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Thursday, July 16, 2009

Loblaw's law

When you have numbers that are really really really big, you can stop talking about probabilities and start talking about laws.

Statistics are often unintuitive for people, and the Monty Hall Problem is a famous example of that un-intuition in action.

And of course, as humans, we distrust machines. And statistics are, so far, the best tool machines have for imitating humans in areas such as language and vision. How we hate these machines, with their statistics.

Yes, so if you are a human reading this, you may feel a certain amount of smugness knowing that language is a built-in feature for you; you can read an article and comprehend with absolute certainty its key points and discuss it in an intelligent and natural way. Even if a computer were able to analyze the same article, it could only offer up soulless suggestions of meaning with varying degrees of certainty, and with little or no actual intelligence.

But as it turns out, you and I live in a crazy universe not governed by laws so much as statistics.

Mandatory Djikstra quote: The question of whether a computer can think is no more interesting than the question of whether a submarine can swim.

On their own, atoms can be in several possible states, each with varying probabilities, where the most probable state has lowest energy.

With more heat added to a closed system such as a gas, the likelihood that each atom is in a higher energy state increases. Each macro state view of the entire system is equally likely, keeping in mind that particles are coaxed from lower micro energy levels with likelihood directly linked to the work performed on the system. All arrangements of molecules in the container are equally likely and they are all interchangeable with one another.

Now if you were to take each of these equally likely macro energy states and group them in buckets by total energy level of the system, the distribution would produce a bell curve. So, looking at this bell curve, you could state with some amount of certainty the likely range of energy (temperature actually).

EXCEPT, that's not how the universe actually works!!

The contradiction arises because the number of particles is on the order of 10^23.

So there's still a bell curve, but it's not really a curve so much as an extremely tall spike with almost zero width and almost no variance. Temperature and entropy and the first and second "laws" of thermodynamics all work because 10^23 is an enormous, enormous number, and the expected value (the center of the curve) is the only value anyone actually ever experiences.

To illustrate I'll create a law right now, named after a brilliant attorney, called Loblaw's law. Loblaw's law states that it is not possible to flip a (normal) coin 10^23 times and produce only heads each time. You may say, well technically that is still possible, so how can it be a law?

Well, technically it is (sort of) also possible for the entropy of the universe to decrease for a few minutes. Technically, it is also possible that heat could flow from a cold object to a hot object. Technically, a broken egg could assemble itself from the floor and leap back into my hand...blah blah blah, Loblaw's law is a law because these events are so unlikely that it they will never happen.

That's the way it goes with ginormous numbers, and that's why we can call them the laws of thermodyamics, although a better name for the field is actually Statistical Mechanics.

Back to computers, the point is that statistics is a strength, not a weakness of computers which may perform natural language processing, image recognition, or other tasks wherein humans naturally have the upper hand. Feed a somewhat sophisticated AI a few petabytes of salient data and I believe suddenly that Dijkstra quote will ring true.

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