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?
Saturday, October 9, 2010
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.
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):
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 (
here).
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.
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.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.
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 (
here).
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 $|\text{PSL}(n,q)| = q^{(n-1)/2}(q^2-1)...(q^n-1)$.
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.
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.
Saturday, October 2, 2010
Determinism
Determinism is the position that all events in the universe come from prior causes, those causes themselves are prior causes of other events, and so forth. This all goes back until the beginning of the universe (or whatever the beginning is, even if there is a beginning). In this way the universe is like a giant celestial clock. It is purely mechanical, one event leads to another, and that is all what happens in the universe. Thus, when it rains tomorrow, this was all predetermined billions of years ago.
Famous determinists include the physicist Albert Einstein, the philosopher Baruch Spinoza, and the mathematician Pierre Laplace. Determinism was popularized by Spinoza who later influenced other people to treat this idea more seriously.
But there are people who do not accept this idea. The determinists are a minority. Indeed, to be a determinist it is necessary for one to reject theism. Theism holds that God gave people the ability to shape their future, if they are obedient they will have a good future, if they are rebellious their future would be unpleasant. Determinists reject this notion because people cannot shape their future, for their future has been predetermined from the very beginning (assuming there even was a beginning). Since a very large majority of the world consists of theists, determinists are a minority. But even among atheists there are some who still reject this idea. I do not know how many atheists take the determinist position, but from my experience it is very common to come across an atheist who consider determinism to be wrong. We will ignore the theistic objections to determinism because theism is silly, and the objections we will consider will come from a secular point-of-view.
I want to give arguments in favor of determinism because I too am a determinist. There is really just one argument for determinism. Things do not just magically happen for no reason. There were laws that were behind some event taking place, the event did not just jump into existence magically. It is rather silly to think that something like this can actually happen. Determinism is the default position to take.
So what are the reasons then that people give in objection to determinism? I will focus on two arguments which are generally used. The first one is that people say that they can choose to act how they want to act. They can decide if they raise their right hand or their left hand. They can just jump around like crazy, surly this cannot be determined if they can choose their behavior, can it? The second objection that people give is that the quantum world operates through random process which we cannot determine, in fact something like the uncertainty principle prevents us from being able to determine the outcomes on the quantum level, and so the universe is not deterministic.
Let us handle these two common objections. The first objection about people choosing to act how they want to act is an easy one to answer. Consider this question. Where do your thoughts to act the way you do come from? Do your thoughts magically and spontaneously jump into existence? No. Your thoughts are determined by your brain. And your brain is determined by mechanical processes that take place in it. Thus, your thoughts themselves are the result of purely mechanical processes that emerge into your thought patterns. Your thoughts are not uncaused causes. In fact, modern neuroscience points to exactly this conclusion. When people pick a choice, say between numbers 1 and 2, an fMRI is able to show within 6 seconds exactly what choice they will take. Thus, their decision is not spontaneous, but rather a result of some mechanical process in the brain that can even be studied scientifically using modern techniques.
The second objection is a lot deeper. I have some thoughts on this objection that I did not really hear other people say. So I will introduce a new word to make clearer what I am trying to say. Let us define "determinable" to mean "a deterministic process that can be computed".
Notice the difference between "determinable" and "deterministic". Determinability is a stronger condition than determinism. Determistic is simply the statement that all events have prior causes in an unbroken chain of cause and events. Determinability is the statement that these events can be computed. There is a difference between Laplacian determinism and Spinozian determism. For Spinoza determinism meant exactly what I meant by "determistic". For Laplace determinism meant a lot more. For him it means that we can compute the determined outcomes. Laplace made this statement clear with his thought experiment of the "Laplacian demon". A creature that knows all the particles and all the information of the universe. Laplace said that such a creature, if given the information and the computing power, would be able to compute all the determined events. Laplace was a mathematician, Spinoza was just a philosopher. It is not a surprise way Laplace had a much stronger form of determinism than Spinoza, because Laplace himself was capable of computing certain deterministic events.
Determinability is not possible, not to its full extent. Laplace did not know this, he lived in the 1700's, he did not know quantum mechanics nor did he know anything about Chaos. In the 1880's one of the great mathematicians, Jules-Henri Poincare' discovered the instability of the three-body problem. The three-body problem was solved (by one of my favorite mathematicians) Leonard Euler. But Poincare' managed to prove that the solutions are not stable, which meant that from a practical point of view it is not possible to compute the outcome of a three-body problem. It is still however deterministic, if we perfectly knew the measurements then we would be able to determine the outcome, but because our measurements are approximations and approximations are not good enough we can never really solve the three-body problem. This is a classical physics problem that does not fall into the category of determinable, but it is nonetheless still deterministic.
The quantum world presents different difficulties which lead to non-determinable events. But just like with the three-body problem it does not mean that these events do not come themselves from prior causes. It just means that we have limitations to what we are able to compute. Thus, what I am trying to say is that the universe is deterministic but not fully determinable.
There is a thought experiment that we can consider that will prove that determinability to its full extent is impossible. Let us assume that determinability is always possible (given enough computing power). If determinability is possible then we can write a book about all the major actions Laplace will ever take in his life. And we will indeed see that Laplace will follow the exact outline of the book. But this determinable result is assuming that Laplace does not know the information which is in the book. Suppose that we planned to determine all the major events Laplace would ever take in his life and we gave him that book, i.e. one of the major events is him reading the book on his major choices in life. Upon reading this book Laplace will violate the outcomes in this book. If the book said that we will buy a house in France he will rebel and purposefully buy a house in Germany to show that the book is wrong. No matter what we write in the book Laplace will be able to read and disobey. So we have something very interesting. If Laplace has no knowledge of our determinable predictions then he will act perfectly in accordance with these predictions. But if Laplace knows about our predictions then he will not act perfectly in accordance with our predictions, he will violate them. It is impossible to give Laplace a book of his major life choices and expect him to follow the book. Therefore, the determinism problem of determining how a person would act if given the information about his actions is impossible. There is no way to compute this. And it is in this way that determinability to its full extent is impossible. This does not mean however that determinism is invalid, it just means that determinability is not always possible.
The last thing I want to talk about is randomness. People object to determinism and say that the universe is just a bunch of random particles colliding with one another, randomness is not determined, so determism is false. These people who use this objection do not understand the meaning of "random".
What does the word "random" even mean? It really depends who you ask. The physicist might answer differently from the mathematician. But I think the mathematician is more accurate in this instance. The physicist will answer that "random" means events which are undetermined, I have seen textbooks that actually describe randomness by this incorrect description (for example, the terrible book on Mathematical Physics by Arfken and Weber). The mathematician's definition is much more precise and useful. "Random", as it means in probability theory, is an event that has an equal probability of happening as all other events. For example, consider a die. The probability of throwing a six is 1/6, which is the same probability as for any other event. That is why we say it is "random". Throwing a die is not an undetermined process. If we can calculate how much force you put into the die, the height of your arm off the table, the angle by which you throw the die, and so forth ... we might (assuming it is not a chaotic problem) be able to determine the actual outcome of the die. The die does not drop uncaused, it is all determined, but it is random nonetheless.
Yes, I agree, the universe is random. However, random does not imply undetermined. It just means that particles simply have equal probabilities of acting one way as another. So in conclusion, the way I describe the universe and solve the problem of determinism is by saying that the universe is a deterministic random but non-determinable universe. I hope this clears up any confusion people have about determinism, if not, well, you can go and kill yourselves because you are rather useless people.
Famous determinists include the physicist Albert Einstein, the philosopher Baruch Spinoza, and the mathematician Pierre Laplace. Determinism was popularized by Spinoza who later influenced other people to treat this idea more seriously.
But there are people who do not accept this idea. The determinists are a minority. Indeed, to be a determinist it is necessary for one to reject theism. Theism holds that God gave people the ability to shape their future, if they are obedient they will have a good future, if they are rebellious their future would be unpleasant. Determinists reject this notion because people cannot shape their future, for their future has been predetermined from the very beginning (assuming there even was a beginning). Since a very large majority of the world consists of theists, determinists are a minority. But even among atheists there are some who still reject this idea. I do not know how many atheists take the determinist position, but from my experience it is very common to come across an atheist who consider determinism to be wrong. We will ignore the theistic objections to determinism because theism is silly, and the objections we will consider will come from a secular point-of-view.
I want to give arguments in favor of determinism because I too am a determinist. There is really just one argument for determinism. Things do not just magically happen for no reason. There were laws that were behind some event taking place, the event did not just jump into existence magically. It is rather silly to think that something like this can actually happen. Determinism is the default position to take.
So what are the reasons then that people give in objection to determinism? I will focus on two arguments which are generally used. The first one is that people say that they can choose to act how they want to act. They can decide if they raise their right hand or their left hand. They can just jump around like crazy, surly this cannot be determined if they can choose their behavior, can it? The second objection that people give is that the quantum world operates through random process which we cannot determine, in fact something like the uncertainty principle prevents us from being able to determine the outcomes on the quantum level, and so the universe is not deterministic.
Let us handle these two common objections. The first objection about people choosing to act how they want to act is an easy one to answer. Consider this question. Where do your thoughts to act the way you do come from? Do your thoughts magically and spontaneously jump into existence? No. Your thoughts are determined by your brain. And your brain is determined by mechanical processes that take place in it. Thus, your thoughts themselves are the result of purely mechanical processes that emerge into your thought patterns. Your thoughts are not uncaused causes. In fact, modern neuroscience points to exactly this conclusion. When people pick a choice, say between numbers 1 and 2, an fMRI is able to show within 6 seconds exactly what choice they will take. Thus, their decision is not spontaneous, but rather a result of some mechanical process in the brain that can even be studied scientifically using modern techniques.
The second objection is a lot deeper. I have some thoughts on this objection that I did not really hear other people say. So I will introduce a new word to make clearer what I am trying to say. Let us define "determinable" to mean "a deterministic process that can be computed".
Notice the difference between "determinable" and "deterministic". Determinability is a stronger condition than determinism. Determistic is simply the statement that all events have prior causes in an unbroken chain of cause and events. Determinability is the statement that these events can be computed. There is a difference between Laplacian determinism and Spinozian determism. For Spinoza determinism meant exactly what I meant by "determistic". For Laplace determinism meant a lot more. For him it means that we can compute the determined outcomes. Laplace made this statement clear with his thought experiment of the "Laplacian demon". A creature that knows all the particles and all the information of the universe. Laplace said that such a creature, if given the information and the computing power, would be able to compute all the determined events. Laplace was a mathematician, Spinoza was just a philosopher. It is not a surprise way Laplace had a much stronger form of determinism than Spinoza, because Laplace himself was capable of computing certain deterministic events.
Determinability is not possible, not to its full extent. Laplace did not know this, he lived in the 1700's, he did not know quantum mechanics nor did he know anything about Chaos. In the 1880's one of the great mathematicians, Jules-Henri Poincare' discovered the instability of the three-body problem. The three-body problem was solved (by one of my favorite mathematicians) Leonard Euler. But Poincare' managed to prove that the solutions are not stable, which meant that from a practical point of view it is not possible to compute the outcome of a three-body problem. It is still however deterministic, if we perfectly knew the measurements then we would be able to determine the outcome, but because our measurements are approximations and approximations are not good enough we can never really solve the three-body problem. This is a classical physics problem that does not fall into the category of determinable, but it is nonetheless still deterministic.
The quantum world presents different difficulties which lead to non-determinable events. But just like with the three-body problem it does not mean that these events do not come themselves from prior causes. It just means that we have limitations to what we are able to compute. Thus, what I am trying to say is that the universe is deterministic but not fully determinable.
There is a thought experiment that we can consider that will prove that determinability to its full extent is impossible. Let us assume that determinability is always possible (given enough computing power). If determinability is possible then we can write a book about all the major actions Laplace will ever take in his life. And we will indeed see that Laplace will follow the exact outline of the book. But this determinable result is assuming that Laplace does not know the information which is in the book. Suppose that we planned to determine all the major events Laplace would ever take in his life and we gave him that book, i.e. one of the major events is him reading the book on his major choices in life. Upon reading this book Laplace will violate the outcomes in this book. If the book said that we will buy a house in France he will rebel and purposefully buy a house in Germany to show that the book is wrong. No matter what we write in the book Laplace will be able to read and disobey. So we have something very interesting. If Laplace has no knowledge of our determinable predictions then he will act perfectly in accordance with these predictions. But if Laplace knows about our predictions then he will not act perfectly in accordance with our predictions, he will violate them. It is impossible to give Laplace a book of his major life choices and expect him to follow the book. Therefore, the determinism problem of determining how a person would act if given the information about his actions is impossible. There is no way to compute this. And it is in this way that determinability to its full extent is impossible. This does not mean however that determinism is invalid, it just means that determinability is not always possible.
The last thing I want to talk about is randomness. People object to determinism and say that the universe is just a bunch of random particles colliding with one another, randomness is not determined, so determism is false. These people who use this objection do not understand the meaning of "random".
What does the word "random" even mean? It really depends who you ask. The physicist might answer differently from the mathematician. But I think the mathematician is more accurate in this instance. The physicist will answer that "random" means events which are undetermined, I have seen textbooks that actually describe randomness by this incorrect description (for example, the terrible book on Mathematical Physics by Arfken and Weber). The mathematician's definition is much more precise and useful. "Random", as it means in probability theory, is an event that has an equal probability of happening as all other events. For example, consider a die. The probability of throwing a six is 1/6, which is the same probability as for any other event. That is why we say it is "random". Throwing a die is not an undetermined process. If we can calculate how much force you put into the die, the height of your arm off the table, the angle by which you throw the die, and so forth ... we might (assuming it is not a chaotic problem) be able to determine the actual outcome of the die. The die does not drop uncaused, it is all determined, but it is random nonetheless.
Yes, I agree, the universe is random. However, random does not imply undetermined. It just means that particles simply have equal probabilities of acting one way as another. So in conclusion, the way I describe the universe and solve the problem of determinism is by saying that the universe is a deterministic random but non-determinable universe. I hope this clears up any confusion people have about determinism, if not, well, you can go and kill yourselves because you are rather useless people.
Friday, October 1, 2010
What is Mathematics?
This question has differing answers to it. Depending on which person you ask. This is what I consider to be mathematics. Mathematics is a subject which is concerned about coming to truth from the use of deduction alone, it is a rationalist constructed universe. It is not subject to empiricism whatsoever, it has no relation to the outside world, it is just the process of pure reasoning.
One can ask then how is mathematics different from what the rationalists did in philosophy? The difference is that mathematics uses formal reasoning and rigorous definitions. In philosophy the definitions are not formal, and the reasoning is not so binary, it is more fuzzy. Which is why pure reasoning has problems outside of mathematics. But within the scope of mathematics reasoning works perfectly. Hence this is why all mathematical theorems end up working and a lot of derived statements from rationalism either make no sense, too ambiguous, or just plain wrong. It was mathematics, in particular Euclid's Elements that inspired the rationalism approach to philosophy. However, the rationalists just do not have the same elegance and precision that the mathematicians always had. (This is not to suggest that one should not use the deductive approach outside of mathematics, only that it is not as clear and it is not foolproof as it is in mathematics).
Mathematics is therefore not a science. It is common confusion to refer to mathematics as a science but mathematics is fundamentally different from science. Science uses the inductive method of experiment to understand the universe. Mathematics does not use the inductive method and it is not necessarily applicable to the universe. It is certainly true that mathematics is the ultimate foundation for many sciences and without it science would not be possible but mathematics itself is not a science. Most mathematicians actually think of mathematics as a form of art. Mathematicians gain a lot of excitement and see beauty in mathematical theorems. It is an art form for them. In many cases a highly obsessive art form.
I divide mathematics into four categories, all of mathematics can fit into one of these (or a few) categories. I just want to mention that my classification of mathematics is entirely my own. However, I found it interesting that other people who classified mathematics classified it into three categories. Three of my categories match with what the other people classified mathematics, I just added a fourth category. I used to classify mathematics into three categories but it seemed to me that three is not enough. Mathematics can be classified as: geometry, analysis, algebra, and combinatorics. The standard classification of math is: geometry, analysis and algebra. I add combinatorics because I think it deserves it own category.
Combinatorics, put simply, is "counting without actually counting". Combinatorics is a category of mathematics that is concerned on determining all the possibilities without actually listing all the possibilities. For example, how many different Texas Hold'em hands are there? This question is not hard. We can simply list all possibilities involving two cards but such is a foolish method of solving a problem. There is a formula (called the "combinations formula" is you are interested to read about it) that gives the answer immediately. Thus, it is possible to determine the answer to this question without actually counting anything. Here is another kind of problem from combinatorics. What is the most number of rooks that can be placed on a chess boards such that no two of which attack each other? Again, we can list the millions of different ways we can put 1 rook, 2 rooks, 3 rooks, ... and see when, no matter how we put the rooks, they always attack one another. This is again a foolish and incredibly time consuming way to solve the problem. Instead we can use the pigeonhole principle. There are 8 coloumns on a chess board. If we put 9 rooks on a chess board then by the pigeonhole principle two of them end up in the same coloumn (since 9>8). In that column the rooks will attack one another. Thus, if we have nine rooks we are guaranteed that no matter how we position them that two must attack one another. Indeed 9 is the smallest such number, it does not necessarily work for 8, if all eight are put into different columns. So we solved this problem without actually listing the myriad of all possibilities. Mathematicians who work in combinatorics are combinatorialists.
Analysis is the study of change. Analysis, I believe, is by far the largest category in all of mathematics. It is incredibly important for applied mathematics. Calculus, what you learned in high school and college is the simplest example of analysis. Calculus worked with finding rates of change and areas of irregular shaped regions. Function and shapes that were always constantly changing. In baby high school math the distance-rate-time problems are unrealistic: "if a man travels at 60 miles per hour for 2 hours how much does he travel?". This is an unrealistic and overly simplistic problem. No one ever travels at 60 miles per hour constantly. People slow down, speed up, stop, the actual speed is variable and it can be complicated. Calculus allows us to solve this problem. Calculus allows us to compute the total distance traveled with variable speed. The problems in calculus are more complicated than in baby math where everything is taken as constant. There is so much more to analysis than just calculus. Differential equations play a big role in analysis. Differential equations are equations that describe the rate of change of a particular phenomenon, or a few phenomenon, solving such an equation gives us an understanding how something changes over time. Mathematics who work in analysis are analysts.
Geometry is the study of space. The word "geometry" is a Greek word that means "measuring the earth". Geometry was inspired by such practical problems but in its modern form geometry is way more abstract than what the ancients have ever envisioned. "Space" can be our three dimensional world. But "space" can also be a highly abstract 10 dimensional world. It can be Euclidean, i.e. a generalization of our world into more dimensions. Or it can be incredibly messed up. Sometimes geometry is synonymous with "topology". The word "topology" means, "study of space". In topology we deform objects into one another without breaking them apart (such as ripping them). From a topological point of view a square and a circle are the same because we can deform a square into a circle. But a sphere is not topologically the same as a sheet of paper because a sphere is closed, we cannot unravel a sphere into a plane without ripping it apart. Modern geometry also takes place on non-flat surfaces. Basic high school geometry that you learn was for the most part plane geometry. Geometry consisting of lines and points lying on some plane. Modern geometry allows to work with curves, not on planes, but on various curved surfaces. Mathematicians who work in geometry are geometers or topologists.
Algebra is the study of structure and symmetry. Algebra is a lot more difficult to explain than the others because algebra is a lot more abstract. The best way I can put it is with some example. Consider a cube. We can ask for the symmetries of a cube. That is, what operations on the cube leave the cube intact? We can rotate the cube about three of its axes without changing how the cube looks. We can also reflect a cube through its center, if we imagine a mirror passing through the center of a cube then reflecting the cube through the mirror will leave it unchanged. Rotations and reflections are symmetries of a cube. Once we have these symmetries we can form something known as a "group" which represents these symmetries and has operations defined between these symmetries. If we can understand what this group is, sometimes we say "determining the structure of the group" then we can reduce problems about symmetry to the solved problem of group structure. I realize that this is not the best way to explain what algebra is about but like I said it is not easy to explain what it is in a single paragraph. I am just trying to give some idea what it is about. Mathematicians who work in algebra are known as algebraists.
This is a basic overview of what mathematics is about. I cannot explain more because that would take a long time to explain. But if you are interested there are books written for people who are interested in understanding what mathematics is.
One can ask then how is mathematics different from what the rationalists did in philosophy? The difference is that mathematics uses formal reasoning and rigorous definitions. In philosophy the definitions are not formal, and the reasoning is not so binary, it is more fuzzy. Which is why pure reasoning has problems outside of mathematics. But within the scope of mathematics reasoning works perfectly. Hence this is why all mathematical theorems end up working and a lot of derived statements from rationalism either make no sense, too ambiguous, or just plain wrong. It was mathematics, in particular Euclid's Elements that inspired the rationalism approach to philosophy. However, the rationalists just do not have the same elegance and precision that the mathematicians always had. (This is not to suggest that one should not use the deductive approach outside of mathematics, only that it is not as clear and it is not foolproof as it is in mathematics).
Mathematics is therefore not a science. It is common confusion to refer to mathematics as a science but mathematics is fundamentally different from science. Science uses the inductive method of experiment to understand the universe. Mathematics does not use the inductive method and it is not necessarily applicable to the universe. It is certainly true that mathematics is the ultimate foundation for many sciences and without it science would not be possible but mathematics itself is not a science. Most mathematicians actually think of mathematics as a form of art. Mathematicians gain a lot of excitement and see beauty in mathematical theorems. It is an art form for them. In many cases a highly obsessive art form.
I divide mathematics into four categories, all of mathematics can fit into one of these (or a few) categories. I just want to mention that my classification of mathematics is entirely my own. However, I found it interesting that other people who classified mathematics classified it into three categories. Three of my categories match with what the other people classified mathematics, I just added a fourth category. I used to classify mathematics into three categories but it seemed to me that three is not enough. Mathematics can be classified as: geometry, analysis, algebra, and combinatorics. The standard classification of math is: geometry, analysis and algebra. I add combinatorics because I think it deserves it own category.
Combinatorics, put simply, is "counting without actually counting". Combinatorics is a category of mathematics that is concerned on determining all the possibilities without actually listing all the possibilities. For example, how many different Texas Hold'em hands are there? This question is not hard. We can simply list all possibilities involving two cards but such is a foolish method of solving a problem. There is a formula (called the "combinations formula" is you are interested to read about it) that gives the answer immediately. Thus, it is possible to determine the answer to this question without actually counting anything. Here is another kind of problem from combinatorics. What is the most number of rooks that can be placed on a chess boards such that no two of which attack each other? Again, we can list the millions of different ways we can put 1 rook, 2 rooks, 3 rooks, ... and see when, no matter how we put the rooks, they always attack one another. This is again a foolish and incredibly time consuming way to solve the problem. Instead we can use the pigeonhole principle. There are 8 coloumns on a chess board. If we put 9 rooks on a chess board then by the pigeonhole principle two of them end up in the same coloumn (since 9>8). In that column the rooks will attack one another. Thus, if we have nine rooks we are guaranteed that no matter how we position them that two must attack one another. Indeed 9 is the smallest such number, it does not necessarily work for 8, if all eight are put into different columns. So we solved this problem without actually listing the myriad of all possibilities. Mathematicians who work in combinatorics are combinatorialists.
Analysis is the study of change. Analysis, I believe, is by far the largest category in all of mathematics. It is incredibly important for applied mathematics. Calculus, what you learned in high school and college is the simplest example of analysis. Calculus worked with finding rates of change and areas of irregular shaped regions. Function and shapes that were always constantly changing. In baby high school math the distance-rate-time problems are unrealistic: "if a man travels at 60 miles per hour for 2 hours how much does he travel?". This is an unrealistic and overly simplistic problem. No one ever travels at 60 miles per hour constantly. People slow down, speed up, stop, the actual speed is variable and it can be complicated. Calculus allows us to solve this problem. Calculus allows us to compute the total distance traveled with variable speed. The problems in calculus are more complicated than in baby math where everything is taken as constant. There is so much more to analysis than just calculus. Differential equations play a big role in analysis. Differential equations are equations that describe the rate of change of a particular phenomenon, or a few phenomenon, solving such an equation gives us an understanding how something changes over time. Mathematics who work in analysis are analysts.
Geometry is the study of space. The word "geometry" is a Greek word that means "measuring the earth". Geometry was inspired by such practical problems but in its modern form geometry is way more abstract than what the ancients have ever envisioned. "Space" can be our three dimensional world. But "space" can also be a highly abstract 10 dimensional world. It can be Euclidean, i.e. a generalization of our world into more dimensions. Or it can be incredibly messed up. Sometimes geometry is synonymous with "topology". The word "topology" means, "study of space". In topology we deform objects into one another without breaking them apart (such as ripping them). From a topological point of view a square and a circle are the same because we can deform a square into a circle. But a sphere is not topologically the same as a sheet of paper because a sphere is closed, we cannot unravel a sphere into a plane without ripping it apart. Modern geometry also takes place on non-flat surfaces. Basic high school geometry that you learn was for the most part plane geometry. Geometry consisting of lines and points lying on some plane. Modern geometry allows to work with curves, not on planes, but on various curved surfaces. Mathematicians who work in geometry are geometers or topologists.
Algebra is the study of structure and symmetry. Algebra is a lot more difficult to explain than the others because algebra is a lot more abstract. The best way I can put it is with some example. Consider a cube. We can ask for the symmetries of a cube. That is, what operations on the cube leave the cube intact? We can rotate the cube about three of its axes without changing how the cube looks. We can also reflect a cube through its center, if we imagine a mirror passing through the center of a cube then reflecting the cube through the mirror will leave it unchanged. Rotations and reflections are symmetries of a cube. Once we have these symmetries we can form something known as a "group" which represents these symmetries and has operations defined between these symmetries. If we can understand what this group is, sometimes we say "determining the structure of the group" then we can reduce problems about symmetry to the solved problem of group structure. I realize that this is not the best way to explain what algebra is about but like I said it is not easy to explain what it is in a single paragraph. I am just trying to give some idea what it is about. Mathematicians who work in algebra are known as algebraists.
This is a basic overview of what mathematics is about. I cannot explain more because that would take a long time to explain. But if you are interested there are books written for people who are interested in understanding what mathematics is.
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