There is but one infinity

I've always been hung up on the notion of countable vs uncountable infinity. It seemed a rather silly thing to say, that you could have more than infinity. I understand the argument for it, but from an intuitive level, it felt wrong. Now, I feel that it is completely wrong. There are only two kinds of numbers, those whose digits can be represented and those whose digits can't. 7, 13, 10000000 all belong to the first category as well as .015, .139, and .00000000000000000000000000000000001. PI, 1/3, and e all belong to the second. The second category of numbers are not really numbers but ideas. They are, in essence, equations. Given a precision, they can produce a number of the first category, but they are not numbers in and of themselves. Another example would be the summation of the reciprocals of all integers, given a precision, we can produce the number, but the summation itself is just an abstract notion. With that we can understand that what we consider to be countable infinity doesn't really exist. It is in the second category and is therefore part of the larger infinity. If you give the countable numbers a precision (limit), then you can discuss those numbers; however, without a limit, the countable numbers are just an idea, much like PI, or the zeta function. In fact, you could think of the countable numbers as a function that, given a number, produces a new number one higher. However, there is a difference between the "idea" of the function and the numbers that the function produces. The idea of the function can go into the infinite, the actual numbers cannot. The same holds true with PI. The idea of PI can go into the infinite, the actual number can't. The problem with the proof of uncountable infinity is that the natural numbers are limited to the finite whereas the irrational numbers are allowed to go to the infinite. When one realizes that the irrational numbers cannot go to the infinite, then the proof becomes nonsense.

The Game

Last Sunday I went to the AFC Championship Game between the Indianapolis Colts and the New England Patriots. It was a blast. The game was amazing and the RCA Dome is an excellent environment. The one thing I noticed is how loud it is there, well, at least when the other team is on the field. You don't get a true idea of the volume through TV or radio. The other thing I noticed were how great the fans were. We were sitting beside a number of Colts fans and they were extremely classy and fun to be around. I have to say that I was surprised at how much fun I had watching the game. I had begun to think that football had become a television sport, but I can safely say that is not the case. It was soooo much better watching it live and in-person. It didn't feel productionalized. It felt raw and real. I had a truly amazing time and I can't wait for next year's game!

Google Books

I didn't know google had a public book storage and viewing mechanism, but apparently they do. Since I've been searching for boolean equation stuffage, I ran across it.

Here's a Dover book I found:
Boolean Reasoning: The Logic of Boolean Equations.

BigTable

Google has been slowly developing tools and technologies to support its unique requirements. I'd like to use this post to catalog information about them.

BigTable is google's answer to the database. Wikipedia has a good article on it here.
MapReduce is google's answer to workflow. The article is here.
Sawzall is google's high level distributed language. The paper is here.
GFS is google's file system. The article is here.

Great CDI Blog

Here's a great blog from someone who understands CDI from the business perspective. An excellent read. Head's up, her name is pronounced "dishay".

Google's video picks

For those of you who want a little substance and are having a hard time finding it with YouTube, check out Google's top 20 video picks here.

Big brother strikes again

Where do we draw the line? Jeff Jonas details the using of a P300 brain response as allowable evidence in court cases.

I would like to have seen more information about what a P300 response is. Is it possible that some people are "malformed" and have this response even when presented with an unfamiliar image? What if it is your living room and, while the blood is unfamiliar, the rest of the information is. There are too many variables for me to enjoy this.

As for going to jail because you thought about committing a crime, that is ludicrous. The difference between humanity and animals is that we can control our actions. I'm not sure we can control our thoughts. Regardless, as humans, we must be judged on what we can control.

Boolean Equation Problem

I'm struggling with the following problem.

I have a series of boolean equations. Only one of these equations can be true at any one time. I need to find an efficient way of determining which of these equations is true. I would also like to know more about these equations, like is it true that only one of the equations can be true and what new equations are needed to make that true.

Here's an example. Let's say that half of the equations deal with the condition where variable A issw equal to the value 'M'. 25% of the equations deal with the condition where variable A is 'm', 25% of the equations deal with the condition where variable A is 'n', 15% where the variable A is '?' and 10% where the variable A is 'N'. The variable B has a situation where 75% of the equations deal with the condition where it is 'M' and the other 25% of the equations are divided among the other choices. There are also variables C, D, E, F, and G. Which of these variables do I evaluate first? What factors should I consider? Are there algorithms that will help me in determining this?

Ch-ch-ch changes

The previous year was a year of changes. I worked for three different companies, in three different cities, one of which required me to move 800 miles away. Now, I'm back where I began and I've learned an awful lot and met some truly amazing people.

One of the first things I learned is that there is nothing more important than your family. I missed being able to pick my kids up from school and taking care of them in the afternoons. I also missed the good friends I had made here. I didn't realize how much I had missed them until I moved back.

Second, I learned that going to church is an important part of your spiritual development. Seems obvious, right? Well, I've recently been going to the First United Methodist Church of Wynne and have really enjoyed both the service and the Sunday School. I've been going there with friends, which really helps.

Third, I have learned a lot, technically. I've learned the value of having business people drive functionality and I've learned the value of architecture that is planned over architecture that evolves. I've learned about J2EE, CORBA, networking, TIBCO, and a whole host of other things. I enjoy learning, so that was fantastic.

Fourth, I learned that to be a good team leader, you've got to be able to stand up for those who are working for you. I've had leaders that I really enjoyed at all the companies -- they were all really positive experiences. Each one had a different style, but when it came time to go to bat, they were there to stand up for me.

Fifth, I learned that if you want to stay challenged in a traditionally business IT shop, you need to challenge yourself. It seems the expectations of those places are low and you can easily "just get by." In fact, you can easily descend into madness. But, to truly excel, you have to challenge yourself to come up with innovative ideas to make their software cost less and be more stable. I think more and more the key to good programming is focusing on stability.

Sixth, I learned that I have to make tradeoffs and I'm thankful for the past year because it helped me understand what those tradeoffs are. Right now, I work from home. It allows me to pick up the kids and help out my wife, who drives an hour to work three days a week. However, it also isolates me from communication with others. I really like the office setting where I can talk to an adult when I run into a roadblock, instead of being isolated from everyone. Therefore, I have to realize the need exists and try to assuage it in a different manner. Eventually, I'll go into the office more (at least 2 days per week) and hope that helps. However, right now I'm not working with a team, so going into the office doesn't really help. Nevertheless, spending the last year in two different office environments helped remind me that it can be a lot of fun and that I need to remember to go into the office more when I feel isolated.

Finally, as I head back to my old job, I realize that the reasons I left it are still there. However, after spending a year studying myself and other companies, I'm am in a much better position to adapt so that those things will no longer bother me. Right now I just want to be appreciative of the great gifts that God has given me. I'll worry about the other things later :-)

Similar Characters

I try to keep track of people who are similar to me: past, present, and fiction. I have three, currently, that I believe fit the bill.

1) House - I've blogged on this before, but Greg House from the TV show House is very similar to what I am like. I think he has let his pride control him more than I have and that accounts for a few differences. But, on the whole, we are very similar.
2) Isaac Newton - We have similar interests, religious beliefs, and personalities. He, of course, was far more brilliant that I ever could be, but we are very similar in our pursuits.
3) Brad Paisley - Yeah, the country singer. It's uncanny how many ideas in common we've had. In fact, before they released his "Two people fell in love song", I commented to my wife that everything that has ever been was because two people had sex. Mine was more crass, but that's beside the point. When my wife heard it on the radio, she immediately came to find me and show me my idea had become a song. It was funny. I still say he has spies that watch me...I need to go now...I have to elude them!

Aliens

I watched a history channel special the other night on UFOs. It had some interesting interviews on it. One thing that was eye opening was the replay of a nasa astronaut saying over the radio that he was looking at the alien spaceship out of his window. He said it very matter-of-factly and very clearly. He was neither surprised nor concerned. I have no problem believing in aliens. However, the Roswell stuff is a bit far fetched. How is it that the aliens can fly billions of miles across the universe and then crash in New Mexico? Doesn't make a whole lot of sense to me.

However, I propose the following. I think the next candidate for president of the United States should run on a platform of revealing all government knowledge of UFOs. I'd vote for that person, I think it'd be cool to know. What about you?

Unfair Infinities

We all know that the cardinality of the set of all integers is countably infinite. To me, it seems unfair that the cardinality of the set of real numbers is uncoutably infinite. It also seems odd that it is unintuitive to most (including me). However, I now understand why it was unintuitive to me. The key point to remember is that mathematicians don't allow integers to have an infinite number of digits. Instead, you can increase the integer by 1, but at any given time it has a fixed number of digits. However, with real numbers, there can, and often does, exist an infinity of digits. For instance, 1/3 is 0.333333.... So, not only do the real numbers extend countably to infinity, they also can have an infinite number of digits. In essence, they are infinite in 2 dimensions, whereas integers are only infinite in 1 dimension. I would imagine that complex numbers will one day be proven to be infinite in 3 dimensions, but who knows.

To be honest, I think it is a bit silly. We're not actually talking about different orders of infinities here, we're only talking about the existence of a function M that will map from one infinite set to another. If we can find that function, then we say the sets have an equivalent cardinality. Otherwise, the cardinality of one set is said to be greater than the cardinality of the other. If we could extract M from mathematics, and instead use an algorithm, then M could simply be, pick a real number, get next biggest integer, rinse, repeat. However, algorithms are not yet mathematical (unless you're Stephen Wolfram).

Readable Code

I think, to structure your code readably, you have to partition your code into different levels. Perhaps the different levels are written in different languages, perhaps not. The top level is the business level. At this level, you should be speaking a mostly declarative language and should be side effect free. Your code should be straightline. The second level is the logical level. At this level, you can have conditions and iterations and anything in the first level. The third level allows anything in the second level plus side effects. Finally, the fourth level allows comparative and arithmetic operations plus anything in the third level. The interesting thing to note, I believe, is that side effects are more readable than comparative and arithmetic code. Of course, it could be that comparative and arithmetic code is less useful than side effect code, so it is pushed to a lower level.

Let's look at an example:

extern int debug;

void foo()
{
int i;
for(i = 0; i < 100; ++i)
{
if(debug)
{
printf("%d\n", i);
}
}
}

This is a level 4 function. It has side effects, comparisons, etc... Let's try to turn it into a full 4-tiered function (yes, you wouldn't do that to something like this, but just for example.)

// level 1 - straightline
void foo()
{
print_numbers_to(100);
}

//level 2 - conditions
void print_numbers_to(int count)
{
if (in_debug())
loop_to(count, print_integer);
}

// level 3 - side effects
void print_integer(int i)
{
printf("%d\n", i);
}

// level 4 - comparisons
int in_debug()
{
return debug == 1;
}

// level 4 - side effects
void loop_to(int count, void (*func)(int))
{
int i;
for(i = 0; i < count; ++i)
{
func(i);
}
}

See how much better that is?

Responses

I put some responses in with comments. However, I would like to say that I was imprecise in a previous statement...it's a habit of mine. My statement should not have been that deficit spending causes war. Cos is entirely correct that war causes deficit spending, not the other way around. You can check the comments for the full scoop, but I was attempting to say that the ability to deficit spend leads to war. If you can't fund the war, you won't have the war (or you won't be successful at it). If you have easy access to funds, then you can easily create a war machine. If not, it is much harder. It is not deficit spending, but the ability to deficit spend. Check the comments ;-)

Response 3

Now, I'd like to respond to a few things I agree with from Cos's reply

1. I really like the idea of the states paying the Senator's salaries. This further reduces the federal stranglehold on our country. That is just pure brilliance!

2. My hope is that people who have a concern for our country would want to be electors. However, now that you mention it, I wouldn't be surprised if party-liners didn't run for electors just to be paid back at a later date by the party nominating them for an office position. I really don't want to enable a lot of restrictions, because that doesn't lead to a naturally balancing system. For instance, we could say that electors can never be publicly elected, but that is just silly in the long run. My hope is that the more educated, politically motivated people would become electors...however that is naive.

Nevertheless, what I realistically expect to happen is that entitlement spending drops dramatically, and socialistic programs begin to decline. When the voters don't elect you based on the entitlements you give them, you no longer give entitlements. Obviously, an elector could run on a platform of entitlements, but it doesn't really help him. He's not getting a huge salary and a plush office for life, so why does he care. His sole purpose is to do what is right for the country, because it is right, not because he's getting compensated. It could be a naive position, and I know it will eventually be corrupted, but I'm hoping that won't be for a number of years, when the system is stable enough to support a number of years of corruption before being reset again.

Response 2

I'd like to respond to a portion of a very well thought out comment from Cosbert Callis:

3) Frankly neither suggestion of the barter system or the gold standard, nor the idea of eliminating the Fed represents an understanding of economics 101.
(as a fiscal conservative, with a BA in Political Science and a Minor in Economics, you would have failed any econ class I was in with these ideas..)

Foreign trade (including deficit spending, which is a form of foreign trade) represents one of the most important natural forces for peace in the world. People just do not make WAR with trading partners. I don't where your original thought inevitably leads to war and inflation. comes from, there is NO, ZERO, NADA in the way of empirical evidence to suggest there is a single iota of truth in that statement.

First, deficit spending is not always a form of foreign trade. Deficits can come in many forms. First, you can sell bonds to your own country. Second, you can have the Fed print more money, thus causing inflation, but giving you the money you requested. Third, you can sell bonds to other countries. I would only classify the last as deficit spending.

Second, deficit spending MOST DEFINITELY causes war. You would have failed any history class I took with your ideas. War's have to be funded in some way. Without funding, you can't fight. There are many ways to get funding, but the easiest is through deficit spending. Do you think the war in Iraq would have happened if we would have had to raise taxes to pay for it? People understand taxes: they understand the effect that taxes have on their pocketbooks. With deficit spending, they don't understand the correlations and are less likely to complain. Therefore, deficit spending is VITAL in starting and maintaining a war machine. Without easily obtainable money, there are far fewer wars.

Third, we didn't always have a national bank. The current Federal Reserve system was established in 1913. Notice that it DIDN'T provide economic stability, as was its mandate. We still went through booms and busts (including the great depression). Before 1913, we had a myriad of different banking styles, including no central bank from 1837 to 1862. Do I think we need banking regulations? Sure. Do we need a central bank? No.

Fourth, a return to the gold standard is independent of the federal reserve system. The Gold standard was not revoked until 1971, almost 60 years after the establishment of the federal reserve. The goal of the gold standard is to reduce inflation. I would say that it did it's job pretty well considering the amount of inflation that has happened between 1971 and today.

Finally, trading partners most definitely go to war. The US traded with Iraq (oil and weaponry). Iraq traded with Kuwait. The US invested in Germany before WWII. The examples go on and on. In a global economy, there is more than one way to get at any resource, so it becomes much easier to bite the hand that feeds you. But, it becomes much harder to bite if you have no teeth. By eliminating easy access to money (via deficit spending through the Fed), we can take out a few rows of teeth.

Fanatics

I love this quote from Ron Jeffries. I found it on a thread in his Agile Forum.

A fanatic is anyone who believes differently from us, and just as strongly.

Response

Mark made a very intelligent comment on my previous post and I'd like to respond to a few of his points.

1. Repeal the 17th amendment:
This sounds good on paper, but what guarantees that the state congress isn't going to simply choose their Senator soley on party lines for just the possibility of future political favors? Even the lowest of state politicians harbors hopes of being president.

I want the state congress to choose their Senator for future political favors. The point is that the people do what is in their best interest, but the state politicians will do what is in their best interest. What is in their best interest is greater state powers. The people's interest is different. If a Senator lessens the state powers, then the state will not vote him back in. Therefore, the Senator must pander to the state in order to be re-elected. Why is that important? Because state's don't care about gay rights, flag burning, and national health care. State's care about schools and roads and infrastructure. All of the stupidity that is pure popular pandering goes away.

2. I like the idea of taking away the presidential election from the idiot masses who decide solely on superficial traits of candidates ("He's a normal guy who I think I could have a beer with." Who the hell wants to have an average person in the most powerful position in the world??). Anyway, I digress. The problem with this is that you're assuming that the electoral college will be composed of people who are more intelligent than the masses as well as unaffiliated with any political party. That's a tough thing to find I suspect.

I do believe that the electorial college will be slightly more educated than the typical voter, but that's not the main benefit. The main benefit is that the electorial college has no constituents. They don't get paid, they shouldn't get bribes (having fewer people will allow more scrutiny on their financial affairs), and they have no real reason to want to get re-elected. Therefore, they won't pick a president because he promises more welfare or free healthcare. People pick a president because he promises entitlements or "No new taxes" or whatever. They vote in what they think is their best interest (unfortunately, they don't understand the consequences). The electorial college could safely ignore those issues and pick a president based on what is best for the country. Since they don't care about re-election, they are free to make more intelligent choices.

3. Let's go back to the barter system. Most politicians really have nothing to offer other than words, so that should sufficiently weaken the ever-growing hegemony those in power represent. :-)

I believe a return to the gold standard is very similar to going back to a barter system. By preventing the devaluation of the dollar because of deficit spending, we can force politicians to tax us explicitly instead of through inflation.

Waiting on the World to Change

I really like John Mayer. Especially his new song. However, I have to disagree with his approach. First, the song:

me and all my friends
we're all misunderstood
they say we stand for nothing and
there's no way we ever could
now we see everything that's going wrong
with the world and those who lead it
we just feel like we don't have the means
to rise above and beat it

so we keep waiting
waiting on the world to change
we keep on waiting
waiting on the world to change

it's hard to beat the system
when we're standing at a distance
so we keep waiting
waiting on the world to change
now if we had the power
to bring our neighbors home from war
they would have never missed a Christmas
no more ribbons on their door
and when you trust your television
what you get is what you got
cause when they own the information, oh
they can bend it all they want

that's why we're waiting
waiting on the world to change
we keep on waiting
waiting on the world to change

it's not that we don't care,
we just know that the fight ain't fair
so we keep on waiting
waiting on the world to change

and we're still waiting
waiting on the world to change
we keep on waiting waiting on the world to change
one day our generation
is gonna rule the population
so we keep on waiting
waiting on the world to change

we keep on waiting
waiting on the world to change


Now, the analysis
The sentiments are good, but he misunderstands a fundamental part of human nature. The elite that are corrupting our society now will hand their corruption to their chosen successors. Just because those successors are from our generation doesn't mean they will be any less corrupt. They will continue to attempt to plunge us into a feudalistic society with them at the helm. The fact that we're "waiting" just makes them smile broader. They know we're powerless to stop them so they're taking full advantage of it. Our freedoms have been tromped upon. Our civil liberties abolished. We have our own version of the SS (we call it Homeland Security - anyone notice how these guys go after child porn? No one complains because we all know child porn is bad, but how is that homeland security? It's not. It's the start of our own version of the secret police.) Why will our generation act any differently? The fringe, artistic types such as John Mayer have never supported war or abdication of civil liberties; however, their generations didn't set things right. Why does John think his will? He's completely wrong. Instead, he's promoting the attitude that suits the would-be dictator's perfectly. He's sitting back and watching it all happen. Eventually, it will be too late to stop it.

What can we do to stop it? We have to take away their power base. Of course, we can't rely on them to do it, because they like their power base. Instead, we have to use the one constitutional outlet that they haven't taken away. We need a 2/3 majority of the states to support certain constitutional amendments.

I propose three amendments that should erode the power base of those in charge.
1. Repeal the 17th amendment. This amendment took away states rights and consolidated power to the federal government by moving the election of the senators to the people. Previously, the states chose the senators that represented them. This made it far less likely that a media campaign could buy an election. It also ensured that the person would actually support the state instead of what made them electable. Now, it is all too easy to support a law because it is "popular". Since you only answer to the people, that is all that matters. The Senate is no more than a balanced House of Representatives. Instead, it is supposed to represent the states. Senators would have a much harder time revoking states rights if they had to answer to the state congress.
2. Make the party nomination of presidential candidates AFTER the election of the electorial college. Furthermore, electors should not be allowed to express allegiance to a political party. Currently, the population, not the electorial college, elects the president. This is NOT how it was designed to work. The writers of the consititution knew that the general population is too stupid to choose the correct presidential candidate. Therefore, the general population should choose a bunch of smart people, who choose the president. This has been lost on today's generation. We have to bring back that purpose. The popular vote for a president should never be taken.
3. We must erode their monetary control of our society. The last amendment should abolish the federal reserve and return us to a gold standard. For the reasoning behind this, I would recommend Griffin's The Creature from Jekyll Island. It is a very informative (and very large) book about the history of the federal reserve, fiat money, and fractional reserve banking. It shows how control of the money system directly leads to a controlling, manipulative, feudalistic government.

We have to stop waiting on the world to change and use what power we have as a people to ensure our future.

Cultural Progression

How do we, as a culture, define our differences with previous cultures. Do we define it by our use of technology? I don't think we can. There are cultures that have advanced technology, yet have similar cultures to their ancestors who didn't have such technology. I think we have to find something that shows motion, not advancement. To that end, I propose we use music. It appears to me, that music is the definition of culture. In other words, by sharing music, you share your culture. By blending music, you blend your culture. Furthermore, music is always changing and adapting. It blends the past into something completely new and unexpected. Music seems to define our progression. Music defines our culture. Explore that concept and let me know what you think.