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.
Monday, September 3, 2012
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.
Saturday, January 14, 2012
One Morning in the North
The recent news coverage of the proposed Northern Gateway pipeline project has reminded me of how much I love the BC north coast. Last year, some friends and I took a phenomenal cruise through the inside passage to Prince Rupert, and then carried on to Haida Gwaii. We disembarked from the ferry in Skidegate (at 6:00 am!) and drove the short distance to Queen Charlotte City before parking at the beach to wait for the town to awaken. I sat on a damp wooden picnic table and counted a dozen eagles while they enjoyed their breakfast in the soft morning sunlight. The air was brisk and invigorating, while by contrast my coffee was hot and soothing. The breeze carried an essence that was, well, different from my urban home. Finally, I understood the phrase "fresh air." It was in that moment, a moment that I will never forget, that I re-prioritized and re-defined "the best things in life."
I wish that I had words that could do justice to the experience that I had that morning. Fortunately, I have some pictures.
Some of these were taken by myself, some by Ginny Casey, and the rest by Jeremy Krogh. I owe many thanks to VIU's ACER project for the opportunity to take this trip.
I wish that I had words that could do justice to the experience that I had that morning. Fortunately, I have some pictures.
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Some of these were taken by myself, some by Ginny Casey, and the rest by Jeremy Krogh. I owe many thanks to VIU's ACER project for the opportunity to take this trip.
Wednesday, January 11, 2012
Joe Oliver vs Evan Solomon
There has been recent political debate regarding the proposed Enbridge Northern Gateway pipeline. This pipeline would carry oil from the Alberta tar sands to Kitimat, BC. This oil could then be very easily traded with countries in Asia Pacific. Canada could stand to gain strong economic growth and trade diversification from the project, which could lead to improved social services such as education and healthcare. However, the risks include hazards to the environmental health of the BC coast, as well as changes to the way-of-life of the Aboriginal people who live there. Additionally, by extracting fossil fuels, Canada is contributing to climate change which has a large portfolio of economic and environmental risks such as increasing global surface temperatures, increased severity of storms, melting of alpine and polar ice, and decreased biodiversity.
In the video below, CBC's Evan Solomon interviews Joe Oliver, the Canadian Minister of Natural Resources. Solomon asks Oliver some very tough but fair questions regarding the public relations tactics used by the government who strongly support the pipeline project. Take a look at the video here.
Although I have formed an opinion regarding this debate, I am not going to state it on here. The point of this article is not to persuade you to agree with my political views. I just liked the types of questions that Evan Solomon asked of Joe Oliver in this interview. They were tough, but they were also clear, fair, and extremely relevant. It is essential to ask these types of questions of those on both sides of this debate in order to hold them accountable to their statements. This helps to preserve the integrity of the information that we, the public, receive from the media.
In the video below, CBC's Evan Solomon interviews Joe Oliver, the Canadian Minister of Natural Resources. Solomon asks Oliver some very tough but fair questions regarding the public relations tactics used by the government who strongly support the pipeline project. Take a look at the video here.
Although I have formed an opinion regarding this debate, I am not going to state it on here. The point of this article is not to persuade you to agree with my political views. I just liked the types of questions that Evan Solomon asked of Joe Oliver in this interview. They were tough, but they were also clear, fair, and extremely relevant. It is essential to ask these types of questions of those on both sides of this debate in order to hold them accountable to their statements. This helps to preserve the integrity of the information that we, the public, receive from the media.
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