Friday, December 31, 2010

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.

Thursday, December 30, 2010

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.

Wednesday, December 29, 2010

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.

Tuesday, December 28, 2010

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.

Monday, December 27, 2010

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.

Thursday, December 23, 2010

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.

Wednesday, December 22, 2010

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?

Tuesday, December 21, 2010

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."

Monday, December 20, 2010

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.

Saturday, December 18, 2010

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?

Friday, December 17, 2010

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.

Monday, December 13, 2010

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.

Saturday, December 11, 2010

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.

Sunday, December 5, 2010

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.

Friday, December 3, 2010

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.

Thursday, December 2, 2010

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.