Monday, December 14, 2015

Math What?

Nature of Math

The saga continues! Is math a science? Is math discovered or invented? I thought I had a solid answer on the former question, but Alas! Mathematics Strikes Back!

The debate on these two questions were brought front and center in my History of Math class. The class was divided on this issue with a large portion of the class believing math was a science. A small portion believed math is completely separate from science. A small few were still agnostic on the issue. The class was less divided over whether math is discovered vs invented. So, I'll start there.

Math Discovered or Invented?
Discovered. Definitely. As our class has discussed and from what Pickover has discussed in The Math Book (blog post), there are math milestones that were worked out by different people simultaneously, but independently from each other (1). Credit for Calculus has been given to both Isaac Newton and Gottfried Wilhelm Leibniz (1). Another example was when hyperbolic geometry was being worked out by three different mathematicians at the same time: Janos Bolyai, Nikolai Lobachevskii, and Carl Friedrich Gauss (4). If math was invented then it would just be a random collection of concepts that were pulled out of thin air by the imaginations of great mathematicians, and it would be hard to make the case that these simultaneous creations by different mathematicians occurred by coincidence. How could they just happen to have the (almost) exact same creation at the same time? That would be like Leonardo DaVinci and Michelangelo painting the exact same Mona Lisa at the same time. Don't Even.  

Furthermore, mathematics is far too ordered, structured, and interconnected to be randomly invented. In BBC Horizon's documentary, The Last Theorem, a connection was established between three seemingly unrelated math topics: elliptic curves, modular functions, and Fermat's Last Theorem (5). To prove Fermat's Last Theorem, Andrew Wiles had to prove that modular forms and elliptic curves are the same, because when someone believed that Fermat's Last Theorem was false, those equations looked like an elliptic curve that wasn't modular (5). In the end, Andrew Wiles proved Fermat's Last Theorem was true by showing elliptic curves and modular forms were the same (5).

The reason behind the order, structure and interconnectedness in mathematics is logic. Jordan Ellenberg in his book, How Not to Be Wrong: The Power of Mathematical, describes math as using common sense, and applying rational thinking or logic to problems (the problems don't have to be mathematical) (4). Eugenia Cheng, in a presentation at Grand Valley State University, explained that "mathematics is the logical study of logical things" (6). Logic is the underlying pinnings that unites all of mathematics together.

A part of mathematics logical structure are axioms. Ellenberg describes axioms as self-evident rules that form a system of mathematics (4). They are used to manipulate elements in the system to prove logical statements about the system, and these logical statements are based on truth tables, the language of logic. When Janos Balyai decided to remove the Parallel Postulate from Euclid's Axioms, he worked out a new geometry - hyperbolic geometry (4). He found that the parallel postulate was neither contradictory, which would make Euclid's geometry false, nor mandatory, which would make it the only possible geometry (4). Hyperbolic geometry is just another geometric world in addition to Euclid's geometry or, more broadly speaking, mathematical space in mathematics (3). When another world was discovered with a different set of axioms, the nature of axioms changed (3). Axioms are more like hypotheses that lay the framework for the world one is in (3).  But even though there is an important difference between hyperbolic and Euclidean geometry, there is overlap because there is overlap of axioms between the two worlds and the logic that is used to prove statements in all of mathematics is the same. Another example is with integers and rings. Integers are a specific type of ring, and depending on which axioms are used, proven statements (theorems) in the integers can be used or slightly modified to prove similar results in rings.

Furthermore, when I think of something that is invented, I think of art and games. When children come up with a new game, they create their own rules. Most of the time, these rules aren't logical. Take Monopoly for example. Why do players get free money whenever they pass Go? That doesn't make sense or even apply to the real world. It's a whimsical rule, and mathematics are not governed by whimsical rules.

Mathematics are governed by logical rules, specifically truth tables and axioms. Axioms, which are the foundation of proof and mathematical theorems, can be compared to rules of a game. With games, rules can be changed and the game can still be played. To continue with the Monopoly example, if the rule of getting money when passing Go disappears, all the other elements of the game remain. But in math, the rules, or axioms can't just change or disappear and leave the theorem or concept fundamentally different. If they do, all the mathematical concepts that relied on those axioms and theorems crumbles (4 and 5). This is the key difference between games (rules without logic) and math (rules with logic). Axioms and truth tables, the logical rules, makes math a discovered endeavor, and rules without logic are of human invention.

Lastly, mathematics is not a solely human invention. If math is truly a human invention, then only humans could do it, understand it, and come up with it. But primates and other animals can be trained to count, while a type of Saharan Ant knows the number of steps and the lengths of their stride to find their way back to their nest (1). But as someone in the class mentioned, if math is discovered, then it has to be found in nature, which would make it a science. And that brings us to the next question.

Is Math a Science?
I was a firm believer that math is not a science. Now, there is room for doubt. I have written a detailed blog post on this topic, so I will only give a brief description of what I already discussed. The pursuit of science is to discover how natural processes work, while the pursuit of math is discovering relationships with numbers. Math is found and can be applied to nature. In fact discoveries in math started with the ancient Greeks describing our 3-D world with geometry (3). But even though math can be found or be used to describe nature, there are abstract ideas that transcends the physical world. The example I gave before was a one-dimensional line expanding to infinity. Furthermore, math's capability to describe our natural world with accuracy does not make it the reason why it works (2). Just because math is accurate in modeling natural phenomenon, it does not mean it's the explanation behind the process (2). There could be a completely different reason that makes the process act that way (2). Accuracy does not necessarily make something true (2).

But then I came across the Mathematical Universe Hypothesis. Max Tegmark is the man who came up with the hypothesis (1). According to Clifford Pickover's explanation of the hypothesis, "...our physical reality is a mathematical structure and that our universe is not just described by mathematics - it is mathematics" (1). If this hypothesis is proven to be true, then mathematics is not just something that can be applied and describe our natural world with accuracy - it is the foundation, the reason, and the process behind how everything in our natural world works. And that would make math a science....and it would make science provable with mathematical proof. But at this point, the Mathematical Universe Hypothesis is just a hypothesis, and the divide between science and math, nature and number, clear. And I can continue believing, a little shakily, that math is not a science.

(1) Pickover, Clifford A. The Math Book. Sterling Publishing Co., Inc. New York, NY. 2009.

(2) Wigner, Eugene. "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." College Course 9. 

(3) Space (mathematics). Wikipedia. 

(4) Ellenberg, Jordan. How Not to Be Wrong: The Power of Mathematical Thinking. The Penguin Press. New York, NY. 2014. 

(5) Simon Singh and John Lynch. Fermat's Last Theorem Documentary. BBC Horizon. 1996. 

(6) Eugenia Cheng's Presentation at Grand Valley State University. Quotation from MathHombre's Tumblr page:

Monday, December 7, 2015

The Spider and the Fly: An Exploration in 3-D Geometry and Graphing

Doing Math

The NRICH asks an interesting math question at this Link. Let's say you're a spider, and you see a fly across a room. What would be the shortest, and therefore fastest, route you would have to crawl to catch the fly?

The spider is in a room, so it's 3-D. The usual 2-D graph of x and y, rise over run, doesn't setup the problem accurately. There is width, height, and length: an x, y, and z coordinates respectively. But even though the spider and the fly lives in a 3-D world, we can break down the problem into a 2-D one, since the spider can only crawl on the wall in a 2-D way - side-to-side (x), up-and-down (y).

We are given that the room is 4 meters wide, 2.5 meters high, and 5 meters long. The spider is in the middle of the smallest wall on one side of the room. The fly is on the opposite, smallest wall and it's position is 1.5 meters high and 0.5 meters away from the adjacent wall next to the window. Below is how I visualized the room, and how I labeled the coordinates. The room is not drawn to scale.

Then, I thought how to move the spider from wall to wall to get to the fly in the shortest distance. If the spider goes towards the back wall, then the spider would have to cross the window to get to the fly, which is 3.5 m away from the fly (4-0.5 = 3.5). So, the better way for the spider to travel is to move towards the front wall. Next, I imagined the fly on the spider's wall in it's (0.5, 1.5) position. Eventually, the spider has to get to the height of 1.5 meters, so I figured it should do so while traveling to the edge of the front wall. That way, the spider will only need to crawl in a straight line to get to the fly. Below is the route indicated by the arrows.
Since the spider is crawling a straight line on the front wall and the opposite wall, then the spider will be crawling a distance of 0.5 + 5 = 5.5 meters for those two walls. I needed to use the distance formula to calculate the diagonal distance the spider crawls to get to the edge of the front wall at the height of 1.5 meters.
Distance Formula

The coordinate of the spot I want the spider to crawl to is (0, 1.5, 5), since x=0 is the edge of the front wall, 1.5 is the height of the fly, and 5 is the z coordinate for end wall. I don't need to use the z-coordinate, 5, because the spider is moving in 2-D, from side-to-side and up-and-down. My calculation using the distance formula is below.
This distance is approximately 2.02 meters. Therefore, the total distance traveled for the spider to reach the fly is 0.5+5+2.02= 7.52 meters. Then the problem asks another question: If the fly starts moving towards the floor (I'm assuming the fly  moving straight down), then at what point should the spider change it's route to catch the fly?

Not knowing when the fly starts moving downwards, and where the spider is on it's route, I decided to redo the problem with the fly already at the bottom of the wall. Then I thought I would do the problem in the same way as above. But then I had an Eureka! moment. I was already breaking down spider's movements or the walls of the room into 2-D. The two end walls had (x, y) coordinates and the front and back walls had (z, y) coordinates. So, I decided to remove the back wall, and flatten the 3 remaining walls into one large wall. Then, the spider just needs to travel diagonally to get to the fly, instead of changing direction to go straight once the spider reaches an edge. Above is how I visualized the new problem (the fly is moving straight downward).

Now, we have just have (x,y) coordinates to deal with. The fly's new coordinate is (9.5, 0) and the spider's new coordinate is (2, 1.25). I used the distance formula again to calculate the distance the spider will travel on the diagonal line. Below is the calculation.

The answer is approximately 10.68 meters. Since, I thought of a new way to look at the problem I decided to re-calculate the distance of the first problem using this method. So, the spider will, again, move diagonally only, and not change direction to go straight. Below is the new visual for the first problem.
So, the spider has the coordinates of (2, 1.25) and the fly has the coordinates of (9.5, 1.5). The calculation for distance using the distance formula is below.
The answer is approximately 10.61 meters. So, it would seem that the best route was my first one. Going diagonally all the way across adds distance instead of shortening it. So, going diagonally over a shorter distance and then going in a straight line to the final destination point is the best route. But the fastest route, instead of going around the room, is to go through the room. So, what if, instead of a spider that has to crawl over a 2-D space (side-to-side/up-and-down), there is a dragonfly that can fly through a 3-D one (side-to-side/up-and-down/forward-and-backward)?

3-D Extension
I was very interested in exploring 3-D geometry, and the first thing I wanted to learn was how to obtain a line in 3-D space. So, I watched David Butler's "Example: Finding the equation of the line in 3D through two points" YouTube Video from Maths Learning Centre UofA channel, and applied it to this problem. My work is below.

Then, I wanted to find the distance the dragonfly would have to travel to reach the fly. My professor explained I could use the distance formula with the z-coordinates added. My work is below:

The square root is about 5.226. So, the dragonfly would have to fly about 5.226 meters to get to the fly.

A great problem, like this one, inspires the imagination. The original problem, even though the setup is in 3-D, asks a 2-D question. The concept that needs to be understood is that a spider can only move in two ways on the wall or plane. The problem can be thought of and solved in different ways, as shown above. So, the original problem has lots to explore, but it also offers more. As soon as I read the problem, my mind went straight to "But what if the spider could fly?", turning the 2-D question into a 3-D one, and then I sought out ways to answer my question. Curiosity is an indication of interest and readiness to learn something new. Offering opportunities for students to imagine "what if" is the best way to engage and encourage their learning. It will motivate them to learn on their own, fostering a sense of responsibility for their education. And as a teacher, that is the most important skill I can encourage in my students.

Figures created using GeoGebra and Microsoft Word. I do not have any affiliation with NRICH, GeoGebra, or David Butler, and none have endorsed my work. I am a college student.