Originally mathematical ideas were based the physical world, specifically that of geometry. The Greeks, Egyptians, and Pythagoreans measured and used number in a practical and tangible sense. The number 2 'existed' as 2 units of length, maybe the side of a triangle, or the length of a column. Well imagine the pythagoreans disappointment when they stumbled upon the idea of the square root of 2. We know this number to be irrational. The way they discovered the problem was when they split a perfect square into two right angled triangles. If each side length was 1 unit, then the hypotenuse of the triangle should be the square root of 2 units in length, there, they drew it, it must exist. Yet when you try to split the number 1 into even pieces, lets say 5. You can't split the square root of 2 the same way, in fact there is no way to split the square root of 2 evenly. Impossible, but true, such is the case with all irrational numbers. Their values go on and on in the decimal world at infinitum.
Euclids geometry also ran into contradictions. For century's mathematicians have tried to prove that if two lines bisect a third line, and if the angles made add to a number less than 180, then the two bisecting lines must intersect at some other point, in other words, they are not parallel. It seems so obvious it should not be an assumption, but a hard fact that can be proved, yet no one can. Lambert tried, but a proof by contradiction later all he came up with was a famous picture called lamberts rectangle.
So, we can't base mathematics in geometry like we thought, there are too many things left unexplained.
Then the smart guys turned to algebra, and straight numbers for numbers sake. Turns out there's a contradiction here to, but I'll spare you the details.
They tried set theory, and the world of abstract algebra, but there were still mathematical ideas that escaped these axioms and proofs. Like this one... "the set of all sets that don't contain themselves" its a pure paradox.
Then logic, logic must be the answer.... nope even after the book Principa Mathematica was written... a book so complex and precisely defined that it took some 6oo pages for the authors to finally assert that 1 plus 1 does in fact equal 2. Well even this book contained an error, a loop hole that missed some critical information.
So the search for a true foundation for mathematics continues. And while the saga continues, most mathematicians, especially those in the applied realms, turn a blind eye to the predicament, because math still models everyday occurences. Engineers still build bridges, using geometry, actuaries still calculate complex algorithms to predict losses for big businesses, professors still cram the ideas of null sets, convergence theorems, and isomorphic relations into the brains of their pupils, and children still count using their fingers, effectively demonstrating the use of abstraction by the time they are 3 years old. Math still goes on, even when no solid foundation exists.
These thoughts led to others...
What other fields claim to have sure foundations? Science claims to find truth, but only in things tangible and earthly, or at least physical. Yet they model perfect situations that can in no way be tangible... have you ever seen a physics rope in real life? No weight, no friction.... ya, didn't think so. Chemists use imaginary numbers to calculate complex equations, but when it comes down to the actual chemicals, no imaginary numbers are actually used. Then there are the Humanities and Social Sciences. I don't know what they claim to be their foundation, but I'm sure whatever it is changes with the opinion of the professor, or teacher year to year. And we all know the foundation of english is the fact that every rule has an exception, which is a contradictory statement in and of itself! So does any area of study really have a true 'foundation' or building block on which all elements of the field can be derived and explained from? And who really cares anyway? Just the crazed, perfect model seekers called mathematicians probably.
And then I thought about religion. My religion specifically, and the truths I hold to be pure knowledge and wisdom. I remember a time when I was frustrated with the blanket statement that covers everything. If we don't know it, its not because it isn't, it's because we can't, or shouldn't know.... yet. There's always that yet, because with faith and diligence, knowledge of all things must surely be attainable. Faith. Faith is the foundation upon which all truth, light and knowledge builds. That I believe.
A foundation as shaky or as sure as the individual.
1.30.2008
1.27.2008
chocolate things I ate today...
cookies, 8
brownies, 1
hot beverage, 0
icecream scoops, 1/2
mints, 4
I'm bad on sundays.
and I'd do it again.
brownies, 1
hot beverage, 0
icecream scoops, 1/2
mints, 4
I'm bad on sundays.
and I'd do it again.
1.26.2008
the human voice
I'd forgotten the beauty of the human voice.
I hear singing every day, from cd's, my roommate's mouth, cute boys trying to impress me on their guitars, random ipods turned too loud in the library,.... everywhere. But tonight I remembered what trained voices, singing in perfect harmony's can do to a torn soul. I remembered the mathematical beauty of chords in tune, and the powerful dynamic of a choir.
Flanders Fields left the audience in a trance. A tension seemed visible, but not uncomfortable. It was a tension of intrigue. I felt as though I was hanging on to every sound as their voices swelled in and out of the dissonant chords. Words can not describe the basses present, but not driving, circular rhythms, the way they wafted in and out of conscious acceptance. And I can't tell you what it meant, only that I heard pain and sorrow, and hope at the same time. I could tell you that the sopranos had perfect, glorious pitch, that transcended this world and brought something of the heavens down to a concert hall, but you wouldn't have been there, and couldn't know to what depth this piece spoke.
It felt like water. Like a huge ocean of water, with its heavy and smooth currents pulling methodically, constant, yet dynamic. Powerful and moving, even when soft. The building seemed transformed. Hollowed and deep. I wish it could have gone on longer, yet I was drained, our mortal bodies it seems can't stay in that hollowed, transcendent state for too long before they tire. It was an undescribable moment, and I think the person sitting to my right missed it.
There just aren't words. And that's why we sing.
I hear singing every day, from cd's, my roommate's mouth, cute boys trying to impress me on their guitars, random ipods turned too loud in the library,.... everywhere. But tonight I remembered what trained voices, singing in perfect harmony's can do to a torn soul. I remembered the mathematical beauty of chords in tune, and the powerful dynamic of a choir.
Flanders Fields left the audience in a trance. A tension seemed visible, but not uncomfortable. It was a tension of intrigue. I felt as though I was hanging on to every sound as their voices swelled in and out of the dissonant chords. Words can not describe the basses present, but not driving, circular rhythms, the way they wafted in and out of conscious acceptance. And I can't tell you what it meant, only that I heard pain and sorrow, and hope at the same time. I could tell you that the sopranos had perfect, glorious pitch, that transcended this world and brought something of the heavens down to a concert hall, but you wouldn't have been there, and couldn't know to what depth this piece spoke.
It felt like water. Like a huge ocean of water, with its heavy and smooth currents pulling methodically, constant, yet dynamic. Powerful and moving, even when soft. The building seemed transformed. Hollowed and deep. I wish it could have gone on longer, yet I was drained, our mortal bodies it seems can't stay in that hollowed, transcendent state for too long before they tire. It was an undescribable moment, and I think the person sitting to my right missed it.
There just aren't words. And that's why we sing.
Do something that scares you every day
I did something scary today. I painted. With watercolor. A frog and some lilly pads. I'm not an artist, and even as I began I was trying to map it all out, in some grid like, systematic way. I learned to just let it go though, and that was a big step for me. I found the actual act of painting extremely theraputic, even if the final 'materpeice' as it were, was not all that eye pleasing. I like that it looks like water.... but the frog looks like a leaf. Oh well, I did it, and that's all that matters. 2 points.
1.25.2008
Enchanted
Apparently it's a movie I'm supposed to go see. It also describes all things magical, spellbinding, or otherwise captivating in some beautiful way. Today I was enchanted...
I was on campus rather late for a top secret meeting that I can't talk about, let alone blog about. dumb campaign rules. As I walked outside, frigid air filled my lungs and invigorated my library intoxicated mind. Anyone so privileged to be on campus that late might know of the magical lights that come out like fireflies in the evening. The seemingly formidable buildings of scholastic torture fade to black, and all that is left to see are the playful spots of light illuminating beautiful trees, walkways, and architecture. It was magical, but I hadn't been enchanted, yet.
I made my way down to home sweet home. I should have stayed in the library to read, but the sound of squeaky gym shoes compliments my study habits so much better. The Richards building, home to athletes, dancers and scholars alike, called to me.
There's a definite smell... and while others may argue with my opinion, I don't find it displeasing in the slightest. I grew up in a basketball gym, watching my dad coach his team every Saturday morning. The Richards building, with all of its sounds and smells feels like home.
I walked down the top floor and stopped at every window. Like some patron of a three dimensional, moving museum of Kinesiology, I analyzed every scene, and searched for words to describe the compositions...
Window 1
"flailing insecurity finds home in a life preserver"
This was the water polo window. And not just any water polo mind you, it was BYU intramural INNERTUBE water polo. I always wanted to play. I wanted to be the goalie who gets a navy blue innertube to set him/her apart from the rest of the team. It adds some decency and respect to a player who has just agreed to wear a swimming suit and sit awkwardly in a tube of air while flailing their limbs in an attempt to maneuver enough to benefit their team in some way.
Window 2
"man vs man .... vs ball, hoop, refs, spectators, girlfriend and shoelace"
Oh basketball. What can I say. Its a step up from church ball here at good old brigham young, but not by much. I watched for a second, groaning at the defense, and lack of jump shot skill on the whole. I laughed out loud when I spotted the all too typical 'high-school-glory-days-guy" Sporting his sweat bands and top of the line kicks, this is the guy that fights the ref more than anyone and believes if the coaches had just looked at him he would be playing on the marriott center floor Saturday night. Oh contraire.
And the there are the girlfriends. Some dressed in fur, and I'm thinking.... its a gym darling, leave it at home with your purse by Prada. Some scream incessantly at the refs, who, let us all remember, are just students dressed in black and white, tired from a day full of class and just trying to make a little money by blowing a whistle a few times a night. I don't understand yelling at the refs. They're part of the game, the rules are different with each ref, you just get used to it and keep playing.
Finally the shoelaces.... ok, these are not always such an antagonist, but tonight they were. I can't begin to describe the elegant dismount, the leap and tumble, the wrenched face and the blood that rushed in to his rosy, embarrassed face. All I can say is touche shoelace, touche.
Window 3
"one plus one is one"
Ok, so this wasn't actually a window. I walked into 278 to grab my clogs, and fell upon the ballroomers. Like always they were fighting for every last square inch of hardwood. We like to joke about the rivalry between dance teams, but honestly I have a lot of respect for the ballroom genre. Tonight I took a second and just took in the scene.
In one corner the recently graduated, and married ballroom king was passing off his skill to the ballroom/math girl. I know her from both area of my life, and I was totally captivated by her and her partners movement across the floor. They make it look so effortless, they're bodies flow like liquid, seamless... until one takes a wrong step and the balance is lost, the flow is cut. But they were there to practice after all.
I watched other couples, and I still laugh at the drama of it all. The head that's held in the most unnatural horizontal positions, the snooty air of elegance, and the ridiculous hand flailing, but even with all of that nonsense, there is something fundamentally captivating and completely beautiful. Two separate bodies coming together in such total unity fascinates me. Dance is such an incredible expression of emotion and thought, and I'll hand it to the ballroomers tonight.
I was enchanted.
I was on campus rather late for a top secret meeting that I can't talk about, let alone blog about. dumb campaign rules. As I walked outside, frigid air filled my lungs and invigorated my library intoxicated mind. Anyone so privileged to be on campus that late might know of the magical lights that come out like fireflies in the evening. The seemingly formidable buildings of scholastic torture fade to black, and all that is left to see are the playful spots of light illuminating beautiful trees, walkways, and architecture. It was magical, but I hadn't been enchanted, yet.
I made my way down to home sweet home. I should have stayed in the library to read, but the sound of squeaky gym shoes compliments my study habits so much better. The Richards building, home to athletes, dancers and scholars alike, called to me.
There's a definite smell... and while others may argue with my opinion, I don't find it displeasing in the slightest. I grew up in a basketball gym, watching my dad coach his team every Saturday morning. The Richards building, with all of its sounds and smells feels like home.
I walked down the top floor and stopped at every window. Like some patron of a three dimensional, moving museum of Kinesiology, I analyzed every scene, and searched for words to describe the compositions...
Window 1
"flailing insecurity finds home in a life preserver"
This was the water polo window. And not just any water polo mind you, it was BYU intramural INNERTUBE water polo. I always wanted to play. I wanted to be the goalie who gets a navy blue innertube to set him/her apart from the rest of the team. It adds some decency and respect to a player who has just agreed to wear a swimming suit and sit awkwardly in a tube of air while flailing their limbs in an attempt to maneuver enough to benefit their team in some way.
Window 2
"man vs man .... vs ball, hoop, refs, spectators, girlfriend and shoelace"
Oh basketball. What can I say. Its a step up from church ball here at good old brigham young, but not by much. I watched for a second, groaning at the defense, and lack of jump shot skill on the whole. I laughed out loud when I spotted the all too typical 'high-school-glory-days-guy" Sporting his sweat bands and top of the line kicks, this is the guy that fights the ref more than anyone and believes if the coaches had just looked at him he would be playing on the marriott center floor Saturday night. Oh contraire.
And the there are the girlfriends. Some dressed in fur, and I'm thinking.... its a gym darling, leave it at home with your purse by Prada. Some scream incessantly at the refs, who, let us all remember, are just students dressed in black and white, tired from a day full of class and just trying to make a little money by blowing a whistle a few times a night. I don't understand yelling at the refs. They're part of the game, the rules are different with each ref, you just get used to it and keep playing.
Finally the shoelaces.... ok, these are not always such an antagonist, but tonight they were. I can't begin to describe the elegant dismount, the leap and tumble, the wrenched face and the blood that rushed in to his rosy, embarrassed face. All I can say is touche shoelace, touche.
Window 3
"one plus one is one"
Ok, so this wasn't actually a window. I walked into 278 to grab my clogs, and fell upon the ballroomers. Like always they were fighting for every last square inch of hardwood. We like to joke about the rivalry between dance teams, but honestly I have a lot of respect for the ballroom genre. Tonight I took a second and just took in the scene.
In one corner the recently graduated, and married ballroom king was passing off his skill to the ballroom/math girl. I know her from both area of my life, and I was totally captivated by her and her partners movement across the floor. They make it look so effortless, they're bodies flow like liquid, seamless... until one takes a wrong step and the balance is lost, the flow is cut. But they were there to practice after all.
I watched other couples, and I still laugh at the drama of it all. The head that's held in the most unnatural horizontal positions, the snooty air of elegance, and the ridiculous hand flailing, but even with all of that nonsense, there is something fundamentally captivating and completely beautiful. Two separate bodies coming together in such total unity fascinates me. Dance is such an incredible expression of emotion and thought, and I'll hand it to the ballroomers tonight.
I was enchanted.
1.21.2008
the first.
I hate starting, but I love newness. hmmm. I just needed to get this first one out of the way, you know? I needed to clear the cyber air that was looming, crying out.... "start your blog, start it now!" So much pressure for the beginning, the first. Hate that. So here we go, a few lines of thought, in extreme stream of consciousness form. Done. It's been started. And now that that's out of the way...... I can really begin.
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