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.