Abstract algebra has become both a favorite hobby and the bane of my existence. The nuances and caveats are so grand, they swallow me up like sink holes. It's difficult to appreciate the great landscape of the subject from deep inside the chasm of a single theorem. The detailed elegance of every piece of the algebraic machine takes my breath away. But putting those pieces together... well that's a different story.
So far, I have struggled to tie the subject together. I puzzle over problems for days, only to find the solution to be, of course, simple and perfect and easy. Slowly, very slowly, these beautiful theorems and quaint definitions are weaving together. The other students in the class ask intelligent questions and correct small errors in the teacher's definitions. They nod and smile and assimilate the ideas quickly and gracefully. I write every word the prof says and pray that later on it will make sense. This course is the biggest mental challenge I have undertaken, and I am sitting at the back of the pack.
My primary goal is not to achieve a high score or out-compete my classmates (although an A would be nice.) This subject is so beautiful and every detail is so precise, that it would be immensely satisfying to open my eyes wide and behold the magnificence of the theory. That's what I want. Not the grade, but that satisfaction. The solution to the grand puzzle. The insight. Thankfully, this stuff is much more important than my big ego.
Friday, September 21, 2012
Friday, September 14, 2012
Adventures in Education: New Beginnings at UVic
A degree wasn't enough. My week now consists of commuting back and forth to Victoria to take undergraduate math and computer science courses. If all goes well, next September I'll be in a Masters program. If not, I get to spend eight months learning undergraduate mathematics from geniuses. Win-win.
I learned very quickly that VIU (my home institution) and UVic (my new school) have slightly different approaches to teaching math. At VIU, in-class solutions are often quite detailed. By contrast, yesterday a prof wrote a solution as follows: "Proof: Induction." This is analogous a recipe for chocolate chip cookies that reads "Method: Bake." The student is still expected to understand all of the details of the proof. A benefit to this method: I must keep up with the reading. The drawback: I must keep up with the reading. Independent study skills? Yes, I think I'll need those. This semester presents an opportunity for substantial personal growth.
When I arrived in my first Abstract Algebra II* class, the front row was full and the desks had been pushed forward as far as they could possibly go. Strange, I thought, since there were only seven students in the class. The keeners in the front knew something I didn't. The prof came in with a few notes scrawled on the back of a receipt and a used letter envelope, and then plunged into an amazing lecture. At least, the parts I could hear were amazing. He was quiet. Mental note: get a seat at the front for every lecture from this point forward. Anyways, this guy knew every detail about the theory he was describing, as if it was his first language. Amazing.
As I adjust to the expectations of a new department, my emotions transition between excitement, intimidation, fear, and hope. I remind myself of these words of wisdom from a VIU math instructor (very, very paraphrased):
In math, it's not about smart you are. Success is the result of being able to keep your head in the game. Get immersed in a problem. Do your best. Sometimes you don't get results quickly. It happens to everyone. It's not your intelligence that will define you; it is far more important to be able to find your way back into a groove when you are lost. We do this because we love it. You have to love it to be good at it, because you have to have the patience to stick with a tricky problem, even if you aren't getting anywhere quickly. If you really want to, and if you are persistent, you will do well.
I'll try my best.
*Abstract algebra is a subject about algebra of objects that may or may not be numbers. These objects could be all of the numbers, certain types of numbers, remainders, symmetries, systems of equations, the arrangements of sets of objects, or something else entirely. It's easier than it sounds.
I learned very quickly that VIU (my home institution) and UVic (my new school) have slightly different approaches to teaching math. At VIU, in-class solutions are often quite detailed. By contrast, yesterday a prof wrote a solution as follows: "Proof: Induction." This is analogous a recipe for chocolate chip cookies that reads "Method: Bake." The student is still expected to understand all of the details of the proof. A benefit to this method: I must keep up with the reading. The drawback: I must keep up with the reading. Independent study skills? Yes, I think I'll need those. This semester presents an opportunity for substantial personal growth.
When I arrived in my first Abstract Algebra II* class, the front row was full and the desks had been pushed forward as far as they could possibly go. Strange, I thought, since there were only seven students in the class. The keeners in the front knew something I didn't. The prof came in with a few notes scrawled on the back of a receipt and a used letter envelope, and then plunged into an amazing lecture. At least, the parts I could hear were amazing. He was quiet. Mental note: get a seat at the front for every lecture from this point forward. Anyways, this guy knew every detail about the theory he was describing, as if it was his first language. Amazing.
As I adjust to the expectations of a new department, my emotions transition between excitement, intimidation, fear, and hope. I remind myself of these words of wisdom from a VIU math instructor (very, very paraphrased):
In math, it's not about smart you are. Success is the result of being able to keep your head in the game. Get immersed in a problem. Do your best. Sometimes you don't get results quickly. It happens to everyone. It's not your intelligence that will define you; it is far more important to be able to find your way back into a groove when you are lost. We do this because we love it. You have to love it to be good at it, because you have to have the patience to stick with a tricky problem, even if you aren't getting anywhere quickly. If you really want to, and if you are persistent, you will do well.
I'll try my best.
*Abstract algebra is a subject about algebra of objects that may or may not be numbers. These objects could be all of the numbers, certain types of numbers, remainders, symmetries, systems of equations, the arrangements of sets of objects, or something else entirely. It's easier than it sounds.
Monday, September 3, 2012
The best things about September
Cool evenings.
Cool mornings.
Hot coffee.
Cinnamon in coffee.
Cinnamon in hot apple juice.
Cinnamon in everything.
Scarves.
New jeans.
Sunny afternoons.
Walks in the forest.
Walks by the beach.
Camping at the beach.
Homework at the beach. (Yes, homework!)
Fresh school supplies.
New courses.
High expectations.
High work ethic.
Classmate reunions.
Potluck parties.
Curried zucchini and apple soup.
Comfort food.
Warm hearts.
Cool evenings.
Cool mornings.
Hot coffee.
Cinnamon in coffee.
Cinnamon in hot apple juice.
Cinnamon in everything.
Scarves.
New jeans.
Sunny afternoons.
Walks in the forest.
Walks by the beach.
Camping at the beach.
Homework at the beach. (Yes, homework!)
Fresh school supplies.
New courses.
High expectations.
High work ethic.
Classmate reunions.
Potluck parties.
Curried zucchini and apple soup.
Comfort food.
Warm hearts.
Cool evenings.
Tuesday, July 24, 2012
Counting to infinity
Riddle me this: if you have the collection of all the positive integers (1, 2, 3, 4, 5, and so on) and you remove all of the odd numbers, is the remaining set of even numbers (2, 4, 6, 8, and so on) smaller than your old set?
I've spent the last three weeks or so thinking hard about infinity. It's a required portion of one of my second year classes, and although we have covered the material and moved on I am still haunted by paradoxes and strange dreams. Heads up: this article may get a little technical and thought provoking.
In the 19th century, a German mathematician named Georg Cantor founded a field of math known as set theory. His new ideas were met with criticism, and he was repeatedly institutionalized for mental illness. (After studying just a few of his ideas, I don't blame him. This stuff would drive anyone mad.) However, David Hilbert welcomed Cantor's revolutionary ideas, and stated that "No one will drive us from the paradise that Cantor has created." The logic used in set theory has been used to prove remarkable results in modern mathematics.
I'd like to share some really interesting ideas with you. Let's suppose that we have an infinite set of objects and some way to count them. For example, if our collection was the set of positive integers (1, 2, 3, 4, 5, and so on), we could agree that there are infinitely many and that we could count them. We would say that this set was countably infinite. If we included 0 and the negative integers as well, we could still count the set. (0, -1, 1, -2, 2, -3, 3, -4, 4, and so on). As long as the set can be put in some sort of list, all elements are guaranteed to appear in the list, and the list goes on forever, then we consider the set to be countably infinite. (Note: some sets of numbers, such as the real numbers, cannot be ordered in such a list. We call these sets uncountable, but we will not deal with these sets here.)
We can find the size of a finite set by counting the number of objects. For example, the set {a, b, c, d, e} has five elements in it, so it has size 5. But what size does an infinite set have? Can there be different sizes of infinities? What does "infinite size" even mean?
Cantor suggested that we could consider two sets to have the same size if we could pair up each element from the first set with a unique element from the second. If each element had a partner and no elements were left over, then the sets are the same size. So if you have a crowd of people boarding a train and each person takes a seat and all seats are taken, then the number of seats is equal to the number of people. Now imagine that the train has infinitely many seats and the line of people is infinitely long. Cantor supposed that we could use the same logic on infinite sets; if there is exactly one seat per person then there are the same number of seats as there are people. (Author's note: I have not yet accepted the use of this logic on infinite sets, but I'll share some results with you anyways.)
Let's look back at our original question. Consider the set of positive integers {1, 2, 3, ...}. Suppose we removed all of the odd numbers from this list. We would be left with the evens {2, 4, 6, ...}. If you take any number from the first list and multiply it by 2, you will get a number from the second list. Also, each integer when multiplied by two will produce a unique even integer. Furthermore, for every even number, you can divide it by two to get a number from the first list. We are pairing up our lists like so:
{1, 2, 3, 4, 5, ...}
{2, 4, 6, 8, 10, ...} => (1,2) (2,4) (3,6) (4,8) (5,10),...
We pair each element from the first set with a unique element from the second set, and no elements are left over. By Cantor's logic, these sets are the same size.The number of positive integers is equal to the number of positive even integers. Oh dear.
This means that you can take a countably infinite set, remove infinitely many objects, and the set will be the same size. In fact, you could add a countably infinite number of objects and the set would be the same size. You could add a countably infinite number of objects a countably infinite number of times, and the set would still be the same size. Holy paradox, Batman!
Some people are not bothered by this result. After all, infinity is infinity, right? Doesn't it make perfect sense that infinity as the same size as infinity? Well, not quite. It gets weirder. I'll leave you with one more strange thought.
It turns out (for reasons that I will not address in this post) that there are different sizes of infinity. Some infinities are bigger than others. For example, there are more numbers between 0 and 1 (including decimals like 0.5, 0.111234, and 0.101010...) than there are integers (0, -1, 1, -2, 2, etc.)
Stay tuned for more insomnia fueled posts about infinity. I'd like to address my issues with Cantor's logic, sets that should be the same size but aren't, and Hilbert's infinite hotel. Check your intuition at the door. It will serve no purpose when discussing infinity.
I've spent the last three weeks or so thinking hard about infinity. It's a required portion of one of my second year classes, and although we have covered the material and moved on I am still haunted by paradoxes and strange dreams. Heads up: this article may get a little technical and thought provoking.
In the 19th century, a German mathematician named Georg Cantor founded a field of math known as set theory. His new ideas were met with criticism, and he was repeatedly institutionalized for mental illness. (After studying just a few of his ideas, I don't blame him. This stuff would drive anyone mad.) However, David Hilbert welcomed Cantor's revolutionary ideas, and stated that "No one will drive us from the paradise that Cantor has created." The logic used in set theory has been used to prove remarkable results in modern mathematics.
I'd like to share some really interesting ideas with you. Let's suppose that we have an infinite set of objects and some way to count them. For example, if our collection was the set of positive integers (1, 2, 3, 4, 5, and so on), we could agree that there are infinitely many and that we could count them. We would say that this set was countably infinite. If we included 0 and the negative integers as well, we could still count the set. (0, -1, 1, -2, 2, -3, 3, -4, 4, and so on). As long as the set can be put in some sort of list, all elements are guaranteed to appear in the list, and the list goes on forever, then we consider the set to be countably infinite. (Note: some sets of numbers, such as the real numbers, cannot be ordered in such a list. We call these sets uncountable, but we will not deal with these sets here.)
We can find the size of a finite set by counting the number of objects. For example, the set {a, b, c, d, e} has five elements in it, so it has size 5. But what size does an infinite set have? Can there be different sizes of infinities? What does "infinite size" even mean?
Cantor suggested that we could consider two sets to have the same size if we could pair up each element from the first set with a unique element from the second. If each element had a partner and no elements were left over, then the sets are the same size. So if you have a crowd of people boarding a train and each person takes a seat and all seats are taken, then the number of seats is equal to the number of people. Now imagine that the train has infinitely many seats and the line of people is infinitely long. Cantor supposed that we could use the same logic on infinite sets; if there is exactly one seat per person then there are the same number of seats as there are people. (Author's note: I have not yet accepted the use of this logic on infinite sets, but I'll share some results with you anyways.)
Let's look back at our original question. Consider the set of positive integers {1, 2, 3, ...}. Suppose we removed all of the odd numbers from this list. We would be left with the evens {2, 4, 6, ...}. If you take any number from the first list and multiply it by 2, you will get a number from the second list. Also, each integer when multiplied by two will produce a unique even integer. Furthermore, for every even number, you can divide it by two to get a number from the first list. We are pairing up our lists like so:
{1, 2, 3, 4, 5, ...}
{2, 4, 6, 8, 10, ...} => (1,2) (2,4) (3,6) (4,8) (5,10),...
We pair each element from the first set with a unique element from the second set, and no elements are left over. By Cantor's logic, these sets are the same size.The number of positive integers is equal to the number of positive even integers. Oh dear.
This means that you can take a countably infinite set, remove infinitely many objects, and the set will be the same size. In fact, you could add a countably infinite number of objects and the set would be the same size. You could add a countably infinite number of objects a countably infinite number of times, and the set would still be the same size. Holy paradox, Batman!
Some people are not bothered by this result. After all, infinity is infinity, right? Doesn't it make perfect sense that infinity as the same size as infinity? Well, not quite. It gets weirder. I'll leave you with one more strange thought.
It turns out (for reasons that I will not address in this post) that there are different sizes of infinity. Some infinities are bigger than others. For example, there are more numbers between 0 and 1 (including decimals like 0.5, 0.111234, and 0.101010...) than there are integers (0, -1, 1, -2, 2, etc.)
Stay tuned for more insomnia fueled posts about infinity. I'd like to address my issues with Cantor's logic, sets that should be the same size but aren't, and Hilbert's infinite hotel. Check your intuition at the door. It will serve no purpose when discussing infinity.
Sunday, June 10, 2012
Rain on a Plastic Skylight
It's early. I've left my fellow adventurers to snooze, and gone outside to blog in an early morning shower. The soundtrack this morning features melodious bird song against the rhythm of raindrops on tin roofs and plastic skylights: an orchestra without a key or time signature.
Last night, my coworker and I walked along a beach on the south side of Malcolm Island at dusk. The surface of the bay was roughened by the weather, and had been etched with lines from boats and winds and physics. The atmosphere moved past us excitedly, but the mountains and the skyline seemed to hold their breath in anticipation of nightfall. A family of ducks, oblivious to the impending darkness, swam and dove in synchrony. Meanwhile, birds and insects softened the harsh silence of paradise after sunset.
We didn't speak much, but in my mind I placed boundaries around the universe of my memories. I realized (as I often realize but choose to forget) that life, although composed of an infinity of instants, is a short thing with a definite beginning and a firm end. There are only so many evenings in which we can feel the sharp bite of a cold ocean around our ankles or the playful hands of a western breeze across our faces. Furthermore, personal philosophy and preference are dynamic; there may only be so many evenings in which we desire these experiences.
How many of those who would risk our coastlines for their bottom lines have stuck their toes in the ocean at dusk? How many? Are they unaware of the majesty of the waterline, or is it simply less impressive than a large number in a spreadsheet?
As I sip the last of my coffee, I listen to the sound of the rain. When I leave, I'll remember the sound of this cold, foggy morning: bird song and rain on a plastic skylight.
Last night, my coworker and I walked along a beach on the south side of Malcolm Island at dusk. The surface of the bay was roughened by the weather, and had been etched with lines from boats and winds and physics. The atmosphere moved past us excitedly, but the mountains and the skyline seemed to hold their breath in anticipation of nightfall. A family of ducks, oblivious to the impending darkness, swam and dove in synchrony. Meanwhile, birds and insects softened the harsh silence of paradise after sunset.
We didn't speak much, but in my mind I placed boundaries around the universe of my memories. I realized (as I often realize but choose to forget) that life, although composed of an infinity of instants, is a short thing with a definite beginning and a firm end. There are only so many evenings in which we can feel the sharp bite of a cold ocean around our ankles or the playful hands of a western breeze across our faces. Furthermore, personal philosophy and preference are dynamic; there may only be so many evenings in which we desire these experiences.
How many of those who would risk our coastlines for their bottom lines have stuck their toes in the ocean at dusk? How many? Are they unaware of the majesty of the waterline, or is it simply less impressive than a large number in a spreadsheet?
As I sip the last of my coffee, I listen to the sound of the rain. When I leave, I'll remember the sound of this cold, foggy morning: bird song and rain on a plastic skylight.
Tuesday, January 24, 2012
Pho on the Go
I live across the street from a Vietnamese restaurant. Every month or so, my man and I will arrive home from work too drained to cook. We put our jackets on and dodge the traffic Frogger-style until we arrive at the doors gates of the small restaurant heaven. Our order is always the same: number 21. Pho tai sach. Beef noodle soup. Pho at its finest.
I sometimes joke about opening up a Vietnamese take-out restaurant and calling it "Pho on the Go." The fatal flaw with this plan is that I don't cook any Vietnamese food.
Last week, I saw a recipe for beef pho in Canadian Living magazine. I decided that I will attempt the dish tonight (if I fail, I will run across the street, order some take out, bring it home, pour it into bowls, ditch the styrofoam containers, and take credit when the man arrives home.) However, I have some reservations about using a Vietnamese recipe from a "Canadian" magazine. What if it is not authentic?
I could use your help. Have you ever made Vietnamese beef noodle soup at home? Do you have any tips or suggestions to enhance the recipe?
Thanks! In the words of Joey, "Mmm, noodle soup."
I sometimes joke about opening up a Vietnamese take-out restaurant and calling it "Pho on the Go." The fatal flaw with this plan is that I don't cook any Vietnamese food.
Last week, I saw a recipe for beef pho in Canadian Living magazine. I decided that I will attempt the dish tonight (if I fail, I will run across the street, order some take out, bring it home, pour it into bowls, ditch the styrofoam containers, and take credit when the man arrives home.) However, I have some reservations about using a Vietnamese recipe from a "Canadian" magazine. What if it is not authentic?
I could use your help. Have you ever made Vietnamese beef noodle soup at home? Do you have any tips or suggestions to enhance the recipe?
Thanks! In the words of Joey, "Mmm, noodle soup."
Tuesday, January 17, 2012
Letters about a Pipeline
Last week, Joe Oliver, Canadian Minister of Natural Resources, published an open letter to Canadians regarding the public proceedings over the Enbridge Northern Gateway pipeline project. Many media outlets picked up on the letter and made it the topic of their news programs. Read the letter for yourself here. Not long afterward, Elizabeth May published a response, which you can read here.
Hold on to your hats, folks. This debate is going to be one for the history books, and it's just heating up.
Hold on to your hats, folks. This debate is going to be one for the history books, and it's just heating up.
Subscribe to:
Posts (Atom)