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April 04, 2005

Probability and Induction

Lets go back to the official definition of induction - the examination of the the particulars or specifics in an attempt to develop generalization. To this process, people assign statistical quantities such as probability. This, as we know is a theoretical jump. Probability just doesn’t apply (nothing is random, and nothing to choose), and it wouldn’t give a meaningful result even if it did.

Theoretical Induction: Lets test out the “Theory of Gravity” (it isn’t exactly a good thing to call a theory, but for goodness sakes, it clearly doesn’t matter). Suppose I let go of an apple. According to gravity, it is going to fall (whether you attribute the falling to gravity or not). No, wait, Gravity doesn’t say that. Gravity says that it would accelerate downward with -9.8m/s^2 or whatever that value is. Falling is a consequence based on all forces applied to the object. You see theories don’t really tell you what exactly is going to happen, but what could happen with the model provided.

Remark — Of course, we expect that theories give pretty accurate results for all (sufficiently) practical purposes, but Gravity explains objects falling on Earth “well” enough.

Class Induction: Okay, suppose I meet tons of white sheep, where by “tons”, I mean “a whole lot”. Induction would tell me all sheep are white. This is not a theory, because it (the final result) wouldn’t lead anywhere (deductively). Now, lets just suppose that I meet a black sheep. The probability of me meeting another black sheep wouldn’t change very much. You would argue that it is an anomaly, but suppose then I meet another black sheep, and yet another black sheep, and yet a whole more black sheep. Then, you would be willing to bet that the next sheep I would meet would be black, even though probability would tell you otherwise.

Remark — You see something: human intuition is more based on recurrent thinking. We believe that things happen in an orderly manner. We predict the future based on recent findings. And while we find quite a few anomalies, in the example above, the colour changes only once.

Posted by Oleg Ivrii at April 4, 2005 06:39 PM

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