Thursday, October 6, 2011

Paradoxes that Aren't

Probability Theory can be very non-intuitive.  The "Monty Hall Problem" is one famous example.  Even stranger is the "Two Envelope Problem:"

'Let us say you are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are offered the possibility to take the other envelope instead'.
It is possible to give arguments that show that it will be to your advantage to swap envelopes by showing that your expected return on swapping exceeds the sum in your envelope. This leads to the logical absurdity that it is beneficial to continue to swap envelopes indefinitely.
 What's the solution?  Read the Wikipedia article if you're curious.  (Hint: it's all in how you frame it.)

Language is messy.  Words are representations of things, not the things themselves.  "The cake is a lie."  Natural language is a super-set of mathematical language, and thus subject to Godel's Incompleteness Theorem, among other issues.

Natural language allows us to formulate the "Omnipotence Paradox:" can God make a rock so heavy God can't lift it?

I was raised by Christian Fundamentalists, in a Church which taught the Bible is the inerrant word of God.  But these were smart Fundies.  Once, our Sunday School teacher put a map of the world on the wall.  A Mercator projection, it made Greenland, and Antarctica, absolutely huge.

Inevitably distortion results from representing three dimensions - the globe - in two dimensions. Distortion also results from using finite human language to represent the workings-out of a Transcendent God.  IOW, "the map is not the territory."  How then can The Bible be inerrant?

Did I mention that there are other types of maps - equal area projections - that don't show Greenland being larger than America?  But they have a different problem.  Whereas the Mercator projection gets sizes wrong, it shows the contours of Greenland's coastline correctly.  Equal area maps get the sizes right, but don't accurately show the coastline.

In short, you need to choose the right map for the right job; if you're planning to sail around the coastline of Greenland, the Mercator projection is the map for you; the apparent size distortion won't be a problem.  And so we see how Scripture, composed of finite language as it is, is still inerrant.  The Christian Life is a raised nail.  The Bible is our hammer!

2 comments:

  1. The problems with 2d representations of the globe are consequences of the Dimension Invariance theorem: bascially you can only preserve *all* aspects of form if the dimension of the model is the same as the thing modeled. Note that absolute size is *not* an aspect of the form: http://mathworld.wolfram.com/DimensionInvarianceTheorem.html

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  2. Yes. And we have a similar dimensional collapse with all the models we use to describe "reality." In other words, is there only physics, or is physics a way to model an underlying phenomena? (As weird as that question sounds, my understanding is that some forms of the Copenhagen Interpretation more or less argue the former) Is God just the words in the Bible, or do the words in the Bible model something richer?

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