Friday, September 13, 2013

Orders of Complexity

If we assume that we, as humans, have no knowledge a priori from our creation, and that we instead are only more or less certain about any given proposition, then we must start to examine the most basic propositions in our world. The goal, in so doing, is not to determine some empirical probability that we have for any specific proposition or type of proposition, on average. Nor should we be naive enough to believe that we might be able to order our confidence in any two specific propositions. Although confidence can be expressed on a continuum from 0 to 1, we resign our practical ability to fall short of precisely determining our internal confidence in any given proposition, much less a strict comparison between two propositions in which we have similar confidence.

Rather, by examining some propositions, we can perhaps discover how our brains operate in practice, by producing a rough classification which distinguishes certain problems from others. Then, we might generally say that we are naturally inclined to be more confident in the truth of some propositions than others. Another way of stating our goal is that we might classify the order of complexity of a given proposition, based on the constituent elements of that proposition - much in the same way as a grammar teacher might be able to evaluate the complexity of a sentence based on the parts of speech, syntax, and vocabulary of which it is comprised.

Such a classification system would certainly not be rigid; there are many gray areas which will plague any proposal for what causes complexity. However, one thing that we will want to avoid is to use observations of human behavior to determine whether or not a problem has been complex. Much human behavior is predicated on an unconscious process: recognition of patterns, reliance on heuristics, and ill-formed reaction might cause us to answer swiftly, even if we should be unsure of ourselves. Instead, we will want to produce orders of complexity which are defined and endogenous to the problem, rather than the problem-solver.

The most basic form of proposition expresses only the existence of some object or concept (hereafter simply referred to as "objects"). We might express the existence of some thing X simply as "X is." Existence is the most basic proposition: Without objects, it is difficult to think of what any other proposition might look like. This is particularly true when considering that "nothing" may be used as a concept which can exist, as "X" may refer to a vacuum.

Above existence, the following constructs are listed in order of increasing complexity. I will briefly describe each with explanation on why it should be more or less complex than its neighbors, where appropriate.

Categorization extends existence to describe some characteristic of an existential object X. For example, "X is red" implies not only that X exists, and that the concept "red" exists, but that X fits the definition of (can be categorized as) "red".

Ownership defines one existential object as dominant. For example, "X has a dollar" implies not only that X exists and that a dollar exists, but that the dollar belongs to (is owned by) X. Ownership introduces complexity above categorization, as it is not a necessary condition that X should have the dollar, and X may become dispossessed of the dollar without any effect on the inherent categorization of X as an object.

Action introduces time, space, and/or logic as an element of consideration. For example, "X moved" contemplates that X existed at one place at one time, and existed at a different place at a different time, without X having substantially changed otherwise, and with X positively being the same object in both instances.

Association also introduces time, space, and/or logic, but goes beyond the existence of X within that framework by providing a relationship between two existential objects, usually (but not necessarily) using a categorization. For example, "X is under Y" implies that X exists, Y exists, the concept "under" exists, and X can be categorized as "under" Y. Similarly, "X and Y are friends" implies that both X and Y exist, the concept of "friendship" exists, and that X and Y are related by friendship.

Causal Relation introduces causality which provided the basis for the discussion of determinism here. The statement "X resulted in Y" consists of an action (or process) X and a post-event state of affairs Y. This goes beyond the concept of action by providing relationship between multiple objects or concepts; in other words, it is the difference between the simple action "X moved" and the more complex "X moved Y".

The list above may not be exhaustive, but can serve as a guide for thoughts of higher-order problems. Complexity may build rather quickly as we introduce many existential objects, associations, categorizations, and causal relations. For example, it might be said that the problem of projecting a country's GDP is dependent upon all actors which exist in the economy, their assets (ownership), their actions, and the resulting causal relations. This is a highly complex problem indeed!

The purpose of this thought experiment is not to provide an infallible framework for how complexity arises, but rather, as a way of thinking about the complexity of problems, and as a rough framework for comparing problems to one another. As a general rule, we should be more certain in propositions as their complexity is reduced: We are relatively more sure that the ball exists than that it is falling due to gravity (even if we are very highly confident in all of these things). A more detailed treatment of this framework will be necessary to provide boundaries and discussion on how we approach problems of science in practice.

1 comment:

  1. 1. This is now my favorite blog. Keep it up!
    2. It seems X & Y must both exist in propositions such as X moved because movement is really a statement of reorganizing the state of matter. Therefore something is only moving if its relationship with other matter changes. But a single point X in a universe devoid of all else can never be said to have moved. Same for color or ownership. Even existing suggests a causality and a relationship. Just existing is a complex event.
    3. Do you think if it were possible to know the state & direction of all material existence: Every proton, every neuron in every brain, every photon, etc. -- would it be possible with mind capable of processing all those things to state with certainty what will be even until the very last reaches of time? Or would you say that the "free will" of rational men is not merely dependent on interaction of matter but also dependent on an immaterial soul and that the will itself cannot be perfectly determined even with perfect information of the material world?
    4. You sound very Catholic. Have you considered attending a traditional Latin Mass? You should check out for one nearby.