## Simple reflections

When you study Algebra, you study the rules of equations and polynomials. Algebra is used when there is some unknown variable or number in an equation.

Geometry deals with the study of shapes like triangles, squares, hexagons, and polygons. 

Calculus, while using techniques taught in both algebra and geometry, focuses on the rates of change of functions and the area and slope of curves.

For example, say you have an equation that, when graphed, is a straight line **************y = 3x - 5.**************

Using the slope intercept formula taught in algebra tells you the slope of that function is 3.

So what do you do if you need to calculate the slope of a parabolic function, like  *******y = x^2 + x - 3?******* This is where calculus comes in.

First, you take the derivative of the function. The derivative of ****************y = x^2 + x - 3**************** is *f’(x)* *= 2x + 1*. Once you have the derivative function, you take the point you want to find the slope of and plug it in. If the point is (1, 3), you’d calculate f’(3) = 2(1) + 1 = 3 (This function is saying “when *x* is *3*, what is *y*?. So at the point (1, 3), the slope of our function is 3. 

Calculus also becomes useful when you want to find the area under a curve. A classic problem is where you have a plot of land that borders a river on one side, and a straight road on all the others. Since this isn’t a proper square, the formula for calculating the area of a square won’t work, but if you have the function of the curve of the river (which I know is unlikely to come on the deed of the property), you can calculate the area.

Another part of calculus deals with the instantaneous rate of change of an object. Let’s say you were going on a road trip and you were traveling at a constant speed of 60mph for the entire trip. If the trip took two hours, it’s easy to calculate that we went 120 miles.

But if your speed *******wasn’t******* constant and instead averaged at 60mph, which meant you were going 70mph at some points and 50 mph at others, you can use calculus to determine how fast you were going at ***any*** point in time on your trip. Pretty cool!

## Reflections on how I learned it

I took Math 104 (Calculus 1) on [Study.com](http://Study.com) as a prerequisite to a Computer Science program I’d like to attend. I was accepted, but just needed to get this course out of the way. It was…daunting…to say the least. I hadn’t taken a proper math class in about a year and the last one I took was more of a logic-math, not pure math.

Surprisingly, I really enjoyed it. I told myself all through high school that I was bad at math (Algebra was the only class I got a C in) and had always been more of a words guy. Turns out I was wrong. I simply didn’t have the desire to get good at math, nor have the resources available to help me when I was stuck. Though I had YouTube, I didn’t really know how to use YouTube for help at the time. The main limiting factor, though, was my desire: there wasn’t any. I had no need for algebra in my life and didn’t see how it would be useful for my career as a video editor/graphic designer. But now I have the desire.

As my curiosity has grown over the last few years, I’ve become more interested in science and especially physics. Most of physics is math. I also am pursuing a degree in Computer Science, which is a math heavy degree. Therefore, in an effort to understand armchair physics better and graduate with a degree in Computer Science, my desire to get good at math sprouted.

[Study.com](http://Study.com) was helpful because after each video I had to take a 5 question quiz. If I got more than one answer wrong, I was forced to retake it in order to get actual credit for the course. This rapid learn-then-test-yourself environment prevents you from fooling yourself by saying you understand something when you really don’t.

The course had 124 quizzes, which I had to get an 80% or better on. When I finished the quizzes, I had to take a proctored exam. The exam was easier than I expected, thanks in no small part to the unlimited practice exams I could take. To study for the exam, I took a practice exam, saw what types of questions I didn’t understand well, study those, retake the exam, and repeat until I felt ready to take the final.

The downside of taking the exam on [Study.com](http://Study.com) is that it takes two weeks to get your official results back because proctors watch the video back to ensure there were no shenanigans. It makes sense, but I feel like they could give you the official score and then come back if there are any problems. I also feel like you could easily train an AI model to exam the videos and flag any odd behavior for a human to double-check.

## In conclusion

I’m excited I did it. I’m excited I learned. I’m excited for the rest of my learning journey. Thanks for following along.

Reflections on Calculus

## Impressions

Captivating, heart-breaking, and awe-inspiring are just a few of the many adjectives I could use to describe this book. The author, Margot Lee Shetterly, did a great job weaving together the lives of “the black women mathematicians who helped win the space race” to show their grit and cant-stop-me attitude mixed with the terrible injustices they had to face.

One thing I realized was that African American’s fought for “freedom” in both world wars but they weren’t offered that same freedom back home. America and the allies painted themselves as a land that loves all and recognized the injustices the Jews were facing in Germany. Yet at the same time, African Americans were still fighting for fundamental freedoms and rights back home. I had never seen that juxtaposition before.

A similar situation happened during the space race. The USSR ensured anyone who so desired could be trained as an engineer. They had waves of people learning the most advanced math, science, and physics. Meanwhile, In America, African Americans were struggling to get a decent grade school education. In 1959 in Prince Edward County, rather than integrate, segregationists defunded the entire county school system. The schools in that district remained closed for five years. “Virginia,” Shetterly writes, “a state with one of the highest concentrations of scientific talent in the world, led the nation in denying education to its youth.”

When the space race took off, Civil Rights activists thought we didn’t have any business reaching out beyond the stars if we couldn’t figure out how to live together beneath them. I see their point.

Lucky for us, Dorthy Vaughn, Mary Jackson, and Katherine Johnson looked these difficulties in the face and persevered. Their work helped John Glenn orbit earth and Neil Armstrong walk on the moon.

## Notes

- Charles Hamilton Houston, one of the NAACP’s top lawyers, reflected on his time in the war in 1942 to the *****************Pittsburgh Courier***************** after he was on a train on the way home and a white man refused to sit by him and a friend. He said, “I felt damned glad I had no lost my life fighting for my country.”
- In 1959, President Eisenhower set the plan for constructing a secret bunker deep under the Greenbrier Hotel
- ‘This is not science fiction’ wrote President Eisenhower in a fifteen-page document called ****************************[Introduction to Outer Space](https://lunar.colorado.edu/jaburns/publicfiles/IntroOuterSpace.pdf).*
- School board’s in white-only districts paid special “school fees” to black families to keep their children in black districts. Virginia colleges did something similar to black graduating students, paying them a “scholarship” to attend college out-of-state.

> But simple luck is the random birthright of the hapless. When seasoned by the subtleties of accident, harmony, favor, wisdom, and inevitably, luck takes on the cast of serendipity. Serendipity happens when a well-trained mind looking for one thing encounters something else: the unexpected. It comes from being in a position to seize the opportunity from the happy marriage of time, place, and chance.
>

**********Star Trek********** wasn’t a show that just made nerds cool. It helped usher in a new era of tolerance in America. Showcasing the different races (and genders) of the universe inspired those who were usually dealt the short end of the stick.

Lieutenant Uhura, a black women and proud citizen of the United States of Africa was fourth in command aboard the spaceship **********Enterprise**********. The actor, Nichelle Nichols, tendered her resignation with the show’s creator, Gene Rodden, after the first season. Devastated, Rodden asked her to take the weekend to think it over.

That weekend just so happened to host an NAACP civil rights fundraiser and a coordinator for the event told Nichols that “one of her biggest fans” wanted to meet her. That fan was Dr. Martin Luther King Jr.

Margot Lee Shetterly, writing in **************Hidden Figures**************, said that Nichols appreciated Dr. King’s effusive praise of the show, but let him know she had decided to leave. Before she even finished, according to Shetterly, King replied:

> ”You can’t leave the show…We are there because you are there.” Black people have been imagined in the future, he continued, emphasizing to the actress how important and ground-breaking a fact that was. Furthermore, he told her, he had studied the Starfleet’s command structure and believed that it mirrored that of the US Air Force, making Uhura–a black woman!–fourth in command of the ship.

**”This is not a black role, this is not a female role,” he said to her. “This is a unique role that brings life to what we are marching for: equality.”**
>

**On Monday morning, Nichols marched into Rodden’s office and asked to tear up the resignation letter.**


Hidden Figures by Margot Lee Shetterly

Let's pretend one day you find magic machine that could grant any lifestyle you asked it for. Professional golfer? Done. Star of *Pulp Fiction*? Say no more. Enlightened writer? Easy. Any lifestyle you wished to live could be yours upon demand.

But like with any magical box or genie, there's a catch: **The only thing it gave you was *exactly* what you asked it for.**

If you asked to be a star of *Pulp Fiction*, it could do that, but who knows what *Pulp Fiction* would become. It certainly wouldn't be the genre-breaking film it was. Perhaps it would be a book about orange juice, but hey, you'd be the star!

You could be a professional golfer, but it would be the year 1100 and golf would be nowhere near the sport it is today. It would take your prompt, but it would not give you the assumed preferences–the things you value and care about–unless you *specifically* asked for every single one.

Let's look at a more concrete example: that of a young couple wishing to get married, buy a house with a white picket fence, and have two kids with a dog. If this couple gave the machine with that criteria for a lifestyle, they would get their house with a white picket fence and two kids, but they might be the only people on their side of the Mississippi. Since they didn't specify they wanted a car–or any other technology for that matter–they wouldn't have it.

But for the purpose of this thought experiment, let's say before the young couple entered their lifestyle preferences into the machine, they had a teacher instruct them on the complications of it:

"Make sure to specify exactly the kind of lifestyle you want to live. If you want to live in Paris in the 1950s, make sure to specify you want to live in the 1950s Paris, just as it was in the real 1950s," says the teacher. "This machine will do *exactly* what you tell it to do with no assumed preferences or inherited human belief systems, so be specific and precise about the things you care about. Don't forget anything!"

Even on a good day, that couple isn't getting their lifestyle exactly as it is today. Maybe they have the technology that they have today, but since they didn't specify that they also wanted the ability to love, they now hate each other.

## Complexity of value
The task of defining your every preference is difficult. If you didn't follow along with that thought experiment, just imagine all the different ideals and preferences you have about your current lifestyle. When I say preferences, I don't mean drinking lattes over americanos, though that is a valid preference. Instead, I mean how you define love, the legacy you wish to leave behind, and your definition of living a "good life." You could try listing things for hours and you're guaranteed to miss at least a couple of preferences. 

This idea, the fact that our preferences and values cannot be explained with just a few sentences, is known as our **Complexity of value**. [LessWrong](https://www.lesswrong.com/tag/complexity-of-value), a popular blog geared towards improving decision-making and reasoning, explains that it's impossible for human values to be "summed up by a few simple rules, or compressed."

This is true because of the [fragility of our human values](https://www.lesswrong.com/posts/GNnHHmm8EzePmKzPk/value-is-fragile). If we all woke up tomorrow and *kindness* didn't exist, the world would be *wildly* different. 99.999% of everything else in the world could be the exact same as it is today, but if we're missing kindness, or love, or humor, the world would cease to be the same. It might not be horrible, or evil, or bad, but it would most certainly be dull. Imagine a world without beauty. Everything, everywhere you look, was just one shade of black or gray. Not black *and* gray, but just different shades of either black *or* gray.

[Eliezer Yudkowsky](https://en.wikipedia.org/wiki/Eliezer_Yudkowsky), a computer scientist who wrote the post cited above explaining the fragility of our value puts it this way:

> Consider the [incredibly important human value of "boredom"](https://www.lesswrong.com/lw/xr/in_praise_of_boredom/) - our desire not to do "the same thing" over and over and over again.  You can imagine a mind that contained _almost_ the whole specification of human value, almost all the morals and metamorals, but left out _just this one thing_ -
> 
> and so it spent until the end of time, and until the farthest reaches of its light cone, replaying a single highly optimized experience, over and over and over again
> 
> Or imagine a mind that contained almost the whole specification of which sort of feelings humans most enjoy - but not the idea that those feelings had important _external referents._  So that the mind just went around _feeling_ like it had made an important discovery, _feeling_ it had found the perfect lover, _feeling_ it had helped a friend, but not actually _doing_ any of those things - having become its own experience machine.  And if the mind pursued those feelings _and their referents,_ it would be a good future and true; but because this _one dimension_ of value was left out, the future became something dull.  Boring and repetitive, because although this mind _felt_ that it was encountering experiences of incredible novelty, this feeling was in no wise true.

Humans are complex. Our values might be even more so.

## Kolmogorov Complexity
If we want to get technical, which I always do, you can say that human values have a high Kolmogorov complexity. [The Kolmogorov Complexity](https://en.wikipedia.org/wiki/Kolmogorov_complexity) of something refers to the size of the shortest possible description of that thing. If the Kolmogorov Complexity is high, like we know our human values are, it's difficult to describe that thing in any shorter way than actually just actually saying that thing. Since our values are very complicated, we are unable to create a rule, theory, or formula to feed to the magical machine and say "Give us this lifestyle with these parameters," because we have to then go and define every single one of those parameters. It reminds me of that classic illustration where someone is blindfolded and they have to relay instructions to someone else on how to make a peanut butter and jelly, but the person making the sandwich does exactly what the boss tells them to do. If they say, "Put the peanut butter on the bread," but fail to mention anything about opening the peanut butter jar or taking bread out from the bag, the sandwich-maker might just put the peanut butter jar on top of the bag of bread and call it good. In this case, the Kolmogorov Complexity of making a PB&J is high.

Inversely, if something is low in Kolmogorov Complexity, its description can be shorter without sacrificing any detail.

Let's pretend you and I went out for Chinese food after a long day at the office. I don't like fortune cookies, so I gave you mine and said you could pick the best one. Here they are:

    dmdmdmdmdmdmdmdmdmdmdmdmdmdmdmdm
    83jsns02u84nlw9jsn48anw2kfmv8940

After reading both of them, I ask you what the first one said. You could say, `"It says dm 32 times"`, which is only a 19 character response and is a simple description. Now I ask you what the second one said. The only real way to describe it is to say aloud the random string of characters exactly as they appear.

So naturally, *the first set is less complex than the second set* because the description of the first set is shorter (without sacrificing any detail) than the actual output. If for some reason I wanted to copy the fortunes, you could have said "Write dm 32 times." That's a very short input for the same output.

Conversely, what you might say for the second string, "Write `83jsns02u84nlw9jsn48anw2kfmv8940` ", is longer than the original output, so the Kolmogorov Complexity is high. It's interesting, though, that the complexity of any string cannot be much higher than the length of the string itself. Even with our high complexity string, the input is only 6 more characters long.

## Why this matters
Calculating the Kolmogorov Complexity of a 32 character strings is not exciting, but when using Kolmogorov Complexity as a model as to how the world works, it gets fun. You start to understand the intricacies of everyday life and see how the world is actually really complicated, no matter what someone else says. It's hard to accurately represent something using just a map or description of that thing. As you start to realize that more and more, you become aware of all the possible, "Oh and this too"'s that are in the world.

An attempt to understand the Kolmogorov Complexity of human values

In 1925, John von Neumann published one of the first papers introducing "game theory" into popular discourse. In it, he explained that most games are just a series of events. Some of those events depend entirely on chance–which von Neumann calls 'draws'– and others on the decision of a player–what von Neumann calls 'steps.'

Ultimately, he concluded a game, as *[The Man from the Future](https://www.amazon.com/Man-Future-Visionary-Life-Neumann/dp/1324003995)* explains, "could be represented simply as the choice by each player of a single strategy (effectively an amalgam of all the strategies they play in the game), followed by a calculation of their respective payouts that accounts for everyone's choices (and which factors in their luck)." 

But he couldn't get any further in his thinking with multi-player games, so he started thinking about games with two players whose payouts equal 0.

Since there are only two players, one *has* to win and the other *has* to lose. What one person wins is equal to what the other person loses. So if Player A wins $5, then Player B has to have lost $5 somehow. The two payouts, when added together equal 0 (5 + (-5) = 0). Therefore, these games were determined to be *zero-sum*: the sum equals zero.

## Why this matters
The terms "zero-sum" and "positive-sum" are popular in the online world I hang out in. The benefits of the world and the creator economy being "positive-sum" or touted and the mindset that the world is "zero-sum" is negative. I always understood what they meant, but didn't know the background behind them. It turns out, John von Neumann helped birth these terms into the world through his earlier work on game theory.

The paper in which he lays out these terms is the one where he describes his 'minimax' strategy. In our example above, Player B wants to minimize the maximum amount of money they can lose and Player A wants to maximize the minimum amount they can win (called the 'maximin' strategy).

People who are described as having a "zero-sum mindset" are those that don't think there can be multiple winners. They'll do anything they can to climb the ladder or make the sale. They are also usually very jealous of other people's success because in their world, if someone else is winning, that means they're losing.

Though that's not how the world really works, so the world can be described as "positive-sum." There is enough winnings for everyone. Thankfully this is the case, because we're a few million years late to the party.
