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?

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?

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.

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.

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.

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?

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?