Burning shadows in the mist
While leaves fall gently
Upon the frozen ground
High above an airliner
Cuts straight across the blue
Breasting the waves of light
That break upon the beach of morning
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The moment of sunrise Burning shadows in the mist While leaves fall gently Upon the frozen ground High above an airliner Cuts straight across the blue Breasting the waves of light That break upon the beach of morning |
| date | 2000-10-25:16:31 |
| Physics | Here's a simple example of the argument from invariance, which may make some of the foregoing clearer. We'd like simple arithmetic to represent certain operations amongst objects. If we take one group of objects and put it into a basket, and then put another group of objects into the same basket, we know that the number of objects in the basket does not usually depend on the order in which we put the groups in. If I put two strawberries into a basket (why are these things always done with apples?) and then put five more in, I expect to get the same number as if I put five in and then two more. Arithmetically, the operation I've described above is addition, and it's symoblized with: 1 + 2 = 3 The property we're interested in here is how the answer changes when we change the order of the operands, and addition is defined such that: 2 + 1 = 3 so that it can represent the addition of real objects in the real world. Because adding objects together produces a result that is invariant under a transformation of the order in which we do things in the real world, we define addition in a way that is invariant under the ordering of its arguments. Most of the time we don't even think about this, precisely because the result of the operation is invariant under changes in ordering of the arguments. With regard to multiplication, if I take five things and pair them up to get ten things, and then take three things an pair them up to get six things, and then add the ten and the six things together to get sixteen things, the result is the same as if I'd added the five and three things together to get eight things, and then paired them up to get sixteen things. "Pairing up" in this case is "multiplying by two." This property of multiplication is described by: (2*5) + (2*3) = 2*(5+3) where the operations inside the brackets are always performed first. This is called "distributivity", and we say that multiplication is distributive. It also commutes. For natural numbers this is all well and good. But what about the integers? What happens when we start dealing with negative numbers? Suppose we are a mathematical genius of a past age, and have just invented negative numbers, and are wondering how to incorporate them into our arithmetic. What we would like to do is extend our definitions of the operations we already understand in a way that preserves the invariances our arithmetic already has, because they represent invariances of the real world. This is how the argument from invariance works in it's creative aspect. Consider just -1 for a moment. We know that: -1*1 = -1 because "*1" is an identity operation -- it always preserves what it operates on. But what about: -1*-1 = ? Should it be +1? Or -1? And why? The answer is -1, because that's what preserves the invariance described by distributivity: -1*(2 - 1) = -1*(1) = -1 and -1*(2 - 1) = -2 + 1 = -1 whereas if we defined it the other way: -1*(2 - 1) = -2 - 1 = -3 but -1*(2-1) = -1*(1) = -1 So to preserve the invariance of the results of multiplication across transformations in the order of operations, we are constrained in how we defined multiplication by negative numbers. This is how we can go from a requirement of invariance, which seems at first to be a purely negative statement, to a positive claim about how our description of the world must be. I have no idea what a mathematician would say to all this -- there are probably much deeper reasons to define things this way, although Barry Hill-Tout, who is a mathmatician, once gave me something like this in answer to my question, "Why is -1 times -1 positive?" The difficulty with making arguments of this kind is in ensuring that you haven't left out any possibilities. You are basically stuck with looking at _every_ possible kind of description, and then somehow winnowing them down to the ones that fulfill your constraints. The case I've shown here is simple because there are only two possible ways of doing it, but in realistic examples just showing that you've covered the whole possible range of solutions is a daunting task. What one normally does is make some auxilliary assumptions to narrow the field, which will have to be subsequently re-examined, for they're always the weak points of any theory. |
| Movies | Beowulf with Christopher Lambert is a better movie than you might think. When I first read the poem, I thought, "This is a role for Arnold." Lambert doesn't quite have quite the physical presence to pull it off, but he's still adequate. The interesting thing about the film is the way it plays with the thesis of the poem -- the oldest Old English work of literature -- which is at once the story of the foundation of the kingdom of Denmark, and a tale of morality in three parts. Beowulf the hero fights three monsters -- Grendel, whose severed arm he displays over the doors of Herot, his battle-hall; Grendel's mother, who is decidedly pissed off at the death of her child; and a dragon, which Beowulf fights in his old age. Each monster is supposed to represent a disordering of a particular aspect of the tripartite soul as described by Plato in the Republic. Grendel represents the disordering of reason, his mother the disordering of spirit, and the dragon represents the disordering of desire. The film takes some pretty serious liberties with the story, mostly in the name of reducing the amount of violence (which is still considerable -- this is Beowulf, after all) and increasing the amount of sex, which is a welcome change from the usual direction these things go in. But by making Grendal's mother into a beautiful succubus, the script combines the concupiscence represented by the dragon in the poem with the irascibility represented by Grendel's mother. Without this understanding of how elements in the poem have been transmuted and combined, the film probably just looks like a lot of violence and a little sex, an unfortunate ratio that can't be changed without straying much further afield from the source material. |