Series and Their Philosophical Significance

6/4/2017
Sunday
Sundown
Sunny

A series is the sum of an infinitely long list of numbers. The most classic example is presented as a paradox attributed to Zeno of Elea:  how can one travel across a room when first one must travel halfway across the room, but before that one must travel half-of-halfway across the room, but before that one must travel half-of-half-of-halfway across the room, and so on? Expressed mathematically, this question asks:  what is the sum of the reciprocals of the powers of two?

A brief study of series reveals that there are two kinds of series, convergent and divergent. A convergent series has an actual value. For example, Archimedes answered Zeno's paradox in 212 BC by finding a value to the sum of the reciprocals of powers of two; namely, the series sums to one (this, of course, reflects the fact that one can indeed walk the length of the entire room). A divergent series does not have a real value, and thus blows up to infinity or negative infinity. An example is the series whose terms are all one, as adding one an infinite number of times will yield infinity*.

The aspect of philosophical significance is the following:  although a divergent series does not have a limit that exists, the divergent series itself does. This mathematical object, a sum of infinitely many elements, exists and can be reasoned about regardless of the existence of its totality. This has the implication that something can exist in terms of its constituents (in the example of a series, a series can exist contingent on the existence of its summands) but taken as a whole, that thing might not exist. God, for instance, in all of His power, might not exist, but perhaps the elements of God do – love, knowledge, goodness.


For further reading:  Zeno's other paradoxes (Achilles and the Tortoise), Riemann Hypothesis, the length of my dick (the series given by measuring the length of my dick is divergent (IF YA KNOW WHAT I MEAN))

*I feel like this sentence is the equivalent of saying something "equals infinity" which is technically problematic, but for lack of better phrasing I'll leave it.

Prototype Albums

4/27/2017
Thursday
Afternoon
Windy

It is well-known that Radiohead's fifth album Amnesiac is a collection of "you weren't good enough" songs from the album Kid A. This puts Amnesiac in a unique situation; when listening to this album, one sees the scratch-work of Kid A, the early drafts. In this way, Amnesiac allows us to learn more about Kid A, one of the greatest albums of the early 2000s.

Something to note is that both albums have the same song, Morning Bell (called Morning Bell/Amnesiac on Amnesiac), one being an older version of the other. The main difference between the versions of the songs is that the song on Amnesiac is in standard time signature, 4/4, while the version on Kid A is in 5/4. I think this is a testament to how Radiohead writes songs. Instead of coming up with the technical aspects of the song, like the time signature, they focus more on the general feeling the song is supposed to elicit; in the case of Morning Bell, futuristic, soft, ghostly.

Why I and you should be taking a poli sci class.

4/13/2017
Thursday
Dusk
Cloudy with a chance of rain

Today in my Poli sci class we had a discussion about Locke that gradually shifted to a conversation about our(the United States) political system. Within the time I spent answering questions and discussing with my the class I had my own internal dilemma about whether or not I wanted to stay in the class.

The answer I came up with was no but I have to tough it out. Most college student reading this (s/o to Nathan probably the only one) might think I am doing this because Poli sci would be the least painful GE, but that would be wrong. I already have the requirements for that section filled; however what I don't have is my requirement to know how and why my government works. You might delude  yourselves with the knowledge from your AP government classes taken in High School, but that's not enough. Most of you probably didn't pay attention half the time and for the time you did how much of it do you remember. All I am saying is that for centuries the greatest change a country could go through would be when the young people educated themselves on Politics and then got involved. Getting an education on our system no matter how faulty or stupid is the first step to changing it.

So that's my rant on why you should at least take a class or two on politics.

(Praise) The Sun

3/23/2017
Thursday
Early Morning
Beautiful

I stood in the sitting area on the bluff by the beach, the one that faces the Sun. It was 6:30 AM, roughly, when two of my fellow students came shivering into the area. They proceeded to document their expedition, taking Snapchats of the sky when the Sun's not even up. Then, they leave, never seeing the sunrise.

Of course, I wonder if this is another marker of the shorter attention spans caused by the heightened use of social media, a facet of instant gratification.

The sunrise was fucking beautiful, by the way. As the Sun rose higher behind the horizon, clouds farther ahead of me were already hit by the sunlight, and it made the edges of the clouds glow like gold. The sky grew more and more orange. In many ways, the sunrise was like a good classical song, or a good handjob: a lot of build-up, so much that when the end is finally reached, the feeling is incredible. And indeed, when that bright, golden sliver peaked above the horizon, it felt holy.

Bonus question to the reader: though I'm somewhat criticizing the masturbatory nature of social media, how masturbatory is it to post the same experience on a blogging website, which is a form of social media?

Nothing is Clear-Cut (e.g. Math)

3/20/2017
Monday
Afternoon
Partly Cloudy, but Chilling

Of course, nothing is clear-cut. Nothing fits nicely into boxes/categories/sets we create for them, but rather things are blurry, multifaceted, and they fall into many different groups. Here is a specific example related to, you guessed it, math.

Consider the fact that the derivatives of the sum of two functions is the sum of the derivatives of the composite functions. Stated a bit more symbolically, given two differentiable functions f and g, we see

(f+g)'(x)=f'(x)+g'(x).

A high-school calculus student would see this and recognize "a-ha! this is the Sum Rule!" On, say, the AP Calculus AB exam, this would be sufficient reasoning behind this fact; it was dictated in the class, and the students have memorized it, either through brute force or through much practice of finding derivatives of functions.

Of course, there is different justification for this fact from different mathematical perspectives. For example, consider Real Analysis:  calculus' rigorous, proof-based older brother. A mathematical analyst will tell you "well, the definition of the derivative is given by a limit, and since we can add limits, we can add derivatives." Moreover, the analyst could relate these limits back to sequences, and reason that the Sum Rule follows from the arithmetic of limits of functions, which follows from the arithmetic of limits of sequences.

From another perspective, consider Linear Algebra. A linear algebraist would tell you that the set of real-valued, differentiable functions form a vector space with the operations of function addition and multiplication of real numbers. Therefore, such a mathematician would say that the Sum Rule follows from the fact that the vector space of differentiable functions is closed under addition.

This is something about math that I like; you learn one thing, and it pops up everywhere, in a bunch of different fields of math. I suppose this tells us that math is less like a library, each field belonging to a bookshelf, each bookshelf containing an exhaustive list of the ideas in the field. Rather, math is like a swamp, where concepts lurk beneath the fields we've constructed.

Think different. Think DASANI.

Underrated Snack

3/7/17
Tuesday Night
Temperate, Windy

There's a lot of value in the concept of chips and dips. Some classic, popular examples are tortilla chips and salsa and pita chips and hummus. Such snacks require no heating, but just putting two different things (chips and dip) in two different bowls, and eating the dry substantial bit (chips) with a flavorful wetness (dip).

Yet, people forget their god damn roots. The simplest of all of these is right in front of us, the most basic of all chips-and-dip variants: bread and olive oil and balsamic vinegar. Dry, spongy bread soaked in savory oil and tangy vinegar is a simple staple, and should be enjoyed in larger variety. We, as a people, need to open our eyes to simplicity. We need to open our eyes to bread, olive oil, and vinegar.

On Seemingly Arbitrary Changes, namely in the Classroom

2/9/17
Thursday
Around noon
Foggy

One of the things that I have been hearing in classrooms and lectures a lot recently is "easing the monotony." For example, my Intro to Real Analysis professor will often ask the class, "What should I name this variable this time?" Or Sal Khan, in one of his Khan Academy videos, will switch up the colors he uses when he is demonstrating a mathematical concept, switching from magenta to teal, for instance.

On the surface, these changes seem arbitrary or inconsequential. That is, in what way are these tiny changes affecting the absorption of the course material? These variables could be named anything (within reason, of course) and the color of the pen could be any color, and one would think that the material would be unchanged. There is a tangible purpose to this, though: breaking the cycle. We all adhere to some sort of routine day after day; as Douglas Coupland once wrote, "Most people can't handle an unstructured life." Such a routine can make one feel trapped, fixed in a treadmill, or a hamster wheel, repeating a structure day by day, mindlessly functioning, robotic. Yet, an important part of the human experience is choice*. Freedom. We should be able to do whatever we want! We should be able to brush our teeth with our left hand instead of our right, we should be able to have a Mocha this morning instead of an Americano, and we should surely be able to name the limit of a function "D" instead of the conventional "L." These arbitrary changes are not useless, but rather expressions of our freedoms as conscious people. Sal Khan and Prof. Zimmerman understand this, and and they choose to break the dreary cycle of monotony with which mathematics is often associated. Sal and Birge, I thank you. And God fucking Bless America.

*The illusion thereof, at least.