It seems that a lot of people have issues with burning books. Burning books is wrong, they tell me. But why? I think that being anti-book burning is something people picked up from being kids. They learned burning books is bad. They learned that burning books is only for the Nazis. They learned one should never burn a book because all books are valuable even if we disagree with them. And so forth.
But I have a question. What about People Magazine? That is valuable intellectual content? I disagree, it is a poison to one's mind and virtue. People Magazine is a waste of ink and paper. And so I find absolutely nothing wrong with burning it.
The Bible is a despicable book. I do not think the Bible (altogether) should be destroyed. Because I do not believe ideas should be destroyed. The material in the Bible has to be preserved because it is important if we want to understand how history turned out to be. If we cannot understand why people did what they did since we do not have what the Bible says then we will not be able to why history turned out the way it did. So I am not saying the Bible (altogether) should be destroyed.
What I am saying is different. Suppose you want to start a fire. But you cannot find a good source for fire. Suddenly you remember that you have a Bible lying around your house that you do not need. Why not put the book to good use, for once, and start a fire with a Bible?
Burning books that you disagree with is perfectly fine. As long as you do not burn actual people. I have no problem with burning the Bible or the Mein Kampf.
Obviously, the only problem with burning a Torah is that it is just too expensive.
Monday, November 8, 2010
Saturday, November 6, 2010
Generalizing Ptolemy's Theorem
Cladius Ptolemy was man who condemned the world to geocentrism. Though he was heavily mistaken in his model, and the Church was further poisoned by his error, he was nonetheless a good geometer and a smart man. The theorem that we will discuss now will be the well-known Ptolemy's theorem.
This theorem is hardly ever studied in high-school math. So we will need to recall what the theorem actually says. Let us remember a simple fact about triangles. If ABC is a triangle then it is always possible to circumscribe this triangle in a circle (which is unique). In fact, we will prove this result later on in this post when we generalize Ptolemy's theorem.
Let us come up with some unusually strange terminology which is never used in any geometry text. Instead of saying "circumscribe" we will say "circumcise". If some shape (like a triangle or quadrilateral) can be circumcised i.e. it is possible to circumscribe it in a circle, we call call this figure "male". If some shape cannot be circumcised in a circle then we will call this figure "female".
If we have a quadrilateral (four-sided figure) ABCD (where the vertices, by convention, are named in cyclic order - this is never an issue with a triangle because the triangle has just three sides) then it is not always possible to circumcise this quadrilateral. To see why not suppose that it was possible to. Then ABC is circumcised in a unique circle. Since ABCD is assumed to have been circumcised in a circle the fourth vertex, D, must lie on this circle. And now just move this vertex around away from the circle. Since the circle circumcising ABC is unique it means the resulting quadrilateral with a moved vertex D is female.
Therefore, by the above observations, we note that all triangles are male. However, not all quadrilaterals are male. Some quadrilaterals are male and some are female. Ptolemy realized a simple way to test if a quadrilateral is male or female.
Ptolemy's Theorem: Let ABCD be a quadrilateral, this quadrilateral is male if and only if $[AC][BD]=[AB][DC]+[AD][BC]$.
In case, it is not clear, the meaning of $[AC]$ means the distance from A to C, and so on. Draw a picture for yourself of this theorem and it becomes very easy to see. Ptolemy's theorem says that a quadrilateral is male only if the sum of the product of opposite sides is equal to the product of the diagonals.
There are many proofs to Ptolemy's theorem. Go to Wikipedia here. There are four proofs given to this theorem. One is geometric (classical), two trigonometric and algebraic (using complex numbers). These proofs are complicated! Look at how long and complicate they are. And it seems that these proofs do not even prove the converse! They just show that male quadrilaterals satisfiy the cross diagonal product formula, thereby only giving a necessary condition.
I have a different proof. A proof that is extremely easy and beautiful. And this proof proves the necessary and sufficient condition simultaenously. The smoother proof of Ptolemy's theorem uses "inversion in a circle" which can be found here. But the circle inversion proof is still geometric. You still need to draw pictures and calculate lengths.
I have a much easier proof of Ptolemy's theorem. This proof is nothing original. All I did was take the circle inversion proof and clean it up even more. I never seen anyone ever did this adaptive proof before, but I think this is by far the easiest proof of Ptolemy's theorem.
Proof: We will use Mobius transformations to prove Ptolemy's theorem. The only Mobius transfarmotion that we will use is "inversion" it is the mapping $M(z) = \frac{1}{z}$. Note we use the variable $z$ to denote that this function is a complex function it takes it values for complex numbers. The Mobius map $M$ has some really nice properties. The property that we want is the following. If $C$ is a circle passing through the origin then $M(C)$ (the image of the circle under the map $M$) is a line. Of course, $M$ is not defined at the origin, so if we want to be a little more formal we should say $M(C \setminus \{ 0 \})$ is a line. But I think this is clear from the context.
If $p$ is some point in the complex plane then $M_p(z) = \frac{1}{z+p}$ is basically the same as the Mobius map above it is just translated to a different point that acts as the new origin.
Let use denote the verticies of a male quadrilateral ABCD as $z_1,z_2,z_3,z_4$ in that order, these points will be the corresponding complex points representing the vertices of our quadrilateral. These points are assumed to lie on a circle. Consider the function $\frac{1}{z-z_1}$. This is a Mobius transformation like we have discussed above. And so it maps circles passing through the origin (in this case $z_1$) to lines. If C is a circle that circumcises ABCD then C is mapped to a line under the function $\frac{1}{z-z_1}$ because C passes through $z_1$. Since $z_2,z_3,z_4$ lie on this circle it means the corresponding points under this function $w_2=\frac{1}{z_2-z_1}$, $w_3=\frac{1}{z_3-z_1}$, and $w_4=\frac{1}{z_4-z_1}$ lie on a line. The point $w_3$ lies in between $w_2$ and $x_4$ because $z_3$ was the vertex in between $z_2$ and $z_4$ (this is why we are careful about the order in which we label the verticies). Since they lie on a line it clearly follows that that distance from $w_2$ to $w_3$ plus the distance from $w_3$ to $w_4$ is the same as the distance from $w_2$ to $w_4$:
$d(w_2,w_3) + d(w_3,w_4) = d(w_4 ,w_2)$
Therefore,
$d\left( \frac{1}{z_2-z_1} , \frac{1}{z_3 - z_1} \right) + d\left( \frac{1}{z_3 - z_1} , \frac{1}{z_4 - z_1} \right) = d\left( \frac{1}{z_4 - z_1} , \frac{1}{z_2 - z_1} \right)$
Combine,
$\frac{d(z_2 ,z_3)}{d(z_2,z_1)d(z_3,z_1)} + \frac{d(z_3,z_4)}{d(z_3,z_1)d(z_4,z_1)} = \frac{d(z_4,z_2)}{d(z_2,z_1)d(z_4,z_1)}$
Clear denominators,
$d(z_2,z_3)d(z_4,z_1) + d(z_3,z_4)d(z_2,z_1) = d(z_4,z_2)d(z_3,z_1)$.
This proves the theorem because $d(z_2,z_3)$ is just [BC], $d(z_4,z_1)$ is just [AD] and so forth. So we get that $[AC][BD]=[AB][DC]+[AD][BC]$. In fact, if you notice we can retrace all our steps backwards. All of these steps in the proof are "if and only if" statements, therefore by just following the proof backwards we get the converse statement.
---------------
What we can do now is generalize Ptolemy's theorem. We can consider a polygon with n sides. The proof is identical with points called now $z_1,z_2,...,z_n$. By using the Mobius transformation $\frac{1}{z-z_1}$ it is clear what the situation is if it is written out algebraically:
$$\displaystyle \sum_{k=2}^{n-1}d(z_k , z_{k+1}) \prod_{j\not = k,k+1} d(z_j , z_1) = d(z_n , z_2) \prod_{j\not = 2,n} d(z_j , z_1)$$.
What this is saying geometrically is not so complicated. Fix a vertex, in this case $z_1$. Then consider two adjacent verticies (not containing $z_1$). Compute their product of their distance and the distances from $z_1$ to all other different vertices. Now sum this product over all such adjacent sides. This resulting number will be equal to the the product of the distances from $z_2$ and $z_n$ and all other vertices from $z_k$ to $z_1$ only if the polygon is male. And this gives us a method to determine which polygons are male.
-----
We can also prove Ptolemy's inequality. Which says that if ABCD is a quadrilateral then $[AC][BD]\leq [AB][DC]+[AD][BC]$. This proof is immediate from our previous work. If ABCD is a quadrilateral then fix a vertex, say $z_1$ and again consider the Mobius transformation $f(z) = \frac{1}{z-z_1}$. This image of the vertices B,C,D under this Mobius map will be mapped to three pair that is not necessarily a line because we are not assuming that B,C,D lie in a circle. Therefore, the distance from f(B) to f(C) plus distance from f(C) to f(D) is greater or equal to the distance from f(B) to f(D). This follows from the triangle inequality. Because since f(B),f(C),f(D) are not necessarily straight they form a circle. So the sum of their sides exceeds the length of the third side. Therefore, we always get $[AC][BD]\leq [AB][DC]+[AD][BC]$ for any quadrilateral. Equality holds only in the case if ABCD is male.
This theorem is hardly ever studied in high-school math. So we will need to recall what the theorem actually says. Let us remember a simple fact about triangles. If ABC is a triangle then it is always possible to circumscribe this triangle in a circle (which is unique). In fact, we will prove this result later on in this post when we generalize Ptolemy's theorem.
Let us come up with some unusually strange terminology which is never used in any geometry text. Instead of saying "circumscribe" we will say "circumcise". If some shape (like a triangle or quadrilateral) can be circumcised i.e. it is possible to circumscribe it in a circle, we call call this figure "male". If some shape cannot be circumcised in a circle then we will call this figure "female".
If we have a quadrilateral (four-sided figure) ABCD (where the vertices, by convention, are named in cyclic order - this is never an issue with a triangle because the triangle has just three sides) then it is not always possible to circumcise this quadrilateral. To see why not suppose that it was possible to. Then ABC is circumcised in a unique circle. Since ABCD is assumed to have been circumcised in a circle the fourth vertex, D, must lie on this circle. And now just move this vertex around away from the circle. Since the circle circumcising ABC is unique it means the resulting quadrilateral with a moved vertex D is female.
Therefore, by the above observations, we note that all triangles are male. However, not all quadrilaterals are male. Some quadrilaterals are male and some are female. Ptolemy realized a simple way to test if a quadrilateral is male or female.
Ptolemy's Theorem: Let ABCD be a quadrilateral, this quadrilateral is male if and only if $[AC][BD]=[AB][DC]+[AD][BC]$.
In case, it is not clear, the meaning of $[AC]$ means the distance from A to C, and so on. Draw a picture for yourself of this theorem and it becomes very easy to see. Ptolemy's theorem says that a quadrilateral is male only if the sum of the product of opposite sides is equal to the product of the diagonals.
There are many proofs to Ptolemy's theorem. Go to Wikipedia here. There are four proofs given to this theorem. One is geometric (classical), two trigonometric and algebraic (using complex numbers). These proofs are complicated! Look at how long and complicate they are. And it seems that these proofs do not even prove the converse! They just show that male quadrilaterals satisfiy the cross diagonal product formula, thereby only giving a necessary condition.
I have a different proof. A proof that is extremely easy and beautiful. And this proof proves the necessary and sufficient condition simultaenously. The smoother proof of Ptolemy's theorem uses "inversion in a circle" which can be found here. But the circle inversion proof is still geometric. You still need to draw pictures and calculate lengths.
I have a much easier proof of Ptolemy's theorem. This proof is nothing original. All I did was take the circle inversion proof and clean it up even more. I never seen anyone ever did this adaptive proof before, but I think this is by far the easiest proof of Ptolemy's theorem.
Proof: We will use Mobius transformations to prove Ptolemy's theorem. The only Mobius transfarmotion that we will use is "inversion" it is the mapping $M(z) = \frac{1}{z}$. Note we use the variable $z$ to denote that this function is a complex function it takes it values for complex numbers. The Mobius map $M$ has some really nice properties. The property that we want is the following. If $C$ is a circle passing through the origin then $M(C)$ (the image of the circle under the map $M$) is a line. Of course, $M$ is not defined at the origin, so if we want to be a little more formal we should say $M(C \setminus \{ 0 \})$ is a line. But I think this is clear from the context.
If $p$ is some point in the complex plane then $M_p(z) = \frac{1}{z+p}$ is basically the same as the Mobius map above it is just translated to a different point that acts as the new origin.
Let use denote the verticies of a male quadrilateral ABCD as $z_1,z_2,z_3,z_4$ in that order, these points will be the corresponding complex points representing the vertices of our quadrilateral. These points are assumed to lie on a circle. Consider the function $\frac{1}{z-z_1}$. This is a Mobius transformation like we have discussed above. And so it maps circles passing through the origin (in this case $z_1$) to lines. If C is a circle that circumcises ABCD then C is mapped to a line under the function $\frac{1}{z-z_1}$ because C passes through $z_1$. Since $z_2,z_3,z_4$ lie on this circle it means the corresponding points under this function $w_2=\frac{1}{z_2-z_1}$, $w_3=\frac{1}{z_3-z_1}$, and $w_4=\frac{1}{z_4-z_1}$ lie on a line. The point $w_3$ lies in between $w_2$ and $x_4$ because $z_3$ was the vertex in between $z_2$ and $z_4$ (this is why we are careful about the order in which we label the verticies). Since they lie on a line it clearly follows that that distance from $w_2$ to $w_3$ plus the distance from $w_3$ to $w_4$ is the same as the distance from $w_2$ to $w_4$:
$d(w_2,w_3) + d(w_3,w_4) = d(w_4 ,w_2)$
Therefore,
$d\left( \frac{1}{z_2-z_1} , \frac{1}{z_3 - z_1} \right) + d\left( \frac{1}{z_3 - z_1} , \frac{1}{z_4 - z_1} \right) = d\left( \frac{1}{z_4 - z_1} , \frac{1}{z_2 - z_1} \right)$
Combine,
$\frac{d(z_2 ,z_3)}{d(z_2,z_1)d(z_3,z_1)} + \frac{d(z_3,z_4)}{d(z_3,z_1)d(z_4,z_1)} = \frac{d(z_4,z_2)}{d(z_2,z_1)d(z_4,z_1)}$
Clear denominators,
$d(z_2,z_3)d(z_4,z_1) + d(z_3,z_4)d(z_2,z_1) = d(z_4,z_2)d(z_3,z_1)$.
This proves the theorem because $d(z_2,z_3)$ is just [BC], $d(z_4,z_1)$ is just [AD] and so forth. So we get that $[AC][BD]=[AB][DC]+[AD][BC]$. In fact, if you notice we can retrace all our steps backwards. All of these steps in the proof are "if and only if" statements, therefore by just following the proof backwards we get the converse statement.
---------------
What we can do now is generalize Ptolemy's theorem. We can consider a polygon with n sides. The proof is identical with points called now $z_1,z_2,...,z_n$. By using the Mobius transformation $\frac{1}{z-z_1}$ it is clear what the situation is if it is written out algebraically:
$$\displaystyle \sum_{k=2}^{n-1}d(z_k , z_{k+1}) \prod_{j\not = k,k+1} d(z_j , z_1) = d(z_n , z_2) \prod_{j\not = 2,n} d(z_j , z_1)$$.
What this is saying geometrically is not so complicated. Fix a vertex, in this case $z_1$. Then consider two adjacent verticies (not containing $z_1$). Compute their product of their distance and the distances from $z_1$ to all other different vertices. Now sum this product over all such adjacent sides. This resulting number will be equal to the the product of the distances from $z_2$ and $z_n$ and all other vertices from $z_k$ to $z_1$ only if the polygon is male. And this gives us a method to determine which polygons are male.
-----
We can also prove Ptolemy's inequality. Which says that if ABCD is a quadrilateral then $[AC][BD]\leq [AB][DC]+[AD][BC]$. This proof is immediate from our previous work. If ABCD is a quadrilateral then fix a vertex, say $z_1$ and again consider the Mobius transformation $f(z) = \frac{1}{z-z_1}$. This image of the vertices B,C,D under this Mobius map will be mapped to three pair that is not necessarily a line because we are not assuming that B,C,D lie in a circle. Therefore, the distance from f(B) to f(C) plus distance from f(C) to f(D) is greater or equal to the distance from f(B) to f(D). This follows from the triangle inequality. Because since f(B),f(C),f(D) are not necessarily straight they form a circle. So the sum of their sides exceeds the length of the third side. Therefore, we always get $[AC][BD]\leq [AB][DC]+[AD][BC]$ for any quadrilateral. Equality holds only in the case if ABCD is male.
Friday, November 5, 2010
Some of My Heroes
I know that after I make this list there are going to be people that I will realize I should have put them on the list. That is why I call these "some" of my heroes.
Issac Newton: When I was much younger Newton was my favorite scientist. He still is. The greatest scientist in all history and the greatest mathematician of his time. I consider Newton to be, without exaggeration, the greatest human being who ever existed second to me, of course. Newton inspired me to pursue science and mathematics. It is amazing how much he achieved considering that he spend like 90% of his time on theology and alchemy. Imagine if he spend all his time on what was actually worthy.
Albert Einstein: My other favorite scientist from when I was much younger. He influenced me also to be interested in physics. I consider Einstein to be the greatest scientist of the 20th century. My theological beliefs are also best identified with Einstein (and Spinoza). Like Einstein I do believe that there is a rational construct in nature that we can observe when we study it. However, I do not need the word "God", for me that word is completely unnecessary. I am an atheist but I do think the universe is ordered in a rational manner, though with no purpose to it.
Thomas Paine: I regard Thomas Paine as the greatest philosopher from the Age of Enlightenment. Christopher Hitchens wrote of Paine that if Paine only written the "Rights of Men" that alone would have earned him the title as the most important Englishmen from the Enlightenment. But Paine did a lot. Paine influenced America. And much more, too much to list here. Paine is my favorite of all the American Founding Fathers. Thomas Paine is important for me because it was Paine who first made me give up religion. I gave up Judaism after I read the "Age of Reason", which was an attack on Judaism, Christianity, and religion in general.
Christopher Hitchens: Of course Hitchens is not anywhere comparable to these great people. But this is not about the great people from history, this post is about my heroes of influenced me, so I can include people who are not so great, just my influences. Hitchens was one of the people who influenced my atheism, a path to intellectual freedom from the tyranny of religion. Hitchens is fantastic. Much better and much smarter than Richard Dawkins. I have never liked Dawkins. And I consider it sad that atheists like Dawkins more than Hitchens, but I think Hitchens is far superior than Dawkins when it comes to arguing for atheism.
Adam Smith: As a good capitalist I cannot possibly not like Smith, he is after all regarded as a father of capitalism (though a term that Karl Marx invented). Smith has been able to set forward the main ideas behind free markets. It is not a surprise that Smith has been wrong on many issues. Today we know much more than he did. But his general ideas are correct and have only been vindicated by the 200+ history after him. The main idea that society cannot be planned but that it is the spontaneous arrangement of individuals working together for their self-interest.
Pierre Fermat: My favorite mathematician. Though he is extremely outdated he is among the most important mathematicians in history that has influenced a lot of mathematics. Including, analytic geometry, calculus, number theory, and probability. Issac Newton had a lot of respect for Fermat and if it was not for Fermat, Newton would have probably not been able to invent calculus.
Evariste Galois: Galois theory is one of the most beautiful concepts of algebra. It is a real beauty to see how group theory and field theory are ultimately connected to one another. It is a tribute to Galois for recognizing this connection. And he did all this work at age 20, the age he died, sadly. Just imagine if he lived a normal life.
Emil Artin: My favorite mathematician of the 20th century. Oh, what a fantastic mathematician. He definitely would have received a Field's Medal award, he just lived before the time the medals were handed over. Since I mentioned Galois above I cannot possibly not mention Artin who revolutionized the classical approach to Galois theory in terms of automorphism groups. Artin was also an excellent mathematician who contributed much to arithmetic.
Thomas Jefferson: My favorite president in history. He greatly influenced me in the idea of decentralizing power. And a general distrust of authority. Whenever it comes to distrusting the state, or reducing state power, I look at Jefferson as an inspiration. I also greatly respect Jefferson for being anti-religious. And respect him (and the other main founders) for setting up the first ever country in history founded on secular principles. It is sad, truly sad, to see the country that was set up as a secular country to have turned into a religious one.
Milton Friedman: Milton beautifully explained in simple terms how liberty, from a practical point-of-view, is superior to authority. It was really clear to follow his economic arguments. At first it seemed impossible to argue against pre-established conventions but Milton had a talent to do it very easily. I respect Milton a lot for not being afraid to challenge his ideas. I have seen a lot of talks and debates that he was in. He was never afraid to argue with other people on these topics. And every time I seen Friedman he was always the superior one, always using such clear arguments. (When was the last time you seen someone like Chomsky argue his ideas with other people, I never seen, probably because Chomsky is too much of a coward to argue with others.)
Carl Gauss: I cannot possibly not put on this list the greatest mathematician from history. Gauss was the last complete mathematician. That is, the last mathematician that was associated with everything there was to be associated with at his time. From an early age till when he died he pursued mathematics (and later physics too). Gauss, for me, is mostly an influence in arithmetic, a subject to which he strongly contributed to.
Baruch Spinoza: My favorite pre-Enlightenment philosopher. He was one of the people who contributed towards my skepticism towards religion. Spinoza nicely refuted the divinity of the Scripture. Spinoza also inspired me to take a materialistic and deterministic approach to nature. After I gave up religion I was not a determinist and I was not a materialist. Spinoza was able to unify all of his ideas together and show how all of the concepts, such as materialism and determinism in Nature are all one and the same.
Bernhard Riemann: Another mathematical favorite of mine. I regard him as one of the most important mathematicians in history. Riemann was a big influence in my life to continue to study mathematics even more.
Terroja Kincaid: You might laugh at me and say it is embarrassing that I choose a YouTube vlogger as a influence in my life, but I do. But TJ (better known as TheAmazingAtheist) did influence me a lot. More specifically back in the old days. Back from four to three years ago, when he was just starting out on YouTube (today he is different, now he is just trying to get popular, so I should be more careful and say the Old TJ as opposed to the New TJ). He was the first person to make me care about defending freedom. I also had a lot of respect for him because he was never afraid to show his weaknesses on camera in front of the whole world.
Lejeunne Dirichlet: Yet another mathematician. He has been one of my favorites because I always enjoyed Dirichlet's results. A lot of his work and his theorems are really wonderful. And he was the first mathematician to use analytic methods in arithmetic when Dirichlet proved the prime progression theorem.
Penn Jillette: It is funny that I choose somebody who went to clown college as an influence. But Penn (and Teller) have been able to advance skepticism and freedom from a very comical approach (as seen on the TV show "Bullshit"). When was the last time you seen people argue against creationism and feature masturbating girls on TV at the same time? He happens to be a real loud-mouth and foul-mouth but a nice guy.
Friedrich Nietzsche: The common stereotype of an insane philosopher. I love people who have to destroy everything that people have ever taken so dear to them. With all values and morals destroyed, with God dead, what is now left? The true potential of the individual. We do not need to bring ourselves down to others, and be their slaves, but we can all be masters.
Leonard Euler: Another mathematician. But this guy is epic. Euler contributed more to mathematics than anyone has ever done so in history. His collected works are in 76 volumes. But what is even more amazing is that for the last 1/5 of his life, he was entirely blind. Every mathematician and everyone who learns mathematics wishes to be the next Euler.
Issac Newton: When I was much younger Newton was my favorite scientist. He still is. The greatest scientist in all history and the greatest mathematician of his time. I consider Newton to be, without exaggeration, the greatest human being who ever existed second to me, of course. Newton inspired me to pursue science and mathematics. It is amazing how much he achieved considering that he spend like 90% of his time on theology and alchemy. Imagine if he spend all his time on what was actually worthy.
Albert Einstein: My other favorite scientist from when I was much younger. He influenced me also to be interested in physics. I consider Einstein to be the greatest scientist of the 20th century. My theological beliefs are also best identified with Einstein (and Spinoza). Like Einstein I do believe that there is a rational construct in nature that we can observe when we study it. However, I do not need the word "God", for me that word is completely unnecessary. I am an atheist but I do think the universe is ordered in a rational manner, though with no purpose to it.
Thomas Paine: I regard Thomas Paine as the greatest philosopher from the Age of Enlightenment. Christopher Hitchens wrote of Paine that if Paine only written the "Rights of Men" that alone would have earned him the title as the most important Englishmen from the Enlightenment. But Paine did a lot. Paine influenced America. And much more, too much to list here. Paine is my favorite of all the American Founding Fathers. Thomas Paine is important for me because it was Paine who first made me give up religion. I gave up Judaism after I read the "Age of Reason", which was an attack on Judaism, Christianity, and religion in general.
Christopher Hitchens: Of course Hitchens is not anywhere comparable to these great people. But this is not about the great people from history, this post is about my heroes of influenced me, so I can include people who are not so great, just my influences. Hitchens was one of the people who influenced my atheism, a path to intellectual freedom from the tyranny of religion. Hitchens is fantastic. Much better and much smarter than Richard Dawkins. I have never liked Dawkins. And I consider it sad that atheists like Dawkins more than Hitchens, but I think Hitchens is far superior than Dawkins when it comes to arguing for atheism.
Adam Smith: As a good capitalist I cannot possibly not like Smith, he is after all regarded as a father of capitalism (though a term that Karl Marx invented). Smith has been able to set forward the main ideas behind free markets. It is not a surprise that Smith has been wrong on many issues. Today we know much more than he did. But his general ideas are correct and have only been vindicated by the 200+ history after him. The main idea that society cannot be planned but that it is the spontaneous arrangement of individuals working together for their self-interest.
Pierre Fermat: My favorite mathematician. Though he is extremely outdated he is among the most important mathematicians in history that has influenced a lot of mathematics. Including, analytic geometry, calculus, number theory, and probability. Issac Newton had a lot of respect for Fermat and if it was not for Fermat, Newton would have probably not been able to invent calculus.
Evariste Galois: Galois theory is one of the most beautiful concepts of algebra. It is a real beauty to see how group theory and field theory are ultimately connected to one another. It is a tribute to Galois for recognizing this connection. And he did all this work at age 20, the age he died, sadly. Just imagine if he lived a normal life.
Emil Artin: My favorite mathematician of the 20th century. Oh, what a fantastic mathematician. He definitely would have received a Field's Medal award, he just lived before the time the medals were handed over. Since I mentioned Galois above I cannot possibly not mention Artin who revolutionized the classical approach to Galois theory in terms of automorphism groups. Artin was also an excellent mathematician who contributed much to arithmetic.
Thomas Jefferson: My favorite president in history. He greatly influenced me in the idea of decentralizing power. And a general distrust of authority. Whenever it comes to distrusting the state, or reducing state power, I look at Jefferson as an inspiration. I also greatly respect Jefferson for being anti-religious. And respect him (and the other main founders) for setting up the first ever country in history founded on secular principles. It is sad, truly sad, to see the country that was set up as a secular country to have turned into a religious one.
Milton Friedman: Milton beautifully explained in simple terms how liberty, from a practical point-of-view, is superior to authority. It was really clear to follow his economic arguments. At first it seemed impossible to argue against pre-established conventions but Milton had a talent to do it very easily. I respect Milton a lot for not being afraid to challenge his ideas. I have seen a lot of talks and debates that he was in. He was never afraid to argue with other people on these topics. And every time I seen Friedman he was always the superior one, always using such clear arguments. (When was the last time you seen someone like Chomsky argue his ideas with other people, I never seen, probably because Chomsky is too much of a coward to argue with others.)
Carl Gauss: I cannot possibly not put on this list the greatest mathematician from history. Gauss was the last complete mathematician. That is, the last mathematician that was associated with everything there was to be associated with at his time. From an early age till when he died he pursued mathematics (and later physics too). Gauss, for me, is mostly an influence in arithmetic, a subject to which he strongly contributed to.
Baruch Spinoza: My favorite pre-Enlightenment philosopher. He was one of the people who contributed towards my skepticism towards religion. Spinoza nicely refuted the divinity of the Scripture. Spinoza also inspired me to take a materialistic and deterministic approach to nature. After I gave up religion I was not a determinist and I was not a materialist. Spinoza was able to unify all of his ideas together and show how all of the concepts, such as materialism and determinism in Nature are all one and the same.
Bernhard Riemann: Another mathematical favorite of mine. I regard him as one of the most important mathematicians in history. Riemann was a big influence in my life to continue to study mathematics even more.
Terroja Kincaid: You might laugh at me and say it is embarrassing that I choose a YouTube vlogger as a influence in my life, but I do. But TJ (better known as TheAmazingAtheist) did influence me a lot. More specifically back in the old days. Back from four to three years ago, when he was just starting out on YouTube (today he is different, now he is just trying to get popular, so I should be more careful and say the Old TJ as opposed to the New TJ). He was the first person to make me care about defending freedom. I also had a lot of respect for him because he was never afraid to show his weaknesses on camera in front of the whole world.
Lejeunne Dirichlet: Yet another mathematician. He has been one of my favorites because I always enjoyed Dirichlet's results. A lot of his work and his theorems are really wonderful. And he was the first mathematician to use analytic methods in arithmetic when Dirichlet proved the prime progression theorem.
Penn Jillette: It is funny that I choose somebody who went to clown college as an influence. But Penn (and Teller) have been able to advance skepticism and freedom from a very comical approach (as seen on the TV show "Bullshit"). When was the last time you seen people argue against creationism and feature masturbating girls on TV at the same time? He happens to be a real loud-mouth and foul-mouth but a nice guy.
Friedrich Nietzsche: The common stereotype of an insane philosopher. I love people who have to destroy everything that people have ever taken so dear to them. With all values and morals destroyed, with God dead, what is now left? The true potential of the individual. We do not need to bring ourselves down to others, and be their slaves, but we can all be masters.
Leonard Euler: Another mathematician. But this guy is epic. Euler contributed more to mathematics than anyone has ever done so in history. His collected works are in 76 volumes. But what is even more amazing is that for the last 1/5 of his life, he was entirely blind. Every mathematician and everyone who learns mathematics wishes to be the next Euler.
Wednesday, November 3, 2010
A Geometry Problem with Circle and Sphere
I came up with a geometry problem that appears to be really surprising.
1) Consider a unit circle (a circle with radius 1 unit). Place $n$ randomly placed points on this circle. Then there exists a point on this circle such that the sum of the distances from this point to all the other points is exactly $n$.
This generalizes in an obvious way.
2) Consider a unit sphere (a sphere with radius 1 unit). Place $n$ randomly placed points on this sphere. Then there exists a point on this sphere such that the sum of the distances from this point to all the other points is exactly $n$.
So in particular, if you have 2010 points on a unit circle or sphere then there is this special point so that the distance sum is precisely 2010.
Note, this problem can be generalized to higher dimensional spheres also. But let us not go there because it is impossible to visualize.
1) Consider a unit circle (a circle with radius 1 unit). Place $n$ randomly placed points on this circle. Then there exists a point on this circle such that the sum of the distances from this point to all the other points is exactly $n$.
This generalizes in an obvious way.
2) Consider a unit sphere (a sphere with radius 1 unit). Place $n$ randomly placed points on this sphere. Then there exists a point on this sphere such that the sum of the distances from this point to all the other points is exactly $n$.
So in particular, if you have 2010 points on a unit circle or sphere then there is this special point so that the distance sum is precisely 2010.
Note, this problem can be generalized to higher dimensional spheres also. But let us not go there because it is impossible to visualize.
Tuesday, November 2, 2010
Voting: How to Feel Good without Doing Anything
I seriously need to stop posting about voting already, it is turning into an obsession. I will make this my last post on voting for a while.
I was in college today (Election Day), I was standing and waiting to be interviewed with a professor. Then I see a woman passing by me, she is happy, with a smile on her face. What can she be so happy about? Maybe her husband screwed her last night, could be, but I think she just came in from voting. She was feeling good about voting. She just came back from voting and had a smile on her face that she voted. Of course, you going to ask me how I can possibly know this. Well, I have a good reason for thinking that. Because she saw me standing there and asked me, "did you vote?". And I said, happily, "No". Then she asked me, "are you registered to vote?". I then said, even more happily and the feeling of superiority over her, "No and I do not want to". You can see how that happy face she had on moments before turned into a disappointed face, she left, and said, "maybe someday you will change your mind".
If she is happy and then she becomes disappointed from knowing someone does not want to vote, it is perhaps a sign that she was happy about the fact that she voted. This is not certain, I am making a conjecture, but I think I am being reasonable here.
Behold, I then had a revelation about voting. Most people vote because they feel good about it. Voting is a way to make yourself feel like you are doing something important. Indeed, the common statist myth is that "we are the government", and if you believe in that, then it makes sense why you are happy to vote. Because you see yourself as acting in justice and virtue.
The true heroes who impact United States for the better are the workers, the entrepreneurs, the capitalists, the people who interact through the market. These are the true heroes that make the country better. These are the people who create jobs, who work, who generate wealth, and innovate, and so forth. These people are making a positive impact. It is true that most of these heroes do not do so because they want to impact the country, most of them do it for their own self-interest, but their actions do positively affect the country.
But what does one who votes actually accomplish? Does he innovate a new product, like the Fleshlight, so that losers who cannot get any pussy finally get some? Does he make cars that allow millions of people to transport themselves? No. Does he give out loans for people that they can use the money to build something? No. Does he create jobs by hiring more workers? No. What does a voter physically do? Nothing. Absolutely nothing.
But the voter feels good about himself because he imagines himself as actually doing something. He believes that he is responsible for the changes that will happen in the country. And since his intentions are good he therefore sees himself as acting positively. Hence, he feels good about himself.
This is the reason why, also, why lots of voters hate the non-voter. Because, by the same reasoning process that they have, they see the non-voter as not caring and not interested in making an impact on the country. So they see him as a person who lives off the country without a concern for his fellow citizens.
What did I do on Election Day? Like George Carlin did, I masturbated.
I was in college today (Election Day), I was standing and waiting to be interviewed with a professor. Then I see a woman passing by me, she is happy, with a smile on her face. What can she be so happy about? Maybe her husband screwed her last night, could be, but I think she just came in from voting. She was feeling good about voting. She just came back from voting and had a smile on her face that she voted. Of course, you going to ask me how I can possibly know this. Well, I have a good reason for thinking that. Because she saw me standing there and asked me, "did you vote?". And I said, happily, "No". Then she asked me, "are you registered to vote?". I then said, even more happily and the feeling of superiority over her, "No and I do not want to". You can see how that happy face she had on moments before turned into a disappointed face, she left, and said, "maybe someday you will change your mind".
If she is happy and then she becomes disappointed from knowing someone does not want to vote, it is perhaps a sign that she was happy about the fact that she voted. This is not certain, I am making a conjecture, but I think I am being reasonable here.
Behold, I then had a revelation about voting. Most people vote because they feel good about it. Voting is a way to make yourself feel like you are doing something important. Indeed, the common statist myth is that "we are the government", and if you believe in that, then it makes sense why you are happy to vote. Because you see yourself as acting in justice and virtue.
The true heroes who impact United States for the better are the workers, the entrepreneurs, the capitalists, the people who interact through the market. These are the true heroes that make the country better. These are the people who create jobs, who work, who generate wealth, and innovate, and so forth. These people are making a positive impact. It is true that most of these heroes do not do so because they want to impact the country, most of them do it for their own self-interest, but their actions do positively affect the country.
But what does one who votes actually accomplish? Does he innovate a new product, like the Fleshlight, so that losers who cannot get any pussy finally get some? Does he make cars that allow millions of people to transport themselves? No. Does he give out loans for people that they can use the money to build something? No. Does he create jobs by hiring more workers? No. What does a voter physically do? Nothing. Absolutely nothing.
But the voter feels good about himself because he imagines himself as actually doing something. He believes that he is responsible for the changes that will happen in the country. And since his intentions are good he therefore sees himself as acting positively. Hence, he feels good about himself.
This is the reason why, also, why lots of voters hate the non-voter. Because, by the same reasoning process that they have, they see the non-voter as not caring and not interested in making an impact on the country. So they see him as a person who lives off the country without a concern for his fellow citizens.
What did I do on Election Day? Like George Carlin did, I masturbated.
Monday, November 1, 2010
Hope that Republicans Win
I do not vote. I never voted. As I said many times before, so I will not get into the reasons again. I am not registered to vote. I do not belong to any party. But I can still say who I rather want to win.
Back in 2008 I wanted the Democrats to win the presidential elections. This was not because I really had anything with them, and also because I was a lot more ignorant back then. I just thought that maybe they are better on civil issues than Republicans. But they are not. I do not see it. The war still goes on. The ban on drugs, even marijuana, is still in place. Gay marriage has not been resolved. The patriot act has not been repealed. So tell me exactly how they vindicated themselves as being pro-civil liberties? They might talk better on these issues than Republicans but as far their results go, they are just as the same.
So the Democrats have not done anything great. Instead they decreased liberties (a political party decreasing liberties, woah, what a surprise, I would have never guessed) with their healthcare bill. Now you need to buy insurance, if you do not, you go to jail. The Democrats care so much about you that if you do not buy what they say then they will throw you into a cage.
That is why I prefer Republicans to Democrats. Republicans have not done anything. On civil issues they are just as bad. But at least they do not expand the size of the government as fast as Democrats do. Between these two parties there are no small government party. Both are big government parties, the only different is that Republicans are just a smaller government party. Therefore, in the long run Republicans are expected to do less damage to the country than Democrats.
This is why I hope Republicans win. If they do win I am not going to be cheering or anything. I am no Republican. I just see them as the lesser of two evils, but I remember that still makes them evil.
In fact, there is a reasonable chance the Republicans would win and then go on to win the presidential elections in 2012. The population has not been happy. Which is actually a good thing. A divided country in conflict is what we need. Having everyone agree and give giant approval ratings to a president is scary, just remember what happened with Bush after September 11.
I never thought I would say this but I am with Bill O'Reilly and Sean Hannity!
Back in 2008 I wanted the Democrats to win the presidential elections. This was not because I really had anything with them, and also because I was a lot more ignorant back then. I just thought that maybe they are better on civil issues than Republicans. But they are not. I do not see it. The war still goes on. The ban on drugs, even marijuana, is still in place. Gay marriage has not been resolved. The patriot act has not been repealed. So tell me exactly how they vindicated themselves as being pro-civil liberties? They might talk better on these issues than Republicans but as far their results go, they are just as the same.
So the Democrats have not done anything great. Instead they decreased liberties (a political party decreasing liberties, woah, what a surprise, I would have never guessed) with their healthcare bill. Now you need to buy insurance, if you do not, you go to jail. The Democrats care so much about you that if you do not buy what they say then they will throw you into a cage.
That is why I prefer Republicans to Democrats. Republicans have not done anything. On civil issues they are just as bad. But at least they do not expand the size of the government as fast as Democrats do. Between these two parties there are no small government party. Both are big government parties, the only different is that Republicans are just a smaller government party. Therefore, in the long run Republicans are expected to do less damage to the country than Democrats.
This is why I hope Republicans win. If they do win I am not going to be cheering or anything. I am no Republican. I just see them as the lesser of two evils, but I remember that still makes them evil.
In fact, there is a reasonable chance the Republicans would win and then go on to win the presidential elections in 2012. The population has not been happy. Which is actually a good thing. A divided country in conflict is what we need. Having everyone agree and give giant approval ratings to a president is scary, just remember what happened with Bush after September 11.
I never thought I would say this but I am with Bill O'Reilly and Sean Hannity!
Lessons from Halloween for Statists
I am not sure if you celebrate Halloween or not, I do not, but there are some nice hidden lessons in Halloween. Lessons which most people probably never realized. These lessons are most important to hardcore statists and central planners.
What I find amazing about Halloween is how many people celebrate Halloween all over the world. It is not just an American holiday. I know Canadians celebrate it. I spoke with a guy from Chile who said his town was celebrating it. It is celebrated, though not to the same extent as in US, in Europe. It is fair to say that is a world holiday.
Halloween is very different from Christmas. Christmas is a religious holiday. Catholics are compelled to celebrate Christmas, for them it is a very important holiday. So I can understand why Christmas is celebrated throughout the world. But Halloween is different. It is not really a religious holiday. I am not familiar with the history of Halloween, it probably started out as a religious holiday, but today it is not a religious holiday anymore.
Halloween is celebrated today for one and one reason only. People like it and so people agree to celebrate it. Halloween is celebrated by voluntary participation. People from all over the world simply agreed to wear costumes and give out candy to kids on Halloween. There was no state law making people do that, there were not central planners that planned for Halloween to take place. Halloween emerged as a holiday all by itself. In fact, if there was a state law mandating some new holiday, people would most likely not celebrate it. Ever heard of anyone celebrating Martin Luther King day? These holidays, like Halloween, are celebrated from the emergence of human interaction, it is a bottom-up arrangement, not a top-down arrangement that central planners have a big giant hard-on far.
By the way, lest statists accuse me of using Halloween as an argument for freedom, I am not. I am not saying, "Halloween exists therefore freedom works". That would be an embarrassingly terrible argument. I am rather trying to point to a world wide agreement between people that took place voluntarily. I am using Halloween as an example of how large scale agreement can happen. Thus, look at my example just as an example, not as an argument.
The second lesson is even more important than this lesson. On Halloween I had this thought. A thought that most normal people would never have, because they do not have such a sick and corrupted mind like I do. I had this thought that I can poison candy. Then when kids knock on my door for candy I can give them poisoned candy. If 50 kids show up, I might be able to exterminate 90% of them.
Do you have any idea how easy it is to poison candy and kill children? It does not even have to be poisoned candy. I can make homemade brownies for Halloween and make them with a poisonous ingredient. Then when children knock on my door to collect candy I will give them my poisonous brownies. They will eat those brownies when they come home and minutes or hours later die.
But somehow this never happens. When was the last time you heard of some mad crazy psychopath poisoning candy for children on Halloween? I never heard of it. Maybe it happened like once or twice, but you never hear about it. If some madman did this it would be all over the news and a national candy scare would soon follow. We never hear about it, so it is likely to assume that the situation I described either never happened or was/is statistically insignificant.
And this is the second lesson that I am trying to illustrate. Just because something terrible can happen does not mean it will. Hardcore statists and central planners have a general mentality about behavior. They think that human behavior would be out of control if there was no law to stop that behavior. I speak from experience here. Back when I was a statist I thought that if there was no laws then the world would be out of control. People would be running in the street and stabbing babies with number two pencils. Pedophiles will be raping children in the street. There would be constant car accidents all over the place. Everyone would steal and lie. And so on and so on.
Or consider this non-violent example. I am not sure how you feel about public sex but it does not bother me. In fact, I think it is great to see free porn shows in the street. But people who against public sex, and in particular people who advocate laws to prevent public sex, tell me that if there was no public sex laws then everyone would be doing it in the street all the time. These Bill O'Reilly's that tell us this non-sense do not have any idea what they are talking about. I like to respond to that argument by saying that there are no laws stopping people from walking in the street with meat cleavers, yet somehow we never see people walk in the street with meat cleavers.
Let us return back to the poisonous candy example. There are no regulators that go from house to house to inspect the candy and homemade food that is made before it can be handed out to children on Halloween. There are no laws for Halloween. But somehow there are no cases of psychopaths poisoning candies on Halloween. As I said, just because something can happen does not mean it will happen.
Again I am not trying to give the impression that my argument is that, "non-poisonous candy is handed out on Halloween therefore statists are wrong and I am correct". That would also be an embarrassingly terrible argument. My entire point of this example is to illustrate the slippery-slope that these hardcore statists create. Statists love to assume the worst, generate fear, and then justify their statism and central control. I wrote back about it more here. I am using Halloween candy as an example of something entirely uncontrolled which does not cause any problems whatsoever.
When I tell a statist that I think there should be no FDA regulating food and drugs, and that any food can enter into the market without having ingredient regulations on it, statists immediately create a slippery-slope. They tell me, "you hate children, if there were no regulations evil businessmen would put bad food that will harm or kill children, what about the children?!".
And maybe they are right in this instance. Maybe when it comes to making profits certain evil businessmen would take advantage of people and put bad food on the market (candy is different, it is a gift, there is no profit in it). I am not trying to say this scenario will never happen. I am just trying to debunk the hugely over-exaggerated blown out of proportion reaction from statists and central planners. Yes, I am sure bad things will occasionally happen if profits are at stake. However, do you not think that statists blow it way way out of proportion? They create a world in which nearly all food and drugs are harmful or poisonous. I find that to be an extreme slippery-slope. As I said, just because bad things can happen does not mean it will necessarily happen. There is no problem with candy. So do you not think that people who exaggerate the danger of unchecked food are creating an extreme slippery-slope?
What I find amazing about Halloween is how many people celebrate Halloween all over the world. It is not just an American holiday. I know Canadians celebrate it. I spoke with a guy from Chile who said his town was celebrating it. It is celebrated, though not to the same extent as in US, in Europe. It is fair to say that is a world holiday.
Halloween is very different from Christmas. Christmas is a religious holiday. Catholics are compelled to celebrate Christmas, for them it is a very important holiday. So I can understand why Christmas is celebrated throughout the world. But Halloween is different. It is not really a religious holiday. I am not familiar with the history of Halloween, it probably started out as a religious holiday, but today it is not a religious holiday anymore.
Halloween is celebrated today for one and one reason only. People like it and so people agree to celebrate it. Halloween is celebrated by voluntary participation. People from all over the world simply agreed to wear costumes and give out candy to kids on Halloween. There was no state law making people do that, there were not central planners that planned for Halloween to take place. Halloween emerged as a holiday all by itself. In fact, if there was a state law mandating some new holiday, people would most likely not celebrate it. Ever heard of anyone celebrating Martin Luther King day? These holidays, like Halloween, are celebrated from the emergence of human interaction, it is a bottom-up arrangement, not a top-down arrangement that central planners have a big giant hard-on far.
By the way, lest statists accuse me of using Halloween as an argument for freedom, I am not. I am not saying, "Halloween exists therefore freedom works". That would be an embarrassingly terrible argument. I am rather trying to point to a world wide agreement between people that took place voluntarily. I am using Halloween as an example of how large scale agreement can happen. Thus, look at my example just as an example, not as an argument.
The second lesson is even more important than this lesson. On Halloween I had this thought. A thought that most normal people would never have, because they do not have such a sick and corrupted mind like I do. I had this thought that I can poison candy. Then when kids knock on my door for candy I can give them poisoned candy. If 50 kids show up, I might be able to exterminate 90% of them.
Do you have any idea how easy it is to poison candy and kill children? It does not even have to be poisoned candy. I can make homemade brownies for Halloween and make them with a poisonous ingredient. Then when children knock on my door to collect candy I will give them my poisonous brownies. They will eat those brownies when they come home and minutes or hours later die.
But somehow this never happens. When was the last time you heard of some mad crazy psychopath poisoning candy for children on Halloween? I never heard of it. Maybe it happened like once or twice, but you never hear about it. If some madman did this it would be all over the news and a national candy scare would soon follow. We never hear about it, so it is likely to assume that the situation I described either never happened or was/is statistically insignificant.
And this is the second lesson that I am trying to illustrate. Just because something terrible can happen does not mean it will. Hardcore statists and central planners have a general mentality about behavior. They think that human behavior would be out of control if there was no law to stop that behavior. I speak from experience here. Back when I was a statist I thought that if there was no laws then the world would be out of control. People would be running in the street and stabbing babies with number two pencils. Pedophiles will be raping children in the street. There would be constant car accidents all over the place. Everyone would steal and lie. And so on and so on.
Or consider this non-violent example. I am not sure how you feel about public sex but it does not bother me. In fact, I think it is great to see free porn shows in the street. But people who against public sex, and in particular people who advocate laws to prevent public sex, tell me that if there was no public sex laws then everyone would be doing it in the street all the time. These Bill O'Reilly's that tell us this non-sense do not have any idea what they are talking about. I like to respond to that argument by saying that there are no laws stopping people from walking in the street with meat cleavers, yet somehow we never see people walk in the street with meat cleavers.
Let us return back to the poisonous candy example. There are no regulators that go from house to house to inspect the candy and homemade food that is made before it can be handed out to children on Halloween. There are no laws for Halloween. But somehow there are no cases of psychopaths poisoning candies on Halloween. As I said, just because something can happen does not mean it will happen.
Again I am not trying to give the impression that my argument is that, "non-poisonous candy is handed out on Halloween therefore statists are wrong and I am correct". That would also be an embarrassingly terrible argument. My entire point of this example is to illustrate the slippery-slope that these hardcore statists create. Statists love to assume the worst, generate fear, and then justify their statism and central control. I wrote back about it more here. I am using Halloween candy as an example of something entirely uncontrolled which does not cause any problems whatsoever.
When I tell a statist that I think there should be no FDA regulating food and drugs, and that any food can enter into the market without having ingredient regulations on it, statists immediately create a slippery-slope. They tell me, "you hate children, if there were no regulations evil businessmen would put bad food that will harm or kill children, what about the children?!".
And maybe they are right in this instance. Maybe when it comes to making profits certain evil businessmen would take advantage of people and put bad food on the market (candy is different, it is a gift, there is no profit in it). I am not trying to say this scenario will never happen. I am just trying to debunk the hugely over-exaggerated blown out of proportion reaction from statists and central planners. Yes, I am sure bad things will occasionally happen if profits are at stake. However, do you not think that statists blow it way way out of proportion? They create a world in which nearly all food and drugs are harmful or poisonous. I find that to be an extreme slippery-slope. As I said, just because bad things can happen does not mean it will necessarily happen. There is no problem with candy. So do you not think that people who exaggerate the danger of unchecked food are creating an extreme slippery-slope?
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