS Detector.

Also worth mentioning is Wolfram|Alpha. It's less practical for finding good viewings, but it has lots of interesting information. Not just about the ISS, not just about satellites, not just about astronomy, it knows about pretty much everything.

So, go out and look at the sky. It's an awesome place.

Wednesday, October 23, 2013

Ode to the ISS

Tonight, I saw a point of light in the sky.

It could have been a burning ball of hydrogen, millions of miles across, and trillions of miles away. But I knew it wasn't a star, because it was much too bright.

It could have been a ball of rock, brightly reflecting sunlight off its clouds of acid. But I knew it wasn't Venus, because it was moving much too quickly.

It could have been a bit of dust, falling to the earth a hundred times as fast as a speeding bullet. But I knew it wasn't a meteor, because it lasted much too long.

It was a structure the size of a football field, made of aluminum, titanium and silicon, over two hundred miles above our heads, always falling, but always missing the ground, current home to six. It was the International Space Station.

Monday, October 14, 2013

Hypothetical Imperatives

In my last few posts about morality, I've been pretty confident about my conclusions. From here on out, it will start getting more speculative. Some of my ideas may have merit, some may not. Please, tell me what you think about what I got right, and what should be changed.

In my last post, I asked how you could derive "ought" from "is". For a certain type of "ought" that's not hard at all. That type of "ought" that is a course of action to accomplish a given goal. For example, if  your goal is to lose weight, you ought to eat less fat. It is possible to empirically observe and test that to see if it effectively achieves the goal. This type of "ought" is called a hypothetical imperative and it's existence is the logical consequence of the existence of goals.

But is that the type of "ought" we're interested in? If we're talking about morality, shouldn't we be talking about something that's objective, and independent of individual goals?

One way of being independent of individual goals is to relate to all goals. If there were such a course of action, you would clearly want to follow it, because it would be a way to achieve your goals, regardless of what your goals are. But it's clear that there is no course of action to accomplish any goal. No matter what course of action you propose, you can come up with a goal counter to it. For example. you can't say that you should never steal to accomplish any goal, because that doesn't effectively achieve the goal of stealing.

Another way of being independent of individual goals is to not relate to goals at all. But in that case, why should you follow such a course of action? That approach seems to fall headlong back into the problem we're trying to answer. There's no apparent way to derive such an "ought" from an "is".

So are we back where we started? Unable to connect morality to reality? Well, not necessarily. We failed to find a special kind of "ought" that in independent on individual goals. But we did find the hypothetical imperative, which is a kind of "ought". Maybe it's the only "ought" we need. Maybe morality can be built entirely out of hypothetical imperatives. Again the question remains: How?

Wednesday, October 9, 2013

The Is-Ought Problem

In my last post, I asked what empirical observations you would be able to make that would be different if morality existed than if it did not. Some of you may have objected that that defies the is-ought problem.

Hume said that you can't derive "ought" statements from "is" statements alone. That is, you cannot determine what is moral or how the world should be just from how the world is. For example, suppose a runaway trolley is heading toward people who are tied to the track. There is a lever, which if pulled will divert the trolley onto another track, where it will cause no harm. Those facts alone are not enough to conclude that you ought to pull the lever. You can only conclude that if you add another "ought" statement to the premise, such as, "You ought to save lives.".

If Hume is correct, then morality is not observable in any way. Which would mean either that morality does not exist, or that morality is meaningless. Either way, it would be completely pointless to talk about morality.

So why are we talking about it? For that matter, why would anyone ever talk about it? No one ever talks about unobservable flergamarms, why would they talk about unobservable morality? Why would they even come up with the idea? Maybe people are just hopelessly confused. Or maybe Hume is wrong. Maybe ought can be derived from is. The question remains: How?

Friday, October 4, 2013

Does Morality Exist?

Does morality exist?

Well, what do you mean by morality? Keep in mind that an answer like "Morality is what is good." or "Morality is what is right." don't really answer the question. That just shoves the ambiguity onto another word. What you need is an explanation that breaks it down and really clarifies it.

If you're not sure how to answer that (and I'm not), the next thing to do is look at what you can observe. What empirical observations would you be able to make that would be different if morality did exist than if it did not?

So, what do you think?

Sunday, September 29, 2013

Meaning

Do flergamarms exist?

Well, what do you mean by "flergamarms"? Or to put it more precisely, what empirical observations would you be able to make that would be different if flergamarms did exist than if they did not?

Flergamarms are invisible, intangible, and in fact, cannot be observed in any way. They have no observable effects, nor are they a logical consequence of something observable. As such, there is no such observation that would be different depending on the existence of flergamarms. So, do they exist?

Well, I wouldn't say they exist, but I'm also wouldn't say that they don't exist. Rather, I'd say that the question is meaningless. I would say that something can only have meaning if it's in some way connected to reality, and to be connected to reality, it has to be observable, or the logical consequence of something observable.

I'm not completely certain about that. Maybe there is some sense in which the term "flergamarm" is meaningful. I can't think of anything that could be though. And even if there is, there's still not much point in discussing such entities.

Sunday, September 22, 2013

The Autumnal Equinox

We live on the surface of a giant spinning ball. From our perspective, everything else appears to move around us as we spin. The sun, the moon, the planets, the stars. As we spin to face the sun, it appears to rise the sky. As we spin to face away is appears to set. It is no coincidence that we spin around exactly once each day. It is our spinning that determines the length of the day.

But we don't just spin in place. As we spin, we move in a circle around the sun. But the way we spin is tilted compared to the way we move around the sun. And the direction we're tilted doesn't change as we move around the sun, which means that sometimes we're tilted toward the sun, and sometimes we're tilted away from it. When you're tilted away from the sun, the days are shorter and the nights are longer, which makes it colder. The winter solstice is the day when you're tilted exactly away from the sun. When you're tilted toward the sun, the days are longer and the nights are shorter, which makes it hotter. The summer solstice is the day when you're tilted exactly toward the sun. It is no coincidence we move around the sun exactly once each year. It is our moving around the sun that determines the length of the year.
Not to scale. Credit: NOAA
Keep in mind that when the northern hemisphere is tilted toward the sun, the southern hemisphere is tilted away, and vice versa. The summer solstice in the northern hemisphere is the winter solstice in southern hemisphere, and the winter solstice in the northern hemisphere is the summer solstice in the southern hemisphere.

But today, we aren't tilted toward the sun, or away from it. Today, we are tilted perpendicular to the sun. Today is the autumnal equinox, the day when the day is equal to the night.

The equinox is a time of change. It marks the midpoint in the transition from the summer solstice to the winter solstice. Beyond that, it is also an inflection point. After the summer solstice, the days get shorter. At first, only a little bit. One day will be only a few seconds shorter than the day before it. But over time, the change increases, until one day will be minutes shorter than the day before it. The solstice is the time of the fastest change. After the solstice, the days will continue to get shorter, but the speed of the change will slow down again.

Wednesday, September 11, 2013

Free Will

If a tree falls in the forest, and no one is around to hear it, does it make a sound? Alice says yes, because it would produce a pressure wave through the air. Bob says no, because it wouldn't be perceived by anyone.

Alice and Bob aren't disagreeing about what happens when a tree falls down, they're disagreeing about the meaning of the word "sound".

Do we have free will?

Well, what do you mean by "free will"?

One notion of free will is that it is the ability to make decisions that are neither deterministically caused, nor random. This concept is called libertarian free will (unrelated to the political party), and it doesn't make much sense to me. To start off with, what's the middle ground there? How can something be neither determined, nor random?

Further, a non-determined decision wouldn't resemble a decision at all. Consider this scenario: you're in a building when a fire alarm goes off. So, you make a decision to evacuate the building. The only reason to evacuate the building was because the alarm went off. If the alarm hadn't played a role in determining your decision, then you wouldn't have made it.

So, that means we don't have free will, right? Well, it certainly means we don't have libertarian free will. But when you say things like "Free will doesn't exist.", people tend to react by saying things like "That means we can't make decisions!".

Which isn't true. Things that are completely physically determined can still make decisions. Consider a simple thermostat. If the temperature is below 18°C turn the heater on, otherwise, turn the heater off. Admittedly, that's in the fuzzy area between "decision" and "reaction", but I don't think it's fundamentally different than an undeniable decision, just simpler. (Which leads to another question of definition. What exactly is a decision?)

The fact that people tend to react that way implies that the libertarian notion of free will isn't the concept most people mean by the term "free will". So, we should find another definition that more closely matches people's free will. Here's my suggestion - Free will is the ability to make very complicated decisions that are determined by a very large number of factors and is very difficult to predict in advance.

Under this definition, we obviously do have free will. This doesn't resolve the issue. I'm still not certain exactly what a decision is. But I think it's a good step forward that helps avoid common misconceptions.

Wednesday, August 28, 2013

Consciousness

What is consciousness? Where does it come from?

I don't know. But I do firmly believe that it is not magic. Whatever it is, it is purely physical.

And I do know a story about how it may have arisen. Don't take it as a scientific explanation. It's not. It's just a story, which might have a hint of validity about it, but then again, might not.

It begins with life. Living organisms capable of reproducing and reacting to stimuli. Organisms that react to stimuli appropriately would be more likely to survive and reproduce than organisms that didn't.

But what response is appropriate can be hard to determine. Is that a predator, or a mate? Should I go towards it, or away from it, or play dead? And so the brain evolved. An organic computer to make decisions.

For a brain to make good decisions, it needs to model the world around it, and predict consequences of different possible actions. If I jump over this hole, can I make it to the other side? If I chase that prey, will I be able to catch it? The more accurately the brain models the world, the better the decisions it can make.

Then a species of social primates evolved. For social creatures, it's helpful to model the others in your group with more detail than other animals, because you interact with them more often. The better your model of them, the more likely you can get them to share their food with you.

But since you're all the same species, they're modelling you too. So the most accurate model of them is to recursively model them modelling you. In other words your brain has to have a model of itself. And that's essentially what consciousness is. It's the brain thinking about itself, trying to predict its own actions before it makes them.

This story doesn't answer everything. Particularly not the hard problem of consciousness. But I think it does a decent job of addressing the easy ones. Assuming, of course, that's it's even remotely true, which it might not be.

Monday, August 19, 2013

Science and Wonder

It is a common notion that science takes the wonder out of life. A prime example is John Keats's poem, Lamia.
Do not all charms fly
At the mere touch of cold philosophy?
There was an awful rainbow once in heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel's wings,
Conquer all mysteries by rule and line,
Empty the haunted air, and gnom├Ęd mine—
Unweave a rainbow, as it erewhile made
The tender-person'd Lamia melt into a shade.
I think this notion is wrong. Science, when properly understood, doesn't destroy wonder, it enhances it.

First, I'd like to clearly separate two relevant meanings of the word wonder. The first is synonymous with awe, the feeling you get when you think, "That's really really cool!". The second is synonymous with curiosity, the feeling you get when you think, "I wonder how that works...". They frequently come together, but they don't have to. It's entirely possible to feel awe at something that you understand completely, or to feel curious about something isn't particularly awe-inspiring.

Science enhances the feeling awe, because it reveals nature, and nature is, well, awesome. The real world is far cooler and more interesting than any fictional world I've ever read about (which is not to say that fictional worlds can't also be cool and interesting). I've written about this before, and given several examples of real awe-inspiring things. Most of those things would never be known about without science. And you can't have a feeling of awe towards something you don't know exists.

Science enhances curiosity in much the same way. Every question answered by science uncovers still more to be asked. Questions you wouldn't even be able to ask before, since you wouldn't have known the concepts they apply to.

I think the reason Keats, and others who make this claim do so because of two mistakes. First, they don't realize that the feeling of awe can be separated from the feeling of curiosity. Second, they don't realize that answering questions you're curious about can uncover deeper questions. If those two things weren't the case, then science would destroy wonder. Fortunately they're not.

Wednesday, August 14, 2013

Law and Morality

People sometimes claim, usually in the context of things like gay marriage or abortion, that you shouldn't legislate morality. Usually, this is in response to the claim that gay marriage (or abortion, or prostitution, or whatever the topic at hand is) is immoral, and therefore should be illegal. But that's a bad response, because morality should be legislated.

Opponents of abortion are actually pretty good about pointing out the foolishness of this stance, by comparing abortion to murder. No one would say "If you don't approve of murder, then don't murder people." or "I don't like murder, but if my neighbor does, who am I to judge?" The response to that is usually that it's ok to have laws against murder because murder hurts people.

But so what? Why is ok to make a law banning something that hurts people? Because hurting people is immoral.

I think the main reason this argument is compelling is that it's usually used against people who are arguing against something that's not immoral. People say, "I think gay sex is gross, therefore it's immoral, therefore it should be illegal!". The argument is clearly flawed, but the flaw isn't with the second "therefore", it's with the first.

The problem is not with saying that something immoral should be made illegal. The problem is with misidentifying what is moral in the first place.

So, what is morality? How do you correctly tell if something is moral or not? That's a topic for another post, or perhaps a book.

Saturday, June 22, 2013

Membranes

As you may be aware, I like making geometrical constructions. Ever since I made an AlDraw app for Android, I've been using it to post constructions on Facebook. The other day I posted this construction, titled "Membranes".
This wasn't the first time I constructed this image, but when I reuse a construction, I still reconstruct it, rather than simply copying the original. But when I did that this time I discovered that I made the original one incorrectly. It wasn't exactly a mistake. I made it the way I had intended to, but I found out when I made a small change, it made the whole thing more elegant.

To explain how, I'll need to explain how I constructed it in the first place. I started with these two circles. The radius of the outer circle is R. The radius of the inner circle is R/2. They are both centered at (0, 0).
Next I drew circular arcs between the inner and outer circles. All three have radius R and are centered on the edge of the outer circle. The red at 0°, the blue at 30° and the green at 60°. I've shown the centers of the circles in the same color as the circles.
Then I added those same circles again, reflected around the x and y axes.
Then I wanted to connect the different arcs, and I want it to look smooth, so I want to make circular arcs that are tangent to the arcs they're connecting. How do I do that?

Well, if two circles are tangent, then a line drawn through the centers of both will pass through the point of tangency. So if you have a circle, and you know what point you want to be tangent, then you can draw a line through the center of that circle and through that point. Then any circle that is centered on that line and that goes through that point will be tangent to the first circle. And if you have two circles like that, and you want to find one circle that is tangent to both, then you can draw two lines. Where they intersect will be the center of the third circle.

So I'll start with the red circles. I drew lines through the centers of the red circles and the points where the red circles intersect the inner circle. Where they intersect is the center of the circular arc that connect the two arcs. Note that each red line comes close to where the other red circle intersects the outer circle, but not quite. Also, each red line comes close to where the blue circles intersect the inner circle, but again, not quite.
The I did the same thing with the blue circles. I drew a line through the center of each blue circle and the point where that circle intersects the inner circle. Note the intersection of the blue lines is close to the outer circle, but not quite on it. Also, each blue line comes close to where the green circles intersect the inner circle, but again, not quite.
The next and final step is where I noticed the mistake. Because the next step I simply drew straight line segments to connect the green arcs. The straight lines are nearly tangent to the green arcs. But not quite.
Now at this point, I could have simply found the circular arcs that would be tangent to the green arcs the same way I did for the blue and red. That would be one way of fixing it. But I wanted those green arcs to be connected by straight lines and for those line to actually be tangent to the arcs. And the only way to do that is the change the radius of the inner circle, which changes where the red and blue circles intersect it, which changes where the connecting arcs need to be. In other words, changing nearly the whole image.

So this time, I started not with the outer circle and inner circle, but rather the outer circle and one of the green circles.
Now I need to find the point on the green circle where a horizontal line will be tangent to it. A line drawn through the center of a circle will always be perpendicular to the circle and hence perpendicular to the tangent line where it intersect the circle. So what I need to do is draw a vertical line through the center of the green circle, and where it intersects the green circle is where a horizontal line will be tangent to it.
Then I can draw the inner circle with the same center as the outer circle and that goes through that point. A little bit of trigonometry shows that the radius of the new inner circle is R, which is approximately .5176R.
Then I drew the other circles the same as before, between the outer circle and the new inner circle.
Again, I drew lines between the centers of the red circles and the points where they intersect the inner circle. But notice, this time they don't pass close to where the other red circle intersect the outer circle, they pass right through it. Also, although the difference is too small to notice at this scale, the red lines also pass directly through where the point where the blue circles intersect the inner circle.
And again, I drew the lines between the centers of the blue circles and the points where they intersect the inner circle. And again, notice that they don't intersect near the outer circle, they intersect directly on top of it. Also, they pass directly through the points where the green circles intersect the inner circle.
And finally, I drew the lines connecting the green arcs, and this time, they're actually tangent.
It's interesting how that one little almost imperceptible change made the whole thing more elegant.






Saturday, June 15, 2013

Superman and the Physics of Collapsing Buildings

I saw the new Superman movie this weekend, and I liked it. But it wasn't perfect, and the thing that bothered me the most was the bad physics. I'm not talking about Superman being able to fly or the Kryptonian terraforming machine being able to increase Earth's mass. That kind of thing is expected in a superhero movie. I'm talking about more everyday physics. The most egregious example is skyscrapers falling over.

It happens multiple times in the movie. Superman throws a bad guy (or a bad guy throws Superman) through a skyscraper, part of the building is damaged, it tips and falls over like a tree. You might be wondering what's wrong with that. After all, trees fall down like that. If you build a tower out of Legos and knock it down, it falls down like that. But large buildings don't fall down like trees or Legos. They don't fall over sideways, they simply fall straight down.

So, why do large building fall down? Because gravity pulls them down. It does not pull sideways, so it doesn't tip sideways. But then why do Legos and trees fall sideways? Because there are other forces at work, namely the internal forces holding them together and in the same shape. Gravity is pulling down, but the internal forces prevent the top from simply collapsing into the bottom, so it falls sideways.
Here's a force diagram of a brick in a Lego tower tipping over. Gravity is pulling down. Normal force is pushing at the same angle the building is tipping. The total force is in blue. The vertical components mostly cancel, leaving the total force going mostly sideways.

But why don't large buildings do the same? Don't they have internal forces too? Well, yes, but they don't scale up. As the building gets bigger, it gets heavier, and gravity pulls more strongly. The internal forces of a large building will be stronger than those of a Lego building because it's made with steel rather than plastic, but it will be weaker relative to the force of gravity. The normal force will still be there, causing it to tip just a little bit, but gravity will dominate, so it will fall almost straight down. The top will simply collapse into the bottom, rather than being pushed to the side.

So why does this matter? It's just a movie, right? That's true, but understanding physics and how forces scale can be important. For example, there was a very well known case where some tall buildings fell down unexpectedly. As physics predicts, the buildings fell mostly straight down. (But not entirely. A lot of nearby buildings were hit by debris.) But a lot of people didn't understand the physics, and thought that the fact that the buildings fell down instead of over meant the buildings weren't brought down by airplanes, but rather by controlled demolition, and thus a conspiracy theory was born.

Monday, May 13, 2013

An Abortion Hypothetical

People opposed to the right to get an abortion like to make the argument that a fetus is alive, has a heartbeat, can feel pain, etc. It's obvious that these features are not sufficient to make the case. After all, cows have all those things, but the same people who complain about abortions don't usually have problems with eating meat.

But there's a related feature (well, more of a complex conglomeration of features) that isn't so obviously irrelevant. Is a fetus a person? That's a more reasonable question. But I still don't think it's relevant. Even if a fetus were undeniably a person, I'd still be in favor of the right to get an abortion. I'll illustrate with a hypothetical.

Suppose one winter's day, you accidentally leave your door unlocked. When you get home, you find that a homeless man has taken up residence. He doesn't pose any direct threat to you. He just takes up space, eats your food, and is generally inconvenient.

So, you decide to kick him out. "But", the homeless man protests, "It's freezing out there, and I have nowhere else to go! If you kick me out, I'll surely die!"

Is it morally prohibited to kick him out? Should you be required to provide him with food and shelter?

Now obviously, this hypothetical has a lot of differences from abortion. Here are the two I find most important.

One, homeless man is most definitely a person. A fetus may have a heartbeat, but this guy can talk. In the case of abortion, the status of the fetus is, at best, ambiguous.

Two, the homeless man is only taking shelter in your house, not your body. If you're allowed to decide who may or may not be in your property, surely you should be allowed to decide who may or may not be in your body.

And that's the important point of the abortion debate. It's not whether or not the fetus is a person. It's about whether women have the right to control their own bodies.

Sunday, March 31, 2013

Jesus

On this day of Easter, let us not forget this important fact: Jesus did not exist.

Now, don't get me wrong. It's entirely possible, perhaps even probable, that there was a guy named Yeshua who came from Nazareth and preached in Jerusalem and was crucified for his beliefs.

But that's not the person that Christians are talking about when they talk about Jesus. They're talking about the son of God, born of a virgin, who performed miracles and rose from the dead. That Jesus never existed. And that Jesus is not the same person as Yeshua. If he is, then Santa Claus lived in Turkey in 300 AD.

And we know next to nothing about the "real" Jesus. The only writings we have that talk about Jesus and were written anywhere close to when he actually lived were written by Christians, and were talking about the miraculous version.

EDIT:
And it's entirely possible the "real" Jesus didn't exist either.

Wednesday, March 6, 2013

Moral Intuition

We usually judge moral theories with our intuition. We see what a moral theory says about a hypothetical scenario and see if that agrees with our intuition. People arguing for a particular moral theory frequently give examples of situations in which that moral theory gives an answer that most people's intuitions agree with. People arguing against a particular moral theory frequently give examples of situations in which that moral theory gives an answer the most people's intuitions disagree with.

But a lot of times the examples used are very unlikely situations. That's not a problem for the moral theory. A good moral theory should work in any situation, likely or unlikely. But it is a problem for our intuition.

Intuition is only useful in circumstances it evolved in.

Consider physics. Humans have pretty good intuition with regards to running, jumping, throwing and other things we've been doing for millions of years. But outside our relatively limited experience, our intuition is virtually useless. There's nothing intuitive about general relativity or quantum mechanics.

Which is why I don't think hypothetical problems like the Trolley Problem are useful in determining what makes a good moral theory (though they can be useful in helping us examine our intuitions). Because such hypothetical scenarios rarely reflect our normal experience, so our intuitions don't necessarily apply.

So then, the question is, if we can't trust our moral intuition, then how should we judge a moral theory?

Thursday, February 7, 2013

Would Gay Marriage Lead to Polygamy?

An argument that opponents of gay marriage make is that legalizing gay marriage will inevitably lead to legalizing polygamy, which would be a bad thing, therefore gay marriage shouldn't be legalized.

As a syllogism, it's not invalid. If the premises are true, the conclusion is too. If gay marriage actually did lead inevitably to polygamy and polygamy actually were a bad thing then gay marriage is a bad thing. But neither premise is true, so the conclusion doesn't follow.

I don't think there's anything inherently wrong with polygamy. Past implementations of it have been sexist and bad, but then again, past implementations of regular marriage have also been sexist and bad.

And gay marriage won't lead inevitably to polygamy because they're two separate issues - and polygamy is significantly more complicated. To change straight-only marriage to include gay marriage, all you have to do is replace every instance of the word "man" and "woman" with "person" and replace every instance "husband" and "wife" with "spouse". It's not like the husband gets certain privileges the wife doesn't, at least, not anymore.

But to generalize marriage to more than two people takes more work then simply replacing "two" with "two or more", because a lot of the ways marriage currently works assumes that there are only two people involved. For example, when one spouse has a medical problem and is incapable of making a decision about what to do, the other gets to decide. How would that work if there's more than one other? What if they disagree? Also, is marriage a transitive property? That is, if A is married to B and B is married to C, does that mean that A is married to C? These are questions that never arise when marriage is restricted to pairs.

I don't have any answers to those questions, but they'll need to be answered to legalize polygamy. And legalizing gay marriage doesn't get us any closer to answering them, which is why it won't inevitably lead to polygamy.

Monday, January 28, 2013

Beware Superstimulus

All animals have evolved instincts and reactions to stimuli they ordinarily encounter. For example, birds protect things that look like eggs, because ordinarily if something looks like an egg, it is an egg. And they'll protect larger eggs rather than smaller eggs, because more resources go into creating a larger egg.

But in a scientist's lab, something that looks like an egg might not actually be an egg. And in such circumstances, birds will protect large non-eggs more than small eggs. They respond to the stimulus even when it no longer indicates what it ordinarily indicates, and the stronger the stimulus, the stronger the response.

We are just as susceptible to this as other animals. In our ancestral environment, sugar and fat were hard to come by, so when it did come by, you wanted to eat as much as you could while it lasted. So sugar and fat taste good, and they keep tasting good even when they are no longer in short supply.

Superstimuli are a large cause of today's obesity epidemic. Ice cream and cheeseburgers have more concentrated sugar and fat than anything a hunter-gatherer could get, so they trigger a stronger response.

Superstimuli apply to more than just food. Models are made-up and airbrushed to look more perfect than any real person ever could. Our modern society is inundated with superstimuli. Movies, television, computer games, the internet, porn. These all provide stronger stimuli than you could ever get otherwise. And when the stimulus becomes disconnected from the benefit it provides, the response does as well.