Beauty of abstract simplicity, again

Last night I wrote about how abstract knowledge makes complicated problems simpler. This morning I remembered my favorite example of using abstract knowledge to solve a complicated problem. Something like this appeared in the ARML test many years back. Here's the problem:

15! = 130767?368000

Now figure out what digit ? is.

Of course, looking this up on Google calculator gives you the answer, but suppose you can't use a calculator. Well, 15! = 15 x 14 x 13 x 12 x ... x 2 x 1. Do you really want to spend all your time computing that by hand?

There are a lot of concrete facts you can name about the number 15! but the only one that's relevant to this problem is that 15! is clearly divisible by 9. This, it turns out, is all you ever need to know to figure out a missing digit. We can use the method of casting out nines to solve the problem in just seconds. We have

130767?368000

Take out the 1 and the 8, a 3 and a 6, the other 3 and 6, and we're left with 7 and 7, which make 14, whose digits sum to 5. Since the number should be divisible by 9, we need ? to be a digit such that ? + 5 is divisible by 9; the only choice is ? = 4, which is indeed the answer you get on Google calculator.

The beauty of abstract simplicity

Gary Davis over at Republic of Mathematics has an interesting hypothesis about how less able mathematics students are making problem solving unnecessarily complicated. He uses some good illustrations to show how an astute mathematics student will take the time to formulate the simplest method of approach. He hypothesizes that a less able student will even choose more complicated examples in studying a particular idea (see his example of multivariable functions).

Excelling in mathematics, or in any theoretical field, requires an ability to make things simpler. It seems to me that what holds people back is that they erroneously equate "abstract" with "complicated," when in fact it's just the other way around. We create abstractions precisely to make simple the things that are complicated.

True Grit and the grace of God

I highly recommend seeing the Coen brothers' new film, True Grit. Then, after watching it, I highly recommend this review of it in the New York Times. I especially enjoyed this summary paragraph:

The new “True Grit” is that rare thing — a truly religious movie. In the John Wayne version religiosity is just an occasional flourish not to be taken seriously. In this movie it is everything, not despite but because of its refusal to resolve or soften the dilemmas the narrative delivers up.

He who wrestles with God

Genesis 25-36

After Abraham comes Isaac, the miracle child, and after Isaac comes Jacob. Jacob is quite different from Abraham. If Abraham has been characterized by his unfaltering faith and humble obedience, Jacob is now characterized by his doubt, his deceit, his lack of courage, and, quite literally, his wrestling with God. The very name "Jacob" indicates that "he supplants;" that is, he supplants his brother, Esau. Genesis is strangely and wonderfully open about the fact that Jacob, later to become Israel, is hardly deserving of the inheritance promised to Abraham. Born as the younger twin with his brother Esau, Jacob is the favorite of his mother, Rebekah. Right from the beginning Rebekah is told that her two sons are actually warring nations, and she takes the side of Jacob. The war starts early with Jacob tricking his brother out of his birthright. It comes to a head when Rebekah helps Jacob to steal his father's blessing from Esau. This leads Jacob to flee from Esau's anger, while Esau goes off marrying Canaanite women just to make his parents angry.

If there's one thing Genesis does well, it's drama.

Abraham, savior of the world

Genesis 12-25

It seems you can't do any justice to the book of Genesis without focusing on Abraham as its key character. The first eleven chapters have told a story of how God created the world and us humans to cultivate it, and how we chose death over life and deteriorated into depravity. God purified the world with a flood to wipe out all living creatures, except for faithful Noah and the creatures he was able to preserve on the ark. Now that this "baptism" has taken place, there are new life and new possibilities for the world; yet humans are by no means perfect. There is a strange ambiguity in this introductory story: was Noah really worth saving? Is not the human race even from Noah just the same sinful race of Adam which the flood was meant to wipe out? Genesis makes no attempt to settle that question. Instead, it goes on to God's new plans. The Bible may bring up more questions than it answers, but its agenda is always forward-looking. The unanswered (and possibly unanswerable) questions must be allowed to remain, while God does a new thing to bring new life to his world.

Baptism of the world

I've set a new ambitious goal for myself: to blog through the entire Bible. Hey, it's been done before. My goal is to force myself to read through it quickly, rather than slowly and in detail. It's very easy to miss the forest for the trees, especially surrounded by an American evangelical tradition which delights in sermons based on two or three verses. There's so much to talk about just in looking at the big picture, rather than obsessing over every last detail. In this discussion I hope to avoid any and all discussion of inerrancy, inspiration, or even biblical authority. Not that those discussions aren't important; I just don't particularly care about them for my purposes here. I really just want to see what happens when I treat the Bible as a whole story, using the knowledge of the particulars which have already been ingrained in me as well as the many various influences that I've had in my intellectual development.

These are all merely notes, not essays, and I though I will try to tie each post together thematically, no attempt will be made to give a definite structure to these notes.

Evolution: Smith before Darwin

In Appendix A of The Fatal Conceit, Hayek has this to say about the development of evolutionary theory:

Though in Hume, and also in the works of Bernard Mandeville, we can watch the gradual emergence of the twin concepts of the formations of spontaneous orders and of selective evolution..., it was Adam Smith and Adam Ferguson who first made systematic use of this approach. Smith's work marks the breakthrough of an evolutionary approach which has progressively displaced the stationary Aristotelian view. The nineteenth-century enthusiast who claimed that the Wealth of Nations was in importance second only to the Bible has often been ridiculed; but he may not have exaggerated so much....

While Smith has been recognised by several writers as the originator of cybernetics (Emmet, 1958, Hardin, 1961), recent examinations of Charles Darwin's notebooks (Vorzimmer, 1977; Gruber, 1974) suggest that his reading of Adam Smith in the crucial year 1838 led Darwin to his decisive breakthrough.

Thus from the Scottish moral philosophers of the eighteenth century stem the chief impulses toward a theory of evolution, the variety of disciplines now known as cybernetics, general systems theory, synergetics, autopoiesis, etc., as well as the understanding of the superior self-ordering power of the market system, and of the evolution also of language, morals, and law (Ullman-Margalit, 1978, and Keller, 1982).

Moral of the story: Read Wealth of Nations.

Hayek vs. Schmitt

Mark Lilla's article in The New Republic on China's interest in Western philosophers contains a concise summary of Carl Schmitt's critique of classical liberalism (HT: Peter Leithart):

Schmitt was by far the most intellectually challenging anti-liberal statist of the twentieth century. His deepest objections to liberalism were anthropological. Classical liberalism assumes the autonomy of self-sufficient individuals and treats conflict as a function of faulty social and institutional arrangements; rearrange those arrangements, and peace, prosperity, learning, and refinement will follow. Schmitt assumed the priority of conflict: Man is a political creature, in the sense that his most defining characteristic is the ability to distinguish friend and adversary. Classical liberalism sees society as having multiple, semi-autonomous spheres; Schmitt asserted the priority of the social whole (his ideal was the medieval Catholic Church) and considered the autonomy of the economy, say, or culture or religion, as a dangerous fiction. (“The political is the total, and as a result we know that any decision about whether something is unpolitical is always a political decision.”) Classical liberalism treats sovereignty as a kind of coin that individuals are given by nature and which they cash in as they build legitimate political institutions for themselves; Schmitt saw sovereignty as the result of an arbitrary self-founding act by a leader, a party, a class, or a nation that simply declares “thus it shall be.” Classical liberalism had little to say about war and international affairs, leaving the impression that, if only human rights were respected and markets kept free, a morally universal and pacified world order would result. For Schmitt, this was liberalism’s greatest and most revealing intellectual abdication: If you have nothing to say about war, you have nothing to say about politics. There is, he wrote, “absolutely no liberal politics, only a liberal critique of politics.”

Since at the moment I'm a bit immersed in Hayek, I felt like this would be a good chance to summarize Hayek's own thoughts in response to Schmitt's. (Hayek himself already attacked Schmitt's ideas several times in The Road to Serfdom.) The above critique seems to be a response to only a rationalistic conception of liberalism, which Hayek scoffed at. If we are referring to the liberalism defended by Hayek, which he took to be the classical liberalism of Locke, Hume, and Smith, then the above critique is simply wrong on every point.

Classical liberalism is not based on the autonomy of self-sufficient individuals. It is built on the study of how people stumble upon political order, rather than arriving there through deliberate design. It would hardly make sense to think of the economy, culture, and religion as autonomous spheres, since these all evolve together--indeed, the functioning of an economy mostly depends on our religious and cultural practices. Yet asserting the priority of the "social whole" would be a worse error still, since, as Hayek says, a "society is greater than the individual only in so far as it is free. In so far as it is controlled or directed, it is limited to the powers of the individual minds which control or direct it." Classical liberalism does not treat sovereignty "as a kind of coin that individuals are given by nature and which they cash in as they build legitimate political institutions for themselves." On the contrary, liberalism mostly rejects the notion that individuals could build legitimate political institutions for themselves; yet still less can it accept that legitimate institutions arise "as the result of an arbitrary self-founding act by a leader, a party, a class, or a nation." Men do not simply speak order into being; it comes about through a slow and often painful process of evolution. (The founding of the United States, for instance, could not have come about except through the inherited tradition of the rule of law in English politics, which itself arose through a long process of trial and error.) Finally, it is simply false that liberalism says nothing about war and international affairs. Its goal has traditionally been to work toward international government; Hayek himself proposed a federal system to govern nations (see The Road to Serfdom, Chapter Fifteen).

One wonders if history could have been much different if Schmitt had better understood the philosophy he so viciously critiqued.

Christmas + Bacon = Awesome

see more Failbook
This is truly inspired.

Hayek's theology

In The Fatal Conceit, Hayek quotes the Bible probably more times than in any of his other works I've read (and really, to even have one quote is more than usual for him). Of course, I should say right off the bat that many of these quotes are meant to be compared against the view he is propounding. For instance, he quotes 1 Timothy 6:10 ("the love of money is the root of all evil") on the way to countering such fears of money. Many times I had to wonder what his motivation was. In his polemic to construe socialism as a new sort of animism, i.e. a return to primitive religion, was he quoting scripture as an affront on socialists, who would have taken this association as an attack on their reason?

A new word for capitalism?

In The Fatal Conceit, Hayek tries to explain how the "extended order" came into being, i.e. the global civilization in which we now live. It is fair to say that, according to Hayek, capitalism is what has allowed the extended order to come into being and continue to exist. Without the spontaneous processes of cultural evolution (see my previous blog post on the matter) human life would never be able to thrive in such large groups.

However, Hayek had trouble with the word "capitalism."

Life needs no justification

The Fatal Conceit is Hayek's final manifesto in defense of capitalism and against socialism. In a fascinating sense, however, it is not a political but a scientific argument. Hayek's thesis is that our moral traditions, which have enabled the survival of as many humans as now exist, have evolved through a process of "cultural evolution" and are not, and never could be, the product of deliberate human reason. This requires some explanation.

Ken Cuccinelli's lawsuit against Obamacare

Ken Cuccinelli, the attorney general of Virginia, just won his case before a federal judge in Richmond on Monday. On Thursday he published an editorial insisting that the case go quickly to the Supreme Court. I think he makes an appropriate argument against the health care bill. Regardless of anything else Cuccinelli might say or do, I have to support him, at least in principle, in this argument:

The health-care law sacrifices the liberty of Americans and abandons the Constitution that protects that liberty. The power Congress claims it has to create the mandate and penalty has no principled limits: If the federal government can order a citizen to purchase a private product such as health insurance in the name of public policy, it can order us to buy anything.

...But is it Christian?

Soon after writing my last blog post, I was reminded by a couple of other blog posts (here and here) that there are many Christians out there wondering what the proper political orientation is from a biblical perspective, and that Hayek's liberalism isn't exactly drawn straight out of scripture (Hayek himself was agnostic). Where does that leave me? Am I being true to my faith, and trying to build a consistent Christian worldview? Or am I simply picking and choosing what I want to believe?

The Party of Life: the liberalism of F. A. Hayek

The postscript to Friedrich Hayek's The Constitution of Liberty is entitled, "Why I Am Not a Conservative." For Americans aware of Hayek's influence on political developments of the past half-century or so, this title may be astonishing. Aren't conservatives the very people most influenced by and most appreciative of Hayek's work? How can he devote a whole essay to why he isn't one of them?

The whole essay is indeed confusing for Americans, and from the tone of the essay one senses that Hayek was exasperated at the confusion Americans have created over the terms "liberal" and "conservative." As I've written before, the two terms scarcely mean in American today what they have traditionally meant; the term "liberal" has probably suffered more distortion even than "conservative." For this reason most Americans reading the essay will probably experience a great deal of confusion over who exactly Hayek is talking about, excepting those who are very knowledgeable about the history of ideas. Nevertheless, this is precisely what makes the essay so relevant. It is, in some sense, Hayek's warning not to do exactly what we have done: we have so closely associated his ideas with American conservatism that we can't tell what he was actually saying.

Government imitates Arrested Development

From The Washington Times:

"Republicans wanted more tax cuts. Democrats wanted more spending. They got both in the tax-cut deal President Obama struck with the GOP, and it's the federal budget that will take the hit.

Just 10 days after Mr. Obama's deficit commission sounded an alarm over the long-term health of the government's finances, the Senate is slated to vote Monday on a tax cut and spending package that will add $857 billion to the federal deficit in the upcoming years, likely setting a new records for red ink."

Wow, that's a new one. Republicans wanted to tax less, Democrats wanted to spend more. So their brilliant compromise is to do both.

It reminds me of that scene on Arrested Development season one when Maeby convinces George Michael that they can just throw away one banana for each dollar they take from the cash register:

Maeby: I can’t believe I volunteered for this. This is my stupidest rebellion ever. (taking money from the register) Hey, you want to go play skee-ball?

George Michael: Well, this is the cash drawer. My dad’s going to come by at the end of the weekend and the number of bananas has to match the amount of money in here.

Maeby: Oh, so it all has to even out?

George Michael: Exactly.

Maeby: Easy. Banana... (throws a banana into a garbage bin...) Buck. (and takes a buck from the register) Banana... Take a buck. (doing it again as George Michael just watches)

Oh, but not to worry. The government knows what it's doing:

"Republicans and Democrats who support the deal point out that the country's short-term economic problems are paramount, and argue that without economic growth, the long-term budget picture would look gloomier still."

Because, as we all know, short-term happiness is the most important key to long-term success.

Symbol and Meaning

I made an abbreviated effort to express my current epistemology of mathematics, in which symbols take on a primary role, to one of my fellow grad students. His response was, "But they're more just than symbols."

Just symbols? I can see where he's coming from. Anyone who's tried to explain mathematics to someone else knows the frustration of trying to express concepts as more than mere symbolic manipulation. As grad students we're expected to teach mathematics to students who are often pretty uninterested in math, and who have often been taught their whole lives that math is nothing but symbols which have no connection to the real world. Trying to convince these students that there is real meaning behind these symbols can be exceedingly frustrating, especially if they care about their grades but not about the material. Students who try merely to memorize symbolic manipulations without understanding any of the concepts will do poorly in a college calculus class, and the worst part is they might not ever know why.

Divine Justice, Human Tyranny

The wind blows where it chooses, and you hear the sound of it; but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit. John 3:8

Is God sovereign over history, or not? Is he sovereign over nature, or not?

It is common for people to cite the horrific events of history as reason either to doubt God's goodness or to doubt his sovereignty, with the result either way that we doubt his existence. It is now also fashionable these days to cite the very mechanism by which human beings came into existence--biological evolution--as evidence that God doesn't exist.

Christians, in response, are prone to answering with theological excuses connecting the fall of Adam with the taint of nature. I won't argue the biblical merits of these excuses one way or another. What I'm interested in is the underlying principle behind such arguments. The complaint is that God doesn't share a basic sense of justice with humans; the defense is that, "Well, yes he does, it's just hard to see it at first."

I question whether this is the proper response to such complaints. Not only does it undermine God's power, but it feeds the human presumption that his own sense of justice is necessarily correct. Yet the same experiences which cause us to question God's justice should perhaps more often cause us to question our own.

Individualism and force

I think I can summarize Hayek's individualism as follows.

There is a false kind of individualism which tells us that we can be free of all force. Taken literally, of course, this is just silly. We will never be free of the force of gravity, or any of the natural forces acting on our bodies at all times. Yet we are tempted to think that what is true of natural forces is not true of cultural forces. We think that individuals can be completely independent of everything except his own will.

This is simply mistaken. In any society, the individual will be subject to pressures from all around him: obviously from people he knows, less obviously yet no less truly from people he doesn't know, and still more indirectly from the institutions which shape his culture. There is no escape from these forces, any more than we can escape the laws of nature.

The only question from a political point of view is, shall the individual be subject to personal or impersonal forces? Shall the individual be coerced into abiding by the will of one or several other individuals, or shall he be free to choose his own path through the set of cultural forces which press against him? By distributing power equally among all individuals, we do not free him from all societal forces, but only from the will of arbitrary rulers.

In traditional terms, we might say man is either subject to the will of other men or subject to the will of God alone. Hayek doesn't see it that way, but it is still helpful to convey the crucial point: the forces to which man must be subject are either personal and arbitrary or impersonal and abstract. These are the only choices.

Americans are largely plagued by false individualism. Our politicians on the Left and on the Right promise us that they will free us from the forces which necessitate difficult choices on our part. The more we buy into this, the more we simply trade impersonal abstract forces for personal ones. Sometimes it's natural to favor personal over impersonal force; impersonal forces are more difficult to understand. But when force is allowed to come from select individuals, then we are all subject to their personal agenda. If these select individuals were divine, this wouldn't be a problem. Since they are not, we will usually suffer for it in the end.

A meditation on progress

In Chapter Three of The Constitution of Liberty, Hayek sets out to justify a measured form of progress. He begins,

Writers nowadays who value their reputation among the more sophisticated hardly dare to mention progress without including the word in quotation marks. The implicit confidence in the beneficence of progress that during the last two centuries marked the advanced thinker has come to be regarded as the sign of a shallow mind. Though the great mass of the people in most parts of the world still rest their hopes on continued progress, it is common among intellectuals to question whether there is such a thing, or at least whether progress is desirable.

Up to a point, this reaction against the exuberant and naive belief in the inevitability of progress was necessary.... There never was much justification for the assertion that "civilization has moved, is moving, and will move in a desirable direction," nor was there any ground for regarding all change as necessary, or progress as certain and always beneficial....

But if the fashionable disillusionment about progress is not difficult to explain, it is not without danger. In one sense, civilization is progress and progress is civilization. The preservation of the kind of civilization that we know depends on the operation of forces which, under favorable conditions, produce progress.

Best "Rickroll" I've ever seen

see more Failbook

Evolution of mathematical symbols

At first glance (at least for me), mathematics seems to be a purely rationalistic endeavor. One sits and thinks about a problem, and then by the sheer power of the individual mind is able to produce an answer. By nothing more than logical deduction, one is able to prove theorems linking certain basic assumptions to interesting and/or useful conclusions. Unlike most other subjects, mathematics is a quest for abstract principles, without any necessary connection to concrete facts.

Yet doing mathematics requires symbols, which are necessarily concrete in their origin and use. The use of symbols in mathematics arises naturally even at its very beginning: counting. There is nothing more fundamental to mathematics than a collection of symbols representing different quantities. What often goes overlooked by those who don't study mathematics is that embedded in those symbols is a theory of numbers. For instance, many people hardly notice that our counting system is a "base 10" or "decimal" system. The number 10 itself is not so significant. What is significant is that somehow everyone knows that when I put a "1" directly to the left of a "0" I actually mean ten. This principle allows us to express remarkably large numbers with relatively few digits; indeed, the value of a number grows exponentially with the length of the number: 10 is ten times 1, 100 is ten times 10, 1000 is ten times 100, and so on. Thus a deep fundamental principle is embedded into the very symbols we use to count things: that all whole numbers are uniquely expressible as a sum of powers of ten, where the coefficients of each power is something between zero and nine.

The use of the mind in society

Friedrich Hayek is primarily known for his contributions to economics and political philosophy. Yet one of his best contributions to both of these fields is an underlying theory of social learning. As an economist, he was interested in how it is that civilizations make use of information which is distributed among many individual minds (see, for instance, Hayek's essay, "The Use of Knowledge in Society"). As a political philosopher, he was interested in how institutions come to be, and how these institutions benefit individuals and allow society to progress. In both cases, Hayek finds that people engaged in free exchanges with one another unwittingly contribute to a process of social learning of which they themselves are mostly unaware.

Reponse to Mike Meredith (and all libertarians out there)

Here is a comment left on my blog by Mike Meredith:

Comment by Jameson Graber on 23 November 2010:
I mostly agree with the views you all are espousing, but I think some commenters here are simplifying the issue far too much. The criticism that the government is “doing things to other people they could never justify if they were in the private sector” is, upon a moment’s reflection, not good enough. The government has the right and even the responsibility to punish offenders of the law, whereas private citizens have no such right. Unless you intend to argue that this is not the way things ought to be (which would be arguing for anarchy) you’d better put forth a more reasonable principle which forbids the government from taking these invasive security measures. The fourth amendment should suffice.

I ran across this post on the Freeman website today and thought a couple of things odd and one humorous. First, how is it that any government has rights? As far as I can tell only people have rights. Second, from whence does government's authority to act derive? I fail to understand why anarchy is implied unless government has power that the people don't. The humor is in supposing that the constitution has force. It hasn't been worth the paper it is written on since the ink dried. Or at least since Marbury vs. Madison in 1803. It has had no force of restraint to speak of on government. And what little it has had diminishes with each passing day.

Mike

And here is my response:

Two years of blogging

Today marks the second anniversary of this blog. This post is the 310th post I've written. I suppose that's an average of one post every two or three days, though I certainly haven't been that consistent.

I don't know that blogging actually does the world a great deal of good, but if it's true that ideas matter, and that thinking critically about what you believe and why you believe it is important, then this blog at least does me a great deal of good. It's motivated me to essentially continue my liberal arts education, reading more about topics ranging from theology to economics to the interaction between science and faith. It's helped me to at least start down the path to developing a more coherent philosophy of life, all the while giving credit to that which has most influenced my thinking.

Some small part of me hopes that because this blog is public, it will have some positive influence on people I have never met, yet remain connected to through this vast incomprehensible network known as the World Wide Web. I have occasionally been surprised to find my blog being referenced in a wide variety of places on the Web, but I have never taken seriously the prospect of other people actually listening to me. Yet if there ever comes a time in my life when people really want to know what I think about things, I hope that this practice will have prepared me to give clear, intelligent answers.

So if you've been reading this, let me say thank you, and I hope you've gotten something out of it. I know I have.

Hayek Against Rationalism

All political theories assume, of course, that most individuals are very ignorant. Those who plead for liberty differ from the rest in that they include among the ignorant themselves as well as the wisest. Compared with the totality of knowledge which is continually utilized in the evolution of a dynamic civilization, the difference between the knowledge that the wisest and that which the most ignorant individual can deliberately employ is comparatively insignificant.

F. A. Hayek
The Constitution of Liberty

Perhaps the single most attractive part of Hayek's philosophy is, in my view, his epistemology. This is not something that the majority of people are likely to care about. Yet whether most people care about them or not, questions about what we know and how we know it are intensely important to the intellectuals who are often most influential in shaping society. The argument for a free society must have, as part of its foundation, an argument for the right kind of answers to these questions.

Modern Spirituality

see more Failbook
These are the haunting religious questions of our time...

The two party system must go

I am a big believer in ideas. That is to say, I believe the ideas people choose to accept in the present profoundly influence how we live in the future. This is what The Road to Serfdom was all about: it traced the influence of ideas on society in Germany as a warning to Great Britain at a time when things looked quite bleak. The book itself influenced future generations after it was written; Ronald Reagan, for instance, listed Hayek as one of his greatest influences (see the Wikipedia article on Hayek), and this in turn greatly shaped American politics. Ideas have consequences for the future of society, and it follows that if we care about our future, we must be concerned with whether or not good ideas are allowed to flourish in the world we live in.

Hayek on Freedom and Democracy

From The Road to Serfdom Chapter Five:

The fashionable concentration on democracy as the main value threatened is not without danger. It is largely responsible for the misleading and unfounded belief that, so long as the ultimate source of power is the will of the majority, the power cannot be arbitrary. The false assurance which many people derive from this belief is an important cause of the general unawareness of the dangers which we face. There is no justification for the belief that, so long as power is conferred by democratic procedure, it cannot be arbitrary; the contrast suggested by this statement is altogether false: it is not the source but the limitation of power which prevents it from being arbitrary. Democratic control may prevent power from becoming arbitrary, but it does not do so by its mere existence. If democracy resolves on a task which necessarily involves the use of power which cannot be guided by fixed rules, it must become arbitrary power.

Hayek on hatred of the enemy

From The Road to Serfdom, Chapter Ten:

It seems to be almost a law of human nature that it is easier for people to agree on a negative program--on the hatred of an enemy, on the envy of those better off--than on any positive task. The contrast between the "we" and the "they," the common fight against those outside the group, seems to be an essential ingredient in any creed which will solidly knit together a group for common action. It is consequently always employed by those who seek, not merely support of a policy, but the unreserved allegiance of huge masses. From their point of view it has the great advantage of leaving them greater freedom of action than almost any positive program.

Makes you think about our own political attitudes, doesn't it? Perhaps I should include the next sentence for impact:

The enemy, whether be internal, like the "Jew" or the "kulak," or external, seems to be an indispensable requisite in the armory of a totalitarian leader.

The road to serfdom is paved with hatred.

Hayek on Social Insurance

From The Road to Serfdom, Chapter 9:

Nor is there any reason why the state should not assist the individuals in providing for those common hazards of life against which, because of their uncertainty, few individuals can make adequate provision. Where, as in the case of sickness and accident, neither the desire to avoid such calamities nor the efforts to overcome their consequences are as a rule weakened by the provision of assistance--where, in short, we deal with genuinely insurable risks--the case for the state's helping to organize a comprehensive system of social insurance is very strong. There are many points of detail where those wishing to preserve the competitive system and those wishing to supercede it by something different will disagree on the details of such schemes; and it is possible under the name of social insurance to introduce measures which tend to make competition more or less ineffective. But there is no incompatibility in principle between the state's providing greater security in this way and the preservation of individual freedom. To the same category belongs also the increase of security through the state's rendering assistance to the victims of such "acts of God" as earthquakes and floods. Wherever communal action can mitigate disasters against which the individual can neither attempt to guard himself nor make provision for the consequences, such communal action should undoubtedly be taken.

What she's really thinking

From The Guardian:

What I'm really thinking: The abortion patient

There isn't a choice: I am an unemployed recent graduate barely able to afford the pregnancy test, with a boyfriend on bar wages. But after the scan, I want the nurse to find some unfathomable medical reason why termination isn't an option, so I'd be justified in keeping a child I don't want to lose but can't really provide for.

...

This is the "right thing to do", as almost everyone has advised. Hopefully soon I can focus on a career and creating the right circumstances eventually to have a child. But the due date for this baby is seared into my mind now. I won't ever be able to forget it.

Yet another example of an honest expression of need. This is why people like Frederica Mathewes-Green write books like Real Choices. This is why, even in the midst of all the intense controversy, people should still care about the abortion issue, ideally in a compassionate way.

One reader points out in the comments:

This is playing right into the hands of the anti-abortion lobby, or the "pro-forced-pregnancy" lobby as I prefer to call them. They put a lot of stock in the "abortion is a terrible tragedy and so traumatic" narrative. For some women it is, for a variety of reasons, and for some it isn't.

What other "choice" is defended so vehemently by downplaying all the pain it causes to those who choose it?

Hayek on self-sacrifice

If I seem to be portraying Hayek's views in moral terms, this is because I want to make the point that the ideals of a free market economy are not simply ideals of efficiency, progress, and prosperity, but also of justice. Without this, there would be no point in defending the free market.

In the fourth chapter of The Road to Serfdom entitled, "The 'Inevitability' of Planning," one of the issues Hayek deals with is the sacrifices that must be made in order to preserve freedom. One of those sacrifices is one which the whole society must make. He willingly admits that sometimes it is possible that market competition can prevent a particular benefit from being afforded to society. He admits "that it is possible that, by compulsory standardization or the prohibition of variety beyond a certain degree, abundance might be increased in some fields more than sufficiently to compensate for the restriction of the choice of the consumer."

Calvin on Civil Government

At last, a conclusion to Calvin's Institutes of the Christian Religion. It's been ten months in finishing, but it has certainly given me an education in Reformation doctrine from the 16th century. If nothing else, reading this enormous work has forced me to listen to the voice of the past, and it has challenged me to rethink my modern assumptions about the way things are.

Nothing could more perfectly illustrate this point than the last chapter of Calvin's work, "Civil Government." It is, of course, appropriate to be talking about this right after an election. However, what I found as I read this chapter was that every other part of Calvin felt more or less familiar except for this part. Based on this experience, it seems to me that modern Christians have kept alive every controversial theological issue except those related to the social order. For if we stop and ponder for a second, I think we will have to agree that without too much trouble we can find in America today Christians with every possible opinion on predestination or infant baptism, and yet you will be hard pressed to find a single American Christian who does not accept liberal democracy as the best form of government. I have my own suspicions about why this is, but I ought to save that for another time. Let's see what Calvin has to say.

A spirit of partisanship

Well, we've just endured another season of political campaigns and made it out alive. As predicted, the Republicans picked up a lot of seats in Congress, enough seats in the House for a majority, and enough Senate seats to take the majority away from the Democrats, leaving neither party with a majority. Everyone is left wondering what will happen in the next two years. Will Obama be able to get anything done with a Republican Congress? Will the Republicans overturn Obamacare? Oh, the possibilities.

Conventional wisdom seems to be that it would just be so great if both parties could set aside their differences and just work together toward the common good. If only "politics as usual" could take a break so we can deal with the "real" issues.

In my humble opinion, this is completely wrong.

Hayek on the need for self-examination

My next little reading project will be to blog through (maybe not so thoroughly as the Institutes) three classic works of F. A. Hayek: The Road to Serfdom, The Constitution of Liberty, and The Fatal Conceit. I've read a bit of Hayek's thought already, mostly through certain essays I was particularly interested in, and I've always found something incredibly attractive about his philosophy. If I could sum it up in one thought, it's that Hayek has a healthy fear of rationalism and presumption. Rationalism, on the one hand, is the desire to only accept those things which you can fully rationally justify to yourself; presumption here refers to the grandiose claim that everything worth accepting can indeed be rationally justified, and thus controlled by enlightened men.

Voting Paradox

Key quote from this video: "While it's entirely rational not to vote...maybe that's not how people decide whether or not to vote."

Happy Reformation Day!

Happy Reformation Day! I finished my reading of Calvin's Institutes, but election day is only two days away, so I decided to write some thoughts on economics instead of blogging on Calvin. I'll have some thoughts on him later in the week. Until then, here's my favorite Reformation Day video!

On traffic lights and economics

Suppose you are sitting behind a long line of cars at a traffic light, and the light turns green. Despite the light turning green, the car in front of you does not move for half a minute. Finally, you are free to drive, yet you are hardly able to move forward before the light is red once more. The light changes again, and you go through the same cycle. By now you'll be late for your appointment. You want to scream at all the cars who were in front of you and wouldn't move.

You think to yourself, "Why is it that when the light turns green, we don't move?" You decide that it would be much more logical if as soon as the light turned green, everyone at the light started moving simultaneously. This would make getting cars through the light much more efficient. Then you wouldn't be late. Driving would be so much better if everyone followed this simple rule that you are convinced they should pass a law requiring it. After all, it would help everyone, in the long run.

Does the good news hinge on bad news?

Is Christianity a guilt-driven religion?

The answer of what I like to call "pop-evangelicalism" is, "No man, it's all about grace."

Grace. Sure. But don't you mean grace in response to...guilt?

The narrative which the evangelical has in mind is still the narrative of Martin Luther--burdened by guilt under the wrath of an angry God, finally liberated by the good news of justification by faith alone. We don't have to keep trying to appease God under the weight of heavy burdens imposed upon us by the Catholic Church. That's good news, right?

Here's the thing: that's still a guilt-driven religion. It's just that you've proposed a different solution to that guilt. Instead of trying to pay of the debt you owe, you are driven to accept the fact that you cannot pay it, confess your guilt, and accept God's forgiveness and cleansing. This proposal would not make sense outside of some shared assumptions about God's wrath and our accountability to him.

Evangelicals often run into the problem that many people aren't starting with those assumptions. Their solution to this problem goes like this: before you tell the good news, you have to tell the bad news. This takes on a couple of forms. One is the fundamentalist bible-thumping technique, where you tell people all the things that they're doing that God hates and will send them to hell. Another is the pop-evangelical method of psychoanalyzing the world around them and convincing us that deep down we're all screwed up. (I read a couple of perfect examples of this here and here.) Whatever may differ in outward appearance, the fundamental principle is the same: make people feel their guilt, so that they may be driven by that guilt to God's forgiveness in Christ. It is the gospel of guilt.

Concerning philosophy and one year olds

rationalism -noun
1. the principle or habit of accepting reason as the supreme authority in matters of opinion, belief, or conduct.

The life of a graduate student, in my experience, is typically spent around peers, around professors, or in solitude. Marriage is common enough in graduate school, but I would venture to guess that the majority of us aren't married. Parents make up a much smaller percentage of graduate students. In this environment it is taken as a truism that having children hinders progress in our studies. (I have heard stunning exceptions to this. One student in religious studies once told me that his advisor recommends having children as a graduate student--then suddenly you will be unable to waste time.)

Well, you'll get no argument from me on the practical implications of having children--they take a lot of work. What I'm interested in is the effect of simply not being around children, of being immersed in a world of professionals. Of course we could go on all day about how this affects our priorities, what it does to our personal development, and so on. But I'm really interested in the intellectual side of it. How does being absent from children all the time affect the way we think, about ourselves and about our beliefs?

Calvin on the Mass, and those other five Sacraments

Lots of Calvin lately. I guess it's like being at the end of a long race. You see the end in sight, so you kick it into gear. There's no reason in particular why I should be reading at a faster pace, except that I did enjoy Calvin on the sacraments, and because I figure, hey, why not finish this whole book before November?

So that brings me to Institutes of the Christian Religion Chapters XVIII and XIX. Some of the most fiery rhetoric in the entire work is in these chapters, which says something about how Calvin views the sacraments.

How "capitalism" works around here

The Freeman tipped me off to a story this week that has stirred in me an even deeper moral outrage against government interventionism. The headline reads:

McDonald's, 29 other firms get health care coverage waivers

Nearly a million workers won't get a consumer protection in the U.S. health reform law meant to cap insurance costs because the government exempted their employers.

Thirty companies and organizations, including McDonald's (MCD) and Jack in the Box (JACK), won't be required to raise the minimum annual benefit included in low-cost health plans, which are often used to cover part-time or low-wage employees.

The Department of Health and Human Services, which provided a list of exemptions, said it granted waivers in late September so workers with such plans wouldn't lose coverage from employers who might choose instead to drop health insurance altogether.

Without waivers, companies would have had to provide a minimum of $750,000 in coverage next year, increasing to $1.25 million in 2012, $2 million in 2013 and unlimited in 2014.

Calvin on the Sacraments

With the help of two reading days this week, which I actually managed to spend reading, I have made a ton of progress on the Institutes. It also doesn't hurt that what I've been reading on the sacraments has been some of the most fascinating material in all of Calvin's work. Truly, there are some rather astonishing passages in these chapters that one could not expect from Calvin if one were judging only on the popular perception of him. From general cultural impressions I seem to have this latent image of Calvin as something of a machine, coldly calculating the true doctrines of Christianity, without regard for feeling or mystery. These chapters on the sacraments, more than any others in the Institutes, completely obliterate this image of Calvin. Instead, we find a picture of Calvin as overcome with the mysteries of God, and wholly inclined toward a direct personal experience of Christ. It is nothing short of a tragedy that this image has been lost in popular Christian consciousness.

Justice and the natural order

A recent article in First Things by Stephen M. Barr entitled, "Fearful Symmetries" argues that scientific reductionism has been misapplied by scientific materialists. Barr decries the "diminished ontological status" of the fundamental "stuff" of the universe in the materialist view of things. In contrast, he expresses the belief that it is precisely by looking at the basic building blocks of the universe that we see its underlying order, and thus we see the Mind of God at work. Working "down" from the complex to the simple does not mean explaining away the beauty of the universe; on the contrary, it means discovering that beauty.

For my brother, and anyone else who has had to edit extraneous punctuation marks

see more funny videos

Christian Intellectual Liberty

In thinking about the relationship between science and religion, and considering very carefully the interaction between my faith and the academic world, it seems good to me to consider the more basic questions (and assumptions) which are really at the heart of the matter, rather than endlessly arguing over creation and evolution, the historicity of the resurrection, and so on. From where I sit, conversations between Christians and non-Christians (especially secularists) can feel rather fruitless, even when they are civil. I suspect this is because we have a hard time identifying what's really important to the people with whom we are conversing.

As indicated by the title of this post, I think one of more fundamental questions that rarely gets addressed seriously is that of intellectual liberty. It is useless to keep arguing whether certain scientific theories such as evolution are compatible with Christianity, if we are not going to address the more basic question of whether or not Christians are free to question old ideas.

Calvin on the Papacy, Part 2

In my last post on Calvin's Institutes, I focused on Calvin's severe critique of the Roman Catholic Church. Here, in discussing Book IV, Chapters IX - XIII, I will focus more on the constructive side of Calvin's criticism. If Calvin so detested the papacy, what was he ready to replace it with? What reasons did he have to support his view?

What's wrong with math education...

see more funny videos

Another sign that our math education is based on mindless application of formulas rather than common sense...

P.S. The teacher is wrong here, not the student.

Free Will and Creativity

In discussions about religious topics in which both atheists and Christians take part, the subject of free will plays an interesting role, in that both sides are internally divided on the issue. Atheists, on the one hand, find it difficult to tell whether a scientific understanding of the universe implies some sort of determinism (quantum mechanics notwithstanding) or whether there really is something to the intuitive belief in free agency. Christians add a theological dimension to this struggle, since on the one hand moral agency is non-negotiable in Christianity, but on the other hand so is God's omnipotence.

I wonder if the question of free will can be framed in more useful terms. It feels like every time the subject arises, we just end up spinning our wheels. Yet the topic also seems important, for a couple of reasons. For one thing, if we're interested in justice, it must be important to know whether someone ought to be held responsible for his choices. For another, isn't it depressing to envision a world in which our choices don't really contribute anything?

Moral Market Principles?

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!

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.

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.

Golden Calf

On Thursday I got to attend a talk by W. J. T. Mitchell at the Forum for Interdisciplinary Dialogue put on by the Jefferson Fellows at UVA. His talk was on idolatry. The tension I found throughout the presentation was between the impulse to counter idolatry and an argument defending it.

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?

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.

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?

• 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.

Contemplate these numbers with me. What might they say about the world we live in?

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.

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.

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.

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.

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.

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.

...

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.

What strength and courage it takes to overcome one's previous assumptions.

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.

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?

• 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

I think it should be pretty easy to get Calvin's central point.

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."

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:

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

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:

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!)

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.