# Math and Philosophy

This is actually from Semester 1 of Grade 10, but I read it again and I’m genuinely proud of it so I’m posting it here so it may be immortalized on the internet.

The link between mathematics and philosophy was my entire inquiry project, so I apologize if you don’t read something incredibly profound about how math is actually grounded in unanswerable questions, just like philosophy. If you are looking for that, please see my inquiry project write-up. Instead, I’m going to go really in depth into specific mathematically philosophical questions.

Language is one of those human concepts that are simultaneously infuriating in its limits and exhilarating in its possibilities. I’ve often felt betrayed by the English language, as I search for a word that perfectly describes what I’m feeling, only to discover that it doesn’t exist. However, as a lover of literature, it’s difficult to stay angry for long. Humans are able to describe nearly everything in our visible universe, and we like to feel that we know everything there is that needs to be known, and then some. But that feeling doesn’t really matter to discoverers and inventors, so it inevitably always gets ignored. Us humans have the concept of infinity, defined as something with no ends and no limits. It just goes on and on and on (like me!), and it is impossible for humans to fully grasp this concept. We rely so heavily on physical representations of ideas, thus making complex concepts, like time travel and infinity, unimaginable to us. We can’t imagine something that doesn’t exist to us, and that is to be expected. So, why do we talk about the infinite so much, when it doesn’t even apply to us?

One perspective is the mathematical side. We use concepts like infinity in, say, graphing, because we logically know that there no boundaries to a function like x + 1 = y. We can represent this on a graph or in a table of values, but all the values of y are impossible to list, because they’re infinite. You can add one to any number an infinite number of times, and you will still get a possible y value. This concept also applies to fractions and negative integers, even irrational numbers. Pi plus one is just π + 1, shortened to 4.1415… Even irrational numbers, that we use so often in mathematics, go on infinitely! An irrational number is defined as a non-terminating, non-repeating decimal, therefore it goes on infinitely. We use these little infinities constantly: pi, phi, certain ratios and fractions. We also find these irrational numbers in nature – the Golden Ratio can be seen almost everywhere in nature, and is directly linked to the Fibonacci sequence. Therefore, this tells us that some form of infinity must exist outside of humans, that there are situations where never-ending and limitless concepts exist.

However, (there’re a lot of those in this little composition) we can also look at infinity philosophically. When we talk about something that never ends, we begin to erase the basic aspects of life itself – as morbid as it sounds, death and decay. If, for example, human beings were infinite, we would either never die or our bodies would never decompose. Either one. But then there’s the idea of souls, or some sort of aspect of humanity that is separate than the total sum of our parts. It is my belief that, if one were to completely clone a living person, give them the exact same memories and emotions as the original, you would still get someone completely different. This new person would not be the old one in all shapes and forms, but that missing aspect is unknown. I’m not sure if this missing aspect is infinite in its existence, but it’s definitely possible. We can also think about infinite time philosophically. Often, we ask ourselves if anything we do is worth it, if it’s all towards some purpose or if it’s even going to remain. Logically, we may know that whatever we make and everything we do will eventually disappear, but we don’t know when. Our brains are hardwired so that the idea of humanity perishing, of us just ceasing to exist, terrifies us. There’s the survival instinct, of not wanting to die, but we also don’t like the idea of putting all this work, effort, and time into the things we’re proud of, and thinking that they will have been for nought. I know I don’t like thinking about it.

Our brains tell us that time isn’t infinite, but science tells us that the universe itself might be. According to multiple theories, proposed by those much more educated than I, there are an infinite number of possibilities in an infinite number of dimensions in an infinite universe. This is described as the multiverse theory, and it gives a certain amount of hope that things are better somewhere in our supposedly infinite universe. Although, this also means that things are worse, but good and bad are all relative and depend on our definitions, so it’s inevitably a language issue, which I touched on earlier, and this whole mind-screwing idea is a topic for another composition. Anyway, according to this theory, literally anything is possible, and that sounds like something you’d see on a poster with a sunset in the background, but it’s also a mathematical goldmine. In inequalities that state x ≥ y, x can be equal to or larger than y, meaning x can be equal to almost anything other than y. However, if y is equal to negative infinity (-∞) x can be anything, including negative infinity. Because infinity (even negative infinity) is, you know, infinite.

Some infinities are larger than other infinities, but that’s per our current definition of infinity. This definition might change with more concepts introduced over time or more “mathematical evidence”, but it always just comes down to language and how we interpret it. When I think of the word “infinity”, I think of the multiverse theory. Other people might think of infinite time, of wormholes, of graphs. The thoughts that come into our minds when we think of infinity are, dare I say it, endless. Does this imply our own thoughts, the little electric shocks zooming through our brains at thousands of miles per hour, are infinite? Who knows? Why do we even care? If they are infinite, why? Why wouldn’t they be? I can come up with multiple answers to each of those questions, but none of them are definitive, which is really cool. This implies that even our most logical field, mathematics, isn’t definitive. It implies that current science isn’t definitive, and it’s all just theory, when you really think about it. Philosophy as we know it has managed to worm its way into every aspect of logic with its unanswerable questions, almost always beginning with “why”. Maybe this explains why we use “y” as a variable. Maybe it doesn’t, but that’s okay. (If the multiverse theory is to be believed, then there is a Sara who knows why, or has at least convinced herself that she does.)

I think what I took away most from these philosophical discussions in math class was accepting that sometimes there isn’t an answer, that math isn’t necessarily the say-all, do-all. It’s mostly about discussion and exploring all these ideas. Maybe it’s naïve of me to think about, but how much could we do if we acknowledged that not everything is clean-cut with a simple answer and a picture? If we did things out of pure curiosity (as I do most of the time), and because we think there might be a positive outcome? Is there any better way of doing things?

# Math and Philosophy

I just really like math class when we get to discuss mathematical philosophy. I think about these kind of things a lot (I have a couple friends in which our entire relationship is based on our mind-blowing conversations, some of which happen at two o’clock in the morning).

I could go on and on and on about all the questions we discussed today and all the concepts that blew our minds. But I really like the Discussion part – we were given a lot of time to just talk through our ideas, and I absolutely love that because it’s how I make everything make sense in my head, by verbalizing it. I tried my best to include everyone in the conversation, but I admit, there were certain points where I would get really excited about what was being said and I would want to respond to that thing as soon as possible.

# Modeling Polynomials

1.

$(x+1)^2$

$(x+1)(x+1)$

$x^2 + x + x +1$

$x^2 + 2x + 1$

2.

$(x-1)^2$

$(x-1)(x-1)$

$x^2 - x -x +1$

$x^2 - 2x +1$

When squaring polynomials, it’s a good idea to write out the whole equation, so you don’t have any exponents. This makes it easier to see all the operations you have to do. Multiply the first value of the first bracket by the first value in the second bracket, then the second value. Then you do the same thing, but with the second value of the first bracket.

3.

$(x+1)(x-1)$

$x^2 + x - x - 1$

$x^2 -1$

With equations like this, you can use the same distributive property used in the previous questions, but you just have to be careful with the signs – it’s really easy to get a completely different answer because you forget to put a negative sign in there.

4.

$(x-1)^3$

$(x-1)(x-1)(x-1)$

$(x^2 - x - x + 1)(x-1)$

$(x^2 - 2x + 1)(x-1)$

$x^3 - x^2 - 2x^2 + 2x + x - 1$

$x^3 - 3x^2 + 3x - 1$

For powers of three, you write the whole thing out, then do the first part, which is the same as squaring a polynomial. Then, you multiply it by itself again! It’s just taking the previous two strategies into one question. Take it step by step, and it’s not nearly as scary as it looks!

1. (x+1)^2

2. (x-1)^2

3. (x+1)(x-1)

4a) (x-1)^3

4b) (x-1)^3

# Everything I Know About Trig (Spoiler: it’s not that much)

Trigonometry is the study of triangles, specifically right triangles, specifically the angles and side of right triangles. It is also a long word that I still don’t know the root of.

Trigonometry deals with the ratios of sine (sin), cosine (cos), and tangent (tan). The shortened versions (as seen in the brackets) can be found on a calculator, and you can figure out angle degrees using them.

Sine is equal to opposite/hypotenuse. Cosine is equal to adjacent/hypotenuse. Tangent is equal to opposite/adjacent.

Opposite and adjacent sides are in comparison to the angle you’re working with, most likely “x”. “X” can never be a right angle, because then the opposite is the hypotenuse, and that just does not work.

There are also a series of right triangles where you don’t need a calculator to figure out the ratio, such as when the adjacent and opposite both equal 1, and hypotenuse is √2.

UPDATE:

Sine, cosine, and tangent, you put in an angle and that gives you ratio. The inverse operations (ex. sin-1), you need to put in the ratio and it will give you the angle. Both ways will help you find the required sign or the angle.

# Garibaldi Lake

What would happen if the Barrier at Garibaldi Lake were to collapse?

Garibaldi Lake is a lake located just north of Squamish, BC, and lies between mountains and volcanoes. Surprisingly enough, the sheer amount of water the lake holds is more dangerous than the active volcanoes.

There is one thing separating the lovely town people of Squamish and approximately 1.2 x $10^9$ tons of water, and that is the Barrier. If it’s important enough to have a Wikipedia page, you better believe it’s pretty vital. The Barrier is about 300 metres thick and 2 kilometres wide, which seems like a lot, but if for some reason it were to break, all the water in Garibaldi Lake would come rushing out and probably swamp any surrounding towns.

There actually used to be a town next to the lake called (you guessed it) Garibaldi, but they had to relocate in 1981 because the dangerous tectonic plates and heavy rainfall made it an incredibly dangerous place to be.

The water in Garibaldi Lake is equivalent to ten times the amount of tap water Americans drink every day.

Calculations:

Interesting note: while trying to find the equivalent, I discovered MYSELF that when Mount Vesuvius erupted in Pompeii in 79 AD, it erupted close to 1.35 x 10^10 tons of debris, which is a lot. About ten times more than Garibaldi. Source.

Latex coding does not support exponents larger than 10. So I apologize for the not-so-fancy format.

Sources:

https://en.wikipedia.org/wiki/Garibaldi_Lake

https://en.wikipedia.org/wiki/The_Barrier

http://www.baycountyfl.gov/water/facts.php