A Few Mathematical Concepts That Blow My Mind (Part Two)

Euler's Identity

This post is the true sequel to the first of the Mathematical concepts that blow my mind series. In this post, I'll talk about Euler's (I recently learned this is actually pronounced roughly how American's would say Oiler's) Identity. Without further adieu, Euler's identity is as follows:

The first time I saw this equation, I was drawn to it; It’s a pretty elegant equation from a mathematical perspective. I mean you have the number e, which is the base of the natural logarithm which is prevalent everywhere in nature. the next thing you see is an exponential operation and in that there’s the number i (-1) which is the basis for all imaginary numbers, then you have a multiplication operation attaching i with the number which is extremely important when dealing with circles and is the basis of measurement in radians. This whole term is being added to the number 1, which is then equated to the number 0. I mean wow, it’s amazing how you can take all of these numbers and principals which are vastly important in their own field of mathematics and join them into one, simple looking equation, which is no more than an inch or two long. For this reason, many people regard this equation as beautiful; some even venture to call it the most beautiful equation ever. When I first came across this equation, I had never heard of the notion of mathematical beauty, but when I saw this, there was just something about it that give it more value than just how one can use it in mathematics. I kind of see it as a rite of passage for the human race, being able to show to the universe that we understand these individual concepts to the point where we feel comfortable combining them all and using them for our gain.

Now, Let’s get down to what this equation actually means from a mathematical point of view. Well this equation is actually a special case of Euler’s formula relating to the field of complex analysis. the general formula states:

note that since x is being used here, we’re assuming radians for the trigonometric portion of this equation. So if we take x to be pi, we’ll have:

and since the cosine of pi is -1 and the sin of pi is 0, this gives us:

after some cleaning up of the equation, this will eventually give us Euler’s famous equation. So, now that we have this equation. What does it actually do? What can we use this for? Well, the first thing that comes to mind is that it can be used to simplify things in differential equations. This is important because differential equations in my opinion are kind of like a cornerstone of physics. without our understanding of differential equations, our knowledge of physics would be severely lacking and no where near as accurate. Some other uses I’ve heard of but don’t really know too much about are it’s ability to allow us traverse circles easily, as well as help explain electronic signals that vary overtime in the realm of electrical engineering since the formula combines sine and cosine.

Alrighty, that’s all for now. I’ll probably post another one of these in the near future because I just keep running into cool mathematical principles, but I don’t exactly want to saturate this blog with math. The next post in this series will most likely be one the continuum hypothesis. Thanks for reading,

-Carlos

Can Our Minds Live Forever?

With technology increasing as rapidly as ever and people like Ray Kurzweil making predictions for us to be finished with reverse engineering the brain around 2030, the question arises, will we be able to recreate digital copies of our brain? if so, what does that mean for all of us? Will we see the start of human's uploading and downloading information directly to/from our brains via a central hub? Will we be able to send mental messages to our friends and transmit raw thoughts and feelings to one another?

Imagine this: say you live in a society where this technology exists. you also like a girl, but can't really bring up a way of telling her how you feel. wouldn't it be awesome if you could transmit to her the sensation of the butterflies in your stomach that you feel when you're around her? after all, this feeling is merely just your brain's interpretation of information received from your nervous system. Imagine your reaction to a beautiful painting for the first time and the emotional response it evoked out of you. Imagine being to tag that picture with this first time feeling, so that you can experience it anytime you want after that. Let's be honest, sometimes words just don't suffice for whatever you're trying to express. Maybe in the future, that will never be a problem because we'd be able to communicate our raw thoughts/emotions to whoever we wish. But what else can we do with these new digital brains of ours?

What if we could "live forever" by just re-downloading another full copy of our brain into a mechanical body once your current one starts to fail? I find this whole concept one of extreme interest because our society would be forever changed by this. Think about what would have to happen if we had the ability to never die. Well, for one thing, Earth would get crowded pretty quickly because people would stop dying but new babies would keep being born. This would definitely force us to begin settling on the other planets/moons in our solar system. Let's think of something else: when you trade your organic brain for the digital representation, is it really you? will your friends see any difference in the way you act? is there more to us than our brains? can we really accurately construct a mind? I don't know the answers to any of these questions. Maybe when we switch from our organic brain to our digital representation, it's really lights out for us, and the birth of an artificial intelligence which already has your memories and acts and learns just like you do, or maybe we actually will wake up on the other side. The answer if any of this will come to pass still remains to be seen, and I won't be holding my breath for it, but I'll be keeping a watchful eye on the development of technologies which might lead up to this.
Thank you all for reading,
-Carlos

Thoughts On Education

About a year ago, I was looking for a good Ted Talk to watch, and I stumbled upon Ken Robinson's 2006 talk, called "Schools Kill Creativity". My view on education was forever changed. If you're reading this and you've never seen that talk, please do so! (and for that matter, watch his 2010 follow-up talk too) It helped me realize something that I notice every day now, that is, in the words of Sir Robinson, "we're educating people out of their creativity". His talk also tells a story of a little girl in a drawing class, who usually doesn't way attention, was hard at work at what she was drawing. the teacher came to investigate and asked what she was drawing. the girl responded, "I'm drawing a picture of god", the surprised teacher says ,"well no body knows what god looks like" to which the girl responded, "well, they will in a minute". now, I don't know about you, but I can't even remember the last time I attempted to picture what god looks like, let alone draw him out on a piece of paper. Why is that?

One thing I've also noticed during my progression through the education system, and something that I still notice in college, is that as people progress through their educational career, they become increasingly afraid of being wrong. In Elementary, everybody raised their hand to answer the teacher's question, when middle school came around, about half of the people were willing to answer the question, and when I was in high school, it pretty much boiled down to 5-10 people per class answering the question. In some of my college classes, it's lucky if someone even gives the question a go. Now, much like in Robinson's talk, I'm not saying to be wrong is to be creative, but if one is unwilling to be wrong, he or she will never be able to give themselves to a creative idea. In my opinion this stigma associated with being wrong is largely the fault of our education system. Fail once or twice in your educational career now a days, and it's like the world is over. It's almost like the world has forgotten that the best lessons learned come right after a failure.

Another thing our education system does, is make amazingly talented people who are passionate and brilliant in their own right think they're inadequate. This goes back to Einstein's quote saying "everyone's a genius. But if you judge a fish on its ability to climb a tree, it will believe its whole live believing that it is stupid" This is because the thing these people were good at wasn't valued by our education system. This is something that I've been guilty of for so long and I still have to struggle to not fall into this. I used to tell my friends who wanted to study something in college they were passionate about things like "Don't do that, you'll never find a job doing something like that". But now I believe that people would be most efficient, doing work for things they feel passionate most about. I also think people would be happier doing the thing they feel most passionate about for a living no matter what amount of money they're making.

I think in every class, no matter what the subject is, should be taught in a matter that evokes the most creativity out of the student as possible. I think we should move more towards this because in the age of the internet, knowledge of cold facts are practically useless, we are moving towards a society that values creativity nearly above all. I believe math should be taught in a way where the computation takes a back seat and the actual development of a problem is focused on. How do we take a real world problem, convert it into a mathematical expression, solve it, and convert it back into a real world solution? the computation is only a single step in the process and it is soon becoming the least important for humans to do. I believe that if we shift our attention towards evoking creativity, our whole world would profit so much more. At the end of Ken Robinson's second talk in 2010 he says, "every day, everywhere, our children spread their dreams beneath our feet, and we should tread softly".

A Few Things I Love and Hate About Humanity and the World We Live In

Things I Love:

There are so many things that I love about this world and so many reasons why I choose to be a part of it every day of my life. 

• One of the main things that gets me out of bed in the morning is that the universe still has so much to teach us, yet we're uncovering its secrets every day. Have you ever thought of college outside the terms of  "I must complete this stage of my life if I am to be successful/make money"? Sometimes (especially when I'm in a math/computer science class) I like to think of college in terms of "x amount of years ago, people had no idea how this thing that I'm learning now worked. It took years for the brightest minds of the time to figure this out, and I'm learning it now, in my teens and early 20's" Think of your next class in that kind of light and I promise you, you'll appreciate whatever you're learning a little bit more. I just love this concept because to me this is humanity at its most cooperative. Everyone's working together to make the next generation smarter, and I get to be a part of that. I pretty much just love feeding my curiosity for the universe
  

• I love how the internet makes it so much easier for creative people to get their ideas out and enables them to actually make a living off their creative work. To some extent, I've experienced this myself. Occasionally, I write poetry and if it wasn't for the internet, I'd have no way of getting these feelings out to the world. They'd be stuck in my room and maybe only read by my closest friends. But I have the ability to post them to numerous amounts of websites, and it's a pretty cool feeling when you get to share the emotion of a poem with a group of people, even if they're strangers
  

• I love that we can use music as a way of bypassing language when trying to communicate an emotional thought or experience. There have been so many times when words just don't do a thought or an emotion justice. I've never felt that kind of barrier when experiencing music. There's just something about it that speaks to me on a deeper level. One of my favorite musicians, Dallas Green said one time in an interview, "I don't want to make people dance, I want to make people cry". There's just something powerful about music and how it can connect us all and reach us on a level so deep, without even really knowing that person.
  

• I love people that are passionate and pure in their hearts. It seems like its these people's missions in life to just make everyone's life a little bit more awesome. This is a rare kind of human, but I've had the pleasure of knowing some of these individuals; some I feel like I've known forever and others I'm dying to make more memories with. I think this is actually one of my favorite things in this list of favorite things. People like this just have the power to transform a shitty day into something so much better if you let them work their magic. I continually strive to be more like these people and make every effort to show them the same kindness

Things I Hate:

•  I hate how we can be so selfish sometimes. You don't really have to go far to experience this aspect of humanity. All of us have been guilty of this at one point or another. Even though you might need what I have more than I do, (I actually won't even miss it) I don't want to give it to you because it's mine and I like the fact that it's mine. Every time I come across a homeless person asking for money, I go through this internal debate of whether or not the money I would give him would help him or hurt him. I guess that's kinda bad on my part for trying to make that kind of judgment call on a person I've never seen before all in a matter of the time it takes him to walk over to my car. Sometimes I give and sometimes I don't, but I should probably give more often. This can also be taken to the large end of the spectrum. How long can you think about the fact that the CEO that makes so much money a year, he/she doesn't even know what to do with it all and the person who makes a dollar a day working his/her fingers to the bone trying to provide for his/her loved ones live on the same planet. It just makes me think if we could all just be a little more kind to everyone around us, it would have such an impact.
  

•  I hate that we feel the need to put down other people to make ourselves seem tall in the eyes of the world. this ranges from the cut-throat politician to the elementary school bully. This is an issue pretty close to my heart. I was bullied in middle school. The people who were doing it probably didn't realize the effect that it had on me, but it made me hate coming to school during the time that this was going on. I also realized that on some occasions in high school, I was the one doing the bullying. It's just a vicious circle of feelings of inadequacy being circulated around. I appreciate what I've learned from these situations, but if I had the chance to go back and help instead of hurt, I'd take it in a heartbeat. Especially now with the age of the internet coming into full swing, the bullying doesn't just stay at school, it's all over the internet and consequently everywhere. Whenever I hear anything on the news about a tragedy involving bullying, it just motivates me to love and encourage the people I'm close to a little bit more. I feel that if all the decent people in the world were to just saturate their environments with love, none of this would be happening.
  

•  I hate how money gets in the way of progress (see my previous post). Think of how much more awesome every country would be if it's politics didn't stink of dirty money. Every time election year comes around in the U.S. I always get so angry that these candidates are out fundraising and worrying about that than the actual issues at hand. I just wish there was a way to separate politics from huge payouts. Call me crazy but I believe when a government official does something, it should be for the good of the people, not so his buddy gets his palm greased with some cash.

Well there you have it folks, just a few things that I love and hate about humanity and the world we live in. I may or may not post more things like this. it depends on my mood, but if you'd like to add your own, just do it in the comments. see ya
-Carlos

Thoughts On Transcending Money

A lot of times I find myself thinking the phrase, "I hate money". I always stop myself and think of how odd that sounds, coming from a person who goes to college partly in hopes of getting a well-paying job, who currently works at a job so he can have money, and is sometimes reluctant to spend his money on certain things. So how can I hate money, but also spend huge chunk of energy and time pursuing it? Well, maybe I don't hate money itself, but what it does to people.

       Why I'm Not a Fan

        Too many times we see and hear about people that put their pocketbooks before actual human livelihood. This happens to people that have all amounts of money. Hell, it happens to me sometimes. How many opportunities have I had to give money to homeless people who need the money a lot more than I do? I could argue because he might be using the money I gave him to buy drugs or alcohol, but in my heart of hearts, I think this is a cop-out answer that a lot of use all too often (that is not to say it doesn't happen, though). Why does money have such a caustic effect on our character? We hear about all these stories of big companies using borderline slave labor to help increase profits. How do these execs sleep at night, knowing that their giving people a nickle a day for hours of intense labor. How do we consumers know that this is going on, but still find a reason to buy their product? Don't get me wrong, I don't have that much of a problem with the concept of a free market, I just have a problem with how we implemented it. I think it's funny how everyone knows that communism looks good on paper but is terrible in its implementation, but people haven't realized yet that its kind of the same thing for capitalism.We just don't see it as well because the people suffering from its effects are oceans away.

        In my opinion, money gets in the way of progress and hinders humanity's ascent into greater things. Think about it: can you think of anything monumental that happened that didn't involve the greasing of some people's palms? Now think about all of the monumental things we could accomplish if we didn't have to worry about money. Just because you can't make a profit off something doesn't necessarily mean that we shouldn't invest time researching the subject and learning more about it. There are certain things that just have intrinsic value in their discovery. Think about when we finally discovered the Higgs particle at CERN's large hadron collider. If I'm not mistaken it has been about a year since the discovery and I haven't really heard of any practical applications or ways of monetizing this discovery. Does this mean we should stop funding these projects? Hell no! These discoveries are unlocking some of the greatest secrets of our physical universe, just because we can't make money off of it, doesn't mean its not important or worth spending time and resources looking into.

 Transcending Money

       When hearing this concept of a society without money, a lot of people think "well, what would you have us do then, go back to the barter system?" Well, not really. We've already transcended the barter system, and I think in the future we will transcend our current system to something totally new. What new system will be put in place? How will it work? I have no freaking idea. But I do know that as the capabilities of technology rapidly increase in an exponential trend (see anything written by Ray Kurzweil), It will become much cheaper to live longer and more luxurious lives. Sure, the rich will still have better versions of those things other people have, but what about when/if humanity's progress finally peaks? There will no longer be faster versions of things, because we've already made them as fast as they can go. The only thing that makes sense to me after that is that we're just going to have a lot of the best things we can make in enormous supplies. When the supply gets high enough, anyone will be able to afford all of the best technology. At this point (maybe even before), I believe money will have outlived its usefulness. I'm pretty sure there are other (more elegant) arguments that lead to a system that must transcend money, but if all else fails, this is kind of like the "brute force" way of getting there. I said that I have no idea what kind of system we would put in place, but I'm hoping for a system where people work and make things because it's what drives them, because they're passionate about it. Even if I wasn't getting paid for it, I would still program and create software, because it's something I love to do, and it's something I'm passionate about. Who really needs more of a reward than giving something to help humanity. I know this sounds kind of Utopian, everybody working their jobs because they like working them, but I really believe that something like this is a possibility of the future.


Alright, That's the end of my rant. Once again, Thank you all for reading.
-Carlos