Science is hard and in general, get two scientists together and you'll have three opinions. But slowly and unsteadily, a consensus emerges, and the human race gets a little wiser.
None of this is news anymore, but I wanted to recognize some really cool results that have come out recently.
One is "just" a simulation -- but a good simulation is a lot of work. This one shows that the ever-victorious Standard Model correctly predicts the mass of atoms within 5% of their measured values. The other 5% is a mystery that those expensive particle collider experiments are designed to solve. The 95% is a pretty impressive triumph for an increasingly awesome theory.
From a theoretical result about very small scales we move to an observational result about very large scales. For the first time, astronomers have succeeded in takingpictures of planets in other solar systems. They're not the greatest looking pictures, but considering that any picture like this was considered impossible a few years ago, I think they're beautiful. We know now, in a visual, almost visceral way, that our solar system is one of many.
I also think it's worth talking about the fact that India's first unmanned lunar spacecraft has successfully entered orbit around the moon. The more active space programs, the better, as far as I'm concerned. NASA desperately needs some competition. And it is a hopeful sign of progress for the human race as a whole when the so-called Western and Non-Western worlds are each capable of and committed to science on this scale. Space exploration should be a human activity, not limited to a specific culture. Practically all science fiction fans agree.
A couple of more links I want to throw in while I'm linking:
Memoirs of a Space Engineer gives a couple of very real, very entertaining anecdotes about the essence of engineering: problem solving.
And The Necessity of Mathematics is an essay by Science Blogger Blake Stacey about the role mathematics plays in science, with some insightful examples that feel like little revelations in themselves.
Sunday, November 23, 2008
Friday, November 21, 2008
Mathematical Grammar
I'm strongly of the opinion that math is nothing more or less than a language. All languages are different, and it's harder to say certain things in some languages than in others. Math is particularly well suited toward making quantitative statements, and it's unusually difficult to contradict yourself in the language of Math... Or rather, it's more obvious when you've contradicted yourself, when you express something in the language of Math, than it is in English.
It takes some work to express qualitative statements like "That leaf is yellow" in Math, but if we resort to less than or greater than signs, or even "approximately equal" signs, we can usually do it:
That may need a little context, but the basic meaning is clear if you speak Math. (I could also have given RGB color values or even hex color codes, but I don't know those off the top of my head. In any case, there are usually lots of ways to quantify a quality like "color.")
Of course, it's usually much easier to understand Math if you provide the necessary context in English or another natural language... Rather like programming languages, which are just a dialect of Math really -- commenting your code makes it a lot easier to read.
But you know, some of the need for context is reduced by certain conventions which are a part of the grammar of Math, but which are rarely if ever formally stated. When students screw these up, it makes their work a lot harder to read or grade. I find it interesting that everyone eventually picks up on these conventions, all over the world, but no one ever really talks about them. For instance, above I used the greek letter "lamda" without saying what it meant. But to anyone who has had any contact with the physics community, "lamda" means wavelength. The fact that I talked about color and gave a number on the scale of the wavelength of light makes the interpretation certain. If you're a physics person, I don't need to tell you that I'm talking about wavelength there, and not some kind of computer color code. I should probably specify that the wavelength I'm talking about is that of the light reflected from the leaf, but the subscript "Leaf" and the context are again probably enough for you to guess that too.
Here are some of the unwritten rules that come immediately to mind:
Letters from the beginning of the Roman alphabet -- these are usually constants, especially if they are upper case.
Upper Case Roman Letters - these also frequently indicate a constant quantity, especially if the letter is from the beginning of the alphabet. Capital "C" and capital "K" are very popular choices to represent a constant because "constant" starts with a "C" in English and a "K" in German. Upper case Roman letters may also stand for matrices or tensors -- Usually you can tell which is which, because if it's a matrix or a tensor, all of the other terms in the equation will also be a matrices or tensors, so that knowing one symbol gives you hints about the others. (Also, I like to put little upside-down caret hats on my matrices. Most people use right-side-up carets, but I like to reserve those for unit vectors.) If you see a capital X, Y, or Z, however, you know it's likely to be a matrix, because those are almost never used for constants. Which brings me to my next rule.
The symbols x, y, z, and t are reserved for position and time - These are pretty much always variables, and nearly always stand for quantities with units of length, or in the case of t, of time. Sometimes students use "x" to stand for other quantities, because it's a popular choice to stand for any unknown quantity in high school algebra classes, but in the world of math-using professionals, this is a bad idea. Using "x" for a quantity that does not have units of length will confuse people unless you make it very clear what it does stand for. And even then, there are usually better, more conventional choices. If, for instance, the quantity is a pressure, use a "p"(preferably lower-case). If it's a temperature, go for the upper case "T," because the upper-case-letter-means-constant rule is not as firm as the lower-case-t-means-time rule. If it's a frequency, use "f" or "nu"... Most things have their own conventional symbols, in other words, specifically to avoid the problem of using "x" for everything. Save that for position. If there really is no conventional symbol for your quantity, use the lower case Greek letter xi. That's one of few letters that doesn't, by convention, already stand for something else (at least not in physics)
If it's a constant with units of length or time, use subscripts -- the unadorned x, y, z, and t are read as variables, but if you stick a subscripted number (not a letter) on them, they will be read as constants. A good choice is x-subscript-0 or t-subscript-1. In fact, almost anything with a suscripted number will be read as a constant, regardless of the other rules in this list.
Letters from the middle of the alphabet are integers -- In particular, do not use upper or lower case "N" to mean something that is not an integer, because it will confuse people. The letter "m", especially in lower case, is almost always an integer too. In computer science, so are "i", "j", and "k". In physics, however, it is not wise to use these symbols for integers unless they are appearing as an index or subscript. That's because these symbols have other meanings already. ("o" is never used because it looks just like zero.)
Lower case i means square root of negative one -- you can't use lower case "i" for any variable, because its meaning is already assigned. (Likewise that of "e" and lower case "pi" which stand for the natural base of logarithms, and the ratio of a circle's circumference to its diameter, respectively.) The exception to the "i" rule is if you are an electrical engineer, in which case you may use lower case "i" for current if you must. (I have no idea why "I" is the conventional symbol for current -- physicists use it too, but in upper case). Electrical engineers use "j" for the square root of negative one. As for "k" -- that is a seriously overused letter. It has seemingly dozens of conventional uses. It is often used for Boltzmann's constant, though I prefer to give it a subscript for that. In my world it is the wavenumber of light. But it may also be the so-called "lattice constant" for a crystal, a related but different concept. People use it for constants because the German "konstant" is spelled with a "k" even though it does not conform to the other rules about constant names. Plus, as I said, it's a popular index name, and used as a subscript usually represents an integer. In short, don't call anything else "k" unless you have to.
Lower case Greek letters are variables -- frequently, but not always. If you're going to use one to stand for a constant, though, I think you should stick a subscript on it, same as with x, y, z, and t. If you're in need of a good name for your variable, pick a lower case Greek letter. Of course, many of them already have conventional interpretations within a given field. As I said, "lamda" is wavlength, "nu" is frequency, "omega" is angular frequency, to me. "Iota" isn't used because it looks too much like "i". Lower case "mu" is a magnetic or electric dipole moment (as well as being the symbol for the prefix "micro"). Both lower and upper case "gamma" stand for decay rates, in my field. Lower and upper case "Psi" stand for quantum wavefunctions. Lowercase "alpha" is an absorption or loss coefficient... Those conventions are specific to my subfield. But there are also some broader rules: "theta" and "phi" are angles or phases to almost everyone. Lower case "delta" stuck in front of another variable means "a small change" in that variable, and upper case delta stuck in front a variable means the same thing, or by itself means "the difference between two quantities" (lower case "d" used in calculus to indicate a derivative.) Epsilon means "a small quantity" usually, as can lower case delta when used by itself. That still leaves a lot of letters, though, and I think most math people in all fields do read lower case Greek letters as variables. Again, I especially like "xi" and "zeta" for new variables I'm introducing because they don't already have conventional interpretations.
Use tildes primes, and subscripts to indicate related quantities -- If you start out with some variable "p" and then you introduce a new variable which is p*e^i*omega*t, a good name for the new variable is "p-tilde" or p with a tilde on top. You can also use "p-prime," which is p with an aprostrophe thing after it, though I prefer to reserve this notation for the derivative of p. A third alternative is to stick a subscript on it. For this example I might use "p-subscript-rot" because I think of this as p-rotating-with-the-field. For some letters you can also get away with switching between a Roman letter and a related Greek letter -- go from "r" to "rho" for instance, or from "b" to "beta."
Capital Greek Letters aren't good variable names -- Okay, you can get away with capital Psi, Theta, Gamma, and a few others, but capital "Sigma" is the summation symbol, and does not stand for a quantity at all. Likewise capital "Pi" (which must be written carefully to distinguish it from lower case "pi") means "multiply all of the terms in this sequence." Capital Delta, as mentioned before can be a variable indicating the different between two quantities, but it is usually a label used before a quantity to indicate a change in that quantity. Many of the other capital Greek letters look too much like their Roman counterparts to be useful.
Then there are all the different "hats" -- vector hats, unit vector carets, dots and double dots to indicate derivatives, bars to indicate averages... And specialized notations like Einstein summation indices, the use of parentheses, square brackets, and curly brackets... I would say these all serve a more or less "grammatical" function. (Then there are things like "bra-ket" notation and quantum field theory's "contraction" notation. Those aren't exactly grammatical, they're just a sort of shorthand.)
I'm sure if I surveyed some friends we could come up with a lot more rules like this, but I don't think anyone ever taught them too us formally. We just learned them, the same way you learn the grammar for any language that every one around you is speaking and writing. The rules are as weird and full of exceptions as the rules of English grammar, which is why mathematicians may not like to talk about them -- they like to pretend math is completely logical. But they serve a real purpose, just like English grammar. You try grading the work of a student who doesn't follow them. It's unreadable.
It takes some work to express qualitative statements like "That leaf is yellow" in Math, but if we resort to less than or greater than signs, or even "approximately equal" signs, we can usually do it:
That may need a little context, but the basic meaning is clear if you speak Math. (I could also have given RGB color values or even hex color codes, but I don't know those off the top of my head. In any case, there are usually lots of ways to quantify a quality like "color.")
Of course, it's usually much easier to understand Math if you provide the necessary context in English or another natural language... Rather like programming languages, which are just a dialect of Math really -- commenting your code makes it a lot easier to read.
But you know, some of the need for context is reduced by certain conventions which are a part of the grammar of Math, but which are rarely if ever formally stated. When students screw these up, it makes their work a lot harder to read or grade. I find it interesting that everyone eventually picks up on these conventions, all over the world, but no one ever really talks about them. For instance, above I used the greek letter "lamda" without saying what it meant. But to anyone who has had any contact with the physics community, "lamda" means wavelength. The fact that I talked about color and gave a number on the scale of the wavelength of light makes the interpretation certain. If you're a physics person, I don't need to tell you that I'm talking about wavelength there, and not some kind of computer color code. I should probably specify that the wavelength I'm talking about is that of the light reflected from the leaf, but the subscript "Leaf" and the context are again probably enough for you to guess that too.
Here are some of the unwritten rules that come immediately to mind:
Letters from the beginning of the Roman alphabet -- these are usually constants, especially if they are upper case.
Upper Case Roman Letters - these also frequently indicate a constant quantity, especially if the letter is from the beginning of the alphabet. Capital "C" and capital "K" are very popular choices to represent a constant because "constant" starts with a "C" in English and a "K" in German. Upper case Roman letters may also stand for matrices or tensors -- Usually you can tell which is which, because if it's a matrix or a tensor, all of the other terms in the equation will also be a matrices or tensors, so that knowing one symbol gives you hints about the others. (Also, I like to put little upside-down caret hats on my matrices. Most people use right-side-up carets, but I like to reserve those for unit vectors.) If you see a capital X, Y, or Z, however, you know it's likely to be a matrix, because those are almost never used for constants. Which brings me to my next rule.
The symbols x, y, z, and t are reserved for position and time - These are pretty much always variables, and nearly always stand for quantities with units of length, or in the case of t, of time. Sometimes students use "x" to stand for other quantities, because it's a popular choice to stand for any unknown quantity in high school algebra classes, but in the world of math-using professionals, this is a bad idea. Using "x" for a quantity that does not have units of length will confuse people unless you make it very clear what it does stand for. And even then, there are usually better, more conventional choices. If, for instance, the quantity is a pressure, use a "p"(preferably lower-case). If it's a temperature, go for the upper case "T," because the upper-case-letter-means-constant rule is not as firm as the lower-case-t-means-time rule. If it's a frequency, use "f" or "nu"... Most things have their own conventional symbols, in other words, specifically to avoid the problem of using "x" for everything. Save that for position. If there really is no conventional symbol for your quantity, use the lower case Greek letter xi. That's one of few letters that doesn't, by convention, already stand for something else (at least not in physics)
If it's a constant with units of length or time, use subscripts -- the unadorned x, y, z, and t are read as variables, but if you stick a subscripted number (not a letter) on them, they will be read as constants. A good choice is x-subscript-0 or t-subscript-1. In fact, almost anything with a suscripted number will be read as a constant, regardless of the other rules in this list.
Letters from the middle of the alphabet are integers -- In particular, do not use upper or lower case "N" to mean something that is not an integer, because it will confuse people. The letter "m", especially in lower case, is almost always an integer too. In computer science, so are "i", "j", and "k". In physics, however, it is not wise to use these symbols for integers unless they are appearing as an index or subscript. That's because these symbols have other meanings already. ("o" is never used because it looks just like zero.)
Lower case i means square root of negative one -- you can't use lower case "i" for any variable, because its meaning is already assigned. (Likewise that of "e" and lower case "pi" which stand for the natural base of logarithms, and the ratio of a circle's circumference to its diameter, respectively.) The exception to the "i" rule is if you are an electrical engineer, in which case you may use lower case "i" for current if you must. (I have no idea why "I" is the conventional symbol for current -- physicists use it too, but in upper case). Electrical engineers use "j" for the square root of negative one. As for "k" -- that is a seriously overused letter. It has seemingly dozens of conventional uses. It is often used for Boltzmann's constant, though I prefer to give it a subscript for that. In my world it is the wavenumber of light. But it may also be the so-called "lattice constant" for a crystal, a related but different concept. People use it for constants because the German "konstant" is spelled with a "k" even though it does not conform to the other rules about constant names. Plus, as I said, it's a popular index name, and used as a subscript usually represents an integer. In short, don't call anything else "k" unless you have to.
Lower case Greek letters are variables -- frequently, but not always. If you're going to use one to stand for a constant, though, I think you should stick a subscript on it, same as with x, y, z, and t. If you're in need of a good name for your variable, pick a lower case Greek letter. Of course, many of them already have conventional interpretations within a given field. As I said, "lamda" is wavlength, "nu" is frequency, "omega" is angular frequency, to me. "Iota" isn't used because it looks too much like "i". Lower case "mu" is a magnetic or electric dipole moment (as well as being the symbol for the prefix "micro"). Both lower and upper case "gamma" stand for decay rates, in my field. Lower and upper case "Psi" stand for quantum wavefunctions. Lowercase "alpha" is an absorption or loss coefficient... Those conventions are specific to my subfield. But there are also some broader rules: "theta" and "phi" are angles or phases to almost everyone. Lower case "delta" stuck in front of another variable means "a small change" in that variable, and upper case delta stuck in front a variable means the same thing, or by itself means "the difference between two quantities" (lower case "d" used in calculus to indicate a derivative.) Epsilon means "a small quantity" usually, as can lower case delta when used by itself. That still leaves a lot of letters, though, and I think most math people in all fields do read lower case Greek letters as variables. Again, I especially like "xi" and "zeta" for new variables I'm introducing because they don't already have conventional interpretations.
Use tildes primes, and subscripts to indicate related quantities -- If you start out with some variable "p" and then you introduce a new variable which is p*e^i*omega*t, a good name for the new variable is "p-tilde" or p with a tilde on top. You can also use "p-prime," which is p with an aprostrophe thing after it, though I prefer to reserve this notation for the derivative of p. A third alternative is to stick a subscript on it. For this example I might use "p-subscript-rot" because I think of this as p-rotating-with-the-field. For some letters you can also get away with switching between a Roman letter and a related Greek letter -- go from "r" to "rho" for instance, or from "b" to "beta."
Capital Greek Letters aren't good variable names -- Okay, you can get away with capital Psi, Theta, Gamma, and a few others, but capital "Sigma" is the summation symbol, and does not stand for a quantity at all. Likewise capital "Pi" (which must be written carefully to distinguish it from lower case "pi") means "multiply all of the terms in this sequence." Capital Delta, as mentioned before can be a variable indicating the different between two quantities, but it is usually a label used before a quantity to indicate a change in that quantity. Many of the other capital Greek letters look too much like their Roman counterparts to be useful.
Then there are all the different "hats" -- vector hats, unit vector carets, dots and double dots to indicate derivatives, bars to indicate averages... And specialized notations like Einstein summation indices, the use of parentheses, square brackets, and curly brackets... I would say these all serve a more or less "grammatical" function. (Then there are things like "bra-ket" notation and quantum field theory's "contraction" notation. Those aren't exactly grammatical, they're just a sort of shorthand.)
I'm sure if I surveyed some friends we could come up with a lot more rules like this, but I don't think anyone ever taught them too us formally. We just learned them, the same way you learn the grammar for any language that every one around you is speaking and writing. The rules are as weird and full of exceptions as the rules of English grammar, which is why mathematicians may not like to talk about them -- they like to pretend math is completely logical. But they serve a real purpose, just like English grammar. You try grading the work of a student who doesn't follow them. It's unreadable.
Thursday, November 13, 2008
The Post Copyright World
So I've thought for a little while that copyright is more or less going away whether we like it or not. How, then, do creative people make money by entertaining us? The same way they did before audio recording or the printing press were invented -- live performances and generous patrons.
But the patrons don't just have to be wealthy aristocrats these days. I think what happens is, you release a couple of titles -- books, albums, whatever, for free. Build up a fanbase. Then you hold the next work hostage. You set a fundraising target, and you release the work once that target has been met. If you have millions of fans, your target can be millions of dollars. All of them will contribute five or ten. If you have a couple hundred fans, your target can be a couple hundred dollars. Someone who's really eager will contribute extra, or talk their friends into joining in, to get the next work out that much faster.
These schemes are starting to show up, and I'm interested to see whether they take off.
Here's a site where you fund investigative journalism. Once a story gets enough funding, the investigation gets done. I might actually try this.
One of my favorite essayists, the Real Live Preacher Gordon Atkinson, has proposed to publish his next book this way.
Maybe I can get research funding this way. Want me to work on this idea for a fiber optic vibration sensor I've got? Send me five bucks. I'll start when I get to $10,000...
But the patrons don't just have to be wealthy aristocrats these days. I think what happens is, you release a couple of titles -- books, albums, whatever, for free. Build up a fanbase. Then you hold the next work hostage. You set a fundraising target, and you release the work once that target has been met. If you have millions of fans, your target can be millions of dollars. All of them will contribute five or ten. If you have a couple hundred fans, your target can be a couple hundred dollars. Someone who's really eager will contribute extra, or talk their friends into joining in, to get the next work out that much faster.
These schemes are starting to show up, and I'm interested to see whether they take off.
Here's a site where you fund investigative journalism. Once a story gets enough funding, the investigation gets done. I might actually try this.
One of my favorite essayists, the Real Live Preacher Gordon Atkinson, has proposed to publish his next book this way.
Maybe I can get research funding this way. Want me to work on this idea for a fiber optic vibration sensor I've got? Send me five bucks. I'll start when I get to $10,000...
Wednesday, November 05, 2008
Celebrations
I've been reading comment threads, and at least one person on Metafilter said just how I feel...
Also from MetaFilter:
I love The Wire. But I've only seen the first four seasons. So far Carcetti's not so bad. What I can't help thinking of what one of the former Baltimore mayors told him being mayor was like. Something like "Every day you go into your office and eat one bowl of shit after another -- it just keeps coming, in beautiful golden bowls". I'm afraid that's what Obama's got in front him him now...
A comment I left here but want to repeat on my own blog:
And Obama himself, as usual, captures the mood:
"AND THEN THE EWOKS ROCK OUT"
posted by middleclasstool at 8:15 PM on November 4
Also from MetaFilter:
"John McCain should concede more often, he's really good at it!"
posted by finite at 8:25 PM on November 4
"Wisconsin is declared in our hearts."
posted by drezdn at 8:51 PM on November 4
"Gun Owner For Obama!! I'm so happy, I won't even go out and shoot randomly into the air in celebration!"
posted by Balisong at 8:53 PM on November 4
"Thank you. Thank you. Thank you.
Sincerely,
Rest of the World."
posted by hoskala at 8:56 PM on November 4
"Truly the end of an error."
posted by punkfloyd at 5:03 AM on November 5
"...
But I'm still thrilled that he won. We now have a president who said "The Wire" is his favorite show and, for some insane reason, this gives me great hope for the future. Absurd really, the things we pin our hopes on.
Just dear god/cthulhu/fsm, please don't let Obama turn into a Carcetti."
posted by pandaharma at 1:04 AM on November 5
I love The Wire. But I've only seen the first four seasons. So far Carcetti's not so bad. What I can't help thinking of what one of the former Baltimore mayors told him being mayor was like. Something like "Every day you go into your office and eat one bowl of shit after another -- it just keeps coming, in beautiful golden bowls". I'm afraid that's what Obama's got in front him him now...
A comment I left here but want to repeat on my own blog:
I had a cool experience Friday night. I was on an overnight bus out of Chicago, and most of the other people on the bus were black. The driver on the PA system should've been a DJ. He was making jokes, telling stories, and wishing us all a smooth ride... And somewhere in there he said "It's gonna be a whole new world out there after Tuesday, you just watch," and the whole (double decker) bus spontaneously cheered, and the little eleven year old girl sitting next to me was grinning so hard I thought she was going to break her face. And the driver went on about how the city of Chicago was just going to go nuts, and what a great thing it was for the city... But he personally was gonna stay as far away as he could get, because the traffic was gonna be a nightmare.
Late night, bus full of strangers, all so excited, all feeling like the world is about to change, and for once, like they're the ones changing it... Pretty unforgettable.
And Obama himself, as usual, captures the mood:
A man touched down on the moon, a wall came down in Berlin, a world was connected by our own science and imagination. And this year, in this election, [a black woman who remembers when black people and women weren't allow to vote] touched her finger to a screen, and cast her vote, because after 106 years in America, through the best of times and the darkest of hours, she knows how America can change. Yes we can.
Tuesday, November 04, 2008
Why I like Barack Obama, Part II
Everything from Part I, which I posted here four years ago(!) still applies. But there's something I wanted to add.
I'm happy that both candidates this year represent a significant improvement over the incumbent. But I'm even happier that one of them is a deeply thoughtful, introspective man. I think it's amazing that this such a man is also the most charismatic candidate in a generation.
I don't agree with Obama on everything. For instance, he favors removing restrictions on abortion. I do not. He favors policies which flirt with protectionism. I think that's a losing battle and hard on the very poor in other nations. I believe the death penalty should be abolished. He believes the system can be fixed. He believes we're going to need to rely more on nuclear power. I'm divided on the issue. I believe it is safe, but we don't really have a solution for disposing of the waste. There's not much of it now, but it's not going away either, and if we rely on it too heavily, it will continue accumulating until it's a problem for our great-grandchildren the way global warming is a problem for us.
But here's the thing. I believe Obama has thought very hard about every one of these issues, and is aware of the other side of the argument in each case. I believe he sees these as complex problems. I believe this because I have seen him reason his way to a conclusion in a debate or a speech (with numbered arguments, no less), and because I have read his first book, which is full of searching and self doubt.
More than I want someone who agrees with me, I want someone who thinks about problems and struggles with them and sees them from all sides.
And that is what I see in Obama. Someone who is not afraid to consider the possibility that he might be wrong. Someone who who would govern with his head, not his gut. I am content to let others make my policy decisions for me, as I must be under a representative government, but I want to know that those people have thought about the decisions they are making more than I have, not less.
And that's what I see in Barack Obama.
I'm happy that both candidates this year represent a significant improvement over the incumbent. But I'm even happier that one of them is a deeply thoughtful, introspective man. I think it's amazing that this such a man is also the most charismatic candidate in a generation.
I don't agree with Obama on everything. For instance, he favors removing restrictions on abortion. I do not. He favors policies which flirt with protectionism. I think that's a losing battle and hard on the very poor in other nations. I believe the death penalty should be abolished. He believes the system can be fixed. He believes we're going to need to rely more on nuclear power. I'm divided on the issue. I believe it is safe, but we don't really have a solution for disposing of the waste. There's not much of it now, but it's not going away either, and if we rely on it too heavily, it will continue accumulating until it's a problem for our great-grandchildren the way global warming is a problem for us.
But here's the thing. I believe Obama has thought very hard about every one of these issues, and is aware of the other side of the argument in each case. I believe he sees these as complex problems. I believe this because I have seen him reason his way to a conclusion in a debate or a speech (with numbered arguments, no less), and because I have read his first book, which is full of searching and self doubt.
More than I want someone who agrees with me, I want someone who thinks about problems and struggles with them and sees them from all sides.
And that is what I see in Obama. Someone who is not afraid to consider the possibility that he might be wrong. Someone who who would govern with his head, not his gut. I am content to let others make my policy decisions for me, as I must be under a representative government, but I want to know that those people have thought about the decisions they are making more than I have, not less.
And that's what I see in Barack Obama.
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