How Large is your Penis?

Sunday, October 31, 2010

Fundamental Theorem of Algebra

This is a classic result of mathematics. If $p(z)$ if a complex polynomial which is non-constant then it has a zero in the complex numbers. There are a ton of proofs of this theorem. I wanted to presents three different proofs. The nice feature of these proofs is that each is from a different area of mathematics: analytic, algebraic, and topological. I have never seen a combinatorical proof, maybe there is one, but I have difficultly imaging how such a proof would be possible.

I also want to say that there is not such thing as a "purely algebraic proof". Because the construction of the real numbers, and thereby the complex numbers, is topological! To develop the algebraic properties of the real and complex numbers we must know the topology of the reals. However, when we say "an algebraic proof" we mean to say that we can prove the theorem just using the algebraic properties of the reals and complex numbers alone without using any other additional non-algebraic facts.

Analytic Proof: The standard analytic proof of the fundamental theorem of algebra uses Liouville's theorem from complex analysis. But that proof is too boring because it is too commonly used. Instead we will give a stronger proof using Rouche's theorem. This is a better analytic proof because it gives a bound on the size of zeros of the polynomial. Liouville's theorem simply asserts a zero exist, it does not put a bound on them. Rouche's theorem will allow us to bound the zeros. Hence this is a better proof.

First let us recall what Rouche's theorem says. We will state it a little less general to keep the discussion simpler. But let f and g be analytic functions on an open set containing the closed disk $z\leq r$ for some r>0. Suppose that f dominates g on the boundary i.e. $|g(z)|<|f(z)|$ on $|z|=r$. Then f has as many zeros as (f-g) inside the disk |z|< r (counting multiplicities).

With this result we can prove the fundamental theorem. WLOG, assume that $p(z)=z^n+c_{n-1}z^{n-1}+...+c_1z+c_0$. If $z\geq 1$ then $z^{n-1} \geq z^k$ for all $0\leq k < n$. Therefore, $|c_{n-1}z^{n-1}+...+c_1z+c_0| \leq C|z|^{n-1}$. Where $C = \Sigma_k |c_k|$. If $r>\max(1,C)$ then on $|z|=r$ we see that $C|z|^{n-1} = Cr^{n-1} < r^n =|z|^n$. Thus, $z^n$ dominates $c_{n-1}z^{n-1}+...+c_1z+c_0$ on $|z|=r$. By Rouche's theorem we immediately see that $p(z)$ has as many zeros as $z^n$ inside $|z|=r$, which is n. Not only did we prove a zero exists, but we have a bound. We know all these zeros must lie inside this disk |z|< r

Algebraic Proof: We need some lemmas before we jump into the proof. The first is that there is no proper odd degree extension of $\mathbb{R}$. If F is a field extension with $[F:\mathbb{R}]$ odd then $F=\mathbb{R}$. The reason is very simple, pick any $\alpha \in F - \mathbb{R}$ (assuming this set is non-empty). Then $[\mathbb{R}(\alpha):\mathbb{R}]$ divides $[F:\mathbb{R}]$ therefore it must be odd. Which means $\alpha$ is a root of a odd degree irreducible polynomial over $\mathbb{R}$. However, all odd degree polynomials over $\mathbb{R}$ have a real zero. This is a standard basic fact about odd polynomials which follows from the intermediate value theorem.

The other lemma that we need is that there is no quadratic field over $\mathbb{C}$. This is because if $[F:\mathbb{C}]=2$ then $F=\mathbb{C}(\alpha)$. Thus, $\alpha$ is a root of a quadratic polynomial over $\mathbb{C}$. But all quadratic polynomials have roots in $\mathbb{C}$ - because the quadratic formula is always applicable in this case.

If K/F is a field extenion the "normal closure" of this extension is the extension N/K where N is defined to be the splitting field of $\{ \min (F,a): a\in K \}$. This in the way is the smallest extension over K that makes N/F normal. If K/F is normal then N=K. It is not hard to show that if $K=F(a_1,...,a_n)$ i.e. $K/F$ is finite, then $N$ is the splitting field over $\{ \min (F,a_k): 1\leq k \leq n \}$, so that N/F is finite also.

Let us begin with the proof. To say that $\mathbb{C}$ always has a zero over any non-constant complex polynomial is the same as saying that $\mathbb{C}$ is algebraically closed. If $A$ is the algebraic closure over $\mathbb{C}$ and $a\in A$ then $[\mathbb{C}(a):\mathbb{C}]<\infty$. Therefore, if we want to prove that $A=\mathbb{C}$ i.e. it is algebraically closed it suffices to prove that $\mathbb{C}$ has no proper finite field extensions.

Let F be a finite field extension of $\mathbb{C}$. Thus, clearly F is also a finite field extension of $\mathbb{R}$. Since the characteristic of $\mathbb{R}$ is zero it is a perfect field which means that $F/\mathbb{R}$ is a seperable extension. However, it is not necessarily normal. To make this extension normal we work with the normal closure instead. Let $N$ be the normal closure of the extension $F/\mathbb{R}$. We will argue that $N=\mathbb{C}$ to complete the proof.

Since $N/\mathbb{R}$ is a normal and separable extension it is a Galois extension. Let $G=\text{Gal}(N/\mathbb{R})$. Now $G = [N:\mathbb{R}]$ which is even because $[\mathbb{C}:\mathbb{R}]=2$ divides $[N:\mathbb{R}]$. Thus, $G$ has a Sylow 2-subgroup by the Sylow Theorems. Let P be a Sylow 2-subgroup.

Let $K=N^P$ i.e. let $K$ be the fixed field under the group $P$. Then $[G:P] = [K:\mathbb{R}]$ (by the degree correspondence of Galios theory). But $[G:P]$ is odd since $P$ is a Sylow 2-subgroup, which means $K$ is an odd extension over $\mathbb{R}$, but then it means $K=\mathbb{R}$ which forces $P=G$. Thus, we have shown that G is in fact a 2-group i.e. a group of order a power of $2$. Therefore, $G_1=\text{Gal}(N/\mathbb{C})$ is a 2-group also (unless it has order $2^0=1$). By Sylow's theorems again we know there is a maximal subgroup M of $G_1$. Hence, $[G_1:M]=2$, if E is the fixed field under M then $[E:\mathbb{C}]=2$. But this is impossible. Hence, $G_1$ must be a trivial group, hence $N = \mathbb{C}$.

Geometric Proof: Here we will assume standard results from algebraic topology. We will assume that the polynomial has the special form $p(z) = z^n + c_{n-1}z^{n-1}+...+c_1 z + c_0$ where $\Sigma_k |c_k| < 1$. The general case follows easily from this special case.

Let $S^1 = \{ z\in \mathbb{C} : |z| = 1\}$ be the unit circle. Then we know that $\pi_1 (S^1,1) = \mathbb{Z}$. Consider the continuous function $f : (S^1,1) \to (S^1,1)$ defined by $f(z) = z^n$. This map induces a group homomorphism $f_*:\pi_1(S^1,1)\to \pi_1(S^1,1)$. Since $\pi(S^1,1)$ is an infinite cyclic group with generator consisting of the loop $[ e^{2\pi i t} ]$ it follows that $f_* [ e^{2\pi i t} ] = e^{2\pi i n t}$. Henceforth, we can think of $f_*$ as acting on the group $\mathbb{Z}$ by multiplication by $n$. The important fact that we need about $f_*$ is that it is a monomorphism (injective).
Now consider the map $g: (S^1,1) \to (\mathbb{C}^{\times},1)$. If we let $i:(S^1,1) \to (\mathbb{C}^{\times},1)$ then $i_*$, the induced homomorphism between homotopy groups, is a monomorphism. This is because there is a retract from $\mathbb{C}^{\times}$ to $S^1$. Since $g = if$ it follows $g_* = i_* f_*$, but $f_*$ was a monomorphism, $g_*$ being a composition of monomorphism is a monomorphism.
Assume that $p(z)$ has no zero then we can define a map $p: B^2 \to \mathbb{C}^{\times}$. Let $h$ be the restriction of $p$ to $S^1$. The map $h$ is nulhomotopic because it extends to a continuous function of the ball $B^2$ which is contractible.
If we define $H:S^2\times I\to \mathbb{C}^{\times}$ by $H(z,t) = z^n + t(c_{n-1}z^{n-1}+...+c_1z+c_0)$ then $H$ is a homotopy between $h$ and $g$. Notice, to prove that this is a homotopy we use the fact that the sum of the coefficients $c_k$ is strictly less than 1.
Therefore, $g$ is nulhomotopic. But this is non-sense because $g_*$ is injective, if $g$ was nulhomotopic then $g_*$ would have to be the trivial homomorphism. We have arrived at a contradiction. Therefore, it must be that $p(z)$ has a root.

I am curious to see more proofs of this theorem. What are some of your favorite proofs of the Fundamental Theorem?

Tuesday, October 26, 2010

Case Against the Death Penalty

My position on the death penalty has changed. There was a time when I used to support the death penalty (not that this means anything, because I once thought drugs should be illegal, I had a lot of foolish ideas back then).

The standard argument against the death penalty has been the moral argument. It is immoral to kill other people, other than in self-defense, therefore the death penalty is immoral since it is not done in self-defense but for revenge. But somehow this argument never got to me. I never been moved by this argument. I always thought about Adolf Hitler or Joseph Stalin. I considered what would happen if these people were sentenced to death. Would I be crying my eyes out for them? No. It would not bother me if these people got executed. Even though I agree that these people should be killed, because it does not achieve anything, I am still not moved by the moral argument.

The other common argument used against the death sentence is that the death penalty is too expensive. This is the argument David Friedman uses against the death penalty on his blog here. But I do not like Friedman's argument. Friedman generally approaches these sorts of questions from a purely economic point-of-view by just comparing cost vs benefits. Friedman seems to agree that if the death penalty is actually cheaper than imprisonment then it should be used instead over imprisonment. So Friedman does not really take a stance on this issue, he simply only cares about the cost associated with killing and imprisoning.

The most persuasive argument against the death penalty that I heard came from TheAmazingAtheist in a YouTube video. Strangely he did not even use this as an argument against the death penalty. He pointed to the hypocrisy of conservatives who say that the government should stay out of the market but have no problem with the government staying in the business of deciding who should be executed, that is real "limited government" for you.

The main argument that I take now is a variation of what was said above. The state should not have the power to execute other people. Because that is giving the state too much power. The only kind of execution that I support is voluntary capital punishment. That is to say, if the criminal decides to be executed rather than imprisoned then he can choose this option. But if the state can decide for people who lives and who dies then this is giving the state too much power. I do not think the state can have this kind of power. And therefore, I do not support capital punishment.

Now I do have one more argument against capital punishment. I do not think anyone ever used this argument before. I call this the philosophical objection to the death penalty. The state is supposed to represent the delegated rights of the people. Of course, in all actuality this is non-sense. There is no consent of the people. There is no "non-consent of the governed". The social contract is a myth. The state does things that people cannot do themselves. And so forth. But ideally the state is supposed to simply be that institution that represents the people and it has its rights only from the delegated rights from the people. Do people have the right to execute other people? No. People have the right to self-defense, but people do not have the right to execute others. But if so then how can people delegate this right to the state? If people have no right to execute then how can they delegate this right? How can someone delegate a right they do not have? Thus, the state cannot have the delegated right to execute people.

If you are interested, Penn and Teller did an episode on capital punishment: here

Monday, October 25, 2010

Wrong to Preach Atheism?

Is it wrong to tell other people that their beliefs are wrong, that there is probably no God, and that they are wasting their only lives on a false belief? Some people would say "yes". In fact, most people would say "yes", including some atheists.

Let us consider some of the arguments against preaching atheism: (1) atheism is just your opinion, religious people have their own opinions, you do not really know who is right; (2) people should keep their beliefs (or lack of beliefs) to themselves, it is not right to push your beliefs on others; (3) religious people are happy, it is not right to take away from their happiness and make their life confusing.

The first objection is one that I have to deal with on a day to day basis. Every day, or almost every day, I have to hear people remind me, "well it is just your opinion", not just on the whole religious issue but on every other issue. I hate it. I hate it because it is an anti-intellectual mentality. Everyone has opinions. So what? It does not mean all opinions are equal. If you have some pain in your stomach and you say "perhaps I should go to the doctor", and I say, "you have a tree growing inside your stomach from all the seeds you ate" then it is clear you do not need to take me seriously. We both have opinions. I had a pretty retarded one but it was an opinion nonetheless. (I blame this anti-intellectual attitude on college. In college you often hear professors tell you how you need to respect other's opinions. But that is another issue that is taking me off topic). Opinions need to be challenged and argued for, the best one should win out. So this objection to preaching atheism is a bad objection, not to mention self-defeating, if people have opinions and they should not be taken seriously, then what you just said is an opinion, but somehow I have to take it seriously? It just makes no sense. This argument is not going to stop me from preaching disbelief and heresy to other people.

The second objection I disagree with. Who said that people should not push their beliefs. (College again?) I hear this all the time, on TV, on YouTube, on online discussions, we should not push our beliefs. But why not? I never hear people say a reason for not doing so. I agree that it is hypocritical for atheists to tell theists "keep your religion to yourself" but at the same time post bus posters in promotion to atheism. But I do not think that religious people should keep their beliefs to themselves. Let us say that you believe in Jesus. And you think Jesus is the greatest person to exist. And you think that Jesus is the truth and the light. And that Jesus will make your life and the world a better place. If you believe in all of that, honestly believe in that, then what kind of a person are you if you keep that to yourself? I would consider you a rather terrible person. You have this great gift that you can share with the world and you keep it to yourself? If you really think that Jesus is great then tell other people about it. Of course, be nice about it, but preach it. There is a difference between preaching your beliefs and enforcing them. I agree that people should not enforce their beliefs, after all that would violate the separation of church and state. But preaching is fine, I encourage it.

The logical conclusion from the third objection is that if drug users are happier using drugs then we should not try to help them and just let them continue taking drugs because otherwise we would be interfering with their lives. Of course, no one agrees with this, at least I hope so. We got no problem with telling drug users that they are destroying their lives, they can live more meaningful lives off the drugs. Religion is not that different. Religious people are destroying their minds. They destroy their individuality by being slaves to an authoritarian God. What kind of people are we if we let them live a delusion and self-destruction? I am not saying that atheists should deprive religious people of their right to be religious. I am just suggesting that atheists should reach out in some way and let the message of a secular life be known. So that maybe some people would listen and change themselves (and the world consequently) for the better.

I once, like a year ago, was searching craigslist under personals. I searched under the title "frum". There were quite a lot of frum gay Jews there looking for gay sex. I responded to one of the ads (if you are wondering, no it did not get anywhere). I send an e-mail saying that he does not have to live a depressing life as frum and gay, I said that the arguments against religion are too strong, if he really studies it honestly I am sure that he would leave religion, and thereby attain a happier life for himself. He got very angry at me in his response by e-mail. Did not speak with me again. But this story does not bother me. Because sometimes when you try to do good to others they will be angry at you for doing this. Their anger and hatred to what I do will not stop me from doing it. Because, in the case of frum and gay, I think that if they really do consider the secular lifestyle then they would have much better life. Lives that they cannot have under Judaism. Lives that they are deprived off and need to live in depression denying such a life. Their resentment to me is no argument against preaching atheism.

Saturday, October 23, 2010

Olber's Paradox

Before I say anything I just wanted to say that I really hate the word "paradox". Because people think it means "something which is self-contradictory". But in actuality the way the word is used, especially in mathematics, it means something which is really crazy and surprising. For example, consider the Birthday Paradox. The paradox is that it is more likely to find two people with the same birthday in a group of 23 random people than not. There is nothing contradictory about this statement. It is just surprising. The Monty Hall Paradox (here) is not some contradiction, it is a puzzle that for some reason confuses a lot of people. The only paradox that I can actually think of at this moment that has a self-contradictory nature is the Russel Paradox. All other paradoxes are just surprising statements. I guess the misunderstanding of the world "paradox" comes from the misapplication of the English meaning of "paradox" to math and science. Paradox has a literary meaning as well, it means that the character of the story does something which is contradictory to what he does (something like that). So the confusion over what paradox means in math and science is confused with its meaning in English literary interpretations. Anyway, let me return back to what I wanted to say.

Olber's Paradox says that the universe must be finite. If the universe was infinite then there would be infinitely many stars in the sky. If there would be infinitely many stars in the sky then the entire sky would be brightly lit up. But instead we see the stars "discretely", meaning, we see a point there, another dot there, and so forth. The sky, when we look at it, is basically all dark with a few bright dots in the sky. Therefore, the universe must be finite.

There are a number of problems with this paradox. The first obvious problem is that the universe can be infinite with finitely many stars. Maybe the universe keeps on going forever but it has finitely many stars? But there are more serious problems with this paradox, even assuming there are infinitely many stars.

The first serious problem with this paradox is more of a scientific objection. That is, people cannot see objects from an arbitrary far distance. The weaker the light source the less of a distance one can see the source. For example, I think that in a extremely dark valley one can see a candle from 30 miles away. But if the candle is too far away then the person would no longer see it. His eyes are not strong enough for collecting such weak signals of light. Indeed, this is the case when one observes the heavens with his eye and when one observes the heavens with a telescope. With your own eye you can see about 6000 stars (if you live outside a large city where the sky has no interference with anything). With a telescope you can see a million. Under a telescope the entire sky brights up. Therefore, we only see a finite number of stars in the sky. There could be an infinite number of stars, but we only see finitely many. Therefore, there is no paradox.

The second serious problem is even assuming there is an infinite number of stars and even assuming we have perfect vision it is still possible for there to be gaps in the sky. It is still possible for there to be an infinite universe, with infinitely many stars, with a perfect vision, and we still see an almost empty sky. To see how this is possible consider the xy-grid. Let the origin (0,0) represent you. Now cover this grid with points (n,m) where n and m are integers. These integers represent the stars in the sky. If you join (0,0), where you at, with a star at (n,m), you get a line. This line is the line of vision. If you remember from high-school math the slope of the line between (0,0) and (n,m) is m/n (assuming n is non-zero). But m/n is a rational number because n and m are integers. This means that all the vision lines have rational slope. If you remind yourself a bit more high-school math the equation of a line passing through the origin at (0,0) with slope r is given by the equation y=rx. This line is a line of vision with slope r. Now if r is a rational number then it will hit one of the points (n,m). Meaning if you look at the night sky with rational slope you will see a star. But if r is irrational then the line y=rx will fail to hit any of the stars at (n,m). This means this line is a clear line of vision i.e. it does not see any star in its path. So we see from this simple example that it is possible for there to be infinitely many stars, and even under the assumption that we have perfect vision, we can still find lines of vision that do not see any star. Of course, this argument is two dimensional, but we can just as easily extend this argument to three dimensions, we just worked in two dimensions because it is easier to follow the argument.

But the biggest problem with this paradox is that we can prove that no matter how we position infinitely many stars in this sky there will always be clear lines of vision. This argument is a little more advanced because it no longer uses high-school math. It uses the important concept of "countable" and "uncountable", that was introduced by the mathematician Georg Cantor at the beginning of the 20th century (which is among the most important ideas of modern mathematics). Think of the xy-grid again. On this grid randomly position the stars anywhere, with (0,0) representing us. The number of these stars, represented as grid points, will be a countable number. Which is infinite, but it is a certain size of infinity. The number of all lines passing through the origin, which have the form y=rx, where r is any number, is also infinite. But it is a different kind of infinity. This kind of infinity is uncountable. The infinity of all lines (representing our line of vision) is a larger kind of infinite than the infinity of all stars. Therefore, there will always have to be lines, an uncountable number in fact, that would be clear lines of vision. And so we will always see gaps in the night sky.

Olber's Paradox is utter fail.

Thursday, October 21, 2010

Fear of the Unknown in Statism

Anyone can make a dictionary (at least I think so, maybe I am wrong, but I think anyone can write their own dictionary) . This means a group of evil people who want to corrupt the language can write up their own dictionary. They can define certain words in an unfair way. Or they can define words in a completely wrong way to confuse people. What incentive do they have? I have no idea, but let us just assume that there is such an evil group of people. But what is very interesting is that I never heard anyone in my life being fearful of such people. I never heard anyone propose to have state regulators control the dictionaries. Why not? Because we already live in a world where pretty much anyone can write a dictionary and put it on the market. Since we live in such a world people are not scared of the thought that some evil people will destroy all language by writing up fake dictionaries. They never even consider this possibility because this is the world they are used to, and in this world this problem never happens.

Wikipedia is not regulated by the state. It is a free encyclopedia that anyone can write on. I even made a few contributions to it myself and they still are up there. This does not mean that anyone can post anything they please on Wikipedia. Wikipedia does have people in charge to recheck the information so that no silly person can post anything he wants to. Wikipedia is the best encylopedia ever. I know there are some haters who like to say that Wikipedia is not reliable. But I have no idea what they are talking about. So far all I used it for it was reliable. I use it mainly for mathematics. And the mathematical articles there are superb, usually written by professional mathematicians. But other information that I found on it was useful also. I am sure there are a errors, but that is excusable, it have over 3 million articles, Britanica does not come anywhere close. However, perhaps Wikipedia somehow decides to post all wrong information. Or consider for instance a "Falsepedia" that becomes a competitor for Wikipedia. But Falsepedia contains a lot of false information, purposely. Why are there no people crying out for state regulations to control internet encyclopedias because perhaps some encyclopedia can purposely put false information and trick the pubic into believing something which is false? The answer is simple. Because unregulated encyclopedias is the world in which we live. This is the world we are used to. And because of this people are generally not afraid of living in an unregulated world of encyclopedias. It probably never even comes to anyone's mind to want to regulate encyclopedias to make sure they do not contain false information.

People can walk in the street with knight armor and swords. I know a guy who dressed up for Purim as a knight and had a sword on him. There is no law against wearing knight armor and having a sword on you. There are sometimes special events in the City when medieval lovers dress up in armor and come with their swords. They are not stopped. And I do not think outsiders are scared by knights in the street. If you saw a knight in the street you would probably be amused, not scared. But consider the following argument against the right to be a knight: "if people were able to dress as knights and have swords they would run through the streets and murder everyone, there would be an all out brutal fight, and what about the children, the children will be killed". But does any person today take this argument seriously? If I told an average citizen this argument he would probably find it stupid. Why? Because this is how the world is today. There are no laws stopping knights. And because this is how things are people are comfortable with the current state of the world and are not bothered by unregulated knights.

Gas oven stoves are dangerous. They can be used as a miniature explosive. The danger is magnified by much in places like New York City were people mostly live in apartments and not houses. I am sure gas oven stoves are regulated (just about everything else today) for safety standards, but it does not prevent some crazy person from deciding to abuse the gas stove and cause a dangerous fire or a miniature explosion. So perhaps gas stoves should be banned and replaced by electrical stoves? Electrical stoves are much safer. But where is the fear of gas stoves? It is absent for the same reason. People already live in a world which consists of gas stoves. So people are comfortable with such a world. Let us say that it can empirically be shown that electric stoves save more lives than gas stoves. Do you really think people will catch on to this new regulation? I doubt it. A Google search on "ban gas stoves" does not even give any helpful suggestions. This is not a fear on people's mind. This is the world we live in, so we are naturally comfortable with it.

But now consider the reverse situation. Consider laws that are in place today but would seem funny to an outside observer. The best example I can think of is fire exit signs. Buildings are required to have signs that point to an exit, in case there is a fire. You need to try to be a little imaginative here. But try to imagine living in a world which had no fire exit signs? What would be the difference between this world and our current world? Hardly anything, except maybe a few more saved people from exit signs (though I even doubt this). But imagine further that you were to walk over to a random citizen from this parallel world and ask him, "I propose for all buildings to have fire exist signs". Would he really care? No. Fire exit signs would be a laughable issue for him. Just like unregulated dictionaries, encyclopedias, knights, and gas stoves are to us. Because in his world no fire exit signs are the way things are. People are used to this kind of world, and so they are unbothered with its current state. But what is really funny to watch is that some people get so defense about fire exist signs. You tell them that you are against a mandate for buildings to install fire exit signs and they get angry, "you only care about money, you would rather save $10 then save people".

But where does this fear come from? People in our current world are not afraid of an evil group of people taking over the dictionaries or making a Falsepedia. But why are many people afraid of living in a world without fire exit signs? I think the answer is because there is a fear of the unknown. People cannot imagine a world without exit signs, so they get scared. They immediately assume the worst. And become defensive against people who want to get rid of laws for fire exit signs.

Statism lives off fear, not rationality. Statists are not statists because of some rational reason that they have developed on their own to be statists (a few are, but those are exceptions). Rather statism is the mainstream idea that lives off people's ignorance and fear. The state does not justify itself rationally, not really. It scares people into thinking what would be otherwise. In a world they cannot imagine. Statists are afraid of the unknown, and the state is a means for them to escape from this fear.

Freedom scares statists. Because freedom is not controlled. Freedom cannot be calculated. Freedom is an unknown world. I have no idea that a free world is like. But it does not scare me. It does however scare statists. Because central planning and central calculation is part of statism. If I was able to know what a free world would be like, or how it should be managed, then that would actually be a pretty good argument in favor of me being a dictator. Rather I do not know. I have no idea what will be in place in the free world. This is what scares statists. For them a comfortable life is filled with knowing the future and being (or more precisely "thinking that they are") able to predict the future.

Knights do not scare people. It is not an unknown world. But if instead of swords and knights you mention guns then it scares people. Well, not here in the US, the US is rather gun-friendly. But in most places in Europe. Tell a European about being pro-guns and he would be scared by you. Because he is a statist. He cannot imagine a world with guns. He will immediately assume the worst. He will assume a world in which people run around and shoot each other. He is a statist, he is afraid of the unknown. But strangely he does not have the same attitude about people owning swords, even though he can make the same argument and say if people owned swords they would run around and kill each other.

Unregulated dictionaries and encyclopedias do not scare people. This is the world as it is. But unregulated (unlicensed) doctors do. I have a Jewish friend who is in law school. I once told him that I am against licensing of doctors, and think that any doctor can enter the market. He is a statist. What does he do? The obvious. He assumes the worst. He tells me that if doctors were unlicensed then there would be evil doctors who would lie and cheat and kill people. In such a world lots of people would be dying all because they are unlicensed. I was not surprised. I tried to convince him otherwise, but he was just as fearful. He was fearful because he cannot imagine a world with unlicensed doctors. And is scared. Though strangely there are unlicensed computer repairmen who walk into your house to fix computers - nobody seems to be scared by this as it is the current state of the world.

Wednesday, October 20, 2010

What Happens When We Die?

I do not think this question is really that hard to answer. I know that a lot of people find it overly arrogant for someone to state what happens when he die. But I am extremely sure of myself. And I do not see what is the problem with being so certain of yourself about what death is if you can give a strong argument that defends your position.

The answer is really boring and dreadfully unexciting. Nothing happens to us. Your soul will no longer exist and you will not feel anything. So sorry if you were expecting something more interesting, even if it was a negative experience. Death is neither positive nor negative. No angels escort you to heaven (though that would be a rather terrible afterlife for me). No demons escort you to hell. No you are not stuck in your body just conscious about everything which is going on from that point till eternity. And you do not become a ghost. It is really rather simple. The experience that you feel now will be gone. Think of it as a dreamless sleep. Sometimes when you sleep you have no dreams. When you wake up after many hours all that time in between when you went to sleep and awoke is what death is like. You did not feel anything. You were not aware of anything. (I know there is going to be this one guy who is going to tell me saying "actually you always have dreams you just forget about them", if you are such a person, please kill yourself, I hate people who have to repeat overly trivial things that everybody knows. You know what I mean by a dreamless sleep.)

My reason is very simple. I could offer explanations involving cognitive neuroscience to why death is simply non-existence. But I have a simpler argument. We all know what death is like. Yes. That is right. We have all experienced death before. People say "you have no right to say what death is like because you never experienced it". But I disagree. We have experienced it. We have been dead for billions and billions of years. How was it like? Your brain did not exist. Your body did not exist. Your soul did not exist. None of it existed. And how was the experience? It was not anything. It was not positive and it was not negative. Peaceful and quiet? Not really. Hard to imagine since he are alive now, but we at least can somewhat comprehend that death is.

Tuesday, October 19, 2010

Why I Hates Adults

When I was a young kid I used to beat up a lot of people. By "beat up" I mean physically beat up. I could have easily been a bully since I was the toughest of all the kids. Though, I was no bully. I never bullied anyone in my life. Nor did I ever bother anyone intentionally. I was a nice kid. Which is why I think some kids started up with me. But unlike other nice kids that just stood silent, I was good with talking with my fists. Violence for me was self-defense.

I remember when I was 14 years old I was at someone's birthday party. One of the other invited boys to the party, whom I never met before, took something of mine. I asked him a few times to give it back. He ignored me. So I had to beat him up. It was a rather funny fight. He threw a punch, but he missed. I took a step forward. And he was not expecting this at all. I smashed my head into his face. He fell on the door, started crying, and ran away. I got my thing back. Some people would say that I took it too far, but I do not think so, I was just defending my property.

Guess what happened a few hours later? We made up. We became friends. I invited him over and he invited me over. Then we did stuff together and hanged out for several years until he disappeared strangely. He was a actually a cool guy.

Now let us consider how disputes take place in the adult world. Two "friends" who were friends for many years invite one another for dinner. At the table one person says, "how can you drink that (referring to soda)?", the other one might say, "I drink what I like", the other says something to the effect of, "well I think it is stupid for people to drink such unhealthy drinks". Then these adults act 'mature' (whatever that word means) and act as if nothing happened. But when they come home they never call one another again.

Or consider a different situation when one adult says, "Republicans messed up this country". The other friend is a Republican. These two friends go back and forth between one another a few times and then they get 'mature'. Which basically means they stop talking and after dinner never talk to one another again.

What I hate about adults is how easily they break up with people. Someone does something they do not like in the slightest and they never talk with one another again. I headbutted another kid in the face. And we made up as if nothing happened. He did not complain about how I hurt him and I did not complain about how he took something of mine. In the adult world you come across these adults who cannot handle situations like this. The smallest offense and they will complain about it for years.

This is why I do not want to "grow up" or get "mature", because I find nothing of value in the adult population. They are rather mean and boring people.

Sunday, October 17, 2010

The Deception of Beauty

I have seen several discussion boards on the topic of beauty. And the general theme of these boards is that, if you think you are beautiful then you are beautiful. If you are confident and act beautiful then you become beautiful. You hear this mantra expressed by people repeatedly over and over again.

But is this really true? No. It feels nice to believe that you are beautiful if you treat yourself as a beautiful person. But in actuality it is a lie. There is no consciously created beauty as many people believe. Beauty is what other people perceive of you, not what you perceive of yourself.

I have a tiny penis. My penis is 5 inches long when erect, pretty small, quite asian. If I think of myself as having a giant penis it is not going to happen. Because "giant" is what other people consider giant, not what I consider giant. "Giant", from my internet experience, is usually 8 inches. Guys like to brag online how giant their cocks are. I wonder how true it is, I doubt it, but an 8 inch cock sounds pretty large. If people think of a "giant" cock as 8 inches then that is the defining property of a giant cock. Under this definition my penis is extraordinary small. If I look into a mirror and tell myself "I have a giant penis" that would not make it giant under the consensus of the masses. My attitude towards my penis would not change the public's perception of my penis.

Beauty is the same thing. What is beautiful is to a large degree based on social consensus (and partly evolutionary). During the Victorian age it was considered to be beautiful to have very pale skin. Today it is considered ugly, to look beautiful today people need to have tanned skin. There were civilizations that respected fat people, because being fat was the sign of wealth and success. In the US and in European countries being fat is considered to be repulsive, especially in the US. The US today glorifies nearly anorexic women on TV, because the message is that thin is beautiful. To be "beautiful" therefore means to be in the norm of what the public consensus is. If in the US to be beautiful means to be well-built, shaved, tanned, and tall then that is what qualifies as beautiful. It does not matter what you think of yourself. All of that is irrelevant. If you are a disgusting piece of meat then that is what other people would think of you.

What I really hate about the topic of beauty is this deception that the beautiful give to the ugly. It is deception, and nothing more. Its goal is to make the ugly people feel good about themselves but in all actuality it is deception. When the beautiful tell the ugly, "you are beautiful on the inside", that is just a statement to make them feel happier about themselves. I agree that internal beauty is more important than external beauty. However, it is deception nonetheless, because the message of such a statement is to say that they are beautiful, which they are not.

It is also very strange when beautiful people tell ugly people, "you just need to be positive about how you look". Easy for you to say, because, ... you are beautiful! You see this on the Tyra Banks show. Tyra is gorgeous and the girls she sometimes has on her program are ugly or fat or just not happy about their looks. And her general message, to summarize, is to be positive about how you look. Beautiful people are not supposed to be lecturing ugly people on how to feel good about themselves. It makes no sense. It is like having a super fat trainer teach you about how to lose weight.

Here is the true. It is the sad truth but it needs to be said. There is no such thing as "internal beauty" (well there is, it just does not express itself to your physical beauty). All of that is just an excuse for you to feel good about yourself. The way the world works is very simple. The hot looking people date hot looking people. And the ugly looking people date the ugly looking people because they have no one else to date (unless they happen to be rich and buy themselves a hot date). If you are fat and repulsive there is not much you can do. You are destined to basically date ugly people. You cannot aspire to "higher levels" (I put it into quotations because for me beauty is so superficial that it is not really a higher level on any rational level) in your repulsive body.

There are three things you can do about your repulsion. First, you can continue to deceive yourself that beauty is what you make it and continue living a lie. Second, you can accept the truth and then work to become beautiful or if you cannot then you can cry about how ugly you are.

Third, and this is what I do, you can raise your middle finger, point it to the beautiful people and tell them, "I do not care how I look. I may be ugly, but I am free". You can recognize that most of beauty is just a social fabrication that changes from time to time, a deception over the eyes of the people. It is unphilosophical and against reason to care for how someone looks rather than who someone actually is. And with that you can recognize the advantages of being ugly (here). You will be happy in this way, you will finally be at ease.

Saturday, October 16, 2010

UnOrthodox Judaism

Jewish bloggers like to pick on Orthodox Judaism. But other forms of Judaism are not much picked on. Rarely do I find Reform or Conservative Judaism being picked on. Or whatever other Liberal form of Judaism that there is. It is understandable why not. Because R/C/L Judaism do not bother people. Reform Judaism is probably a lot of fun, I am sure they do not care much if you are a observant or not, they welcome you regardless. And they do not share the "I am better than you and you are beneath me" mentality that I happen to clearly express in my personality, well not just me, but Orthodox Judaism as well. It is not so much really how silly Orthodox Judaism is that it gets picked on but really that is sometimes a very mean religion. So I decided to make a post against UnOrthodox Judaism in its honor.

The thing about UnOrthodox Judaism that I do not like is that the UnOrthodox appear to me to be lost in between the Jewish world and the real world. As much as you may hate the Orthodox Jews they have their own world in which they are perfectly comfortable at. The UnOrthodox Jews do not have their own world. They levitate between the normal world and the Jewish world. They never find their own place to stand on in the world.

Let me try to make it clear what I mean by this. My synagogue is an Orthodox synagogue. Men dress with suit, ties, and they wear hats (fedora hats). It is very Orthodox. There is not much talking at shul, they generally get angry at people who talk. And when they celebrate Shabbos or some Jewish holiday it feels that they belong in this world. These Jews are so perfectly comfortable in this world.

In a strange way these Orthodox Jews look normal. Not normal by outside standards but it feels normal when you are around them. After Shabbos or yom tov when shul is over and Jews stand in the street and speak with one another they do not look strange compared to the outside world (at least to me). You see a group of gentleman dressed appropriately in the street talking with one another. That looks okay. And even if you know they are Juden you still nonetheless see them as looking okay because you can identity yourself and them as separate.

Even if these Juden were to take off their hats (which, oh my science, never happens in the street) they would look okay - I know it sounds strange but I cannot explain it any better. Because their big black yarmulkes mark their own identity. And outsiders can immediately see the difference between them and these Juden.

But this sense of personal identity that I have a difficult time trying to describe (maybe I cannot describe it, it is more of a feeling, maybe you know what I am trying to say) is absent from UnOrthodox Jews. And not necessarily UnOrthodox Jews, but even sometimes Modern Orthodox Jews. Because if you go to an UnOrthodox Jewish temple (they say "temple" not "shul") you often find people there in jeans. Or t-shirts. Or whatever other casual outfit (on weekdays). And that just does not feel normal. It looks funny. Because you do not get a sense that these UnOrthodox Jews have their own identity. You get the sense that these Jews are lost in between the real world and the Jewish world and they want to combine the two together in a harmony (but that harmony never works).

Or consider Jews who wear their titzis out of their clothes, wear decorated yamulkes, and wear casual clothes, like jeans. It looks so stupid! Maybe I am just used to how Orthodox Jews dress, but when I see those kinds of Jews I just think to myself how stupid they look. I know, I know, Orthodox Jews look silly too with their tefillin. However, my point is that Orthodox Jews make it feel "right". Because they somehow manage to give the message that this is exactly who they are. The UnOrthodox Jews are just lost in between worlds.

It looks so ridiculous wearing jeans, titzis out, a knitted yarmulke with your name on it, and a t-shirt as a Jewish outfit. It gives the message that you are part of the normal world and Jewish at the same time. And it just does not go together.

The same is true when UnOrthodox Jews celebrate the Shabbos. When Orthodox Jews celebrate the Shabbos it feels "right". Like this is what needs to be done. When UnOrthodox Jews celebrate the Shabbos with their funny dressed members, with weird dancing, and sometimes music playing, it feels unnatural.

I guess what I am trying to say is that you have natural actors and unnatural actors. The unnatural actors make it feel like they are acting, the natural actors make it seem as if it is real. Orthodox Jews are the natural actors. UnOrthodox Jews just look so ridiculous, even by Jewish standards.

This is not supposed to be some sort of defense of Orthodox Judaism. Rather this is making fun of UnOrthodox Judaism. They are lost without their own identity.

Thursday, October 14, 2010

Liberal Lunacy 5: Complete Inheritance Tax

There are people who propose a 100% inheritance tax. They say that people should earn what they can earn themselves, not be given gifts that give them an advantage other people do not. When rich parents give their children millions of dollars it is unfair for poor people, because the rich have an advantage over the poor. Therefore, there must be a 100% inheritance tax that takes away this money from the rich so that their children should earn their money just like everyone else. That is what is fair.

A problem with such a law is how will this law actually be enforced? Let us say that an old man knows he is going to die soon. So instead of writing a will to give his money as inheritance he simply gives out his money to his children as he is still alive. That way when he dies his children already received all the money. The complete inheritance tax just makes it a minor inconvenience for rich people to give out their money. Thus, the goal of preventing the children from inheriting millions from their parents is not achieved. It will still happen. The only way to prevent the rich from giving money to their children is to regulate the way rich people can spend their money. I am sure I do not have to explain why such a system is a totalitarian system. So such a egalitarian goal is incompatible with a free society.

But the method of enforcing such a law is not even the main problem with it. The major problem with such a goal is that it is impossible to make the world fair. Some people are born extraordinary smart. Some people are born really dumb. The smart ones have it easier to learn than the dumb ones. It is certainly unfair that there are smart people and dumb people. What must the state do? Must the state keep the smart people out of universities because it is unfair for the dumb people? Or must the state accept exactly 1/2 smart and 1/2 dumb people because that is fair? Or consider really beautiful people. Some people are ugly. The beautiful people would be able to find physically attractive people to sleep with, the ugly would have a big difficultly here. That is unfair. Or consider healthy people. Some people hardly ever get sick. Some are constantly stricken with disease. Is that fair? What must the state do? Must the state now forcefully inject into healthy people diseases so that they can be sick too because that is fair?
There is nothing fair about this world whatsoever. The amount of money people have is actually one of the more fairer things in the world. Because rich ugly sick people would give up their money to look attractive and be healthy. They do not value their money as much as beautiful people value their beauty or healthy people value their health.

It is certainly true that rich children will have it easier in this world, but it is also true that children from intelligent partents would also have it easier in this world. Must the state prevent professors from educating their children because it is unfair for children who did not come from intelligent families? Some kids are taught to play the violin by their parents and have a valuable skill as a gift from their parents. Most kids do not have this talent. That is unfair. But must the state prevent parents from educating the kids from playing the violin because it is unfair for other children?

Money, just like everything else in the world, is unfair. But it is probably the most fair thing out of all unfair things that exist. Because a person with a few million dollars will blow it in his lifetime if he is a dumb person, and will have nothing to pass down to his kids. Money, unlike health and beauty and talents, is temporary unfairness.

It is highly inconsistent for people who complain about the rich giving money to their children as unfair and ignore all the other injustices that take place in this pathetic world on a daily basis.

Here is a video about inheritance (Milton Friedman): Here.

The Case Against the 911 Truth Movement

So you expect me to believe that the government was responsible for 911 and lied to its citizens about it? Some people get all offended by this, "how dare he say something like this about our government, the government would never do something like this". Of course, these people are stupid because all governments exist on lies. There is no such thing as a honest government, if it were honest, it would have difficulty existing from the anger of the people.

I am not offended by such a statement, on the contrary, I find it hilarous. Too hilarous to take it seriously. Because you expect me to believe that a giant clumsy central agency that has difficulty running the post office, that has messed up public education, that wastes billions of dollars on programs which are not necessary, that is so highly inefficient, that this agency can successful plot a brilliant attack and trick the entire world into believing that the people responsible for the attacks were someone else? The non-sense detector in me immediately lights up when I hear this, how can a government which cannot run a public school make this master plan happen so well?

The truthers probably are looking for an excuse to hate the US government. But you do not need to invent excuses, the excuses are all around you. Just look. They killed millions of people. They enslave about a million in iron cages. They steal money from the current generation through taxation. They steal from the poor by the process of inflation. And they steal from the unborn by the evil process of borrowing money. And lots of that money is not even used on you, it is used to fund its own empire and power structure, and to kill those who get in its way. I think that is all one needs to realize to hate the US government, and unlike the 911 truth movement, this is actually true.

Tuesday, October 12, 2010

Questions for Honestly Frum

A few days ago I made a response to Jezuzfree...,oh, sorry, I meant Honestly Frum here. It was not really so much of a response as it was an excuse to give him the middle finger. Though it did contain some reasonable points amidst all the insults. I later left a comment on Honestly Frum's blog saying that I made a response and he can read it if he wishes to read. Without any surprise at all, Honestly Frum deleted my comment and said, "Spinoza, I won't respond because you do not deserve a response." So I really doubt that Honestly Frum would answer my questions that I want to ask him.

Yesterday Honestly Frum made a post about Yehuda Levin, here. I knew about Yehuda Levin for a long time now. I have seen his YouTube channel and videos, and I have heard some of his talks on Jewish radio. For those of you who do not know who Yehuda Levin is, he is like the Pat Robertson of the Orthodox Jewish world.

In my earlier response post to Honestly Frum I made the argument that Honestly Frum is not that deviated from the extreme "right-wing" (I do not know why these people are described as "right-wingers", but whatever, I will use the term everyone else seems to be comfortable with using even though I think a different term is suitable) form of Judaism. I said that Honestly Frum likes to present himself as some moderate and reasonable Orthodox Jew but in all actuality he is similar to the right-wingers of Orthodox Judaism that he complains about.

I will demonstrate my proposition about what I said of Honestly Frum. It will become apparent in these questions.

Honestly Frum, if you are reading this, then can you answer these questions? Begin the answer to every one of these questions as "yes" or "no". Then you may, if you wish, give a paragraph explanation to why you take the position that you take. Do not skip any of the questions.

1) Do you agree that the anti-homosexual movement is driven by religion? If not then can you explain why all anti-gay movements turn out to be religious?

2) Do you agree that the self-hatred that homos feel for themselves is the result of religion? If not then can you explain why this self-hatred is usually absent from secular or atheist homos living in secular communities?

3) Do you agree that if two good, honest, loving, and charitable men (or women) in the privacy of their own home have buttsex (or scissor) with one another then they have committed a greater evil than what Bernie Madoff did? If not then explain why under Torah and Talmudic Law the punishment for buttsex is death (a more severe form) while the punishment for theft is double-payment (a less severe form)?

4) Do you agree that if two men (or women) love each other and express their love in a sexual act then they are an "abomination"?

5) Do you believe that the Torah is the perfect word of God passed down exactly like it is to Moshe and that what it says regarding everything is perfectly moral?

6) Do you believe that if there was a Sanhedrin in place today, and Jews were ruled under Biblical and Talmudic Law in (Biblical) Israel, then it would be appropriate to kill two men who have buttsex with one another if they were brought to court by two witnesses as described in detail by the Talmud?

7) Do you believe that when Meschiach comes and the Jews are restored as a nation in the land of Israel that the Jews will set up a Sanhedrin court system that will judge people as described in the Bible and the Talmud?

8) Do you believe that the law system set up by the Jews of the generation of Meschiach would be a moral system of law?

9) Do you believe that it would be the responsibility of the state (the Jewish state under Meschiach) to carry out punishments and executions for homosexuals who want to live together ?

10) Do these questions make your uncomfortable?

11) Do you agree that Yehuda Levin would answer questions #3 through #9 as "yes" and that if you answer these questions as "yes" also then you are not much different from Yehuda Levin except in the difference that Yehuda would answer "no" to #10 and you would answer "yes".

Let us see if Honestly Frum is really honest.

(I do not expect any response back. He will use the excuse that my last post was "inappropriate" and that he does not want anything to do with me. Even though I stood away from some butt sex jokes in this post to remove any excuse from preventing Honestrly Frum to responding to these questions.)

Sunday, October 10, 2010

Martingale Betting System

Ever since the very beginning when people started to gamble (which was probably man's second hobby, the first was masturbation) people tried to devise methods of gambling, various systems, that they believed would make them win at gambling games. All of these systems are flawed. None of them work. The basic idea is that if the casino has a small advantage over you, then you will lose money, not matter how you play.

We will demonstrate the failure of betting systems by focusing on a popular system known as the "Martingale". The idea is the following. Suppose you are playing blackjack. You bet 10. If you win, then you won 10. If you lose then you bet again, but this time 20. If you win, you win 20 minus the 10 you lost earlier, which is a net gain of 10. If you lose then your net lost if 30. But if you lose then the next game you bet 40. If you win then your net gain would be 40 minus the 30 you lost earlier which works out to be 10. If you lose your next lose would be 70. But then next game you bet 80. And this process continues. The general rule is that if you lose you double your bet. The idea of the Martingale is based on that eventually you will win. And when you win you will get back all your lost money. If you start win 10 then your bets will be: 10,20,40,80,160,320,640, .... It is highly unlikely for a casino to consecutively to win all those games. Thus, you expect, with extreme likelihood of winning back your lost money.

It is often useful to consider extreme examples when working on a problem. Here we will consider the example when the player has an infinite amount of money and the other case when the casino has an infinite amount of money.

If the player has an infinite money of money the Martingale always works. This follows by the infinite monkey theorem. The idea is simple enough. A monkey hitting a typewriter randomly will eventually type out a book. Because no matter how unlikely it is for a monkey to randomly type out a book it still has a chance happening. So eventually is must happen. So this means the Martingale system will always work, provided the probability of winning a game is positive. Even if the casino has a 99% advantage over the player the Martinage system will eventually, by the theorem, make the player win all his money back.

But there are two obvious problems with such an extreme example. The first one is that no one has an infinite amount of money (not even the Federal Reserve, the counterfeiting that they do has to come from paper, there is a limited amount of paper). The second one is that what would be the point of gambling if you have an infinite amount of money? The point of gambling is to try to make some profit. If you have an infinite amount of money and you gain a thousand dollars you still have an infinite amount of money. The feeling of danger and risk is also absent from gambling. If you have an infinite amount of money and you lose a million, you have not really lost anything. Gambling with an infinite money supply is pointless because the person no longer values his money, in general the more a person has of something the less he values it. However, this thought experiment was just suggested to see what is going on with the Martingale system.

So let us then suppose that the player does not have an infinite money supply, but that he has a very large money supply. Let us assume you have 10 quadrillion dollars. And you want to buy up the entire casino, which is worth, say 100 million dollars. So you bet 100 million dollars. If you lose you keep doubling your bet. The infinite monkey theorem is no longer applicable here because you have a finite supply of money. But because it is so unlikely for the casino to win every single game your winning chances are overwhelmingly on your side that you would win up the entire casino. In this case the casino would not bet against you. It is unfavorable for them to bet. Because they are in a losing position.

Now consider the extreme case when the casino has an infinite supply of money. Ignoring the issue that no casino would have an incentive to be a casino given that it had an infinite amount of money, let us just pretend that there was such a casino. Such a casino would always play against a player with a finite amount of money. The player, even if he has 10 quadrillion dollars, will probably win billions or trillions of blackjack hands with the Martingale. However, the infinite monkey theorem is now applicable on the side of the casino this time. So eventually the player would have to disastrously lose every single game that he has played. So the casino would lose trillions of dollars, but eventually they will get lucky and win, at once, more trillions of dollars back. Why this happens? Well we will explain the reasoning for this below.

So now we need to consider actual situations when both the player and the casino always have a finite amount of money. Let $b$ be the bet a player bets in a game (say the red/black game on the roulette wheel, to make it as simply as possible). For most people $b$ will be little money, like 10. Not many people in the world have enough money to bet 1000 dollars and double it many times along the way. Let $n$ represent the number of games the player is able to play. Because it is assumed that the player has a finite amount of money there is a maximum number of games he can play before he loses all his money by being dreadfully unlucky. For example, if he has 1000 dollars with him and he bets b=10, then he can only play games with bets: 10,20,40,80,160,320. Therefore, n=6, he can only play six games. He cannot play seven because he does not have enough money for the seventh game, that would be 640, but that is too much. Let $p$ be the probability that the player wins. On the classical roulette wheel there are 26 red slots, 26 black slots, and 2 green slots. The chances of guessing the right color between red and black is almost 50%, except the casino has a slight advantage, because your chances are at 26/54 = 48%.

What are the possible ways to win on the roulette wheel under the Martingale? You can win your first game (W). You can lose your first game but win on your second game (LW). You can lose your first game, lose the second game, but win on your third game (LLW). Or (LLLW). And this keeps on going (LL...LW) until you are at your n-th game, the last possible game you can play. What are the possible ways to lose on the roulette wheel under the Martingale? There is only one way. When you lose all your possible money by doubling bets, i.e. the situation is (LL...L) where you played n-games.

So your winning events are: (W),(LW),(LLW), ... (LL...LW). Where the last one contains n total games. Your losing event is just: (LL...L) where the number of times L appears is n-times.

What is the probability that you would win? It is any one of the possibilities (W),(LW),.... happening. To compute this probability it is a lot easier to look at the reverse problem. What is the probability of losing everything? It is just (LL...L). The probability of this is $(1-p)^n$. The reason for this is because you have probability p of winning, so that your probability of losing is (1-p). Therefore, losing n-times in a row is just $(1-p)^n$ by just multiplying (1-p) by itself n times. This means the probability of winning, in any one of these winning events, is $1-(1-p)^n$.

The probability that you would win under Martingale on the roulette wheel with probability p=.48 is therefore given by $1-(.52)^n$. Notice that if $n$ is a large number, say n=10. This quantity works out to be .998, your chances of winning the bet b are at 99.8%. The higher $n$ the smaller $(.52)^n$, the smaller $(.52)^n$ the closer and closer $1-(.52)^n$ is to 1. Therefore, as $n$ goes to infinity the probability of winning $1-(.52)^n$ goes to 1. However, in our finite case the number $1-(.52)^n$ is very close to one, almost one, but not quite.

A gambler would look at these numbers and scream "success, the Martingale works!". But he is missing one very important point. When you win, you win b. But if you are to lose the game, you are to lose disastrouly. It means you would lost b in the first game, 2b at the second game, $2^2b$ at the third game, and so on until you would lose $2^{n-1}b$ on the last n-th game. In total you would lose $(1+2+2^2+...+2^{n-1})b$. Now using the identity that $1+2+2^2+...+2^{n-1}=2^n - 1$ we see that if you were to lose you lose $(2^n-1)b$. This is an exponential lose. You win little, but you lose exponentially.

A better way of understading how you win or lose is by looking at the expected value. Say that you are playing a game where your chances of winning at 1/10. This means if you bet a dollar you essentially get back 10 cents, and lose the other 90 cents. Your expected gain/lose in such a game, if you bet b, will be $-(9/10)b$. If you were playing a game where your chances of winning at 1/2, then your expected gain/loss would be 0. The expected gain/lose can be calculated by: (win on bet) TIMES (probability of winning) MINUS (loss on bet) TIMES (probability of losing).

Let us try to figure out the expected value for our Martingale system. When you win, you win $b$, at probability $1-(1-p)^n$. When you lose, you disastrously lose $(2^n-1)b$, at probability $(1-p)^n$. The expected gain/loss is therefore, $b(1-(1-p)^n) - (2^n-1)b(1-p)^n$.

It is interesting to consider the case when p=1/2, i.e. the game is fair. In this case, $b(1-(1-p)^n-(2^n-1)b(1-p)^n$ works out to be $b(1 - (1/2)^n) - b(2^n-1)(1/2)^n$. Now expand everything out. To get, $b-b(1/2)^n - b + b(1/2)^n = 0$. The expected return under the Martingale system in a fair game is 0. All this means is that what you win, but eventually after many many games you will be lose, and you would be back to no overall winnings.

Let us consider when the expected gain is positive. So we need to solve the inequality $b(1-(1-p)^n) - (2^n-1)b(1-p)^n >0$. Dividing by b we get $1-(1-p)^n-(2^n-1)(1-p)^n>0$. Substract 1 from both sides and multiply by -1, but remember when you multiply by a negative number the inequality is switched. So we get $(1-p)^n+(2^n-1)(1-p)^n<1$. Divide by $(1-p)^n$ to get $1+(2^n-1)<(1/(1-p))^n$. This works out to be $2^n<(1/(1-p))^n$. For the left hand side to be less than the right hand size it means the base of the exponent must be smaller, so that $2<1/(1-p)$. Multiply to get $2(1-p)<1$. Divide to get $(1-p)<1/2$. Multiply by -1 and switch inequality to get $p-1>-1/2$. Add 1 to both sides to finally get $p>1/2$. So as long as the probability is greater than 1/2, i.e. the game is favored for you, you are expected to earn a gain by the Martingale system. Of course, if you have an advantage there is no need to to use this system at all. If however, $p<1/2$ then your expected gain would be negative. Which means you would expect to lose money with this system.

These simple calculations show that the expectation of using the Martingale system is negative in casinos (since they all have an advantage over the player). And so the Martingale system is doomed to fail.

The Martingale is a short-term system. If you just want to win 100 briefly in a casino then go ahead and use this system. But do not expect on being able to make a living doing this. Since eventually you will lose it all in one unfavorable game. Long-term effectiveness of the Martingale is just as effective as standard betting. Do not use it, unless you want a short-term win once in Las Vegas or Atlantic City (assuming you are not banned from entering the casinos).

Saturday, October 9, 2010

Give Money to Homeless?

A lot of my Jewish friends say that I should not give money to the homeless because the homeless are homeless because they choose to be homeless, not only should I therefore not help them, they would probably spend that money on drugs if I was to give it to them. I have even heard a Rabbi tell me that I should not give them food because the city has places for them to go if they really wanted to get food. Are most Jewish people correct on this issue?

I am not sure where I stand on this issue, it has been bothering me for a long time. I am not sure because I agree with much of what Jews told me, but I just disagree with their conclusion. I do think that a lot of homeless are homeless because they are failures on life. But at the same time I do think there are people who had unlucky events happen in their lives that made them end up as homeless. I also very strongly believe that the restriction to enter the labor market (from regulations) leads to more unemployment (like the minimum-wage law) and other labor regulations, thus the extremely poor have a tough time finding a job. In this sense the homeless are homeless not through a fault of their own, but a fault of policies that prevented them from being employed.

So I am kinda split on the issue of whose fault is it? I do think there is some personal responsibility, because everyone has to take responsibility for his actions, but the question is to what extent are the homeless responsible? Lots of my Jewish friends are under the impression that it is entirely the fault of the homeless for being homeless. But I do not really think so. I think that in many cases this is true. But in many other cases it is not.

When I see a homeless person I give him the benefit and I assume that he is homeless by events that made him unfortunate. I do not know why I do that. Perhaps, it is my faith in people. I guess this is one of my weaknesses - to have faith in people (though this faith was really shaken when I learned that the top selling PC game of all time was the Sims). If I have money to give I usually give it.

But let us consider the worst-case scenario. Let us consider the homeless person is homeless entirely through his own fault, and let us also assume that he will spend that money to buy himself drugs. But even in this case I am not bothered. Because I see compassion and altruism as virtuous. And so even if someone does not deserve help I still think humans need to learn to be compassionate and altruistic. Giving money to homeless, even if it is their own fault, is a sign of compassion and altruism, which is why I still think that we should give money (or food) regardless.

But what about the objection that they may buy drugs with it? My question is, who cares if they buy drugs? Let them buy drugs and enjoy their depressing lives. Besides do you really think they buy drugs with that money. Somehow I doubt that. Drugs are expensive (the fault of the failed war on drugs). Do you really think that the dollar they get from you will be enough to buy something? Even if twenty people give them a dollar that is not enough money for them to get a supply of drugs. I am skeptical to the idea that homeless people use your money to get drugs for this reason.

This is what I really think about the homeless situation. I think that most ordinary people are not comfortable with homeless people next to them. Homeless are outcasts and so normal people do not want the homeless near them. People also assume that homeless are dangerous or they would rob you, so they have an added negative feeling from this falsely perceived threat. Now the anti-homeless attitude is very clear. Since people do not like them they invent excuses not to give them money. So they invent the excuse that the homeless are all failures and that they just wants drugs. But maybe I am wrong. I do not know.

Should we give money to the homeless?

Friday, October 8, 2010

Response to Honestly Frum

Honestly Frum made a most recent post about a gay Jewish couple which is going to marry . This is my response.

I have been reading Honestly Frum, from time to time, for like a half-a-year now. And I hate him. I hate him because he is the JezuzFreek777 (a Christian YouTuber, that perhaps some people might know of) kind. He tries to show himself off as some moderate Jew who happens to be reasonable, but in all actuality he is on the extreme side. If you read his "About Me", he says, "pointing out the ever growing extremism of the religious right". What is the only possible implication of that statement? Obviously, that he views himself as some moderate and reasonable Jew, not like the other Orthodox Jews he distances himself from.

Now there is nothing wrong with extremism. I am sure lots of people would call me an extremist. But here is the thing, I do not hide it. I do not hide what I really am and what I really believe. Honestly Frum does not do that. He is an Orthodox Jew, so he must accept the Torah's law that gays must be killed if they have buttsex. That is not moderation. See, at least Jewish Philosopher, as retarded as he is, and as evil as he is, does not hide his extremism. Honestly Frum does. That is what I so hate about him. So Fuck you. Fuck you, fuck you, fuck you. Fuck you and sit on my middle finger. Fuck you and the horse you rode in on.

Of course, he is not going to respond to me. Because I used, oh my science, nivel peh. Oh my science, these words, these words, they scare me! Run!

I am a douchebag. People who read some of my posts know that. I am a terrible person. And I am a pretty pathetic person. But I do not hide it. I do not pretend as if I am someone who I am not. This post alone testifies to this fact. But Honestly Frum why not be honest too, tell everyone that you want to retain the tribal Jewish ways without trying to seem moderate.

Anyway, let me get to the actual point I want to discuss.

Orthodox Jews, especially the fox-like ones like Honestly Frum, give the illusion that they reach out and help the homosexuals. They will tell you that they supported the event at YU. They will tell you that they support the Statement of Principles that was signed. But the reality of the situation is obvious to any outside observer to Judaism. It is Judaism itself that causes homosexuals to be depressed. It is Judaism to blame on their depression. (To be fair it is religion in general, not just Judaism, but I am concentrating on Jewish gays). What Judaism does is slaps the homo's face, makes them cry, and them pretends to reach out and care. The truth is that if Judaism never existed (and religion in general) there would be no anti-homosexuality movement.

Honestly Frum sees the marriage of a gay couple as a path downhill of morals in people. I have a question for you? What are you some 40 year old guy? People your age (I get the impression that you are in your 20's) are not supposed to be complaining about the decline of morality in our society. That is something for the 40 year olds to do. That is their job. That is for the Bill O'Reillys and the Sean Hannitys to complain about. The job of the 20 year olds is to fool around and make the 40 year olds angry.

But seriously now, what is there to be afraid of. The slippery slope that you are talking about is not a slippery slope, it is progress. There is no slippery slope. All what is happening is that people are slowly realizing the non-sense that is taught in Judaism and are abandoning it. The world is not worse, people are not worse, people are not meaner, people are not greedier, just because this progress is happening. So what can you possibly have against it?

Pre-marital sex. Oh my science, teenagers doing their normal biological functions and enjoying themselves, how terrible! Honestly Frum, pre-marital sex is great! You ought to try it sometime. You seriously need to get laid. In fact, the only good kind of sex is pre-marital sex. Marital sex is only good for the first week, or two weeks, then you realize that you must have sex with the same woman until you die, and she will get ugly soon. That is when you realize that marital sex was one big mistake. Long live pre-marital sex.

You ask, when acceptance starts when does it end? The answer is, it does not end anywhere. Social progress, or what you consider to be a slippery slope and the decline of our morals, is precisely accepting those people who are different. Today people have an issue with homosexuality. Tomorrow they will learn to accept them. The day after tomorrow they will learn to accept pre-marital sex. And this will continue going. And there is no stopping it now. Orthodox Judaism will fight as hard as it can to put a stop to this, but in the end they will lose, as it is losing today.

The point of all of this is that you are not some moderate. You belong to Orthodox Judaism and you want to see it kept the same way as it is. You just know how to act moderate. Just like Artscroll. Their series often look moderate, reasonable, and respectable, but really it is not. It is just Orthodox Judaism dressed up in more attractive clothing.

Thursday, October 7, 2010

Response to Tal Yarkoni

I am responding to this blog that I came across online. The author is a doctor of psychology, but this is irrelevant. He will probably never see my response in his life but I decided that I respond to it regardless. He wrote (about things that we used to believe but does not believe anymore):

That libertarianism is a reasonable ideology. I used to really believe that people would be happiest if we all just butted out of each other’s business and gave each other maximal freedom to govern our lives however we see fit. I don’t believe that any more, because any amount of empirical evidence has convinced me that libertarianism just doesn’t (and can’t) work in practice, and is a worldview that doesn’t really have any basis in reality. When we’re given more information and more freedom to make our choices, we generally don’t make better decisions that make us happier; in fact, we often make poorer decisions that make us less happy. In general, human beings turn out to be really outstandingly bad at predicting the things that really make us happy–or even evaluating how happy the things we currently have make us. And the notion of personal responsibility that libertarians stress turns out to have very limited applicability in practice, because so much of the choices we make aren’t under our direct control in any meaningful sense (e.g., because the bulk of variance in our cognitive abilities and personalities are inherited from our parents, or because subtle contextual cues influence our choices without our knowledge, and often, to our detriment). So in the space of just a few years, I’ve gone from being a libertarian to basically being a raving socialist. And I’m not apologetic about that, because I think it’s what the data support.

Let me begin by saying that I think it is great to finally see a person admit that he is not for freedom and staying out of people's lives. Often when people deny freedom they never admit it. He starts of by saying, "if we all just butted out of each other’s business and gave each other maximal freedom to govern our lives however we see fit". The implication I get from here is that Tal no longer thinks that we should entirely stay out of other people's lives, well at least he is honest about it.

Then he says, "I don’t believe that any more, because any amount of empirical evidence has convinced me that libertarianism just doesn’t (and can’t) work in practice, and is a worldview that doesn’t really have any basis in reality." Okay, so where is your argument? You make the statement that freedom does not work, but you leave it as a dangling statement. You never explain this point.

I heard the argument, "I am a realist! I am a realist! I just go with what the empirical evidence says!" way too much that it makes me want to vomit at this point. It is an empty statement. It does not mean anything. Because I can play the same game and claim I am a realist. Besides, it is an empty statement for the reasons that I do no not want to explain again (

Furthermore, what Tal said is entirely wrong, liberty does work, and it works more efficiently (for the most part) than anything else in history. But unlike him I will offer a quick defense of this statement. I will even supply a historical argument because he claims to be a realist, so he will probably be unimpressed with rational arguments in favor of liberty. Consider America shortly after the Civil War. From about 1860's to 1910's. There are very few examples in history of libertarian societies but during this time period America was the closest it ever was to a libertarian society. The economy was for the most part laissez-faire. I am not sure about civil liberties, I know this was always a problem through out US history, but if there were suppression of civil liberties then it was more of a state and local suppression than federal suppression. But as far as economics goes America was very much a laissez-faire economy. This was the time period of American history that saw the greatest rate of increase. This was the time period when the immigrants were all rushing to come here. Were they coming here because it was so terrible, why would they come here to be exploited? The standard of living was rising. This was also the time that saw great charitable contributions of private citizens. And it "worked", whatever that means to you. While it is true that life today is better than in the past, to conclude that this is the failure of laissez-faire would be nothing but a post-hoc-propter-ergo-hoc fallacy.

Time to move on to when he says, "When we’re given more information and more freedom to make our choices, we generally don’t make better decisions that make us happier; in fact, we often make poorer decisions that make us less happy." I am not sure if you know this but liberty is not about happiness, it is about, umm, let us see, what is the word?, liberty. Liberty is not always happiness and happiness is not always liberty. Freedom includes the freedom to fail. And freedom means that people have the right to be douchebags. The KKK certainly makes a lot of people unhappy. But we do not ban them because we recognize their right to assemble and free speech, despite the fact that most people would be happier if the KKK did not exist.

Besides, happiness sucks. What do happy people ever do? If people are always happy they hardly achieve anything. If hunter-gatherer primitive men were happy with their lives they never would have developed agriculture and they would have never developed fire. They would be happy with their lives. People act because they are unhappy with the way things are. People create new inventions that make our lives better precisely because people are unhappy with the current state of the world. Happiness is counter-productive. Happiness is counter-philosophy. Happy people talk about the weather, and their family, their kids, and what trips they like to take on their vacation. Oh my science, how exciting! I wish I can live such an unbelievable exciting life. So happiness sucks, basically what I am saying. Pursuit of happiness that is fine, that is what keeps people motivated and interesting, but not actual happiness.

Friedrich Nietzsche criticized the utopian socialist state by saying that if this state was to actually come to be then it would kill the human drive for innovation and greatness. As Nietzsche said, "one must have chaos within oneself to give birth to a dancing star". Discomfort and struggle is what makes people great. People have to work hard to make themselves great. Being given a gift of an easy life does not make people great. It is in this way that a utopian socialist state would destroy much of human greatness and struggle.

All I am saying is that it seems that your goal is to make as many people as you can happy. Which just seems like such a boring society to live in. Would you really want to live in a world like that? Heaven sounds rather boring to me. I know that you do not actually believe in an actual utopian socialist state, because you seem smart enough not to believe in silly utopian fantasies, though I am curious to know. If you could snap your fingers and create a utopian socialist state in where everyone or nearly everyone would be happy and have what they desired, would you actually want to live in such a world? Or would you rather live in a world were people have to struggle, have to work, but they would eventually be able to make their lives a little better. Would you choose happiness over the pursuit of happiness?

Next point Tal says, "In general, human beings turn out to be really outstandingly bad at predicting the things that really make us happy–or even evaluating how happy the things we currently have make us." I entirely agree with you here, you probably know this better than I do because you are a psychologist after all. People are basically dumb, and most of us have no idea what we really should want. Often people cannot make good choices for themselves. But by what madness can you possibly conclude that if people cannot make good choices for themselves that other people can make better choices for them? If people cannot make good choices for themselves, then it is insanity to propose that other people (who cannot make their own good choices) make better choices for other people. Is this not what you try to imply? I know you do not actually say it, but you most certainly imply it. You do imply about taking away the freedom of the people to fully control their lives. You do say that people cannot make good choices for themselves. And you do say that you are a "raving socialist", the only implication from this that I get is that the state should make certain choices for other people. But then you have others making choices for others, which is an insane proposition given what you said about human choices. Capitalism sucks. That certainly is true. Because the whole world sucks, and everything that has to do with the world sucks. But how do you come to defend socialism by saying that capitalism sucks? True, capitalism sucks, but it sucks far less than any other system in the world.

I am willing to bet by chopping off my tiny little penis that Tal calls himself a "social democrat". Because self-identified proud socialists today love to call the system they support as "Social Democracy". What Tal said about people making poor choices can be extended to people making poor voting choices. If people cannot make their own choices for the better, then certainly they would make poor voting choices. That rationally follows. So does this means that people should not be able to vote? I doubt that Tal would say "yes" because that would negate his social democracy. Tal's argument about the stupidity of people (which I do agree with) can be turned around and used to deconstruct the stupidity of democracy. This makes Tal inconsistent.

I think I will end it here because I am not sure what else he is talking about.

Wednesday, October 6, 2010

Sizes of Linear Groups

Let F be a finite field with q elements. In fact given q (a power of a prime) there is just one such field (up to isomorphism of course, but algebraists are blind to isomorphic structures). Thus, it is well-defined (up to isomorphism, but again algebraists are supposed to be being to isomorphic structures) to define GL(n,q) to be the set of all $n\times n$ matrices over the field $\mathbb{F}_q$ that are invertible. It is easy to see that GL(n,q) is a group under matrix multiplication. This group is referred to as "the general linear group".

We can also define SL(n,q) to be the set all those elements in GL(n,q) which have determinant 1. Clearly, this is a subgroup of GL(n,q) because it contains the identity elements, which has determinant one. And if A,B are matrices with determinant 1 then AB is a matrix with determinant one by the formula $\det (AB) = \det (A) \det (B)$. This subgroup is referred to as the "special linear group".

We can define the following map $\phi : \text{GL}(n,q) \to \mathbb{F}_q^{\times}$ by $A \mapsto \det A$. By the property of determinants it follows that $\phi$ is indeed a group homomorphism (here we are treating $\mathbb{F}_q^{\times}$ as the multiplicate group of the field of the non-zero elements). This map is clearly onto, because for any $a\in \mathbb{F}_q^{\times}$ let A be an $n\times n$ diagnol matrix with 1's everywhere on the diagnol except for one entry which is replaced with $a$, such a matrix has determinant $a$. By definition $\ker (\phi) = \{ A \in \text{GL}(n,q) | \det (A) = 1\}$, but this is precisely the special linear group.

By the first isomorphism theorem (refer to here) the above group homomorphism gives us the isomorphism $\text{GL}(n,q)/\text{SL}(n,q)\simeq \mathbb{F}_q^{\times}$. For our present discussion this isomorphism will not be important for us. What will be important is that because the field, the general linear group, and the special linear group are finite, we get the following combinatorical statement $|\text{GL}(n,q)| = |\text{SL}(n,q)|\cdot (q-1)$.

What we would like to do is determine the sizes of the general and special linear groups. The above formula tells us that if we can determine the size of the general linear group then the size of the special linear group is immediate.

Let A be an $n\times n$ matrix over a field $\mathbb{F}_q$. This matrix A is invertible if and only if $A\bold{x}=\bold{0}$ has only the trivial solution where $\bold{x},\bold{0}$ are vectors in $\mathbb{F}_q^n$, by basic linear algebra. Let $\bold{c}_i$ be the i-th column of the matrix as a vector, and let $x_i$ be the i-th coordinate of $\bold{x}$. Then $A\bold{x}=\Sigma_i x_i\bold{c}_i$. Thus, the condition that $A\bold{x}=\bold{0}$ has a trivial solution is equivalent to saying $\Sigma_i x_i\bold{c}_i=\bold{0}$ has only a trivial solution for $x_i$. Thus, $A$ is invertible if and only if the columns of A are linearly independent when viewed as vectors.

Now we will determine the size of GL(n,q). Let A be an $n\times n$ matrix. For it to be invertible its first coloumn cannot consist of zeros. Because a zero vector in any set of vectors makes the set linearly dependent (unless the vectors belong to the zero-vector space, which is trivial and uninteresting). So we have any choice we want for the first n coordinates in the first coloumn except all zeros. This is $q^n-1$ choices. Consider the second coloumn. It can be anything except a multiple of the first coloum - because remember two non-zero vectors are linearly dependent iff they are multiples of one another. There are $q$ multiples of the first coloumn and $q^n$ choices for the second coloumn which means there are $q^n-q$ choices for the second coloumn such that the matrix still stays invertible. Now consider the third coloumn. It cannot lie in the spam by the first two coloumns. The spam formed by the first two coloumns is $\alpha \bold{c}_1+\beta\bold{c}_2$ where $\alpha,\beta$ vary over $\mathbb{F}_q$, and by linear independence distinct values for these constants determine distinct vectors. Thus, there are a total of $q\times q=q^2$ vectors that lie in the spam of the first two coloumns. The third coloumn must be linearly independent from this one, and there are $q^n$ choices which means the are $q^n-q^2$ choices for the third coloumn to be chosen so that the matrix stays invertible. Continuing in this manner we arrive at $|\text{GL}(n,q)| = (q^n-1)(q^n-q)...(q^n-q^{n-1})$. To make the formula more convenient we can pull out $q^k$ from $(q^n - q^k)$ to get $q^k(q^{n-k}-1)$. So that $|\text{GL}(n,q)|=q^{n(n-1)/2}(q-1)(q^2-1)...(q^n-1)$. This immediately tells us that $|\text{SL}(n,q)| = q^{n(n-1)/2}(q^2-1)...(q^n-1)$.

Actually there are two more linear groups that are worth considering. Those are the "projective general linear group" and the "projective special linear group". Recall that the "center" $\text{Z}(G)$ of a group $G$ is the subgroup $\{z : zg = gz\}$. The center of a group is always a normal subgroup. So we can consider the general linear group modulo its center. This is the projective general linear group $\text{PGL}(n,q)$. Its order will be the order of the general linear group divided by the order of the center. But we know what the center is, we determined the center back here. So the center is the non-zero multiples of the identiy matrix. There are $(q-1)$. Now we just need to divide the order of the general linear group by (q-1). Dividing by $(q-1)$ we get the number, $|\text{PGL}(n,q)| = q^{n(n-1)/2}(q^2-1)...(q^n-1)$.

Finally we need to determine the size of the projective special linear group. By a similar argument we can show that the center of the special linear group is $kI$. But $k$ cannot vary of any element of $\mathbb{F}_q^{\times}$. Because we require $\det (kI) =1$. Notice that $\det(kI)=k^n\det (I) = k^n=1$. Therefore, $k$ must be an n-th root of unity in the finite field $\mathbb{F}_q$. The number of roots of unity will depend on $n$. From basic field theory we know that $\mathbb{F}_q^{\times}$ is a cyclic group. Therefore, the number of solutions to $x^n=1$ is $\gcd(n,q-1)$. Therefore, if we set $d=(n,q-1)$ then $|\text{PSL}(n,q)| = \frac{1}{d}q^{n(n-1)/2}(q^2-1)...(q^n-1)$. So in the special special case when $n$ is relatively prime to $q-1$ we get that $|\text{PSL}(n,q)| = q^{(n-1)/2}(q^2-1)...(q^n-1)$.

Tuesday, October 5, 2010

Math and Virginity

There seems to be a connection between doing math/being a math major and being a virgin. It is a common stereotype that mathematicians and math majors are virgins. This stereotype, like most stereotypes, are based on truth. Of course, not all math majors you would ever come across in college are virgins. But a far majority of them are.

I saw a funny statistic online here. I am not a fan of statistics, for many reasons. We can ask, for example, that these represents the rates of students that actually participated in this survey. It is likely to assume that the non-virgins are more likely to participate in the survey. Besides math majors always have non-normal students which have something mentally wrong with them. These are the kind of students that will probably not participate. So we must ask if these rates really represent all the students. I doubt it, in particular for math majors, I still think an 83% virginity rate for math is too low. But whatever, this statistics confirms the old stereotype about mathematicians that never get any pussy.

I know a lot of math majors. Some of them are normal people and get pussy. But most do not. Not only do they not get pussy but they have never been kissed by a girl. One of my good friends is even scared to talk with girls. I know he is not gay or asexual, because he watches straight porn. But he is just scared to talk with them. In fact I know several math majors that fit this kind of description, nice people, friendly, want to meet a girl, but just are too scared. I remember I had a young college professor that I looked up online. I managed to find him on some forum from years back saying that he is scared to talk with girls. (I have a different problem. I am not scared of girls. They are scared by me. My dangerous personality scares them, and my physical repulsion drives them away).

So I really think there is something up with doing math and being a virgin. Other kinds of people do not have the same struggles. Physicists and other scientists do not go through the same hassle as mathematicians go through, in general, to find some pussy.