The Math You Weren’t Allowed to Do

You probably learned a bunch of things in school math about what you can and can’t do. When you were a first grader, perhaps you learned that that you can’t subtract 5 from 2, but later on, you learned about negatives: 2 – 5 = -3.

You also might have been told that you can’t divide 2 by 6, but then you learn about fractions. And by now, you are no doubt an expert at splitting 2 pizzas among 6 people.

Even so, there is much more that you may not have known…

1. You Can’t Count to Infinity

Actually, you can. It can be done via ordinal numbers.

You start out counting by

1, 2, 3, 4, 5, 6, 7,… 700,… 30,000,000, etc.

When you played a “what’s the highest number?” game with someone, every time you said a number, they countered by saying your number plus one, that is, unless you said infinity. Because infinity plus one is still infinity, right?

Here is where ordinals come into play. The ordinal number \omega (ω, omega) is defined as the first number after ALL of the positive integers. No matter what normal number they might say, whether it’s ten billion or a googol, the ordinal number \omega is far, far larger. It is practically infinity.

But then you can add one to it, and it becomes an even bigger number. Add two, and it becomes even bigger.

\omega, \omega+1, \omega+2,..., 2\omega,...,3\omega,...,n\omega,....,\omega^2,...

What the heck is going on? If you count an infinite number of numbers after omega, you get two omega? Is this two times infinity? And then three omega? And then omega squared?

It turns out to keep on going. Eventually you will get \omega^\omega, and then \omega^{\omega^\omega}, etc. And then you reach \Omega (big omega), which is larger than all things that can be written in terms of little omegas. And then you can make bigger things than that, with no end.

So the next time someone claims infinity is the largest number, you can confidently reply, “infinity plus one.”

2. You Can’t Divide by Zero

Actually, under certain conditions, you can.

The field of complex analysis is largely based around taking contour integrals around poles. Another word for pole is singularity. And another word for singularity is something you get when you divide by zero.

Consider the function y = 1/x. When x is 1, y is 1, and when x is 5, y is 1/5. But what if x is 0? What happens? Well, 1/0 is undefined. However, if you look at a graph, you see that the function spikes up to infinity at x = 0.

What you do in complex analysis is integrate in a circle around that place where it spikes to infinity. The result in this case, if done properly, is 2\pi i. It’s quite bizarre.

3. You Can Only Understand Smooth Things

Actually, there is much theory on crazy, “pathological” functions, some of which are discontinuous at every point!

The image above is kind of misleading, as it is a graph of the Cantor function, which is actually continuous everywhere (!), but nonetheless manages to rise despite having zero derivative almost everywhere.

There is another function with the following properties: it is 1 whenever is x is rational and 0 whenever x is irrational. Yet this function is well understood and is even integrable. (The integral is 0.)

Then you have things that are truly crazy:

The boundary of that thing is nowhere smooth, and is one of the most amazing things that have ever been discovered. Yet it is generated by the extraordinarily simple function z^2 + c, which most people have seen and even studied in school.

4. You Must “Do the Math” and Not Draw Pictures

Actually, math people use pictures all the time. The Mandelbrot set (the previous picture) was not well understood until computer images were generated. There is no such thing as doing the math in a “correct” way. Some fields are quite based on pictures and visualizations.

How else would anyone have thought, for example, that the Mandelbrot set would be so complex? Without seeing that in pictures, how would we have realized the fundamental structure behind the self-similarity of nature?

Yeah, that’s a picture of broccoli. Not a mathematical function. Broccoli.

5. If It Doesn’t Make Sense, It’s Not True

Actually, many absurd things in math can be perfectly reasonable.

What’s the next number after 7?

8, you say. But why 8? What’s wrong with saying the next number after 7 is 0? In fact, I can define a “number” to only include 0, 1, 2, 3, 4, 5, 6, and 7. Basic operations such as addition and multiplication can be well defined. For example, addition is just counting forward that many numbers. So 6 + 3 = 1, because if you start at 6 and go forward 3, you loop back around and end up at 1.

Even weirder is the Banach-Tarski Paradox, which states a solid sphere can be broken up into a finite number of pieces, and the pieces can be reassembled to form TWO spheres of the exact same size as the original!

I hope this was understandable for everyone. May the reader live for ω+1 years!

People Who Agree With You for the Wrong Reasons

Yes, I’m attempting to write a post that talks about both politics AND religion. Yep, the two most heated things that always lead to flame wars on the Internet. In one post.

The Two Types of Disagreeing

We’ve all had those moments we just flat-out disagreed with everything someone said. No matter how many facts we throw at them, they never seemed to listen. And they probably thought the same about us. The argument turned into full-blown war, and we were ready to start throwing punches at each other.

But we’ve also had those debates where we disagreed with them, not in a hostile way at all, but in a calm, mature, intellectual manner. We realized some of the things we said were wrong, and so did they. And while we still had our differences in the end, we felt more connected and felt that we had uncovered some truth out of it. This is the good type of disagreement.

The Two Types of Agreeing

There’s the intellectual style of agreement as well, the good kind. You try to teach your kid about gravity, but she is skeptical, so you encourage her to try to throw a ball so high that it won’t return, to disprove gravity. She quickly learns that no matter how hard she throws it, it will always fall back down. Finally, she ends up agreeing with you, having learned a valuable lesson out of it.

Then there’s the wrong kind of agreement. The kind when once she becomes skeptical of gravity, you only assert that it’s true and don’t give any reason or evidence for it, and you say “Believe it, or else.” Of course, this example is pretty silly because no one needs to threaten someone else to get them to believe in gravity—there is overwhelming evidence for it everywhere on Earth. I’m really setting this up for matters which have no evidence or are misunderstood.

Agreeing Due to Party Alignment, Not Due to Facts

I’m going to go with politics first, and then religion. I consider myself to be a moderate liberal, but I usually don’t care about politics that much. However, sometimes when people talk about politics in stupid ways or completely misunderstand their political party, I do care. I don’t want them making misguided decisions in the ballot.

It’s common for liberals to criticize conservatives for outrageous claims, but many of these liberals don’t understand that they themselves also make outrageous claims. They say how bad conservatives are, but then when someone asks them what has Obama done in the last 4 years, they are silent. Not that Obama hasn’t done anything—he’s done quite a bit. But some of these liberals are just clueless about their own party and seem to vote Democrat just because their friends do or because they think Obama is charismatic.

These people annoy me greatly. They might agree with me, but for all the wrong reasons. For instance, I know people who like to make fun of Rush Limbaugh, despite never having read anything he wrote and never listening to anything he said, and rely instead only on what other people said of him. If you disagree with Rush Limbaugh because you disagree with his views, that’s fine. I respectfully disagree with much of what he says. But if you disagree with him just because it’s cool to disagree with him, then that is pathetic.

It seems that respect is all but forgotten in this era. I can disagree with someone but still understand what they are saying, and admit that some parts of what they say are correct. But respect doesn’t seem to be mainstream anymore. Case in point, in Obama’s 2008 election victory speech, he began in a noble manner by making a respectful statement about McCain’s campaign. But what did the crowd do? It booed him very audibly. And in McCain’s defeat speech, when he congratulated Obama, he got loudly booed by his crowd as well.

What has politics become, a spectator sport where you boo the other team, or boo anyone who says shows respect to the other team? If anything, Obama and McCain’s respect for each other in that moment of the election gives me some hope for the American political system. However, the behavior of the crowd does not.

Agreeing Due to Authority, Not Due to Evidence

Now for religion. I am an atheist. I don’t believe in god for the same reason I don’t believe in Santa or an Invisible Pink Unicorn or the Flying Spaghetti Monster or an invisible fire-breathing dragon. Simply, I believe it is childish and immature to believe in something that has zero evidence, just because other people believe in it.

That said, I am not claiming that Christianity is inherently bad. Despite its numerous provocations, injustices, and wars, I do not know where the world would be right now had Christianity not existed. Without its teaching of generosity and kindness to primitive cultures (and then enslaving them), civilization may not be as advanced as it is. However, given that we have already reached an early Space Age, where technology and the search for knowledge can unite us in place of the mass belief of ancient myths, I question whether Christianity will be of use for much longer.

So if you tell me, “I am a Christian because the moral system is wonderful,” then that is great. But if you say, “I am a Christian because there is evidence that God exists,” then I will facepalm, because that is like saying, “I believe in the Invisible Pink Unicorn because there is evidence that the Invisible Pink Unicorn exists.”

On the other side, people who believe in atheism might agree for the wrong reasons, though not usually, as they tend to be more open-minded. Saying “I am an atheist because there is no evidence of God” is perfectly fine, but saying “I am an atheist because Christianity is evil” is not a valid reason. However, I don’t know of anyone who actually believes that, so as far as I know, there are no atheist “extremists” like there are religious extremists.

So as far as this section goes, I cannot really talk about the atheist side as there are no examples of belief for the wrong reasons that I know of. Instead, I can try to empathize with the religious side and think about what they would consider to be belief in god for the wrong reasons.

The first one is probably believing in God for fear of ending up in Hell or some other divine punishment. That would be a terrible reason to believe in something, simply out of fear of threat for not believing in it. This is one reason I have a problem with Pascal’s wager (the other being that it can just be applied to other religions, forcing the player to have no good choice).

The second is argument from authority. People shouldn’t believe in God just because other people said they should; they should find it on their own. Despite how silly this sounds to me, at least I find it more noble than blindly following the will of other people. In practice, however, it seems most people are led into Christianity through authority, from their parents or community when they are young and vulnerable.

I mean, if someone is nonreligious but suffers a crisis when they are 30, and chooses to accept a religion to cope with it, that is fine. In fact, hooking people up to mythical virtual realities is a valid method these days of dealing with trauma. The real world is too harsh, so they can more easily cope in a fantasy world. But if a kid is forced to accept a religion when they wouldn’t know better, that is an entirely different thing, and is just wrong. (I agree with Bill Nye’s take on this.)

This would be entirely opposite of the gravity case presented in the “Two Types of Agreeing” section. When a kid learns about gravity, if she is skeptical she can try to disprove it by throwing a ball so high into the air that it does not come down. But the more gravity works, the more accepting she becomes. Whether she thinks gravity is true is determined by her own experiences.

However, if she is skeptical of religion, there is nothing she can do to disprove it, since anything could be justified by some made-up explanation, and this is probably very confusing for a young mind. Whether she thinks religion is true is determined solely by the statements of others, i.e. authority figures.

If this forcing of views on a child concerned any subject other than religion, it would be called brainwashing. Yet when it’s religion, it’s not considered brainwashing, and—quite disturbingly—it’s actually considered by some to be education.

My writing of this section is inspired by Carl Sagan’s skeptical philosophy and Bill Nye’s recent video that was linked above.

Disagreeing and Agreeing

I’d rather someone disagree with me using the truth, rather than have someone agree with me based on a lie. Both in politics and religion, a shallow agreement based on lies is valueless, ridiculous, and devoid of morality.

This is often why there are such heated debates in both of these subjects on the Internet, where multiple people can chime in on both sides. The “Democrat” side of a forum thread might be extremely polarized within itself, and so is the “Republican” side. Thus, instead of there being a straight back-and-forth debate, there is a jumbled web of personal insults and baseless accusations. This would be avoided if people were actually knowledgeable and knew what they were talking about and as well as what other people are talking about. This is why knowledge and respect should be taught, not whatever is causing them to resort to insults.

In the case of religion, religious people generally don’t use logic, so even the terms “agree” and “disagree” begin to lose meaning. That’s why a religious debate usually never ends up being a peaceful debate. It always becomes derailed because logic itself is missing from one side, so it isn’t really a debate at all. It is a lecture where the student is willfully ignorant. At least that’s what happens in the case of an atheist vs theist debate. I can only imagine the horror of what a theist vs theist-of-a-different-religion debate on the Internet would be like, e.g. a Christian vs Muslim debate.

In the political system of the United States, I am somewhat hopeful. But I have almost no hope at all for the current education system. Until something like “Logic for First Graders” is taught—lies, misunderstandings, and ignorance will always be the face of our country.

Still on Winter Break Mode

Academically, this has not been a good semester so far. I’m behind in readings for almost every class, and for the first time in my life (besides Math 2230 last semester), I am having trouble understanding some of the lectures. It’s not because they are hard, just that I am not reading, and thereby not learning the background with which to understand them.

Now that I think about it, only one of my classes last semester was mostly textbook based. The rest were highly lecture based. That suited my learning style well—I’m really good at absorbing what people say. This semester, it has become the opposite, in that only one class is not textbook based. And I guess I have not gotten used to reading textbooks yet.

Which leads me to the hypothesis, that I am still on winter break mode.

No, it’s definitely not just a hypothesis. It’s a thesis. More and more, I’m becoming estranged from education. I’m just coasting along right now. It’s kind of ironic because in high school I never suffered from senioritis. Maybe senioritis is an essential part of life. Maybe you have to get it at some point so you won’t have it later. Almost like a vaccination effect against a real disease. Because the problem is that I have senioritis right now, in my second semester of freshman year at college.

But if it is true that I am merely on winter break mode, that still doesn’t explain why this winter break was so much less productive than other breaks I have had. For instance, over the last summer break I read like 10 books; over winter break I read only 1. And it was a really short book at that.

So you do what you normally do in science: you dig back another layer. This leads me to the new hypothesis, that I am still on pre-winter break mode. What was pre-winter break? Finals.

That’s right. Once classes ended last semester, I went on coast mode because I’m not the type of person who studies for tests. But there’s still something unexplained. Even before classes ended, I was on coast mode. What happened right before that? Thanksgiving.

Of course, just saying Thanksgiving doesn’t explain a thing. No, it’s what happened right before Thanksgiving, and what I was doing throughout the month of November that paradoxically put me into this state: NaNoWriMo.

If doing NaNoWriMo taught me one thing, it was how to prioritize some activity ahead of schoolwork. I became pretty adept at it, but perhaps too adept, that it carried over into everything I did. Right now it’s just hard for me to look at a textbook for more than five minutes. I’m trying though.

Also, I have a very extensive project planned for the month of March: editing the NaNoWriMo novel. Perhaps I just need some closure on it. I’m in the “done with first draft” state, but it’s nowhere near finished. Also, I’m considering giving the actual text some more publicity this time. Just a thought.

A Chinese Kid’s Response to “Chinese Parenting”

An article from the Wall Street Journal went viral on the Internet earlier this month, sparking passionate debates from the American and Asian-American public. Titled “Why Chinese Mothers Are Superior,” it is an excerpt from a book by Yale law professor Amy Chua, now known as the “Tiger Mother.” In essence, the article makes the argument that strict Eastern parenting is superior to lenient Western parenting.

The arguments in a nutshell:

  • Chinese parents are stricter than American parents.
  • Chinese parents stress their kids’ education a whole lot more.
  • Drill/practice builds on itself: the better a child is at some activity, the more fun it will become, and the more he or she will want to improve (a “virtuous circle”).
  • “Chinese parents can get away with things that Western parents can’t.” E.g., make their kids get straight A’s. Control their extracurricular activities. Make violent threats. Insult them openly.
  • The Chinese system produces superior results, is better.

When I first read the article over a week ago, I was honestly not sure what to think. I let it drift around in my head for a little while. But if anything, my initial reaction was a respectful disagreement. Chua’s points made perfect sense on their own, but something seemed lacking, as if there was some side of the argument that was kept hidden. I looked back at the article looking for signs of a one-sided argument, but saw that it does, to some degree, highlight pros and cons for both parenting styles. So what was missing?

A few days later I showed the article to my own Chinese mother. I was very interested in what she thought of it; indeed, my suspicions were confirmed. She called the totalitarian strictness in it “ridiculous.” And now I think “ridiculous” is the perfect word to describe the article.

I feel very blessed to have a Chinese mother who is NOT like Amy Chua. Granted, I’ve seen my parents’ “Chinese” side. But to me, they are more like Western parents. They obviously wanted me to get A’s, but if i got a B on anything, it wasn’t the end of the world. They gave me control over what extracurricular activities I did and how much effort I put into any of them.

Eventually I became a chess aficionado, though my parents had never pushed me into chess. In fourth grade I saw some friends playing chess and was intrigued by the game enough so that I joined the school chess club. I think that was the first time in my life that I made an important decision for myself. Sure, it didn’t seem that significant then, but for a while in my life I was a chess person. I traveled, I went to national tournaments, I shook hands with grandmasters. It was my passion.

I don’t think it would ever have become a passion if my parents had forced me to play it.

Which brings up an interesting fact, that my parents had forced me to play the piano starting in first grade, as well as do a whole slew of other things. I suppose I was okay—not brilliant, but not terrible either, as I faired well in competitions. But in fifth grade I got tired of it. Sitting at the piano for an hour was becoming a daily chore, instead of a hobby. Did I enjoy it? Did I want to play it? Did I even make the decision in the first place that I wanted to play it? No, no, and no.

Now don’t get me wrong, I think the piano is wonderful. But my 11-year old self, for some reason, didn’t see it that way. So he rebelled. And there was no “virtuous circle” of feedback that Chua was talking about. This boy saw no meaning in the high marks that he was receiving at piano competitions. He couldn’t enjoy it; the piano wasn’t for him.

So he told his parents that he wanted to quit, but of course this didn’t work. (Try this one with Chinese parents.) For weeks, tension was built. Then he started to make excuses, pretended his hands hurt, but eventually his mother stood next to the piano and forced him to play. (Think Amy Chua-style here. Actually that’s kind of scary.) But the only playing he did was the frustrated banging of random keys, creating a symphony of cacophonous sounds. He wouldn’t touch that piano again for two years.

That was the second time I had made some important decision in my life.

In retrospect, was it a good decision? I’m not really sure. In chess, there’s a saying, “A bad plan is better than no plan.” Is a bad decision likewise better than indecision? This quote certainly wasn’t running through the little boy’s head at the time, but he must have felt that way. I think making a decision in the first place was the correct choice.

As to the title of this post, I have just turned 19, which could make me a “kid” depending on the context, but really, it’s that 11-year old’s rebellion to sitting in front of the piano.

After the piano incident, my parents became Western parents. When I said that they tolerated B’s and allowed me to do what I wanted—that was AFTER this event.

Not that I cared less about education; in fact, it was quite the opposite. I became learning-focused instead of grade-focused. In class, I would be the one asking bizarre questions about material that seemed only remotely connected to the curriculum, but I never asked such a cringe-inducing question as “What percent of the grade is this assignment?” or “Is this for a grade?” or “Is this going to be on the test?” or, my favorite one yet, “Is there extra credit?”—and by the way, I’ve heard these countless times in high school from my Asian peers.

I remember on one AP Chemistry test I was supposed to get a 94 but my teacher (who is brilliant by the way) accidentally misgraded it and gave me an 88. I couldn’t have cared less about the 6-point discrepancy; I let it go. And yet, I knew many people who wouldn’t hesitate to fiercely argue with their teachers that their 96 should actually be a 97 for some wrong answer that could remotely be correct.

My parents did become “Chinese” once again when it came to the PSAT and SAT. In reality, American colleges look for a lot of other things besides standardized test scores. But my parents were used to a system, which still exists in China today, that cared solely about one score on a national exam.

So of course they tried to enroll me in some PSAT/SAT prep course (which would have cost $$$), but I told them I would refuse to go if they did, and that they should spend their money more wisely. I told them that I would compromise by doing PSAT review on my own with practice books. Of course my Chinese parents weren’t pleased with this, so they made me this $100 bet, initially as a threat. If I made National Merit, I would win. Otherwise, I would lose and also have to take a prep course for the SAT. Yeah, pretty ridiculous, right?

So they bought a plethora of PSAT review and practice exam books, and made me do all of them. And here is where I lied to my parents. I did only the first practice exam, just to see what it was like, but that was enough. For the other bajillion of them, since the PSAT is all multiple-choice, I just copied the answers from the back of the books, strategically bubbling a few wrong answers as to appear inconspicuous. After finishing the entire test in five minutes, with still hours left, I would read novels or program on my TI-84 calculator (neither of which my parents had much encouraged me to do).

I won the bet, without any dedicated “studying” for it, and eventually got into a nice college which I am enjoying. I’m still undecided about my major, and my parents are pretty Western about it; they don’t mind what I study, as long as I can enjoy it and excel in it.

Chinese parenting may be better for clawing higher grades, but Western parenting seems to be better for creativity, initiative, and all that other stuff that actually matters.

Mom and dad, if you are reading this, you may be startled about my not actually doing the PSAT and SAT reviews. But know that I meant the best.

And Amy Chua: you are, as my mother said, ridiculous.