## 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.
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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 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.

## 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.

## 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.