I'm not sure whether the thought I am thinking now is well formed enough for me to explain it, but I'm going to give it a try. I just sat down with a nice cup of jasmine tea and was about to do some proofreading for work when, as is so oft the case, I decided to briefly take a glance at my Facebook.
Now, I have had an off-and-on relationship with Facebook for years. I was one of the first people in my age group back in middle school to get a Facebook account, at a time when nearly all of the people on the site were in college -- but back then, I was taking a lot of dance classes and performances with the Allegheny College Dance Department, so many of my friends were that age. Then, when I did my year abroad in high school, it was a great way to keep in touch with folks, and to share parts of my experience with large groups of my friends or acquaintances at once. Then, after I got back, there was about a year in college when I simply deactivated my account, not liking the "timesuck" that it was.
But now, after this and that and oh so many awkward moments of "You don't have a Facebook? I guess I can email you, but I never check my email..." and simply wanting to see the pictures of my friends on vacation and the children of extended family - well, I'm back on it again. But I've noticed a new trend in my news feed of late (that is to say, the past year or so) that I find troubling.
First off, a few caveats. 1. The things I am about to list may be specific to those people I happen to be "friends" with on Facebook. 2. I may be extremely sensitive and pick up on things no one else would think were an issue. 3. Well, I'm sure there are more. I'm just going to tell you what I think anyway.
There seems to be a big wave of negativity these days. I don't just mean that bad things are happening in the world and people are upset about them, but rather there is a desire to post about things that make people upset, hoping to have their outrage validated by others. For example, there are often internet articles that are fads; they go viral and zoom around from one person to another over the course of a few days and get a lot of "buzz". Now, most mainstream "click me" articles, even if they're supposed to "make you cry" or assure you "you won't believe what happens next" - well, they are frequently problematic. Whether it's racist, sexist, classist, whichever-ist - there are usually some problematic undertones. Sometimes they are obvious, sometimes you really have to dig for them -- and someone will. And I think there's a difference between commenting when someone else posts said article and saying (KINDLY) "I think this article has a few issues because...." and between posting that article (as many of my Facebook seem to acquaintances do) simply out of bile. Just so they can say "Take a look at this disgusting exhibit of the patriarchy" - or whichever the enemy du jour is.
I don't know why this bothers me so much - but I find that every time I read over my news feed, I end up unhappy. And I feel there's a difference between responding truthfully and critically to other people and the world, and bringing up something you are angry about merely out of spite, encouraging more anger, hoping for that validation... I just feel it multiplies. It makes me not want to look at Facebook anymore, but maybe that's not such a bad thing.
A few more caveats at the end - I am not advocating pretending everything is jolly and ignoring all the bad things in the world - absolutely not. Also, it's quite possible that after having attended an extremely liberal women's college in the Bay Area, I have an above average number of vocal, angry activists on my news feed.
If you are at all interested in a discussion of similar topics in the form of a podcast, I highly recommend going to this webpage and clicking on the podcast called 'Our Computers, Ourselves'. It's a great listen - talks of technology and how we feel about it and what we do to it vs. what it does to us. Go for it. :)
Friday, May 15, 2015
Friday, April 17, 2015
Spring 2015
Here are a few pictures of the lovely spring week we've been having here - taken in downtown Mainz as well as the University's botanic garden.
Sunday, April 12, 2015
Shadow of Habits Past
I made it through finals for this semester and through the orientation for a new part-time job I have for the next semester. Whew!
And recently, I've been thinking a lot about routines and habits. I've been thinking about the positives and the negatives of each. Habits can be so small and also so big. I mean, there's the habit of checking Facebook every morning or your email, or it could also be the "habit" or tradition of celebrating birthdays and the new year. (More about this thought in a later post!)
Routines and habits have a lot of good aspects. When I'm studying or have a project to work on, nothing makes me productive like a good schedule or routine. For me, that particular set of circumstances is a cup of a particular kind of tea, classical music, and the ritual of lighting a small candle on my desk (if I'm working at home). But then again, nothing makes me feel that distinct kind of hollowness inside like being in the rut of habits and feeling powerless to get out of them.
Just these past few days have introduced me to some habits that I thought were lost. There's that pesky old foe of nail-biting, which I have battled for years, but where it only takes the pressure of an exam or a particularly difficult piece of work to dismantle my resolve. But another habit surfaced that I hadn't witnessed in a while – but I still call it a habit because it is my reaction to a specific set of circumstances.
The circumstances: excitement/stress about an activity (frequently, but not always, travel) very early the next day, for which my presence is absolutely necessary. Now, this doesn't include early morning classes or things of that nature. It's more like transatlantic flights or, in this circumstance, picking up my sister from the airport! In these situations, I wake up at some ungodly hour, am thoroughly convinced that every clock in my apartment/room is showing the incorrect time, and proceed to try to begin that activity that is supposed to happen. This happened most memorably during my first month of Freshman year, when I was a newbie on the crew team and we had practice every day at 5 a.m. and on the eve of one particularly important practice (I believe choosing who would be able to row in an upcoming race) I attempted to go to practice at 2 in the morning (fully dressed in my workout clothes, etc.) and was only stopped by someone who lived in my hall who happened to be coming back from a party while I stumbled sleepily out of my room. She managed to convince me to go back to bed and I remembered nothing of the incident until our eyes met at dinner the next day and it all came rushing back.
Okay, so maybe that's more of a quirk than a habit – but it is a repeated set of behavior. Sometimes habits make us keep to our plans, make us productive and comfortable in our environment. But sometimes routines make us blind to everything that isn't in our sights for the day, or make us forget the individual importance of the parts of our routine – like how good that cup of coffee tastes, instead of just drinking it without thinking.
But every now and then, I feel like the wires in our heads just cross somehow and we get that breach of routine, the breaking of a pattern, and it's like a packet of potential energy gets released when something unexpected happens. Usually, it results in surprise (meeting your friend on the subway you always take to work in the morning, for example). But sometimes, when you accidentally mess up a routine, the only reaction is hilarity. Such as yesterday, when I came home from picking up Rachel from the airport, came into the apartment and put my keys down on the counter. We greeted C and made some tea, hung around and were about to go out for a walk later, and my keys were nowhere to be found, least of all on the small nail in the wall near the door where they generally hang. Since then, we have been through every coat pocket, every shoe (they sit under the coats), and every trash can. We've searched under the couch, desk, kitchen table, behind the oven – everywhere. We looked and looked and found nothing. Now today, while we were making an apple rhubarb crumble, C and I were discussing making copies of all my keys since they were nowhere to be found, and I reached up to grab an oven mitt from the small nail that protrudes from the side of our cupboard – and lo and behold, there are my keys. The action of hanging the keys on a small nail in the vicinity of the front door was carried out, but it cracks me up. Putting phones in the refrigerator or losing your glasses while they are on your face – or pushing your "glasses" up your nose while you are, in fact, wearing contact lenses. We are so evolved, and yet sometimes, so thoughtless. Or maybe it's just me. But I don't think so.
Also, it's so lovely to have my sister here.
And recently, I've been thinking a lot about routines and habits. I've been thinking about the positives and the negatives of each. Habits can be so small and also so big. I mean, there's the habit of checking Facebook every morning or your email, or it could also be the "habit" or tradition of celebrating birthdays and the new year. (More about this thought in a later post!)
Routines and habits have a lot of good aspects. When I'm studying or have a project to work on, nothing makes me productive like a good schedule or routine. For me, that particular set of circumstances is a cup of a particular kind of tea, classical music, and the ritual of lighting a small candle on my desk (if I'm working at home). But then again, nothing makes me feel that distinct kind of hollowness inside like being in the rut of habits and feeling powerless to get out of them.
Just these past few days have introduced me to some habits that I thought were lost. There's that pesky old foe of nail-biting, which I have battled for years, but where it only takes the pressure of an exam or a particularly difficult piece of work to dismantle my resolve. But another habit surfaced that I hadn't witnessed in a while – but I still call it a habit because it is my reaction to a specific set of circumstances.
The circumstances: excitement/stress about an activity (frequently, but not always, travel) very early the next day, for which my presence is absolutely necessary. Now, this doesn't include early morning classes or things of that nature. It's more like transatlantic flights or, in this circumstance, picking up my sister from the airport! In these situations, I wake up at some ungodly hour, am thoroughly convinced that every clock in my apartment/room is showing the incorrect time, and proceed to try to begin that activity that is supposed to happen. This happened most memorably during my first month of Freshman year, when I was a newbie on the crew team and we had practice every day at 5 a.m. and on the eve of one particularly important practice (I believe choosing who would be able to row in an upcoming race) I attempted to go to practice at 2 in the morning (fully dressed in my workout clothes, etc.) and was only stopped by someone who lived in my hall who happened to be coming back from a party while I stumbled sleepily out of my room. She managed to convince me to go back to bed and I remembered nothing of the incident until our eyes met at dinner the next day and it all came rushing back.
Okay, so maybe that's more of a quirk than a habit – but it is a repeated set of behavior. Sometimes habits make us keep to our plans, make us productive and comfortable in our environment. But sometimes routines make us blind to everything that isn't in our sights for the day, or make us forget the individual importance of the parts of our routine – like how good that cup of coffee tastes, instead of just drinking it without thinking.
But every now and then, I feel like the wires in our heads just cross somehow and we get that breach of routine, the breaking of a pattern, and it's like a packet of potential energy gets released when something unexpected happens. Usually, it results in surprise (meeting your friend on the subway you always take to work in the morning, for example). But sometimes, when you accidentally mess up a routine, the only reaction is hilarity. Such as yesterday, when I came home from picking up Rachel from the airport, came into the apartment and put my keys down on the counter. We greeted C and made some tea, hung around and were about to go out for a walk later, and my keys were nowhere to be found, least of all on the small nail in the wall near the door where they generally hang. Since then, we have been through every coat pocket, every shoe (they sit under the coats), and every trash can. We've searched under the couch, desk, kitchen table, behind the oven – everywhere. We looked and looked and found nothing. Now today, while we were making an apple rhubarb crumble, C and I were discussing making copies of all my keys since they were nowhere to be found, and I reached up to grab an oven mitt from the small nail that protrudes from the side of our cupboard – and lo and behold, there are my keys. The action of hanging the keys on a small nail in the vicinity of the front door was carried out, but it cracks me up. Putting phones in the refrigerator or losing your glasses while they are on your face – or pushing your "glasses" up your nose while you are, in fact, wearing contact lenses. We are so evolved, and yet sometimes, so thoughtless. Or maybe it's just me. But I don't think so.
Also, it's so lovely to have my sister here.
Wednesday, February 18, 2015
Walking across the campus.
As I mentioned in my last post, this is the "Lernphase". There are only a few courses offered at this University of 30,000 students that actually have exams during the first part of the semester. The vast majority just have a final exam or a final paper due during the semester break (it's a joke, of course, that it's called a break). But because attendance is not required in most lectures (seminars and other smaller courses, sure, but not most lectures), there's at least as many students on campus during the Lernphase if not more than in the regular semester, as all are making the pilgrimage to one of the several libraries or cafés on campus.
Everyone is in a haze of exams. Studying habits like constant tea consumption and the listening of dramatic instrumental music are contagious. Diagrams like this one are flying around Facebook:
And I'm part of the craziness. However, with only one final this time, I have a relatively calm few weeks ahead of me - just a bunch of algebra, which I'm alright with.
There are two non-exam-related things I wanted to mention today. First of all, as I got to campus today I worked for a while on the other side of the University from the math building (a more cozy library than the math one, and since I was only there for the atmosphere and not for the books, it didn't matter that there weren't any math books around). Then after lunch, I made the trek to the back of the campus and as I went, I heard snippets of different conversations, since I (for once) didn't have my headphones on:
In front of the building where most foreign language classes are held:
"Yeah, there's a lot of exchange students in that lecture..."
-
In front of the main library:
"Really? Sub-saharan? I thought that was..."
-
In the absolute center of campus near the music building:
"I have my first exam tomorrow..."
Then a trend starts:
-
In front of the main cafeteria, on the other side of which is the math building:
"Yeah, that's what I said - positiv-definit and symmetric..."
-
On the other side of that cafeteria:
"And we have to calculate Eigenvalues too, right?"
-
And finally, in the elevator of the math building.
"Of course it's obvious if you are considering it over the complex numbers. But as soon as you move to a more general ring, then...."
I just couldn't help grinning to myself. That last conversation was actually happening in English (the others I translated), with heavy non-German accents on either side (grad students, I'm assuming) and it went on to name a bunch more terms that I know vaguely from hear-say but have no real clue what they are mathematically, and just the fact that that kind of research is going on down the hall from where I'm prepping for my one little Algebra exam makes me so happy.
And finally, C and I have started a new project with our fantastic neighbor. We've decided to watch all the movies on the American Film Institute's Top 100 list. Actually, we decided to do this a while ago but we haven't managed to start until last weekend. Let me just say - 100 films is a lot of movies. A few I've seen on the list, but not all. We're starting at the bottom, so theoretically, they'll just get better and better. The list was apparently constructed based on criteria such as: awards won (i.e. Oscars or similar), popularity over time (including DVD/VHS sales as well as box office), critical recognition, historical significance, and cultural impact.
In any case, we began on Saturday night with Ben Hur, Film No. 100, and Toy Story (No. 99) on Monday. We decided to keep certain tallies with each of the films, though the same things might not be counted - for example, for Ben Hur we counted the number of beers drunk during the 3.5 hour spectacle (5) but for Toy Story, we counted cups of coffee (4). Ben Hur also got +3 Jesus points, since none of us expected to see Jesus because all we knew about the film was that there was a chariot race. I shall be sure to keep you posted about our progress there! We'll have to make hay while the sun shines, or rather watch films during our semester break, since if we watched only one every week it would take us at least a two years to get through all of this. We shall see!
Everyone is in a haze of exams. Studying habits like constant tea consumption and the listening of dramatic instrumental music are contagious. Diagrams like this one are flying around Facebook:
There are two non-exam-related things I wanted to mention today. First of all, as I got to campus today I worked for a while on the other side of the University from the math building (a more cozy library than the math one, and since I was only there for the atmosphere and not for the books, it didn't matter that there weren't any math books around). Then after lunch, I made the trek to the back of the campus and as I went, I heard snippets of different conversations, since I (for once) didn't have my headphones on:
In front of the building where most foreign language classes are held:
"Yeah, there's a lot of exchange students in that lecture..."
-
In front of the main library:
"Really? Sub-saharan? I thought that was..."
-
In the absolute center of campus near the music building:
"I have my first exam tomorrow..."
Then a trend starts:
-
In front of the main cafeteria, on the other side of which is the math building:
"Yeah, that's what I said - positiv-definit and symmetric..."
-
On the other side of that cafeteria:
"And we have to calculate Eigenvalues too, right?"
-
And finally, in the elevator of the math building.
"Of course it's obvious if you are considering it over the complex numbers. But as soon as you move to a more general ring, then...."
I just couldn't help grinning to myself. That last conversation was actually happening in English (the others I translated), with heavy non-German accents on either side (grad students, I'm assuming) and it went on to name a bunch more terms that I know vaguely from hear-say but have no real clue what they are mathematically, and just the fact that that kind of research is going on down the hall from where I'm prepping for my one little Algebra exam makes me so happy.
And finally, C and I have started a new project with our fantastic neighbor. We've decided to watch all the movies on the American Film Institute's Top 100 list. Actually, we decided to do this a while ago but we haven't managed to start until last weekend. Let me just say - 100 films is a lot of movies. A few I've seen on the list, but not all. We're starting at the bottom, so theoretically, they'll just get better and better. The list was apparently constructed based on criteria such as: awards won (i.e. Oscars or similar), popularity over time (including DVD/VHS sales as well as box office), critical recognition, historical significance, and cultural impact.
In any case, we began on Saturday night with Ben Hur, Film No. 100, and Toy Story (No. 99) on Monday. We decided to keep certain tallies with each of the films, though the same things might not be counted - for example, for Ben Hur we counted the number of beers drunk during the 3.5 hour spectacle (5) but for Toy Story, we counted cups of coffee (4). Ben Hur also got +3 Jesus points, since none of us expected to see Jesus because all we knew about the film was that there was a chariot race. I shall be sure to keep you posted about our progress there! We'll have to make hay while the sun shines, or rather watch films during our semester break, since if we watched only one every week it would take us at least a two years to get through all of this. We shall see!
Tuesday, February 3, 2015
Forgot to mention!
I realize that the article to which I will be linking in just a few sentences is already taking the internet and podcast discussions by storm, but I want to make sure that I also say a few words about it. I'm talking about the article "Not a Very P.C. Thing to Say" by Jonathan Chait. This article discusses a phenomenon that I frequently find hard to explain to folks who didn't go to a liberal women's college in the Bay Area. It's about the phenomenon of P.C. culture, particularly in academia in the US currently.
Please do read this article if you haven't already, but while you do, keep in mind one important thing: Chait is a white man (he says as much in the article). In addition, I have read several criticisms of the article as well, all mostly focusing on the lack of research into the claims Chait makes, instead covering ground mostly based on anecdotes that illustrate his point. Take all of this with a grain of salt, maybe a tasty grain of salt since it's the first time I've ever heard anything written about the phenomenon. My head has been reeling with arguments for and against the article ever since I read it and I'm about to Skype with a dear, dear friend and fellow Mills graduate tonight to talk about it some more. If you feel particularly flummoxed afterwards and want a little more discussion of it, I also encourage you to listen to a podcast from Slate.com about it. Slate (yes, notoriously liberal-slanted, bear this in mind as well as you listen) offers a great deal of interesting (and free) podcasts and the one to which I regularly listen is called the Political Gabfest. Last week's issue (the podcast title is "The 'Can You Buy a President for $889 Million?' Edition") has the three correspondents discussing three topics - the Koch brothers' enormous financial commitment to the 2016 election, the supreme court facing an upcoming decision on the death penalty as well as Chait's article. You can skip to the end to just hear the latter, but if you feel like giving the podcast a chance, I encourage you to listen to the whole thing. I've linked to the page where you can stream the podcast, or you can find it in iTunes. If you feel like pulling an Emily, put it on while you do the dishes and then forget that you are doing the dishes and stare into space while you listen. :)
If you have any thoughts about it and feel like sharing, please do.
Please do read this article if you haven't already, but while you do, keep in mind one important thing: Chait is a white man (he says as much in the article). In addition, I have read several criticisms of the article as well, all mostly focusing on the lack of research into the claims Chait makes, instead covering ground mostly based on anecdotes that illustrate his point. Take all of this with a grain of salt, maybe a tasty grain of salt since it's the first time I've ever heard anything written about the phenomenon. My head has been reeling with arguments for and against the article ever since I read it and I'm about to Skype with a dear, dear friend and fellow Mills graduate tonight to talk about it some more. If you feel particularly flummoxed afterwards and want a little more discussion of it, I also encourage you to listen to a podcast from Slate.com about it. Slate (yes, notoriously liberal-slanted, bear this in mind as well as you listen) offers a great deal of interesting (and free) podcasts and the one to which I regularly listen is called the Political Gabfest. Last week's issue (the podcast title is "The 'Can You Buy a President for $889 Million?' Edition") has the three correspondents discussing three topics - the Koch brothers' enormous financial commitment to the 2016 election, the supreme court facing an upcoming decision on the death penalty as well as Chait's article. You can skip to the end to just hear the latter, but if you feel like giving the podcast a chance, I encourage you to listen to the whole thing. I've linked to the page where you can stream the podcast, or you can find it in iTunes. If you feel like pulling an Emily, put it on while you do the dishes and then forget that you are doing the dishes and stare into space while you listen. :)
If you have any thoughts about it and feel like sharing, please do.
So, February.
Okay, so don't get me wrong - nothing terrible has happened to me in February, but it is a Tuesday in February as I write, and I always have to think of that quote on such occasions.“It was a Tuesday in February. Many of my life's most awful moments have taken place on Tuesdays. And what is February if not the Tuesday of the year?” -Stephen Fry, in Moab is my Washpot
No, so far, my February has been fairly reasonable. I have two more weeks (including this one) of classes and then the "Lernphase" (studying phase) begins before exams. As I have explained in earlier posts, the German university system (at least in Math) is based on one grade from the final exam, so it's around this time of the semester that a little ripple of panic goes through the lecture halls and study rooms as students realize that the are soon to be responsible for every lecture and piece of homework that has passed through their ears and eyes since the beginning of October. Due to a complicated set of circumstances induced from the switch of University systems, though I was in five courses this semester, I am only required to take one exam, and that's possibly in the very class that I feel most confident. So as I said, February's not been so bad.
We've been enjoying a mixture of Fall, Winter, and Spring weather here, including sudden flurries with the biggest snowflakes I've ever seen:
 |
| A ten minute flurry, photo taken just outside the Mainz train station. |
 |
| The happiest leaf I have ever seen. |
Thursday, December 18, 2014
To Infinity and the Next Kind of Infinity
I can tell it's been a while since I've written because normally, when I type 'bl' into my navigation bar in my browser, I can immediately hit 'Enter' and I will be directed to Blogger, the automatic completion of 'bl'. However, when I typed 'bl'+'Enter' just now, I was directed to another website, one having to do with the programming of linear algebra in Python - a website I have been frequenting more, well, frequently than this one in recent weeks!
I'm sitting with a mug of tea the size of my face in a toasty warm cafe right now. Tomorrow is officially the last day of classes before Christmas break, when I will be flying back home to the States. I have one more workshop tomorrow and then I'm done - no classes at all today, though I have some homework to finish.
All in all, it's been a good semester so far, and the last few weeks in particular. I have found people to work with in nearly every class and am starting to recognize more and more of the faces (and some of the names) in my classes, which is harder than I thought. I never quite realized when I was at Mills just how close-knit our community was. Even though I didn't necessarily like all of my classmates, I knew every single one of them, first name and last, and what year they were in, what major (I'm talking about math folks, for the most part here) -- here, I have no idea. But in some classes, that's getting better - people are getting to know me or at least recognize me in particular in my Fourier Series seminar, because I am the only girl and gave my presentation last week. I can't believe how little I thought about the friend-making process when I was a kid. It was just something you did (not that it was always easy!). But now that I've moved from place to place so frequently and had to go through this process so often, I take special note of it.
As everyone is about to start traveling to see family for the holidays, I want to throw some food for thought out there. Something to make your mind whizz while you pull off the highway because the snow is swirling too much to drive, or while you sit through your four-hour layover in London (which I will be doing), or just while you tune out at the dinner table when the age-old and much-repeated family conversations come up again.
I want to talk briefly about infinity. If you've ever been friends with a mathematician, or mathematician-in-training, or just someone who had a decent math teacher at some point in their life and paid attention at least some of the time -- you might already be aware that there are several types of infinity, or levels of infinite-ness. I'm bringing this up today because yesterday in Algebra, I learned something that I didn't know about infinity and it surprised me so much, I haven't been able to stop thinking about it since.
Okay. So, where do we start? First, one very basic but still (in my opinion) super-cool fact about infinite is as follows: if you have an infinite set of things (like grains of sand, for example) and you take any portion of that away -- so now you have two piles of grains of sand. At least one of those piles must be infinite. Either you took an infinite amount away and left a finite amount, or you took a finite amount away and the rest is infinite. Or both piles could be infinite. But you can't divide it into two finite piles. Because the sum of two finite piles of things would still be finite. Does this impress you? Or give you a new way to think about infinity? It certainly made my head spin when I first heard it.
Alright. Now let's get a little bit more mathematical. First of all, some terminology you probably haven't thought about since elementary school (I swear they teach us this there - I still remember the diagram). The Natural Numbers (abbreviated N) are the numbers {1,2,3,....} -- the 'counting numbers' as they are sometimes called. (Depending on what context you are in, the Natural Numbers may also include 0, but not always). Then there's the Integers (abbreviated Z (from Zahl, the German for 'number') ), which is like the Natural Numbers but reflected across zero - i.e. you get the negative ones, too. Positive and negative whole numbers {...,-3,-2,-1,0,1,2,3,...},this time including zero for sure. Still no fractions allowed. The fractions are part of what we call the Rational Numbers (abbreviated 'Q' ((probably for 'quotient', but I'm not sure)). Rational Numbers by definition are all those numbers that can be written as "a/b" where both a and b are integers and b is not zero. You may have noticed that all of these sets of numbers that I've listed kind of fit neatly inside each other - the Natural Numbers are part of the Integers, who are part of the Rational Numbers -- who are in turn part of the Real Numbers (abbreviated 'R'), the last bunch I'm going to talk about here.
You've probably guessed who the Reals are by now - all the rest you're familiar with. Yes, fractions and integers but also strange decimals that go on and on and never terminate. Some of these decimals can actually be represented by a fraction, take .3 repeating for example which we know is 1/3. Others can't be. One really famous example of this is pi (which is actually particularly interesting for even more complex reasons). But it's not the only one.
So, clearly, the Rationals fit inside the Reals but there's a lot of Reals that aren't Rationals - so that inclusion only goes one way. In set-theory notation, we write this inclusion of sets like this:
So, I think we agree about this chain of inclusions up there. But now we have to notice - there are an infinite number of elements in each set in that chain. An infinite number of counting numbers, integers, rationals and reals. And yet, there's more integers than counting numbers, and more rationals than integers, and more reals than any of the rest. So, we do have a hierarchy of infinite-ness.
The way that Z is more infinite than N kind of makes sense - there's like, twice as many integers as counting numbers. The positive ones and then all their negatives. But what about Z and Q? Think about how many rational numbers there are just between 0 and 1. 1/2,1/3,1/4,...., 1/512,... -- a LOT. Like, really a lot. So, Q is way more infinite than Z. Then just think about R. Is your head hurting?
One way that mathematicians discuss or categorize infinity is by the idea of countability. This idea is not all that complicated - a set of things is countable if we can organize all the things in a line of some kind and give every item in the set a number denoting its place in line. So, finite sets are obviously countable - we know how many there are in total, so clearly we can count them. But infinite sets can also be countable. We might have to count for ever and ever, but we could count them, theoretically.
I'm sitting with a mug of tea the size of my face in a toasty warm cafe right now. Tomorrow is officially the last day of classes before Christmas break, when I will be flying back home to the States. I have one more workshop tomorrow and then I'm done - no classes at all today, though I have some homework to finish.
All in all, it's been a good semester so far, and the last few weeks in particular. I have found people to work with in nearly every class and am starting to recognize more and more of the faces (and some of the names) in my classes, which is harder than I thought. I never quite realized when I was at Mills just how close-knit our community was. Even though I didn't necessarily like all of my classmates, I knew every single one of them, first name and last, and what year they were in, what major (I'm talking about math folks, for the most part here) -- here, I have no idea. But in some classes, that's getting better - people are getting to know me or at least recognize me in particular in my Fourier Series seminar, because I am the only girl and gave my presentation last week. I can't believe how little I thought about the friend-making process when I was a kid. It was just something you did (not that it was always easy!). But now that I've moved from place to place so frequently and had to go through this process so often, I take special note of it.
As everyone is about to start traveling to see family for the holidays, I want to throw some food for thought out there. Something to make your mind whizz while you pull off the highway because the snow is swirling too much to drive, or while you sit through your four-hour layover in London (which I will be doing), or just while you tune out at the dinner table when the age-old and much-repeated family conversations come up again.
I want to talk briefly about infinity. If you've ever been friends with a mathematician, or mathematician-in-training, or just someone who had a decent math teacher at some point in their life and paid attention at least some of the time -- you might already be aware that there are several types of infinity, or levels of infinite-ness. I'm bringing this up today because yesterday in Algebra, I learned something that I didn't know about infinity and it surprised me so much, I haven't been able to stop thinking about it since.
Okay. So, where do we start? First, one very basic but still (in my opinion) super-cool fact about infinite is as follows: if you have an infinite set of things (like grains of sand, for example) and you take any portion of that away -- so now you have two piles of grains of sand. At least one of those piles must be infinite. Either you took an infinite amount away and left a finite amount, or you took a finite amount away and the rest is infinite. Or both piles could be infinite. But you can't divide it into two finite piles. Because the sum of two finite piles of things would still be finite. Does this impress you? Or give you a new way to think about infinity? It certainly made my head spin when I first heard it.
Alright. Now let's get a little bit more mathematical. First of all, some terminology you probably haven't thought about since elementary school (I swear they teach us this there - I still remember the diagram). The Natural Numbers (abbreviated N) are the numbers {1,2,3,....} -- the 'counting numbers' as they are sometimes called. (Depending on what context you are in, the Natural Numbers may also include 0, but not always). Then there's the Integers (abbreviated Z (from Zahl, the German for 'number') ), which is like the Natural Numbers but reflected across zero - i.e. you get the negative ones, too. Positive and negative whole numbers {...,-3,-2,-1,0,1,2,3,...},this time including zero for sure. Still no fractions allowed. The fractions are part of what we call the Rational Numbers (abbreviated 'Q' ((probably for 'quotient', but I'm not sure)). Rational Numbers by definition are all those numbers that can be written as "a/b" where both a and b are integers and b is not zero. You may have noticed that all of these sets of numbers that I've listed kind of fit neatly inside each other - the Natural Numbers are part of the Integers, who are part of the Rational Numbers -- who are in turn part of the Real Numbers (abbreviated 'R'), the last bunch I'm going to talk about here.
You've probably guessed who the Reals are by now - all the rest you're familiar with. Yes, fractions and integers but also strange decimals that go on and on and never terminate. Some of these decimals can actually be represented by a fraction, take .3 repeating for example which we know is 1/3. Others can't be. One really famous example of this is pi (which is actually particularly interesting for even more complex reasons). But it's not the only one.
So, clearly, the Rationals fit inside the Reals but there's a lot of Reals that aren't Rationals - so that inclusion only goes one way. In set-theory notation, we write this inclusion of sets like this:
So, I think we agree about this chain of inclusions up there. But now we have to notice - there are an infinite number of elements in each set in that chain. An infinite number of counting numbers, integers, rationals and reals. And yet, there's more integers than counting numbers, and more rationals than integers, and more reals than any of the rest. So, we do have a hierarchy of infinite-ness.
The way that Z is more infinite than N kind of makes sense - there's like, twice as many integers as counting numbers. The positive ones and then all their negatives. But what about Z and Q? Think about how many rational numbers there are just between 0 and 1. 1/2,1/3,1/4,...., 1/512,... -- a LOT. Like, really a lot. So, Q is way more infinite than Z. Then just think about R. Is your head hurting?
One way that mathematicians discuss or categorize infinity is by the idea of countability. This idea is not all that complicated - a set of things is countable if we can organize all the things in a line of some kind and give every item in the set a number denoting its place in line. So, finite sets are obviously countable - we know how many there are in total, so clearly we can count them. But infinite sets can also be countable. We might have to count for ever and ever, but we could count them, theoretically.
Think about the integers. If I want to set them up in a line, I can start at zero and arrange them like this: {0,1,-1,2,-2,3,-3,…} always alternating from the right side to the left side of zero, getting a number and its negative. If I arrange them like this, I can assign each one their place in the ‘line’ if you will. I know that when I count like this, I’m not missing any of them - I’ll find all of them with this method and all will get a number for their place in line. Thus, the integers are countable.
But what if we tried this method with R, the real numbers? I can start at zero and then… what’s the next one to pick? Even if I just picked 0 and .1 — well, between 0 and .1 there’s .01, .001,.011,.0001 — all of these things! I can’t set them up in a line where I don’t miss any and where everyone has a place in line. The reals are not countable. So, we call them uncountably infinite. And that’s a bigger kind of infinite than the countable kind.
So, now, we’re finally set up for what blew my mind yesterday. Now, I must have been living under a mathematical rock because this is not a new discovery, but everything is new to us at the point when we first learn it, and this was new to me yesterday — the rationals are countable.
Not only that, but the proof is really easy. I’m linking to one version here because I think I’ve probably talked your ears off. Once you see the proof or just think hard about the rationals, it does actually make sense — or maybe it even seems obvious. But I had always assumed they were uncountably infinite. Pretty neat stuff.
Now, that fact about the rationals - specifically the relationship between the rationals and the reals, given that one is countable and one is not- leads to some interesting conclusions in Abstract Algebra, which is why we were talking about it yesterday in class. I might go into that some time - but not today. Now, it's time to go and turn in my algebra homework.
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