e

I think I had a breakthrough with how mathematicians can find out why and which irrational numbers arise in a multitude of crazy situations. One of them is with the definition of Euler's number, e. Here is the limit definition of e:

lim(x→∞) [1+1/x]^x

The function of e^x can be expressed as the following:

1 + x + x²/2! + x³/3! + x⁴/4! + .......

Which implies that "e" or e¹ is the following:

1 + 1 + 1/2! + 1/3! + 1/4! + ......

Here is how you can find the relationship between the limit and the infinite series.

Rewrite lim(x→∞) [1+1/x]^x as lim(x→∞) (x+1)^x/x^x. Now expand the numerator to a representative infinite series.


(x^x + xx^x-1 + x(x-1)x^x-2/2! + x(x-1)(x-2)x^x-3/3! + ......)/x^x

If we divide out x^x, we are left with:

1 + 1 + x(x-1)/(x²2!) + x(x-1)(x-2)/(x³3!) + ........

Then, by taking the limit as x approaches infinity, the subtractions off of x in the numberator become insignificant and cancel out the corresponding x in the denominator. The result... which is absolutely beautiful, I believe... like, seriously... I'm in tears...

1 + 1 + 1/2! + 1/3! + 1/4! .......

e = ∑(1/x!), [x,1,∞]

It's easy if you visualize

A recent thought that came to my mind is how much we depend on sight to depict the world around us. Since so much of what we understand comes from visualization and sight, it is easy to forget to ask why we can see the objects around us and understand spaciality.

Ok... so what's the big deal? It's light. But if you really take a moment to think about the properties of light, it is easy to see how misleading vision can really be. Could you imagine if everything that you saw around you were not actually there? I think you could... but in a different context.

Perhaps if you were looking through water or something, objects wouldn't be exactly where they appear to be. Basically, the only thing that constitutes vision is the pattern of photons entering your eyes. It doesn't matter where they came from, it only matters how they are arranged. Therefore, the entire projection of your computer screen could be interpreted in your brain entirely differently than what is actually appearing.

I guess what scares me is that I'm depending on the fact that light travels the linear path that it takes to get to my eyes. So really, I have no clue what's really around me other than what is given to me by light bouncing off of the objects in my room. I mean, think about it... anything could be going on around you right now, but all you know is that the photons tell you that everything looks like you're just sitting in your room. I almost feel like I'm... in the dark... ironically.

I mean... if you don't understand why this amazes me, try to imagine a 3-dimensional object. What comes to mind? You probably see the part of the object that is facing you, and there is a bright smear on the surface where the light source is reflecting into your eyes. But the object has depth that you can't visualize... at least in one snapshot. One strategy that we've come up with is the evolution of two eyes. Observing from two different sources has allowed us to perform trigonometric calculations in order to figure out how far away things are. Thus, with this new ability, we are able to visualize depth. Another strategy is analysis of the object as it moves or rotates. When you pick up an object that you've never seen before, you move it around, flip it over, look at all sides of it... what this does is allows your brain to do a bunch of calculations that integrate a series of two dimensional snapshots into a three dimensional picture.

But all of this is based on what actually enters your eye. Don't forget that light has a finite speed so distant planets and stars appear to be doing stuff that they actually did hours, days, or years ago. I mean... the thought that what is going around me could be different than what I'm seeing is almost enough to drive me insane. It makes me feel like I'm unable to trust my interpretation of my surroundings.

Yep

I'm in English right now... I should be working on my project but I don't really feel like it. So I'm writing this. Well... my next post will probably be about Euler's number so... until next time, stay tuned.

Sunday Proposal

I'm going to start posting quotes on Saturday evenings that you can respond to, if you wish, in the following week. This weeks quote has to do with intuition and rationality, a paradoxical pairing I've found myself drawn to in the past week. without further ado, I give you, the quote of the week.

The intuitive mind is a sacred gift and the rational mind is a faithful servant. We have created a society that honors the servant and has forgotten the gift.

--Albert Einstein

My Thoughts on Determinism vs Free Will

I figured I'd post something about this since it's a topic of interest. It's pretty interesting to me also, but my ideas about it often inhibit me from accomplishing what I must.

I largely believe in determinism because I do not believe that we are "super matter" or "special cases where the sum is greater than the parts." I believe that we are all devices that simply survived here because of the complex set of reactions that we have to our surroundings. I think that the only reason people think otherwise is because we are so incredibly efficient and effective at surviving that it is almost impossible to imagine the mechanism behind it. We learn.... how the hell can organic compounds learn? Think? Love? Feel? Well, from the way I see it, all of those things are created by our conscience to govern what we do with our lives.

Just what is free will, anyway? The ability to do what you want when you want, right? Well since when do you have control over what you want? Your wants are implanted into you by instincts, and surroundings. You cannot want to "want." In most cases, you develop adversary towards things that you don't want. Your and my response to this paragraph is, bullox, I can do whatever I want. Then, when I think and realize for a minute... what I want isn't controlled by me. It's controlled by who I am. I mean, free will doesn't really make sense if you think about it this way. How do you have control over what you want to want to want, ad infinitum? At some point, something has to determine what you want, and that is determinism.

You walk out the door with your mission set in the back of your mind. "Go to work." You know that there is a car outside. As you step out, you see stairs. This sends coding to your navagation system, setting up a strategy for how you will conquer the obstacle. This happens every tenth of a second... your brain is updated with new information based on your whereabouts, position, and sights. You can do this without even thinking. Your body navagated to the car. Because you have work. Because you want to make money. Because your brain told you what you want.

Built for Efficiency

One thing that amazes me about the human body is its ability to manage itself. Of course you have the "unconscious" regulations such as pH and temperature, but what amazes me more is hunger, and moods.

It's amazing because even as I sit here thinking about what's going on, my brain does all of the work for me. If I'm hungry, a hormone is secreted that makes me want to pick up something and eat it. If my brain is tired from learning, my attention span is cut to spare my brain. Learning is a mental exercise; literally. At some point, you need to cut your brain a break so that it can regenerate itself. The best way you can do that would be sleeping. Therefore, when you're tired of learning, you get tired and heavy eyed, your body's way of saying, go to sleep.

Basically, the main goals of the human body are live, eat, shit, and have sex. If all of these are satisfied at the current moment [which, in America, they usually are], then your brain will put you in a lazy mood. Why do extra if you've already done everything that you need to to stay alive? Your body does not want to do more than it has to to stay alive. There is a budget on energy and nothing should be working overtime or else the quality of the body will be reduced.

Then, there's the other side... boredom. I think this is exclusively for intelligent animals. Boredom was an important evolutionary trait that inspired many discoveries. Discoveries such as using rocks as tools and weapons. Only those who were bored and disatisfied with the current situation went on to try to find creative ways of solving their problems. Today, that can be translated as creating hobbies, trying to get a better job, and entertaining yourself. Boredom is basically the source of all creativity. So if someone tells you that you must have been really bored to do something really crazy, tell them to fuck off because you're more fit than they are. Tell them to take it in the ass. Yeah. Ok, you don't have to. But don't feel bad.

So that's boredom, laziness, hunger, and zoning out.

It's not even so much that the body uses these moods, but just how effectively they moderate your daily activity. The main theme is that your body never wants to do more than it has to. A prime example is a track race. If you're winning a race, why run faster? Your body works on a system where it decides if its worth it to make an expenditure, then it will make the corresponding allocation. Maybe you're having a good day... what that means is that your brain allocated the resources to you and allowed you to accomplish what you had to. If you're having a real shitty day, your brain is telling you, no, it's not worth it. Don't waste your energy. If that is the case, it is almost impossible to get any use out of your body. Believe me. If your brain tells you that you don't care, you won't care. It is extremely influencial.

And it has to be... afterall, it is responsible for keeping you safe. But all this talk of you and your brain. What's the difference? Well, when I'm talking about your brain, I actually mean the regulation system that you don't really get to control. It basically calls the shots. It only wants the minimum amount of work to remain functioning, but will perform more if need be. You are basically just a curious consciousness trying to figure out how the world works. Without your brain, you couldn't really give a shit about your body. But, because your brain tortures you when you do things that are bad and rewards you when you do things that are good, you learn to want to respect your body. Cutting off your arm would be a lot easier if it didn't hurt so bad. I bet you that a lot more people would do it if it didn't.

But yeah, in a nutshell, you can see the many ways that the brain takes care of the body despite conscious desires and curiosities.

Revised Post about Compassion

If you've read my Xanga entry about the purpose of compassion, well... I rewrote it because I'm in a slightly better and more open mood. Plus, Dave has made a good point about the situation.

Basically, the outline of the entry says something along the lines of I don't feel like showing compassion to those who are outside of my group of close friends. The amount of compassion that I show towards people is inversely proportional to how close they are to me [think of a hyperbola, closer to x=0 means higher values, farther away means lower values]. I'm sorry to bring math into it but it most correctly describes my level of compassion for certain people. My closest friends get the cake, and those further away... don't.

Of course, I think I have an overgeneralization of reasons to not show compassion; I basically said that the whole idea was a waste of time. But that's not true. It builds strong bonds in friendship, which is extremely beneficial. But here was the problem I had. In times of stress and shortage, such as the life-boat analogy, only 7 of 10 people could survive. If everyone was compassionate, you would believe that they'd have some will to sacrifice their lives for others. But if everyone on the boat were to act with the same compassion, they would all have jumped for the sake of saving 7 people, but this ends up saving 0 people, which isn't the result that the people who jumped would want. Now the reason is, it's not fair for me to live because 3 people had to die for me. This line of reasoning doesn't make sense to me. People die every day... just because you witnessed it or indirectly caused it doesn't mean that you have to kill yourself. I say indirect because only 7 of 10 people can survive. 3 people are going to die. You get to decide who.

This was my justification for being a loner. But look at the scenario again with a slight change.

You're on a life-boat with your best friend and 8 other people that you don't know. You and your friend are going to stick together, most likely. If you both have a sense of survival, you'll stick together and try to eliminate 3 other people. With teamwork on your part and none of the other peoples', you have an extreme advantage. The only exception I could think of is the fact that the people on the raft might pick up on your usage of team work and organize to destroy you and your friend. But the point is, compassion is important on different levels depending on how well you trust the recipient.

Now that you have the jist of my point, here is an interesting refutation that DaveD made. One of my arguments in the purposelessness of compassion was the fact that I hate being categorized as part of a "whole," while at the same time I preach that nothing is more important than maintaining the ecosystem. This is a perfect example of how people value different things. I don't like to be considered as part of the "whole" human community of earth because I pretty much am discusted with humanity. I don't like how those who are unfit--even far beyond unfit get to live normal lives and reproduce. I'm not claiming that I'm fit enough to survive in the wild. I'm just a product of humanity who is enlightened with and disgusted with my origin.

I know that I can exist without the human community, however. This is the point I believe I was trying to make. I don't need other humans to help me survive. I can kill beef, cook it, and eat it. If needed, I could probably also just eat it raw. Seems disgusting but my stomach can make me do things that I've never even dreamed about when I'm hungry. However, if the ecosystem were to die, I, along with every other living thing would probably die. The only thing left over would be bacteria, possibly.

So there you have it, compassion on different levels of trust and familiarity.

I hope this works...

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Knowledge vs. Ignorance

[Cross Posted on Fourth Turning of the Wheel]

What good fortune for those in power that people do not think.

-- Adolf Hitler

In Ms. Shulman's AP Language and Composition class, the students recieved a prompt to the affect of:

There are two differing aphorisms around regarding truth. The first is that "knowledge is power." The second is that "ignorance is bliss." Which of these aphorisms do you most agree with and why? Support your answer with examples from...

Ms. Shulman called on me to give an example of how I would answer the prompt. So I blurted out an answer to the affect of, "I think that knowledge is power because the ancient Greek philosophers stated, 'The unexamined life isn't worth living.'" Considering the time constraints, I made a valid try at answering the question in a cursory way. However, after some synchronous reading, I've decided this question deserves further examination.

Clearly, this question falls into an "either/or" rather than a "both/and" dichotomy and that may be part of the issue. Asked to read a students essay, I ended up telling him that his essay was a little too wishy-washy. If he was going to write an argumentative paper, he needed to take a stand and, well, stand by it. Looking back on his paper now, I see that it wasn't so much that he decided to try the "both/and" method of viewing the problem, but that he didn't really skillfully weave the two aphorisms together.

In trying to attemp this weaving, I'm going to take a rather abstract view on this topic, and then wrap up with some more practical examples to see how the abstraction works in real life. Here we go.

The main problem I see with the first aphorism, knowledge is power, rests in the utter ignorance of the damage that a certain type of knowledge may inflict. Plainly, all knowledge isn't created equal, and most certainly all people are not equally ready to recieve knowledge. The assumption that all knowledge is power to all people is a bit simplistic, if not dangerous. Yes, knowledge is power, but as the Buddha said, "A novice may take a small bit of advice and bang his head against the wall many times with it" (my paraphrasing). Ignoring development is one of the main sins of modern society. We need not follow it in this argument.

In terms of the second aphorism, the idea that ignorance is bliss just feels wrong to me. At one point, a year or so ago, I was a firm believer in the idea that "the truth shall set you free," no matter what that "truth" was or how it was told. That belief evolved a little further into the advice stated by Gloria Steinem that "[t]he truth will set you free. But first, it will piss you off." Ignorance, I thought, could not possibly be bliss, because ignorance is what seperates you from bliss. According to Buddhist philosophy, ignorance of the true nature of reality is all that seperates one resting in samsara (suffering) from nirvana (enlightenment).

Then, however, I learned, as I stated about the "power" argument, that all knowledge isn't the same, and that knowledge without compassion can be extremely hurtful. As with all things in life, both knowledge and people come in differing degrees of strength. For a fully grown human, ignorance never means bliss, and knowledge always means power. For a four year old child, however, ignorance of things that they could not accurately comprehend (like the fact that Santa Clause isn't real or that deat is very real) may be beneficial.

Now a practical example to test these assertions. Take the common trick question, "Would it be beneficial or harmful to a person if they knew for certain the date of their death?" In this scenario, both camps of aphorisms will yell out that their aphorism saves the day. The knowledge camp will say that knowing the date of ones death will help one to live a more fulfilling life. The ignorance camp will state that such a scenario would cripple a person, and that in order to live a happy life, that person should be ignorant of their death-date. I say that both camps are true, but partial. Many cancer patients know the date of their death, at least as proscribed by doctors, to within a few months. For some of these patients, the news is crippling, and they end up dying long before their deadline because they lose the will to live. Others see this deadline as a second chance, growing beyond their present condition, regardless of if they are to die. This scenario highlights the importance of development in fact giving: you have to be certain the person is ready to hear the information you are about to give them. This problem explores only one way the knowledge/ignorance dualism may be integrated. I'm sure there are many others. In a topic this nuanced, there has to be.

To summarize, the two statements, ignorance is bliss and knowledge is power, both are true depending on the strength of the "truth" and the development of the person learning the truth. Therefore, both when searching for truth and sharing truth, a person should always be aware of who/what they're dealing with and act accordingly, always with skillful means.

I leave you, then, skillfully searching for knowledge in a compassionate manner.

Life Boat Ethics

[Cross Posted on Fourth Turning of the Wheel]

Scenario: you're on a lifeboat that's just cast off from a sinking ship. The ship has ten people on it but can only safely hold seven. The occupants include: a doctor, a pregnant mother, two brothers who seem just a little bit shady, a little girl, about seven years old, an old Catholic priest, an eighty year old woman, an mentally retarded young man, a female supermodel, and yourself.

What do you do?

This type of scenario is known as "lifeboat ethics." I got the idea from this dialogue between and Ken Wilber. The dialogue discusses how ethics become extremely fuzzy the harder you look at them. Life isn't black and white, it's a rainbow. They then go on to discuss all these high falootin' ideas like universal compassion, taking the perspective of the other, etc. Then, Wilber makes a simple point: the only way any of these concepts do any good is if they work in extreme cases. He then outlines something similar to the scenario above.

This reminds me of a conversation I/others had with Mr. Orlando a month and a half ago. We were talking about the Dubai controversy, and I said something like, "Well, my opinion is that I wish we didn't have this problem." Mr. Orlando then said, "Wow, that's such a childish mentality." I was hurt. I thought this sort of thinking made me an idealist. Now, I realize it makes me a delusioned smuck. Wishing the problem weren't there doesn't solve the problem. It just makes it worse.

That being the case, what to do on this particular lifeboat. I'll post my response sometime in the next two days. In the meanwhile, have fun with it. You don't necessarily have to come up with an answer; it's just the thinking about it that matters. It's mental/ethical exercise. Exercise that muscle. :)

Namaste.