Saturday, September 28, 2013

Complex Problems: The Multiplication of Doubt

In my last two posts which build the foundation of my epistemological framework, I have described how the complexity of problems can be defined through building a hierarchy of constructs and relationships in the order of their complexity. There, I briefly describe the most basic elements with a rudimentary explanation for its complexity ranking as a standalone element. To follow, I described the most basic element, existence, and the general concept of abstraction, which allows us to discuss doubt as a concept which is the abstraction of certainty.

Let us build on the previous discussion here of the nature of knowledge as being inherently probabilistic. Let the certainty of the truth of any given proposition X be represented as Pr(X) in standard statistical notation. Suppose X is some basic proposition with the lowest level of complexity: "The spoon exists," for example, where a spoon is observed on a table in front of you. Having experienced the spoon in front of you, and indeed, currently experiencing it in front of you, you would have supreme confidence in the existence of the spoon. You could not provide a logical proof of the spoon's existence, but you could say that you are as close to absolute certainty in this proposition as one may be; in other words, the limit of Pr(X) approaches 1 in this case. For ease of demonstration, let's suppose Pr(X) = 0.9999.

Now let us extend this to proposition Y: "The spoon is silver." Proposition Y contains Proposition X as an element (the spoon must exist; in order for it to be silver, it must necessarily be), and also introduces a categorization of the spoon as "silver". This categorization requires that the concept of "silver" is well defined as a category which may describe or contain the object in question. Thus, while Proposition Y does not seem like much of a leap from Proposition X, it entails a certain degree of greater complexity which must be accounted for.

Suppose that the concept "silver" is defined with the same strength as the spoon's existence; that is, if we were to define a proposition Z, "Silver exists", then we would have Pr(Z) = Pr(X) = 0.9999. Further, suppose that propositions X and Z are independent. Then Proposition Y, being the union of Proposition X and Proposition Z, would be well-defined statistically as: Pr(Y) = Pr(X ∪ Z) = Pr(X) × Pr(Z) = 0.9999 × 0.9999 = 0.99980001.

The complexity introduced by incorporating multiple basic elements into Proposition Y, while seemingly insignificant, must necessarily introduce a greater degree of doubt: If each individual element has a certainty less than absolute (1), then any multiplication of two individual elements will result in a number less than either of the individual elements.

It should be noted that in this simple example, the basic propositions X and Z are likely much more certain in our minds for the purposes of any decisions we need to make - perhaps as much as 1 - (1 × 10e-15) or more. Consider all of the elements that are around you, even in a very safe and controlled environment, that you must account for, which make up the mental framework of your reality. These doubts must multiply quite a bit in order to get to a problem that we find non-trivial in our daily lives; many multiplications of doubt must occur in order to approach even a 1% doubt. 

Nevertheless, we can seem overwhelmed when information is in great supply, even if the answer to the problem at hand is quite easy. Consider a restaurant menu with 5 simple options, and contrast this with a "Chinese menu" with hundreds of options, each with customizable elements. Simply increasing the number of available options makes the problem inherently more complex, thus introducing a greater degree of uncertainty in our process. This is a clear representation of the multiplication of doubt caused by problem complexity.

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