Sunday, September 25, 2005
Dr. MontyHall (or, How I came to stop worrying and love the irrationality)
I swear that the Monty Hall problem proves that there is a deep logical inconsistency in the universe. The problem goes as such: A contestant is on a game show, and can pick from one of three doors. Two doors have a goat behind them, and one has a car. The contestant picks a door, and one of the doors which contains a goat (and is not the door which he picked) is opened. He can then either stay with his door, or switch to the other non-revealed door. Which is he better off doing?
For an example, there are doors A, B, and C, and Contestant Qof (Roman and Greek letters are so restrictive in mathematics). Qof picks door A. Door C is revealed to have a goat. Should Qof stick with A, or switch to B?
It seems that with one door down, Qof has a 50/50 chance, right? You'd LIKE to think that. Turns out that reality has a surprise for us (and by reality, I mean mathematics). Here's how it really works. Qof has a 1/3 chance of getting things right with his/her (we really need a neutral, but personal, 3rd personal pronoun) choice of A. Now, once C has been opened, Qof could stay with that 1/3 chance of A. Alternatively, you could say that Qof had 2/3 a chance of missing with picking A. Given this and the fact that C has been shown to be a goat, B must be the prize, and there is a 2/3 chance that by switching to B Qof will be riding home in style.
It seems like there is something fishy here. What about the earlier argument? Why doesn't that work, as switching doors could be considered to be equivalent to picking one of two doors? Alternatively, it seems as though one could end up with two different probablities for the same door, depending on whether one starts with it and stays with it, or picks another door and then switches to it. However, after running a few million trials on a computer program (quite literally), it turns out that the solution is that switching doors doubles one's chances of winning. This brings one down to an ultimatum: Either accept the mathematical explanation of the Monty Hall problem, or face the fact that reality is inconsistent at its core.
Given the obviousness of this choice, I think that maybe I should invest more of my time in Continental philosophy.
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3 comments:
I would sooner suspect math (not your math, Math of failing to correspond to reality (in this instance or in general) than I would suspect reality of being contradictory.
But you may disagree, and I could be wrong. :)
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So, you get to choose between (A) Christianity, (B) Sokka Gakkai, and (C) Zoroastrianism. You choose religion alpha and religion gamma is revealed to be Zoroastrianism. Do your chances double of eternal bliss if you switch to religion beta?
You didn't know that? Geez, even I knew that....and no, I'm not kidding. I read it in one of that super-IQ-genius-woman's articles in a newspaper. :)
I don't have your e-mail address; could you send it my way, please?
Speaking of irrationality, my logic prof yesterday went through a series of relational predicate logic translations about God and causality, using the opportunity to point out that there were certain contradictory propositions we had to pick between while constructing our theological metaphysics, if we want to be consistent about it.
For example, (x)Cgx and ~(Ex)Cxx are inconsistent. (pretend 'E' is an existential quantifier, since I don't have that in this font!)
But, while she legitimately pointed out that anything follows from logically contradictory axioms, she maintains the position that we shouldn't expect theological metaphysics to be logically consistent because of the nature of the subject matter (!). She prefers to "embrace the irrationality."
But really, either (i) all true propositions that refer to God are logically consistent, or (ii) it is not the case that all true propositions that refer to God are logically consistent. Either (i) it is the case that God's nature is self-contradictory or (ii) it is not the case that God's nature is self-contradictory.
I was feeling uncertain for several weeks about how open minded I should be regarding the universal applicability of logic to reality (at least its basic rules such as the law of non-contradiction), but then I was at TU and had a conversation with your uncle in his office about it, which made things rather clearer in my head again.
At least, it seems to be clearer...but then there's no such thing as Cartesian certainty!! :)
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