I got a comment a few posts back about Graham's number and whether it larger than a googolplex of towers or some such thing. Let me first say sorry for not replying to that; I thought I had, but apparently just thought about it for a little and forgot to write anything up. The answer is I have no idea. I don't even know where you would begin trying to prove something like that. If you read the wikipedia link about Graham's number, then you actually know more about it than I do, since I didn't finish reading it originally and then forgot all about it until now.
If you don't feel like reading about what it is, suffice it to say that it is a positive integer that was given as an upper bound to a problem in Ramsey theory, which is a branch of graph theory. I don't really know much about it, but it seems like an interesting branch of mathematics, albeit one that I don't think about much. As for how large it is, I think only the word unfathomably suffices to describe it. Something like there aren't enough particles in the universe to save it in digital form.
In order to keep this post slightly less low-content, I'll mention that what amazes me is that even though numbers like Graham's number are basically impossible to get a handle on, in math we routinely deal with things that are much, much larger than that. For as large as it is, it's still a finite number (which is the same as a finite ordinal [sort of]!), and we deal with sets that are infinite all the time.
I think we sort of lose track of how big infinite is because we think of sets like the natural numbers, which are infinite, but we think of them in forms like
{1,2, ...}
which sort of hides the fact that there are huge things in there and it's almost easy to forget that.
Now, to blow your mind a little bit, which are there more of, non-negative integers, or non-negative even numbers? Intuitively, there are more non-negative integers since all even non-negative integers are non-negative integers. But consider the function f(n) = 2n. It turns each non-negative integer into an even non-negative integer, and it does it in such a way that we don't repeat anything and every even non-negative integer gets hit. So there must be the same number of them.
So it's natural to think that maybe there are only finite and infinite things, and in the literal sense that is true: something is either finite or NOT finite, but in a sense it isn't true. There are lots of infinities. For example, consider the real numbers. If you don't know what that is, suffice it to say consider the set of all sequences you can make of 0's and 1's. Now, if that set is the same size as the positive integers (or non-negative integers, or just integers, or even rational numbers), then we can count them, and we can make a new sequence by starting with the first sequence and setting the first entry of the new sequence to be the opposite of whatever the first sequence is. That is, if the first sequence has a 0 in the first place, choose a 1, and if it had a 1, choose a 0. Then move on to the second sequence and do the same thing with the second entry of the new sequence.
If you keep doing this for each sequence, you'll get a new sequence that differs from every sequence in one place, so can't be an element of the original set of sequences, but is clearly just made of 0's and 1's, so must be an element of the original set. So, there's a contradiction and you must have a bigger set than the natural numbers. In math terms, we say that the real numbers form an uncountable set.
Can we get even bigger? Yeah, just consider the set of all subsets of the real numbers. Then consider the set of all subsets of that set, and so on and so forth. Cantor proved that you always get a bigger set by doing this, so that there is no largest set. It gets even crazier than that, but I guess that's enough food for thought.
Monday, June 20, 2011
Sunday, June 19, 2011
Hey Look What I Made
It's noodle salad. It's the first time I made this kind of noodle salad, which uses Italian dressing and not mayonnaise. So that's pretty good. My buddy Phill had a cookout yesterday so that is what I made and people liked it. Not much else happening.
Wednesday, June 8, 2011
Take a Load Off, Fannie
I realize I haven't posted in weeks, but I've been busy. Fortunately, that should all be done now and I get to go down, Miss Moses and wait on the judgment day. Won't you stay and keep Anna Lee company? So, maybe there will be a content post soon. Looking forward to weddings and Illinois.
Sunday, May 8, 2011
Planet Waves
It looks like I will have to follow through on my threats of pointlessly reviewing semi-obscure Dylan songs because the comments have been Slow Train Coming (it sounds like slow in coming; expect more terrible puns if I don't get satisfactory comments).
So, I've decided to write a brief review of Dylan's 1974 album, Planet Waves. It will be brief partially because there isn't that much to say about it. It's nice sounding because he's got The Band backing him up again and their version of "roots rock" is always pleasant and interesting, with them switching instruments and having multiple moving parts all at once, but lyrically there isn't much there. Most of the songs feel like they were written on a lark and they don't really address anything deep or have the layered meanings and references of John Wesley Harding. Anyway, here is my track by track review:
"On a Night Like This" - A strong opener because the band knows how to use accordion and have fun with a lighthearted song.
"Going Going Gone" - Sort of nondescript. The best line is probably "all that's gold isn't meant to shine," which is hardly up to his usual standards.
"Tough Mama" - Another upbeat number that The Band gets to have fun with. There's a little bit more imagery here. In a way it reminds me of his earlier song "Love Minus Zero/No Limit" in that it's a step up from most love songs which are bland and not descriptive. You would think if you were so in love with somebody to write a song, you would have plenty of reasons to enumerate, but usually all you get is blah about nice hair or eyes or something. Dylan steps it up here by painting you a picture of his "Tough Mama," though it doesn't come off as genuine as the earlier work. He gets bonus points for this line, though : "Today on the countryside it was a-hotter than a crotch."
"Hazel" - He's played this one in concert, so I have a live version, I think from The Band's Last Waltz concert, but other than them having played it together, I can't think of why he chose this song over any of his others. It's just a typical love song to "Hazel," who never really comes off as a real person to me. There's some nice piano driving the song, though, so that's fun.
"Something There is About You" - More fun stuff going on in the background from the Band, and this time it's under something at least fairly interesting. I don't know what's with the strange structure of the title, but something there is about it that I like. Also sort of notable for a mention of Dylan's childhood in Minnesota, which he never really talks about, "rainy days on the Great Lakes, walking the hills of old Duluth." :)
"Forever Young" - I'm sure everyone has heard this song. Most people are fans of the first version that appears on this album (the next track is another arrangement of the same song), but I like the second one better. Lyrically, it's a decent track and maybe the album's strongest, but I can't see it actually working as a lullaby, which is supposedly the intention. If you're going to write an ineffective lullaby, you might as well have the Band go crazy behind you, I figure.
"Dirge" - "I hate myself for loving you" and whatnot. It's pretty good, and more Dylan imagery. "Doom Machine," "just a painted face on a trip down Suicide Road," etc.
"You Angel You" - a song with "dummy lyrics," in Bob's own words. It's catchy, at least.
"Never Say Goodbye" - this song and its bass line keep getting stuck in my head, so there's that.
"Wedding Song" - Closing the album on a down note seems kind of odd, but it's not as if he didn't do that with "Highway 61 Revisited," "Blonde on Blonde," and "Desire." On the other hand, those albums were far less bouncy, so it's kind of odd. This closing track is ok, but not nearly up to the closing tracks of those albums. Meh
Alright, well that's it. Maybe the next post will be a bunch of nonsense about model categories.
So, I've decided to write a brief review of Dylan's 1974 album, Planet Waves. It will be brief partially because there isn't that much to say about it. It's nice sounding because he's got The Band backing him up again and their version of "roots rock" is always pleasant and interesting, with them switching instruments and having multiple moving parts all at once, but lyrically there isn't much there. Most of the songs feel like they were written on a lark and they don't really address anything deep or have the layered meanings and references of John Wesley Harding. Anyway, here is my track by track review:
"On a Night Like This" - A strong opener because the band knows how to use accordion and have fun with a lighthearted song.
"Going Going Gone" - Sort of nondescript. The best line is probably "all that's gold isn't meant to shine," which is hardly up to his usual standards.
"Tough Mama" - Another upbeat number that The Band gets to have fun with. There's a little bit more imagery here. In a way it reminds me of his earlier song "Love Minus Zero/No Limit" in that it's a step up from most love songs which are bland and not descriptive. You would think if you were so in love with somebody to write a song, you would have plenty of reasons to enumerate, but usually all you get is blah about nice hair or eyes or something. Dylan steps it up here by painting you a picture of his "Tough Mama," though it doesn't come off as genuine as the earlier work. He gets bonus points for this line, though : "Today on the countryside it was a-hotter than a crotch."
"Hazel" - He's played this one in concert, so I have a live version, I think from The Band's Last Waltz concert, but other than them having played it together, I can't think of why he chose this song over any of his others. It's just a typical love song to "Hazel," who never really comes off as a real person to me. There's some nice piano driving the song, though, so that's fun.
"Something There is About You" - More fun stuff going on in the background from the Band, and this time it's under something at least fairly interesting. I don't know what's with the strange structure of the title, but something there is about it that I like. Also sort of notable for a mention of Dylan's childhood in Minnesota, which he never really talks about, "rainy days on the Great Lakes, walking the hills of old Duluth." :)
"Forever Young" - I'm sure everyone has heard this song. Most people are fans of the first version that appears on this album (the next track is another arrangement of the same song), but I like the second one better. Lyrically, it's a decent track and maybe the album's strongest, but I can't see it actually working as a lullaby, which is supposedly the intention. If you're going to write an ineffective lullaby, you might as well have the Band go crazy behind you, I figure.
"Dirge" - "I hate myself for loving you" and whatnot. It's pretty good, and more Dylan imagery. "Doom Machine," "just a painted face on a trip down Suicide Road," etc.
"You Angel You" - a song with "dummy lyrics," in Bob's own words. It's catchy, at least.
"Never Say Goodbye" - this song and its bass line keep getting stuck in my head, so there's that.
"Wedding Song" - Closing the album on a down note seems kind of odd, but it's not as if he didn't do that with "Highway 61 Revisited," "Blonde on Blonde," and "Desire." On the other hand, those albums were far less bouncy, so it's kind of odd. This closing track is ok, but not nearly up to the closing tracks of those albums. Meh
Alright, well that's it. Maybe the next post will be a bunch of nonsense about model categories.
Wednesday, May 4, 2011
Can You Guess Who?
Can you guess who I saw tonight? The answer is Salman Rushdie. This wasn't like that episode where Kramer thought he saw Rushdie but it was just some dude named Salbas. He came here to talk about writing and novels and crap. He was kind of hilarious and almost made me want to read one of his books. Too bad I'm busy doing a bunch of math. But it's alright.
Monday, April 18, 2011
Tuesday, April 12, 2011
Ain't No Good
Every other Wednesday there is a seminar that lasts about an hour at most by a grad student in our department. The only people that come are other grad students because faculty have better things to do than listen to us talk about research or often just definitions of stuff that they probably know better/don't work with, and who else would come to a talk about math?
Anyway, I haven't given a talk and don't really plan on it, at least not any time soon, despite the girl who is in charge of it asking me multiple times to present just to avoid some of the less...enjoyable...people from talking more than once. I've only recently actually considered doing it at all since I've produced exactly sin(0) work until now, and the idea of presenting old problems or just introducing people to some area they aren't familiar with seems somehow more pointless than category theory (get it, because we generally want to move away from thinking of objects like sets, etc. by what "points" they contain and move towards thinking about the maps between such objects...?). But this quarter I am taking an independent study with a professor, pending official approval from the department head and whatnot on a form I had to fill out (for the question about how my grade would be evaluated, he said to write "by professor." haha).
Forgive my abuse of punctuation.
Anyway, it isn't research, at least not yet, but it's sort of close in a way, and may get there if I can master the small object argument (don't ask), so it seems more worthy of presenting. The main allure of it, though, is that if I talk about what I am doing to any other student, literally no one knows what I am talking about. Here's an example:
If a functor from a model category to a category with a class of weak equivalences that is closed under the 2-of-3 axiom takes trivial cofibrations between cofibrant objects to weak equivalences, then it preserves all weak equivalences between cofibrant objects.
That's called Ken Brown's Lemma, apparently. The wonderful thing about it is that it simultaneously says so much and so little. So, I may tell her that I'll do it, but I have to consider it more and make a bit more progress first. Also, I'm always wary of her invitations; she'll pull out your feathers for her brand new hat, and when she's done that, she'll feed you to her cat.
Anyway, I haven't given a talk and don't really plan on it, at least not any time soon, despite the girl who is in charge of it asking me multiple times to present just to avoid some of the less...enjoyable...people from talking more than once. I've only recently actually considered doing it at all since I've produced exactly sin(0) work until now, and the idea of presenting old problems or just introducing people to some area they aren't familiar with seems somehow more pointless than category theory (get it, because we generally want to move away from thinking of objects like sets, etc. by what "points" they contain and move towards thinking about the maps between such objects...?). But this quarter I am taking an independent study with a professor, pending official approval from the department head and whatnot on a form I had to fill out (for the question about how my grade would be evaluated, he said to write "by professor." haha).
Forgive my abuse of punctuation.
Anyway, it isn't research, at least not yet, but it's sort of close in a way, and may get there if I can master the small object argument (don't ask), so it seems more worthy of presenting. The main allure of it, though, is that if I talk about what I am doing to any other student, literally no one knows what I am talking about. Here's an example:
If a functor from a model category to a category with a class of weak equivalences that is closed under the 2-of-3 axiom takes trivial cofibrations between cofibrant objects to weak equivalences, then it preserves all weak equivalences between cofibrant objects.
That's called Ken Brown's Lemma, apparently. The wonderful thing about it is that it simultaneously says so much and so little. So, I may tell her that I'll do it, but I have to consider it more and make a bit more progress first. Also, I'm always wary of her invitations; she'll pull out your feathers for her brand new hat, and when she's done that, she'll feed you to her cat.
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