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Information theory and digitization

The second part of the Totality Axiom asserts that structure is the only aspect of essence that can be communicated. It can be regarded as a restatement of Shannon's definition of information as something that allows us to reduce the number of states a system may be in. For example, suppose we know that a flag must be red, blue, green or yellow. Then it can be in any of four states. Each state corresponds to a different color. If we are now told the flag is green we have reduced the four possible states to a single state. The amount of information transferred is that needed to reduce four states to one state. This requires a number between one and four. One can think of structure as defining the state of an object. In mathematics this is easily made explicit. One can assign a unique integer to every possible finite mathematical object. Defining a finite structure then requires nothing more than selecting the number that indexes it.

Shannon's definition of information applies equally to everything we can communicate. We can always measure the information communicated in terms of how much we have reduced the number of possible states. We can always communicate that information as a number that selects possible states as long as both the sender and receiver agree on which numbers correspond to which states.

Shannon's definition has had enormous consequences in the digitization of media. Everything we see and hear can be encoded as a sequence of numbers on a CD of DVD. These sequences preserve the structure of the sound or image. In the case of a CD the numbers represent sound pressure level at a given instant. By recreating the sequence of sound pressure level we recreate the original sound. That is what a audio system does with the numbers on a CD. In images the numbers represent the intensity of the three primary colors, at a given point in an image and instant in time. Recreate the intensity levels of these colors at the correct location and time and you recreate the image. That is what a DVD player connected to a television does.

Digitization has made Shannon's definition a practical reality of immense importance. Digitization is an example of the difference between structure and essence. The sequences of numbers on a CD contain all of the structure in a Bach Sonata, but none of the essence. The numbers on the CD only come to life when a conscious listener experiences the effect of the sound reproduced from the numbers on the CD.


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