Hello, all. Right now, I want to give you a quick spin through some of the things I've been learning about in my liberal arts education right now. It will be quick because I am exhausted and each ticking of the clock takes me closer to my alarm going off in the morning, but I miss this, so here I go!
1. From Algebra: don't panic. I'm just going to explain one elementary concept. Please understand that what we are actually doing in class involves a lot more than this (and I'm convinced that some of it could be explained to non-math people, but I'm not sure if the one direction, me writing --> you reading format of this blog would be adequate to explain the more layered concepts), but this is an idea that I only heard about for the first time in this class, and I thought it was quite interesting.
You might recall from elementary school that certain numbers (certain integers, actually - integers are all the whole numbers, negative and positive, including zero) have the property of being prime. That means that those certain numbers have only 1 and themselves as factors (factors being things you multiply together to get that number). I'm sure most people know the first few prime numbers: 1, 2 (weird, huh? the only even prime number... :D ), 3, 5, 7, 11, 13, 17, 23... etc, etc. There are lots of interesting facts about primes. First of all, they seem to be distributed in a rather arbitrary way throughout the integers - 7 and 11 are further away from each other than 11 and 13, but then then 13 is farther away from 17 than that, etc. Several branches of mathematics deal with the properties of prime numbers, looking at them from different ways and in great detail. You may have also learned in school that every integer can be written as a product of prime numbers - called a number's prime factorization. One way to look at this is as the prime numbers as the letters of the alphabet, and the whole set of all integers (from negative infinity to positive infinity) as all the words we can form with those letters. In this sense, the primes are the building blocks of the integers. Atoms. Cool, right?
The other neat thing about primes is that I think they are a fantastic medium through which to explain one basic structure of a proof. You're about to prove something. Like, PROVE something, in the mathematical sense. Are you ready? It's gonna feel awesome.
Let's prove that there are an infinite number of prime numbers.
---- hold on a sec. Think about what that means. It's always good to understand what you're proving before you try to prove it. So, we want to show that there's an infinite number of prime numbers - that means that the further out you go on the number line, if you will, you'll always find another prime number - and another after that, another after that. There is no positive integer so big that a prime number can't be found that's bigger than it.
So, we've understood the question. Now we can prove it. Let's assume for the sake of contradiction (you're learning awesome math language now) that there only finitely many prime numbers. Let's say there's n of them. (Sneaky math talk, using a letter to represent a number. It's not fancy, it's just notation, okay? Don't get intimidated by little ol' n.) Since each of those numbers is prime, but I don't know individually what each of them are, I'm just going to call them p1, p2, p3,...., pn. Now I have a group of n (a number n) prime numbers. With me so far?
I'm also assuming that THAT'S IT. There's no prime number bigger than pn, the last prime number. But hold on a second.
What if I took the product of all of those little p's -- p1*p2*p3*.....*pn?
If I multiply them all together, I get some big number - some big number that's divisible by p1, p2, p3 --- by all of the p's, since they are all in that product. So you know what I'll do to that product? I'll ADD ONE TO IT.
So, we have (p1*p2*p3*.....*pn)+1. And you know what? That thing I just wrote? That number is prime. Its only factors are 1 and itself, since I used up all of the rest of the primes in the product.
--> there are an infinite number of prime numbers. You give me a finite number of them, I can find you the NEXT one. And I can do that no matter how big the finite number you give me is. QED.
Doesn't that feel awesome?
Okay, but that's not the concept I wanted to share with you. No, with all that as sort of background information, I want you to think about something else: say you have two numbers like 7 and 12. Yes, 7 is prime, 12 is not. If we look at each of those numbers, we notice that they don't have any factors in common other than 1. And we have a name for numbers like that. We call them relatively prime. Just think of all the fantastic things we can learn about them! :)
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2. I must say, 1 ended up being quite a lot longer than I'd planned! I was just having too much fun. :) Okay, so besides numbers and their properties, what have I been thinking about? In my RSS course, I have been learning about the idea of dependency and the welfare state - the idea that welfare actually encompasses any redistribution of state wealth (collected through taxes) through resources for citizens. However, we don't think of all welfare the same way. In fact, a lot of welfare programs aren't associated with the hot-button word "welfare" at all. We don't consider Social Security to be a "government hand-out", just for one small example. There's something quite odd in that system. Welfare, in the minds of most people, goes to single mothers and lazy people who don't get off their asses to work. Is this a correct thing to think? No. Is it my automatic to the word 'welfare'? Yes, unfortunately. That just goes to show part of a recurring theme in my class that the rhetoric, the vocabulary we use to discuss a subject greatly influences our understanding of the topic itself.
(If I had more time, I could tell you so much more! I have a long weekend coming up - maybe I will then!)
3. I'll combine my two film-related classes. In music, we have discussed recently the very huge change around the 50s and 60s in cinema when the film genres and film industry itself had become established enough that films could reference other films. We see this in the form of parodies all the time. We know film and film music conventions so well that it doesn't even seem so important to talk about it, but it is! Why is it that we laugh when we see someone sitting at a table eating a bowl of spaghetti with music so dramatic it could have played in Jaws? Because we know that type of music is designed for a film like Jaws, not for a film about someone eating noodles. That transition took a while, and the results are really fascinating once you look for them.
In my other film course, we have recently shifted from studying the genre of Formalism in film to Realism. Formalism encompasses about 80% of the films that are made worldwide. Formalism is about entertainment, for the most part. It's about the narrative, it's about providing a conclusion at the end of the narrative that feels like a conclusion - whether it's happy or not. Blockbuster films are formalist, almost 100% of the time. In these films, so much emphasis is placed on the technical aspect of it that we end up having what is called invisible editing, among other things - technique that is so clean, sound design, music, acting, lighting, cinematography - everything done so seamlessly that we don't notice it at all in our willing suspension of disbelief in the theater. We only see the narrative. And it delivers what we want - plot, plot, satisfying conclusion. Fun, entertaining movies. (Batman, Harry Potter, Casablanca, Psycho, etc.)
Realism, however, refers to films that are a bit more ambiguous. Most of the time, there are no noble characters in realist films - no innocent ones either. Everyone is flawed. These movies examine the human condition and what we are apt to do under less than perfect circumstances. On a strictly technical level, one difference between formalism and realism can be seen in the fact that there are very few close-up shots in realism films. In formalism, we quite often get shots that are so close, we have no doubt about what in the scene we are supposed to be focusing on. The editing effectively tells us what is important. This isn't the case in realism. In realism, we see the whole shot - the whole room with characters in it, or the whole house, road, tent - whatever it may be. The camera holds itself back - the viewer is told "You make your own decision. What's going on here? Why? Is it right or wrong? Is there even a right or wrong?" Realism films are quite often more about character study than about the furthering of a plot. (City Lights, Memento, Grapes of Wrath, The Master, etc.)
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Whew! I got caught up in writing. I need to sleep! I hope you found that at least a little interesting. :) I'll write again soon.
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