I'm greatly intrigued by a recent pair of blog posts on economics. One is by the progressive economist Paul Krugman, the other a response to Krugman by Austrian school economist Steven Horwitz. One of these economists argues that the market economy is essentially an amoral structure, while the other argues that morality is inseparable from economics. Try to guess which one said which!
Thursday, September 30, 2010
Pro-life orthodoxy
A recent article from LifeSiteNews.com, a comprehensively right-wing news site, tries to nail Obama for hypocrisy:
Following reports of widespread skepticism over his professed Christianity, President Obama on Tuesday invoked the teachings of Jesus Christ as the inspiration for his public agenda, which he called part of an "effort to express my Christian faith" - and in his next breath defended the legalized killing of unborn children.
Wednesday, September 29, 2010
Responding to Mike Lux...four months later
You know, I never actually think of this blog being read anywhere else but on my own laptop. This is why it never occurred to me that when I wrote a critique of an article about progressives and Christianity, there would actually be a response! You can read the original article here, my critique here, and the response here. Who knew I'm actually a "conservative writer" worthy of a thorough response? In any case, sorry Mike Lux, if I had known you responded way back in May, I probably would've responded sooner. I'm sure people have been just dying to know what I have to say.
I thought it would be a good exercise, after four months of further reflection on economics and political philosophy, to respond to Mr. Lux, point by point. Not that I feel compelled to enter into a full scale debate with him; it just seems as good a way to sort out the relevant issues as any. I think what you'll see is that Mr. Lux has not been able to cross the fundamental divide between the way he and I see political life. Although I have great admiration for him to speak up for a Christian progressive political philosophy, I just don't think he's been successful at questioning his basic assumptions about what the issues really are.
I thought it would be a good exercise, after four months of further reflection on economics and political philosophy, to respond to Mr. Lux, point by point. Not that I feel compelled to enter into a full scale debate with him; it just seems as good a way to sort out the relevant issues as any. I think what you'll see is that Mr. Lux has not been able to cross the fundamental divide between the way he and I see political life. Although I have great admiration for him to speak up for a Christian progressive political philosophy, I just don't think he's been successful at questioning his basic assumptions about what the issues really are.
Monday, September 27, 2010
Golden Calf
Specifically, the picture on the left (which is the work of Nicolas Poussin) became a platform on which Mitchell built a critique of the Second Commandment. There doesn't seem to be anything immoral going on in this picture, says Mitchell. The people are just enjoying themselves in front of a "totem" they have made. So maybe Moses should just drop his tablets and join the fun, eh?
Sunday, September 26, 2010
Molecular Individualists
Sheldon Richman over at the Freeman has written an article, not so much defending the Tea Party movement, but attacking their detractors. His criticism is sound. The Tea Party detractors, he observes, are defending a view of political life in which the individual must submit their livelihood to a core group of experts and aristocrats. These detractors are, he says, "anti-anti-authoritarian."
If there's one thing that has me interested in libertarian political philosophy, it is this critique of central authority. It is not that we don't need central authority (Richman comments on how ironic it is that the label "anarchist" is applied to those who think the government should limit its powers to those given by the Constitution!). Rather, central authority simply ought to have a limited sphere of influence. It ought to appear self-evident that nothing can be worse for a free society than for every part of life to be politicized.
If there's one thing that has me interested in libertarian political philosophy, it is this critique of central authority. It is not that we don't need central authority (Richman comments on how ironic it is that the label "anarchist" is applied to those who think the government should limit its powers to those given by the Constitution!). Rather, central authority simply ought to have a limited sphere of influence. It ought to appear self-evident that nothing can be worse for a free society than for every part of life to be politicized.
Friday, September 24, 2010
Numbers
One of the most profound thing about numbers is how absolute they are. That is, they are absolute in their relationship to one another. "Two plus two is always four" is kind of a standard thing people say when they want to make a reference to the fact that certain truths are absolute.
I believe that by meditating on certain numbers we can better understand the world we live in. With our minds we can pierce through the chaos and confusion of a world where words often fail to have absolute meaning, and reach for a world of fact and reason. It is here that we will find liberty.
So what numbers am I meditating on?
I believe that by meditating on certain numbers we can better understand the world we live in. With our minds we can pierce through the chaos and confusion of a world where words often fail to have absolute meaning, and reach for a world of fact and reason. It is here that we will find liberty.
So what numbers am I meditating on?
- 730 billion - that's 730,000,000,000 - the number of dollars the federal government will spend on Social Security in 2011
- 491 billion - that's 491,000,000,000 - the number of dollars they will spend on Medicare
- 297 billion - that's 297,000,000,000 - the number of dollars they will spend on Medicaid
- 549 billion - that's 549,000,000,000 - the number of dollars they will spend on the Department of Defense
- 719 billion - that's 719,000,000,000 - the number of dollars they will spend on national security, including defense plus other agencies
- 3.834 trillion - that's 3,834,000,000,000 - the number of dollars they will spend total
- 1.267 trillion - that's 1,267,000,000,000 - the number of dollars they will spend over-budget, that is, the deficit they expect to accrue for the year 2011.
- 15.299 trillion - that's 15,299,000,000,000 - the expected GDP of the United States in the year 2011
- 10.498 trillion - that's 10,498,000,000,000 - the anticipated public debt owed by the United States in the year 2011.
Tuesday, September 21, 2010
Religion influencing abortion views
LifeNews.com reports on a recent Pew poll that gives the shocking revelation that religion has some influence on American opinions about abortion.
Let's think about this. Is it really good for the pro-life cause to place itself more and more firmly in the conservative religious camp? Do we really need to cloud the arguments of plain reason with a storm of religious vocabulary that is foreign and unnatural to many American minds?
But then, the questions that polls ask are so ill-defined. What does it mean when someone responds that religion is the most important factor in determining her views on abortion? Does that mean that person accepts whatever the Church teaches on the subject? Does it mean that person reads the Bible critically and accepts whatever she comes up with herself? Does it mean that her religion causes her to believe in certain principles that lead her logically to accept the pro-life stance on abortion?
Sometimes, I think the only hope for democracy is to ban all polls.
More pertinently, I think it's dangerous for the pro-life cause to draw its primary support from the teachings of religion. My issue is not with Christians having Christian ways of looking at the world; rather, my issue is finding a common language to talk about abortion. For most of the world, the language of evangelical Christianity is foreign, and it does not help so much as hinder dialog. We cannot afford to remove ourselves from critical conversations going on in the world at large.
Most disturbing to me is the way activist sites like LifeNews seem to pander to the religious right for the sake of winning elections. This seems like sacrificing the long run for a few short gains. Religious sentiment may be very useful at getting a certain constituency to the polls, but this does not change cultural beliefs or practices on the issue of abortion.
So what about that 55%? Are there any pro-life activist groups willing to listen to them?
I mean, besides the obvious.
A new Pew poll released on Friday shows Americans continue to say their religious beliefs have been highly influential in shaping their views about social issues, including abortion. The way in which religious beliefs play a role in shaping abortion views is more strong [sic] than for other political issues.Unless I am simply being uncharitable (and I confess I have been a bit grumpy in the past few days) LifeNews.com may be too nearsighted to realize the flip-side of the statistic they just quoted. That is, 55% of abortion opponents said religion isn't the most important influence on their opinion.
On the issue of abortion, 26% overall say religion is the most important influence on their opinion, including 45% among abortion opponents.
Just 9 percent of those who support legalized abortion say religion affected their conclusion about it.
Let's think about this. Is it really good for the pro-life cause to place itself more and more firmly in the conservative religious camp? Do we really need to cloud the arguments of plain reason with a storm of religious vocabulary that is foreign and unnatural to many American minds?
But then, the questions that polls ask are so ill-defined. What does it mean when someone responds that religion is the most important factor in determining her views on abortion? Does that mean that person accepts whatever the Church teaches on the subject? Does it mean that person reads the Bible critically and accepts whatever she comes up with herself? Does it mean that her religion causes her to believe in certain principles that lead her logically to accept the pro-life stance on abortion?
Sometimes, I think the only hope for democracy is to ban all polls.
More pertinently, I think it's dangerous for the pro-life cause to draw its primary support from the teachings of religion. My issue is not with Christians having Christian ways of looking at the world; rather, my issue is finding a common language to talk about abortion. For most of the world, the language of evangelical Christianity is foreign, and it does not help so much as hinder dialog. We cannot afford to remove ourselves from critical conversations going on in the world at large.
Most disturbing to me is the way activist sites like LifeNews seem to pander to the religious right for the sake of winning elections. This seems like sacrificing the long run for a few short gains. Religious sentiment may be very useful at getting a certain constituency to the polls, but this does not change cultural beliefs or practices on the issue of abortion.
So what about that 55%? Are there any pro-life activist groups willing to listen to them?
I mean, besides the obvious.
Monday, September 20, 2010
Well, that's good to know
Sue Blackmore, in a friendly article entitled, "Why I no longer believe religion is a virus of the mind," has this to say about her recent discovery:Are religions viruses of the mind? I would have replied with an unequivocal "yes" until a few days ago when some shocking data suggested I am wrong.What strength and courage it takes to overcome one's previous assumptions.
...
The idea is that religions, like viruses, are costly to those infected with them. They demand large amounts of money and time, impose health risks and make people believe things that are demonstrably false or contradictory. Like viruses, they contain instructions to "copy me", and they succeed by using threats, promises and nasty meme tricks that not only make people accept them but also want to pass them on.
This was all in my mind when Michael Blume got up to speak on "The reproductive advantage of religion". With graph after convincing graph he showed that all over the world and in many different ages, religious people have had far more children than nonreligious people.
...
So it seems I was wrong and the idea of religions as "viruses of the mind" may have had its day.
Instead of "virus," it sounds like we can now conclude religion is more like a "bacterium of the mind."
Religions still provide a superb example of memeplexes at work, with different religions using their horrible threats, promises and tricks to out-compete other religions, and popular versions of religions outperforming the more subtle teachings of the mystical traditions. But unless we twist the concept of a "virus" to include something helpful and adaptive to its host as well as something harmful, it simply does not apply. Bacteria can be helpful as well as harmful; they can be symbiotic as well as parasitic, but somehow the phrase "bacterium of the mind" or "symbiont of the mind" doesn't have quite the same ring.She has a point at the end, there. Although "Symbiont of the Mind" might make a great name for a math rock band.
One has to marvel at how this "shocking data" displayed in the form of a few graphs can shift a scientist's ardent beliefs about religion, while thousands of years of theological contemplation and discussion are irrelevant. But then again, this is coming from someone who admits she once had to let go of her earnest belief in the paranormal.
I guess what I have to get used to in this whole "dialog" is that there are plenty of smart people who aren't worth taking seriously.
Sunday, September 19, 2010
Calvin on the Papacy
Oh, man, it's time to lay the smack down.
That's right, today's blog entry from my reading of Calvin's Institutes of the Christian Religion comes after having read chapters IV - VIII in Book IV. Basically all of these chapters (and a few chapters after these) unify around one central theme: the corruption of the papacy.
The whole work up until now has been pretty sharp in its criticism of numerous theological beliefs and religious practices (not all of which come from the Roman Catholics). Yet this part of the Institutes is by far the most polemical. Why don't we just review some of the chapter titles?
How have I been confronting this, as a modern reader? Generally speaking, the way Calvin deals with theological controversy has made me uncomfortable. It's hard in my context to think in such black and white terms about the "papists."
And yet, this section of the book has honestly been profoundly eye-opening. It's quite astonishing to read someone with first-hand experience of that time talk about the corruption of the papacy. Chapter V, in particular, reviews some unbelievably corrupt practices in the church: lawsuits over pastoral ordination, "beneficed" and hired priests, bishops who did nothing for their parishes but collected money from them, priests engaged in all kinds of immoral behavior--my favorite part was about children being appointed as bishops! For all the theological points that the Reformers were eager to make, I wonder if anything close to the Reformation would have ever occurred had it not been for the sheer corruption of the church at that time?
It sounds as if even most Catholics of our day will admit that what was going on back then was an abomination, so perhaps this is nothing new. But I did learn a lot more than this. Calvin develops a pretty deep historical argument in defense of a Reformed view of church government, as well as an impressive amount of historical evidence that originally the bishop of Rome had nothing like his current place in the Roman Catholic Church. I have heard a lot of Catholic and even Eastern Orthodox apologists talk as if Protestants think the church was just corrupt from the time of Constantine until finally the Reformation came along. That's certainly not a fair reading of Calvin.
Although I don't have time to list the relevant quotes from this chapters, Calvin's view of the early church is nuanced and respectful. It is accurate to say that he finds much of the church in decline from around the 6th century onward, but he also finds that corruption crept in slowly, rather than immediately. I think an interesting aspect of his argument to consider is his treatment of Leo I and Gregory I. He uses the words of both of these figures at times to defend his argument against the papacy (see, e.g., Ch. VII Sec. 4). Yet at other times he highlights their weaknesses; I believe at one point he calls Leo a man of "ambition." It is clear in my mind that Calvin had a very nuanced view of all these early Christian leaders. You can see his theology at work here: we are all corrupt to some degree or another, and are continually depending on God's grace to sanctify us. It should be no surprise that this goes for leaders as well.
Chapter VIII is where the theological battleground is still present today. The Roman Catholic Church, as I understand it, would still more or less claim that the Church has some degree or other of infallibility--that is, the Tradition (with a capital "T") of the Church is sacred and without error. To this Calvin responds that the Church is only infallible insofar as it perseveres in its charge to uphold the Word of God. It is the "pillar and ground of the truth" in the sense that it is called to preserve the truth of Scripture--in a sense he is saying that the Bible is the truth, and the Church is its pillar and ground. For Calvin, any proclamation that comes from outside of Scripture can't be authoritative.
Essentially Calvin's doctrine of individual sanctification applies to the Church, as well. The Church is constantly bearing corruption, and must constantly seek God's grace to heal it. There is nothing truly infallible about the Church itself. To be honest, this is something I agree with perhaps more than anything else in all of Calvin. This is why I am a Protestant--the Church is simply not infallible.
There is one comment that comes in Section 15 of Chapter VIII that I found particularly interesting:
For Protestants, the Bible functioned as a source of freedom long before the U.S. Constitution was written. What was written could not be annulled by power structures created by mere human beings. There is certainly something liberating about having the written word transcending human tradition. I have written at other times about the problems lurking underneath this freedom, but I have to say that at Calvin's moment in history, it makes sense that he clung to the supremacy of the written Scriptures over the tradition of the Catholic Church.
Interesting and provocative stuff. I think next time I post on Calvin, it might be "Calvin on the Papacy, Part II."
That's right, today's blog entry from my reading of Calvin's Institutes of the Christian Religion comes after having read chapters IV - VIII in Book IV. Basically all of these chapters (and a few chapters after these) unify around one central theme: the corruption of the papacy.
The whole work up until now has been pretty sharp in its criticism of numerous theological beliefs and religious practices (not all of which come from the Roman Catholics). Yet this part of the Institutes is by far the most polemical. Why don't we just review some of the chapter titles?
- Chapter V: The Ancient Form of Government Was Completely Overthrown by the Tyranny of the Papacy
- Chapter VII: The Origin and Growth of the Roman Papacy Until It Raised Itself to Such a Height that the Freedom of the Church Was Oppressed, and All Restraint Overthrown
- Chapter VIII: The Power of the Church with Respect to Articles of Faith; and How in the Papacy, with Unbridled License, the Church Has Been Led to Corrupt All Purity of Doctrine
How have I been confronting this, as a modern reader? Generally speaking, the way Calvin deals with theological controversy has made me uncomfortable. It's hard in my context to think in such black and white terms about the "papists."
And yet, this section of the book has honestly been profoundly eye-opening. It's quite astonishing to read someone with first-hand experience of that time talk about the corruption of the papacy. Chapter V, in particular, reviews some unbelievably corrupt practices in the church: lawsuits over pastoral ordination, "beneficed" and hired priests, bishops who did nothing for their parishes but collected money from them, priests engaged in all kinds of immoral behavior--my favorite part was about children being appointed as bishops! For all the theological points that the Reformers were eager to make, I wonder if anything close to the Reformation would have ever occurred had it not been for the sheer corruption of the church at that time?
It sounds as if even most Catholics of our day will admit that what was going on back then was an abomination, so perhaps this is nothing new. But I did learn a lot more than this. Calvin develops a pretty deep historical argument in defense of a Reformed view of church government, as well as an impressive amount of historical evidence that originally the bishop of Rome had nothing like his current place in the Roman Catholic Church. I have heard a lot of Catholic and even Eastern Orthodox apologists talk as if Protestants think the church was just corrupt from the time of Constantine until finally the Reformation came along. That's certainly not a fair reading of Calvin.
Although I don't have time to list the relevant quotes from this chapters, Calvin's view of the early church is nuanced and respectful. It is accurate to say that he finds much of the church in decline from around the 6th century onward, but he also finds that corruption crept in slowly, rather than immediately. I think an interesting aspect of his argument to consider is his treatment of Leo I and Gregory I. He uses the words of both of these figures at times to defend his argument against the papacy (see, e.g., Ch. VII Sec. 4). Yet at other times he highlights their weaknesses; I believe at one point he calls Leo a man of "ambition." It is clear in my mind that Calvin had a very nuanced view of all these early Christian leaders. You can see his theology at work here: we are all corrupt to some degree or another, and are continually depending on God's grace to sanctify us. It should be no surprise that this goes for leaders as well.
Chapter VIII is where the theological battleground is still present today. The Roman Catholic Church, as I understand it, would still more or less claim that the Church has some degree or other of infallibility--that is, the Tradition (with a capital "T") of the Church is sacred and without error. To this Calvin responds that the Church is only infallible insofar as it perseveres in its charge to uphold the Word of God. It is the "pillar and ground of the truth" in the sense that it is called to preserve the truth of Scripture--in a sense he is saying that the Bible is the truth, and the Church is its pillar and ground. For Calvin, any proclamation that comes from outside of Scripture can't be authoritative.
Essentially Calvin's doctrine of individual sanctification applies to the Church, as well. The Church is constantly bearing corruption, and must constantly seek God's grace to heal it. There is nothing truly infallible about the Church itself. To be honest, this is something I agree with perhaps more than anything else in all of Calvin. This is why I am a Protestant--the Church is simply not infallible.
There is one comment that comes in Section 15 of Chapter VIII that I found particularly interesting:
And I should not seem too quarrelsome because I insist so strongly that the church is not permitted to coin any new doctrine, that is, to teach and put forward as an oracle something more than the Lord has revealed in his Word. For sensible men see how perilous it is if men once be given such authority. They also see how great a window is opened to the quips and cavils of the impious if we say that what men have decided is to be taken as an oracle among Christians.What if we view the political trend toward constitutional government in light of this sentiment? The written word is often the source of freedom for people. Indeed, that has always been the case in the United States, where our rights as citizens are enshrined in a written document that transcends the will of politicians. (We can only hope that these rights will continue to be enforced.)
For Protestants, the Bible functioned as a source of freedom long before the U.S. Constitution was written. What was written could not be annulled by power structures created by mere human beings. There is certainly something liberating about having the written word transcending human tradition. I have written at other times about the problems lurking underneath this freedom, but I have to say that at Calvin's moment in history, it makes sense that he clung to the supremacy of the written Scriptures over the tradition of the Catholic Church.
Interesting and provocative stuff. I think next time I post on Calvin, it might be "Calvin on the Papacy, Part II."
Saturday, September 18, 2010
Training skeptics
This semester my teaching experience has been much different than last year. Whereas last year I taught a "throw-away" calculus course designed to fulfill a requirement for certain majors (but not designed to prepare students for any mathematics beyond calculus), this semester I'm teaching an honors course in multivariable calculus specifically designed to train students for higher level mathematics. Hence one of the essential parts of the course is teaching students to write proofs.
There is a certain level at which young mathematicians regard "proofs" almost as a specific area of math. They've learned algebra, trigonometry, calculus, and now they're learning "proofs." If they continue on in mathematics, they can't continue to think this way for long. Proof is not a part of mathematics. Proof is mathematics.
On the list of ways in which we teach mathematics badly in American schools, I would certainly add that we teach students that writing is not a part of mathematics. Writing is part of everything. If you can't write it, you can't communicate it. Perhaps part of the problem is that most people (even those who teach mathematics in high school and below) don't even realize mathematics needs to be communicated. Why would you need to convince someone that something is true in mathematics? Hasn't it all been figured out already? I kid you not--I would not be surprised if half the people I meet don't realize there's still being research done in mathematics. As in there's still math we don't know!
Since we teach students to do math without actually writing sentences, we also implicitly teach them that math is nothing more than a series of arcane symbolic manipulations that magically results in an answer. Many students will simply never grasp the idea that mathematics is really a set of questions that real people have asked out of genuine curiosity, and have answered purely through deductive reasoning. No magic necessary.
My students are a little sharper than that, but many of them still have a rough transition to make. They know how to get the answer, but they're totally new to actually writing mathematics. Thus if they were asked to prove that if x is a number other than 1, then 1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1), they might give me an argument that looks like this:
writing a check mark beside the last line. It seems like a legitimate way to argue, because the symbolic manipulations are exactly what they've been trained to do all through their previous math education, particularly when they're trying to solve equations. And the symbolic manipulations generate something that's true! So it must be right.
There is a sense in which this argument is correct. If the student were to indicate that each line is logically equivalent to the line before it, then I suppose the argument would work. But it still wouldn't feel like good style. Everyone who is trained in higher mathematics (or in logic) understands this basic fact about a good argument: you don't start with what you're trying to prove. You start with your hypothesis, then argue step by step to the conclusion.
A reasonable argument of the proposition I just mentioned might simply reverse the lines of the poorer argument I gave, but that would be rather clunky. A good argument would actually use words! It would be much easier and clearer to simply write the following:
Here's what I tell my students: try to write as if you're trying to convince the most skeptical person in the world. Every line you write has to be 100% convincing. Starting with what you're trying to prove will never satisfy a skeptic, because a skeptic knows that you can get anything to be true if you just start by assuming it to be true. For instance, let's say I want to prove that -1 = 1. Well, my argument would be rather short and sweet:
Since what I got in the end is clearly true, the argument must work, right? Well, of course not, because -1 does not equal 1. But that's basically how students will argue when they're first starting to write proofs in mathematics. They've never had to convince a skeptic before. All they had to do was convince their teachers, who really just wanted to see a bunch of symbolic manipulations that magically resulted in an answer. In other words, all they had to do before was computations. Now they actually have to write arguments.
I have a great deal of uneasiness about the whole process of teaching good mathematical writing. I can tell them all day what's wrong with their proofs, and they might write little notes to themselves to try to figure out what I want to see on their homework. But that's the last thing I want them to learn. I don't want them to be able to convince me. I want them to be able to convince anyone who understands the symbols being used. There's no precise way to say this, but what I'm really going for is that they would be able to convince reason itself. Somehow they have to transcend the personal motives of finding acceptance from their teacher and getting good grades. They have to develop an innate desire to critique their own arguments, and write something about which they can confidently say, This is simply irrefutable.
Of course, mathematics is never really exactly like that in the real world of research. Mistakes are made. In the rush to get results published, sometimes mathematicians have indeed overlooked important details. But members of the mathematical community are constantly attempting to hold each other to that rigorous standard of irrefutable proof. This principle is emulated in all the sciences, yet mathematics has the privilege of working by pure logic. There is no "methods" section of a research paper in mathematics.
What an interesting sort of community this is, though. We're not a community of experimentalists, who argue over the analysis of data. We're not a community of theologians or philosophers, who argue over the meanings of words or doctrines. In fact, arguments between mathematicians are rarely over the actual content of what they study. We can argue over what the best method of proof is (though such arguments can be absurd), or we can argue over what we think the solution to a problem will be (though such arguments are rendered somewhat meaningless once the problem is actually solved). (There are also plenty of arguments between mathematicians mainly concerning their own egos; in that our community is not at all unique.)
If all the theologians or philosophers in the world were asked to reach a conclusive solution to a problem in one of these disciplines, they would only succeed in dividing into several camps. Yet do the same thing for all the mathematicians in the world, and all of them, working entirely independently from one another, can still reach the same solution. There will be no disagreement on the final result. Of all the other academic disciplines, the scientific community is most similar to this; yet the scientific community routinely has to correct past theories.
Does that make the mathematics community somehow ideal, which the whole world ought to emulate? Not at all. Mathematics progresses through skepticism; society in general does not. Part of skepticism is the ability to say, "I don't know." In the case of mathematics, we can say that for quite a long time--many famous problems have taken centuries to solve. It would be absurd to advocate the kind of rigor that mathematics demands as a way to unite all people. There are plenty of decisions in life that need to be made now. There are plenty of questions of profound importance which demand incomplete answers. I believe it was an insight of Friedrich Hayek that if we only accepted what we could justify on purely rational grounds, no human institutions could exist.
In short, mathematics is a luxury sport (like everything academic). That's not to say one must come from a wealthy family to do it. But it is certainly something that society could never have developed if it had never moved beyond the point of just doing enough to survive. More than that, it fundamentally arises out of a certain kind of dissatisfaction with mere acquaintance with the world around us. Mathematics can be characterized by a strange and relentless longing to see the inherent logical connections between ideas.
That is basically what I'm trying to train my students into. It should not be surprising if some of them never get it. If this innate desire is in you, you don't really need your teacher to justify it. On the other hand, if this desire isn't in you, then I wonder if you can ever really understand more than what it is that your teacher wants to see on your homework.
The point is that if a student wants to really do mathematics, he has to be skeptical of himself, first of all. He has to understand that it doesn't really matter whether or not he has the basic gist of a problem. He has to learn to expect a kind of precision of himself that can only be based on a relentless desire to see clearly the logical connections between ideas. If one is genuine about this, it really demands a great deal of humility. It demands the ability to say, I don't know until I have truly seen. And when one has seen clearly, then one has to humbly accept that there is no denying what one has seen. Yet while this is a humbling experience, it is simultaneously empowering, as it grants the ability to prove irrefutable claims.
Even as I teach students this, I am still learning it myself. It is an endless process. Yet I have to believe there is some inherent value in it. On the other hand, whether or not mathematics has value is hardly a mathematical question.
There is a certain level at which young mathematicians regard "proofs" almost as a specific area of math. They've learned algebra, trigonometry, calculus, and now they're learning "proofs." If they continue on in mathematics, they can't continue to think this way for long. Proof is not a part of mathematics. Proof is mathematics.
On the list of ways in which we teach mathematics badly in American schools, I would certainly add that we teach students that writing is not a part of mathematics. Writing is part of everything. If you can't write it, you can't communicate it. Perhaps part of the problem is that most people (even those who teach mathematics in high school and below) don't even realize mathematics needs to be communicated. Why would you need to convince someone that something is true in mathematics? Hasn't it all been figured out already? I kid you not--I would not be surprised if half the people I meet don't realize there's still being research done in mathematics. As in there's still math we don't know!
Since we teach students to do math without actually writing sentences, we also implicitly teach them that math is nothing more than a series of arcane symbolic manipulations that magically results in an answer. Many students will simply never grasp the idea that mathematics is really a set of questions that real people have asked out of genuine curiosity, and have answered purely through deductive reasoning. No magic necessary.
My students are a little sharper than that, but many of them still have a rough transition to make. They know how to get the answer, but they're totally new to actually writing mathematics. Thus if they were asked to prove that if x is a number other than 1, then 1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1), they might give me an argument that looks like this:
1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1)
(x-1)*(1 + x + x^2 + ... + x^n) = (x-1)*(x^{n+1} - 1)/(x-1)
(x + x^2 + ... + x^{n+1}) - (1 + x + x^2 + ... + x^n) = x^{n+1} - 1
x^{n+1} - 1 = x^{n+1} - 1
There is a sense in which this argument is correct. If the student were to indicate that each line is logically equivalent to the line before it, then I suppose the argument would work. But it still wouldn't feel like good style. Everyone who is trained in higher mathematics (or in logic) understands this basic fact about a good argument: you don't start with what you're trying to prove. You start with your hypothesis, then argue step by step to the conclusion.
A reasonable argument of the proposition I just mentioned might simply reverse the lines of the poorer argument I gave, but that would be rather clunky. A good argument would actually use words! It would be much easier and clearer to simply write the following:
Observe that (x-1)*(1 + x + x^2 + ... + x^n) = (x + x^2 + ... + x^{n+1}) - (1 + x + x^2 + ... + x^n) = x^{n+1} - 1. Now divide both sides of the equation by (x-1) to obtain the desired conclusion. QED
Here's what I tell my students: try to write as if you're trying to convince the most skeptical person in the world. Every line you write has to be 100% convincing. Starting with what you're trying to prove will never satisfy a skeptic, because a skeptic knows that you can get anything to be true if you just start by assuming it to be true. For instance, let's say I want to prove that -1 = 1. Well, my argument would be rather short and sweet:
-1 = 1
(-1)*(-1) = 1*1
1 = 1 (check mark!)
I have a great deal of uneasiness about the whole process of teaching good mathematical writing. I can tell them all day what's wrong with their proofs, and they might write little notes to themselves to try to figure out what I want to see on their homework. But that's the last thing I want them to learn. I don't want them to be able to convince me. I want them to be able to convince anyone who understands the symbols being used. There's no precise way to say this, but what I'm really going for is that they would be able to convince reason itself. Somehow they have to transcend the personal motives of finding acceptance from their teacher and getting good grades. They have to develop an innate desire to critique their own arguments, and write something about which they can confidently say, This is simply irrefutable.
Of course, mathematics is never really exactly like that in the real world of research. Mistakes are made. In the rush to get results published, sometimes mathematicians have indeed overlooked important details. But members of the mathematical community are constantly attempting to hold each other to that rigorous standard of irrefutable proof. This principle is emulated in all the sciences, yet mathematics has the privilege of working by pure logic. There is no "methods" section of a research paper in mathematics.
What an interesting sort of community this is, though. We're not a community of experimentalists, who argue over the analysis of data. We're not a community of theologians or philosophers, who argue over the meanings of words or doctrines. In fact, arguments between mathematicians are rarely over the actual content of what they study. We can argue over what the best method of proof is (though such arguments can be absurd), or we can argue over what we think the solution to a problem will be (though such arguments are rendered somewhat meaningless once the problem is actually solved). (There are also plenty of arguments between mathematicians mainly concerning their own egos; in that our community is not at all unique.)
If all the theologians or philosophers in the world were asked to reach a conclusive solution to a problem in one of these disciplines, they would only succeed in dividing into several camps. Yet do the same thing for all the mathematicians in the world, and all of them, working entirely independently from one another, can still reach the same solution. There will be no disagreement on the final result. Of all the other academic disciplines, the scientific community is most similar to this; yet the scientific community routinely has to correct past theories.
Does that make the mathematics community somehow ideal, which the whole world ought to emulate? Not at all. Mathematics progresses through skepticism; society in general does not. Part of skepticism is the ability to say, "I don't know." In the case of mathematics, we can say that for quite a long time--many famous problems have taken centuries to solve. It would be absurd to advocate the kind of rigor that mathematics demands as a way to unite all people. There are plenty of decisions in life that need to be made now. There are plenty of questions of profound importance which demand incomplete answers. I believe it was an insight of Friedrich Hayek that if we only accepted what we could justify on purely rational grounds, no human institutions could exist.
In short, mathematics is a luxury sport (like everything academic). That's not to say one must come from a wealthy family to do it. But it is certainly something that society could never have developed if it had never moved beyond the point of just doing enough to survive. More than that, it fundamentally arises out of a certain kind of dissatisfaction with mere acquaintance with the world around us. Mathematics can be characterized by a strange and relentless longing to see the inherent logical connections between ideas.
That is basically what I'm trying to train my students into. It should not be surprising if some of them never get it. If this innate desire is in you, you don't really need your teacher to justify it. On the other hand, if this desire isn't in you, then I wonder if you can ever really understand more than what it is that your teacher wants to see on your homework.
The point is that if a student wants to really do mathematics, he has to be skeptical of himself, first of all. He has to understand that it doesn't really matter whether or not he has the basic gist of a problem. He has to learn to expect a kind of precision of himself that can only be based on a relentless desire to see clearly the logical connections between ideas. If one is genuine about this, it really demands a great deal of humility. It demands the ability to say, I don't know until I have truly seen. And when one has seen clearly, then one has to humbly accept that there is no denying what one has seen. Yet while this is a humbling experience, it is simultaneously empowering, as it grants the ability to prove irrefutable claims.
Even as I teach students this, I am still learning it myself. It is an endless process. Yet I have to believe there is some inherent value in it. On the other hand, whether or not mathematics has value is hardly a mathematical question.
Sunday, September 12, 2010
Baptizing babies
This morning I witnessed two adorable little children get baptized. I'm guessing one was over a year old, maybe even closer to two. I'm not sure why parents wait a little while and then have their children baptized as infants, but it seems to happen quite often at my church. (Maybe it's logistical? I've noticed our church only has so many baptisms per year.)
This little girl was particularly squirmy. At first she was clinging to her father, but then she got bored and started crawling around toward the communion table. There was a particularly adorable moment when she crawled over very close to the pastor while he was carefully describing the deep theological significance of baptism. She looked up at him with a wide-eyed gaze, he looked down at her with a smile. His smile said something like, "These deep mysteries are for you, little girl, even though you don't have a clue what's going on."
I've seen both sides of the endless debate over infant baptism. Both sides can extensively quote the Bible and make one theological pronouncement after another. That's all well and good. For that reason, I don't think the issue will ever be truly settled in my mind. I figure, if God wanted to make sure we knew such things, He'd have figured out a way to make it so. Since He clearly hasn't, I can only assume He prefers we have a sense of humility (and humor) about the whole thing. We really can't take ourselves too seriously when it comes to such debates.
Convictions aside, as a matter of personal taste, I would say I prefer the practice of baptizing our children as infants. That's what I realized as I watched that adorable, squirmy little girl this morning.
It's quite simple, really. I am that squirmy little girl. I wake up every day and pray that I'll have wisdom enough to at least act like I know what I'm doing, but God knows I'm really just winging it. The more I explore intellectually and theologically, the more I realize I'm just a child. Chances are the voice of God can be heard everywhere and at all times, but I'm too easily distracted by the prospect of crawling around on the floor to understand what's going on.
It's quite lovely, I know, to see someone baptized as an adult, making a personal profession of faith and consciously deciding to join the church. But I have to be honest. Part of me suspects that the whole scene is a bit deceptive. Everything is made to look as if someone has finally found the meaning of life after years of living a life of either meaninglessness or self-deception or both. Now that he's finally figured out what it's all about, he's going to be baptized to confirm it. After this he'll really know why he's here, and what his mission in life should be. After this, even if he makes mistakes, at least he's got the road map, so he'll get where he's going eventually.
If that's the Christian life, then I've missed the boat. I don't have it figured out. If I have a road map, I don't think I can read it. I know Jesus is the answer, but what was the question? I've learned enough theology to be able to spit back the language of the kingdom, but I get the feeling like it's all baby-talk from my Father's point of view. Only I can't imagine it's so adorable. If we really ought to wait until children are old enough to understand what's happening before being baptized, then I'm probably too young to be baptized.
But then, I just seem to be one of those people who keeps asking questions after everyone else is already convinced. I'm too squirmy, I guess. Like that little girl, I suppose I have the tendency to ruin perfectly good pious reflections on Christian doctrine. I know it's irritating, but I just can't help myself.
A lot of people say it's terrible to baptize a child before she's had the chance to decide what she believes for herself. That's making the assumption that being baptized means you believe x, y, and z. What if baptism isn't so much a sign of intellectual opinions, as it is a sign of love? What if baptism is a symbol of the fact that God loves you even when you're squirmy, and don't have it figured out? I guess I would prefer to think of it that way.
And since it's my blog, I'm allowed to just leave it at that.
This little girl was particularly squirmy. At first she was clinging to her father, but then she got bored and started crawling around toward the communion table. There was a particularly adorable moment when she crawled over very close to the pastor while he was carefully describing the deep theological significance of baptism. She looked up at him with a wide-eyed gaze, he looked down at her with a smile. His smile said something like, "These deep mysteries are for you, little girl, even though you don't have a clue what's going on."
I've seen both sides of the endless debate over infant baptism. Both sides can extensively quote the Bible and make one theological pronouncement after another. That's all well and good. For that reason, I don't think the issue will ever be truly settled in my mind. I figure, if God wanted to make sure we knew such things, He'd have figured out a way to make it so. Since He clearly hasn't, I can only assume He prefers we have a sense of humility (and humor) about the whole thing. We really can't take ourselves too seriously when it comes to such debates.
Convictions aside, as a matter of personal taste, I would say I prefer the practice of baptizing our children as infants. That's what I realized as I watched that adorable, squirmy little girl this morning.
It's quite simple, really. I am that squirmy little girl. I wake up every day and pray that I'll have wisdom enough to at least act like I know what I'm doing, but God knows I'm really just winging it. The more I explore intellectually and theologically, the more I realize I'm just a child. Chances are the voice of God can be heard everywhere and at all times, but I'm too easily distracted by the prospect of crawling around on the floor to understand what's going on.
It's quite lovely, I know, to see someone baptized as an adult, making a personal profession of faith and consciously deciding to join the church. But I have to be honest. Part of me suspects that the whole scene is a bit deceptive. Everything is made to look as if someone has finally found the meaning of life after years of living a life of either meaninglessness or self-deception or both. Now that he's finally figured out what it's all about, he's going to be baptized to confirm it. After this he'll really know why he's here, and what his mission in life should be. After this, even if he makes mistakes, at least he's got the road map, so he'll get where he's going eventually.
If that's the Christian life, then I've missed the boat. I don't have it figured out. If I have a road map, I don't think I can read it. I know Jesus is the answer, but what was the question? I've learned enough theology to be able to spit back the language of the kingdom, but I get the feeling like it's all baby-talk from my Father's point of view. Only I can't imagine it's so adorable. If we really ought to wait until children are old enough to understand what's happening before being baptized, then I'm probably too young to be baptized.
But then, I just seem to be one of those people who keeps asking questions after everyone else is already convinced. I'm too squirmy, I guess. Like that little girl, I suppose I have the tendency to ruin perfectly good pious reflections on Christian doctrine. I know it's irritating, but I just can't help myself.
A lot of people say it's terrible to baptize a child before she's had the chance to decide what she believes for herself. That's making the assumption that being baptized means you believe x, y, and z. What if baptism isn't so much a sign of intellectual opinions, as it is a sign of love? What if baptism is a symbol of the fact that God loves you even when you're squirmy, and don't have it figured out? I guess I would prefer to think of it that way.
And since it's my blog, I'm allowed to just leave it at that.
Saturday, September 11, 2010
Postmodernism and Liturgy
Last night Tim McConnell at the Center for Christian Study made the comment that in an era of postmodernism, many Christians, including evangelicals, are returning to more liturgical forms of worship, prayer in particular. In context, "liturgical prayer" means prayer that is not spontaneous, but rather written by someone else, to be prayed by an entire group of worshipers. The explanation he gave was reasonable. A revival of liturgical prayer is a reaction against the postmodern notion that your identity and beliefs must come entirely from within; there is nothing in the outside world to guide you. Liturgical worship is a firm response against this. Liturgy helps define our identity and beliefs from outside ourselves. We aren't simply making it up as we go along.
I encounter the word "postmodernism" most often among evangelicals, who tend to overuse the term. They talk about it most often like it is a prevailing ideology which is reshaping the entire way society is structured. It isn't. In fact, what evangelicals have to deal with the day is not so much the influence of thinkers like Derrida as the influence of their own tradition. "Postmodernism" should really be called "post-evangelicalism."
It was evangelical Christians who taught generations of believers that true faith is about a personal choice, that it must come from within, and that it is not a matter of reason, but of the heart. In terms of liturgical practices, anything that seemed like it could be done by rote was rejected in favor of spontaneous, emotionally charged response to God. Worship music became increasingly improvisational. The Lord's supper became a once a month or less event (you wouldn't want it to become less special, would you?). The sermon, even as it solidified as the centerpiece of evangelical liturgy, became something to evoke individual response; at worst we see in our own day sermons used as self-help motivational speaking, or simply as entertainment.. The logic behind all of these moves is the same: God can only be known inside, rather than outside. Any external means we use must simply provide a platform for a "genuine" response to God, where "genuine" always means something that comes from within.
This shows up in apologetics, too. Science, it is said, can tell us many things; but there is another, spiritual dimension of experience, and that is where faith is necessary. But if science studies the outside world, then what realm of experience is left for faith? The inside world, no doubt--the world of thoughts and feelings, where I define myself because no one else can tell me who I am. That world may really be important, as one can learn by spending some time alone with one's thoughts. But it is an awfully small space for faith to live.
Why, then, are Christians so surprised at postmodernism? If transcendent truth is all found in an individual's mind, then how transcendent is it, really? When you rely on personal choice for your epistemology, don't be surprised at philosophical pluralism.
What is postmodernism, anyway? As far as I know, it's simply a rejection of the idea that an individual observer can be objective. I think this view can taken to absurd lengths, but overall, I'm inclined to agree with it. No individual observer can be truly objective, because what we see always comes through the filter of our cultural heritage and personal experiences. Evangelical Christianity, then, cannot be an objectively true system of beliefs, so long as it remains individualistic.
No individual observer can be truly objective; but we were not meant to be merely individuals. Biblical faith is based on collective, not personal, experience. All of the Israelites together heard God speak from Mount Sinai (Ex 20). Hundreds of the disciples saw the risen Christ together (1 Cor 15:6). Real, collective experience is the basis of our faith. More than these experiences, however, we also have experienced the presence of Christ together in the Lord's supper, in singing songs of worship, and in listening to the Scriptures taught. These are rituals, rather than singular experiences; but as we experience them together, we learn more and more what they mean, and how God interacts with us through them.
Personally, I get frustrated with the individualism of evangelical theology, and I think lots of others do, too. For all the evangelical talk about how God loves you and you don't have to earn salvation, they sure make it feel like you do. What if I come to church, and I'm not moved by the sermon, I don't particularly care for the music, I don't feel the urge to wave my hands in the air while I'm singing, and I don't feel like my quiet time that morning was all that spiritual? What if I find myself incapable of conjuring up feelings inside that Jesus is close to me? Evangelical theology--not evangelical people with bad attitudes, not Christians who are judgmental or rude, but the very ideas they hold dear--simply destroy people who find that they don't have the same personal experience as others. I expect to experience Christ personally; yet I find my own personal experience is as small as I am, a mere human trying to make sense of the world. At least one of my assumptions must be wrong.
A liturgy that brings worshipers outside of themselves can alleviate these dangers. I don't have to conjure up "authentic" prayers within myself. I pray the words that Jesus gave us to pray. Or I pray the words that centuries of devout believers have prayed, carefully constructing them out of a multitude of experience. I sing hymns that have been selected after centuries of effort to make beautiful music to God. I let all of these things come from outside of myself. I hear them, consider them, repeat them, absorb them. If I am receptive, I will be changed. It might not feel supernatural, but it is God's grace at work.
The process doesn't need to happen one way. As I experience and absorb the liturgy, my own reflections on it will become present in conversations I have with other Christians. I will have the opportunity for my own response. I don't need to lose my individuality to be part of a whole.
The bigger picture is that spirituality is not simply a dimension of personal experience. If our liturgical practices reflect an attitude that God exists nowhere except in the mind, then we will have a hard time making our faith relevant in public life. If, on the other hand, our liturgical practices connect us with a collective memory of the presence of Jesus Christ in the flesh, then we will find ourselves witnessing more faithfully to his presence now. It is one thing to say, "I had a personal experience of Jesus." It is quite another thing to say, "I worship the risen Lord alongside 2000 years of believers." Which of these is a more powerful testimony to the power of God?
I encounter the word "postmodernism" most often among evangelicals, who tend to overuse the term. They talk about it most often like it is a prevailing ideology which is reshaping the entire way society is structured. It isn't. In fact, what evangelicals have to deal with the day is not so much the influence of thinkers like Derrida as the influence of their own tradition. "Postmodernism" should really be called "post-evangelicalism."
It was evangelical Christians who taught generations of believers that true faith is about a personal choice, that it must come from within, and that it is not a matter of reason, but of the heart. In terms of liturgical practices, anything that seemed like it could be done by rote was rejected in favor of spontaneous, emotionally charged response to God. Worship music became increasingly improvisational. The Lord's supper became a once a month or less event (you wouldn't want it to become less special, would you?). The sermon, even as it solidified as the centerpiece of evangelical liturgy, became something to evoke individual response; at worst we see in our own day sermons used as self-help motivational speaking, or simply as entertainment.. The logic behind all of these moves is the same: God can only be known inside, rather than outside. Any external means we use must simply provide a platform for a "genuine" response to God, where "genuine" always means something that comes from within.
This shows up in apologetics, too. Science, it is said, can tell us many things; but there is another, spiritual dimension of experience, and that is where faith is necessary. But if science studies the outside world, then what realm of experience is left for faith? The inside world, no doubt--the world of thoughts and feelings, where I define myself because no one else can tell me who I am. That world may really be important, as one can learn by spending some time alone with one's thoughts. But it is an awfully small space for faith to live.
Why, then, are Christians so surprised at postmodernism? If transcendent truth is all found in an individual's mind, then how transcendent is it, really? When you rely on personal choice for your epistemology, don't be surprised at philosophical pluralism.
What is postmodernism, anyway? As far as I know, it's simply a rejection of the idea that an individual observer can be objective. I think this view can taken to absurd lengths, but overall, I'm inclined to agree with it. No individual observer can be truly objective, because what we see always comes through the filter of our cultural heritage and personal experiences. Evangelical Christianity, then, cannot be an objectively true system of beliefs, so long as it remains individualistic.
No individual observer can be truly objective; but we were not meant to be merely individuals. Biblical faith is based on collective, not personal, experience. All of the Israelites together heard God speak from Mount Sinai (Ex 20). Hundreds of the disciples saw the risen Christ together (1 Cor 15:6). Real, collective experience is the basis of our faith. More than these experiences, however, we also have experienced the presence of Christ together in the Lord's supper, in singing songs of worship, and in listening to the Scriptures taught. These are rituals, rather than singular experiences; but as we experience them together, we learn more and more what they mean, and how God interacts with us through them.
Personally, I get frustrated with the individualism of evangelical theology, and I think lots of others do, too. For all the evangelical talk about how God loves you and you don't have to earn salvation, they sure make it feel like you do. What if I come to church, and I'm not moved by the sermon, I don't particularly care for the music, I don't feel the urge to wave my hands in the air while I'm singing, and I don't feel like my quiet time that morning was all that spiritual? What if I find myself incapable of conjuring up feelings inside that Jesus is close to me? Evangelical theology--not evangelical people with bad attitudes, not Christians who are judgmental or rude, but the very ideas they hold dear--simply destroy people who find that they don't have the same personal experience as others. I expect to experience Christ personally; yet I find my own personal experience is as small as I am, a mere human trying to make sense of the world. At least one of my assumptions must be wrong.
A liturgy that brings worshipers outside of themselves can alleviate these dangers. I don't have to conjure up "authentic" prayers within myself. I pray the words that Jesus gave us to pray. Or I pray the words that centuries of devout believers have prayed, carefully constructing them out of a multitude of experience. I sing hymns that have been selected after centuries of effort to make beautiful music to God. I let all of these things come from outside of myself. I hear them, consider them, repeat them, absorb them. If I am receptive, I will be changed. It might not feel supernatural, but it is God's grace at work.
The process doesn't need to happen one way. As I experience and absorb the liturgy, my own reflections on it will become present in conversations I have with other Christians. I will have the opportunity for my own response. I don't need to lose my individuality to be part of a whole.
The bigger picture is that spirituality is not simply a dimension of personal experience. If our liturgical practices reflect an attitude that God exists nowhere except in the mind, then we will have a hard time making our faith relevant in public life. If, on the other hand, our liturgical practices connect us with a collective memory of the presence of Jesus Christ in the flesh, then we will find ourselves witnessing more faithfully to his presence now. It is one thing to say, "I had a personal experience of Jesus." It is quite another thing to say, "I worship the risen Lord alongside 2000 years of believers." Which of these is a more powerful testimony to the power of God?
Tuesday, September 7, 2010
It can't be true!
Yesterday I had the joy of sharing Cantor's famous diagonal argument, proving that the real numbers form an uncountable set. This semester, instead of teaching my own class, I get to be a TA for an honors Calculus III course, which goes through many advanced topics to prepare students for higher level mathematics. As a result, I get to share moments like these, in which students go from being calculators to being real mathematicians.
Mathematics finally means something to a student when he encounters a proof of something that he can't believe. I had at least one student express his inability to accept the diagonal argument after I gave it. Of course, the proof is irrefutable--it is logically sound, and despite the large number of attempts made every year by crank mathematicians, it can never be overturned.
That's part of the sheer beauty of mathematics. It shows how pure logic can yield surprising results. Why are we surprised by things that are logically irrefutable? This is not a mathematical question, but a question about the human spirit; yet in some respects it is one which only the mathematician can encounter. No one else can experience quite the same thrill, or frustration, at coming face to face with those results of pure logic that seem to break down every sense of intuition you ever had.
Why does Cantor's proof frustrate students so?
The result itself frustrates people in general, at least in my experience. When you explain that the real numbers can never be counted, even if you counted them for all eternity, even if you go all the way to infinity, they do not believe you. Or they don't understand what you even mean. The young student of mathematics, on the other hand, can see and accept each line of the proof; yet much like any other person, he is intuitively troubled by the result itself. This manifests itself in the form of complaints about the proof; yet what it always seems to come down to is that the result is incomprehensible on any intuitive level.
Why is this? I don't really think that it is because the logical steps are hard to follow. Indeed, I think that even a person relatively untrained in mathematics can understand each step in the proof. What I think people can't really and truly deal with are the starting assumptions themselves. To embrace those assumptions takes imagination.
What is infinity, in the first place? To the extent that it touches upon our every day experience, infinity can actually be something quite small. The national debt, for instance, is for all practical purposes, infinite. No single human being can fathom having control of that much money. If I had $13 trillion in my bank account, I would simply never run out of money. I could spend $5000 every second for the next 80 years and still not run out of money.
Infinity, then, is a guarantee: there's always something left. The truth is, though, 13 trillion can be a very small number of some things. Avogadro's constant is somewhere on the order of a trillion trillions; yet this ridiculously large number is equal to the number of molecules in a measly 32 grams of oxygen. Numbers in this universe get so ridiculously large that our minds stop distinguishing between them all. (This has unfortunate political consequences, as people have yet to truly comprehend how astronomical their own governments' spending really is.)
So then, one who is willing to stretch out with the imagination can make this assumption: for every number n, there is always a number n + 1. You can always go, as Christopher Guest on Spinal Tap might put it, "one louder." We have formalized the guarantee of infinity. We have assumed the existence of an infinite set of numbers. One, two, three, four, five, ... ten, ..., twenty, ... thirty, ... one hundred, ... one thousand, ..., one million, ..., one trillion, ... why should it ever end?
But that isn't the only kind of guarantee one seeks in this world of computation. We also seek the guarantee of being really close to something. For instance, there is this ugly business of finding the circumference of a circle. A circle, of all things! A beautiful, simple shape--nothing could be simpler, really. And yet, the ratio between its circumference and its diameter is such a monster of a number that we are forced to give it a Greek name--"pi"--and leave it at that.
Or are we? We all heard in grade school that pi = 3.14... People sometimes wonder if I have all the digits of pi memorized, although some more sophisticated folks know that you can't memorize all of them but still wonder how many I know. Well I know up to about eight, I think: 3.1415926... And I've heard school children will sometimes have contests to see how many digits of pi they can memorize. Yet no matter how many digits they memorize, they will never have actually gotten pi. Never. Not in a million billion trillion years--not ever.
But they are getting closer and closer. How much closer is rather easy to quantify. If I guess pi = 3.14, then I am within one-hundredth of pi--that is, pi is between 3.14 and 3.15. If I guess pi = 3.1415926, then I am within one 10 millionth of pi. That's pretty close. Never exactly right, but closer and closer. So that's my guarantee. I can always get closer.
There are plenty of other common ratios in the real world which have no exact finite decimal representation--in fact, no repeating pattern ever emerges in their decimal expansions. The square root of 2 is one of them--this is simply the length across the diagonal of a square. The square roots of most numbers are the same way, and these can all be represented using common, everyday shapes. In each case, we do have a guarantee, as we did with pi. We can get closer and closer by taking more and more decimal places. That is, we can take the ratio between two ordinary natural numbers and be as close as we want to be to a weird number like pi.
Now what if I imagine that there is a number at the end of every conceivable decimal expansion? I say to myself, "Look, I can just start typing numbers after a decimal point, and there's no stopping me." In fact, just like at me as I cough up numbers right now:
.011923985461928384093745601092983562390523984691348612340213901098234908123984...
Try it! It's kind of therapeutic, actually.
Now I stretch out my imagination and put my faith in a new assumption: that this process of picking new digits can continue on forever and ever, and that no matter how the digits are picked at each step, the result can rightfully be called a number. I don't know what that number is, but I know that I'm getting closer and closer to it with every arbitrary choice of a new digit. Just as it took imagination to take on the assumption that there is always one more number, so it also takes imagination to embrace this new assumption. It means formalizing a guarantee.
It is, in a sense, an act of faith. Implicitly it means trusting that guarantee to have some sort of meaning. Otherwise, what would be the point of studying the logical results of that guarantee?
If you've gone with me this far, if you have enough faith to believe that after every number n there's always n+1 and that any decimal that can be continued on forever should be considered a number, then congratulations! You have, more or less, just embraced what mathematicians call the real numbers. If not, don't worry. Chances are your world can do without such big numbers. Even the national debt is probably higher than you'll ever need to imagine.
Young students of mathematics have generally been unwittingly indoctrinated into having faith in the real numbers. Duh, there's always a bigger number, and of course every decimal is a number. Why would we have to "imagine" that? Why, indeed! The ancient Greeks didn't even believe in zero.
Our educational system teaches these principles from a very young age. Enter pi in on your calculator. See? A decimal comes up! And those digits can keep going and going... Thus the youth are catechized into the traditions of their elders, unaware that without imagination, none of the structure in their mathematical universe could ever have arisen.
No wonder young mathematicians are so shaken by Cantor's proof! Indoctrinated into the assumptions which the proof begins with, these young minds are totally unprepared to handle the consequences of those assumptions. For as soon as the mind willingly submits to the two assumptions I have fleshed out here, then simple logic reveals a truth that is so staggering that it has drawn downright hostility from philosophers and logicians ever since Cantor first made his argument.
The claim is simple. Take all of those counting numbers: 1, 2, 3, 4, 5, 6, 7, ...
And I mean all of them, all infinitely many of them, because you know there is always one more! And now to each one of those counting numbers, assign some decimal, like this:
Here's what happens: you'll never get all of the decimals!
Never! Never ever! No matter how cleverly you chose which decimals to match to each of your counting numbers! Even though you always have one more counting number--that is, even though there are infinitely many counting numbers--you still don't have enough. The decimals are to the counting numbers what the national debt is to your savings account.
And the argument is quite simple: just read down the diagonal of that list you just made. Take the first digit of the first number, change it to another digit, and write it down. Take the second digit of the second number, change it, and it write it down next to the first one you just wrote. Do the same with the third, and the fourth, and so on. You'll get a new decimal. Mine would start to look like this:
Now, is this new number in your list? No! It can't be. It's not the same as your first number, because the first digit is different. It's not the same as your second number, because the second digit is different. It's not the same as your third number, because the third digit is different. And so on, even for every single counting number.
That's Cantor's proof. As shocking as the result is, the proof is nothing more than simple logic. Yet logic has to build on certain assumptions, and it's really those assumptions that set up this amazing result. It must have been that Cantor was the first person to fully buy into all of those assumptions. He was a mathematician of true faith, and true imagination.
And also true bravery. When I say that Cantor fully embraced those assumptions about numbers, I mean that he was even willing to embrace the logical consequences of those assumptions. That is faith. And without it, there can be no progress.
I'm allowed to say such things, because it's my blog. But I seriously wonder, how many assumptions do we take for granted, yet without being willing to accept their logical consequences? It is often only when someone shows you what the logical consequences are that you're able to see what the assumptions actually mean.
This, to me, is what's so liberating about mathematics. Whatever faith you have in your assumptions will be thrown to the fire to be tested. You must seek out the logical consequences of whatever you start with. And if you are able to overcome your initial fear, you might just find that the universe is a much grander, more majestic, and more mysterious place than you had ever imagined.
Mathematics finally means something to a student when he encounters a proof of something that he can't believe. I had at least one student express his inability to accept the diagonal argument after I gave it. Of course, the proof is irrefutable--it is logically sound, and despite the large number of attempts made every year by crank mathematicians, it can never be overturned.
That's part of the sheer beauty of mathematics. It shows how pure logic can yield surprising results. Why are we surprised by things that are logically irrefutable? This is not a mathematical question, but a question about the human spirit; yet in some respects it is one which only the mathematician can encounter. No one else can experience quite the same thrill, or frustration, at coming face to face with those results of pure logic that seem to break down every sense of intuition you ever had.
Why does Cantor's proof frustrate students so?
The result itself frustrates people in general, at least in my experience. When you explain that the real numbers can never be counted, even if you counted them for all eternity, even if you go all the way to infinity, they do not believe you. Or they don't understand what you even mean. The young student of mathematics, on the other hand, can see and accept each line of the proof; yet much like any other person, he is intuitively troubled by the result itself. This manifests itself in the form of complaints about the proof; yet what it always seems to come down to is that the result is incomprehensible on any intuitive level.
Why is this? I don't really think that it is because the logical steps are hard to follow. Indeed, I think that even a person relatively untrained in mathematics can understand each step in the proof. What I think people can't really and truly deal with are the starting assumptions themselves. To embrace those assumptions takes imagination.
What is infinity, in the first place? To the extent that it touches upon our every day experience, infinity can actually be something quite small. The national debt, for instance, is for all practical purposes, infinite. No single human being can fathom having control of that much money. If I had $13 trillion in my bank account, I would simply never run out of money. I could spend $5000 every second for the next 80 years and still not run out of money.
Infinity, then, is a guarantee: there's always something left. The truth is, though, 13 trillion can be a very small number of some things. Avogadro's constant is somewhere on the order of a trillion trillions; yet this ridiculously large number is equal to the number of molecules in a measly 32 grams of oxygen. Numbers in this universe get so ridiculously large that our minds stop distinguishing between them all. (This has unfortunate political consequences, as people have yet to truly comprehend how astronomical their own governments' spending really is.)
So then, one who is willing to stretch out with the imagination can make this assumption: for every number n, there is always a number n + 1. You can always go, as Christopher Guest on Spinal Tap might put it, "one louder." We have formalized the guarantee of infinity. We have assumed the existence of an infinite set of numbers. One, two, three, four, five, ... ten, ..., twenty, ... thirty, ... one hundred, ... one thousand, ..., one million, ..., one trillion, ... why should it ever end?
But that isn't the only kind of guarantee one seeks in this world of computation. We also seek the guarantee of being really close to something. For instance, there is this ugly business of finding the circumference of a circle. A circle, of all things! A beautiful, simple shape--nothing could be simpler, really. And yet, the ratio between its circumference and its diameter is such a monster of a number that we are forced to give it a Greek name--"pi"--and leave it at that.
Or are we? We all heard in grade school that pi = 3.14... People sometimes wonder if I have all the digits of pi memorized, although some more sophisticated folks know that you can't memorize all of them but still wonder how many I know. Well I know up to about eight, I think: 3.1415926... And I've heard school children will sometimes have contests to see how many digits of pi they can memorize. Yet no matter how many digits they memorize, they will never have actually gotten pi. Never. Not in a million billion trillion years--not ever.
But they are getting closer and closer. How much closer is rather easy to quantify. If I guess pi = 3.14, then I am within one-hundredth of pi--that is, pi is between 3.14 and 3.15. If I guess pi = 3.1415926, then I am within one 10 millionth of pi. That's pretty close. Never exactly right, but closer and closer. So that's my guarantee. I can always get closer.
There are plenty of other common ratios in the real world which have no exact finite decimal representation--in fact, no repeating pattern ever emerges in their decimal expansions. The square root of 2 is one of them--this is simply the length across the diagonal of a square. The square roots of most numbers are the same way, and these can all be represented using common, everyday shapes. In each case, we do have a guarantee, as we did with pi. We can get closer and closer by taking more and more decimal places. That is, we can take the ratio between two ordinary natural numbers and be as close as we want to be to a weird number like pi.
Now what if I imagine that there is a number at the end of every conceivable decimal expansion? I say to myself, "Look, I can just start typing numbers after a decimal point, and there's no stopping me." In fact, just like at me as I cough up numbers right now:
.011923985461928384093745601092983562390523984691348612340213901098234908123984...
Try it! It's kind of therapeutic, actually.
Now I stretch out my imagination and put my faith in a new assumption: that this process of picking new digits can continue on forever and ever, and that no matter how the digits are picked at each step, the result can rightfully be called a number. I don't know what that number is, but I know that I'm getting closer and closer to it with every arbitrary choice of a new digit. Just as it took imagination to take on the assumption that there is always one more number, so it also takes imagination to embrace this new assumption. It means formalizing a guarantee.
It is, in a sense, an act of faith. Implicitly it means trusting that guarantee to have some sort of meaning. Otherwise, what would be the point of studying the logical results of that guarantee?
If you've gone with me this far, if you have enough faith to believe that after every number n there's always n+1 and that any decimal that can be continued on forever should be considered a number, then congratulations! You have, more or less, just embraced what mathematicians call the real numbers. If not, don't worry. Chances are your world can do without such big numbers. Even the national debt is probably higher than you'll ever need to imagine.
Young students of mathematics have generally been unwittingly indoctrinated into having faith in the real numbers. Duh, there's always a bigger number, and of course every decimal is a number. Why would we have to "imagine" that? Why, indeed! The ancient Greeks didn't even believe in zero.
Our educational system teaches these principles from a very young age. Enter pi in on your calculator. See? A decimal comes up! And those digits can keep going and going... Thus the youth are catechized into the traditions of their elders, unaware that without imagination, none of the structure in their mathematical universe could ever have arisen.
No wonder young mathematicians are so shaken by Cantor's proof! Indoctrinated into the assumptions which the proof begins with, these young minds are totally unprepared to handle the consequences of those assumptions. For as soon as the mind willingly submits to the two assumptions I have fleshed out here, then simple logic reveals a truth that is so staggering that it has drawn downright hostility from philosophers and logicians ever since Cantor first made his argument.
The claim is simple. Take all of those counting numbers: 1, 2, 3, 4, 5, 6, 7, ...
And I mean all of them, all infinitely many of them, because you know there is always one more! And now to each one of those counting numbers, assign some decimal, like this:
1: 0.123846130865861902034109826394384...and so on. Except, don't just do it how I just did it--do it however you want! Be completely arbitrary!
2: 0.988349102039136358923403948190234...
3: 0.238463985658654865435864238653428...
4: 0.843565643854843643874398340954893...
5: 0.458430938230483240893291293048239...
6: 0.789234923902309238942394823094872...
7: 0.097138428094217089340873078320683...
...
Here's what happens: you'll never get all of the decimals!
Never! Never ever! No matter how cleverly you chose which decimals to match to each of your counting numbers! Even though you always have one more counting number--that is, even though there are infinitely many counting numbers--you still don't have enough. The decimals are to the counting numbers what the national debt is to your savings account.
And the argument is quite simple: just read down the diagonal of that list you just made. Take the first digit of the first number, change it to another digit, and write it down. Take the second digit of the second number, change it, and it write it down next to the first one you just wrote. Do the same with the third, and the fourth, and so on. You'll get a new decimal. Mine would start to look like this:
0.2996455...All I did was add 1 to each of the "diagonal elements" of my list. You can't add 1 to a 9, but you can just change 9 to 8, and the same idea holds.
Now, is this new number in your list? No! It can't be. It's not the same as your first number, because the first digit is different. It's not the same as your second number, because the second digit is different. It's not the same as your third number, because the third digit is different. And so on, even for every single counting number.
That's Cantor's proof. As shocking as the result is, the proof is nothing more than simple logic. Yet logic has to build on certain assumptions, and it's really those assumptions that set up this amazing result. It must have been that Cantor was the first person to fully buy into all of those assumptions. He was a mathematician of true faith, and true imagination.
And also true bravery. When I say that Cantor fully embraced those assumptions about numbers, I mean that he was even willing to embrace the logical consequences of those assumptions. That is faith. And without it, there can be no progress.
I'm allowed to say such things, because it's my blog. But I seriously wonder, how many assumptions do we take for granted, yet without being willing to accept their logical consequences? It is often only when someone shows you what the logical consequences are that you're able to see what the assumptions actually mean.
This, to me, is what's so liberating about mathematics. Whatever faith you have in your assumptions will be thrown to the fire to be tested. You must seek out the logical consequences of whatever you start with. And if you are able to overcome your initial fear, you might just find that the universe is a much grander, more majestic, and more mysterious place than you had ever imagined.
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