The Horrible power of the Dialectic.

I've been thinking a lot about the concept of information. As I study, I realize that which launched the information age was the idea that information is objectively real.

First, its a good word: "In-Form-ation", the making of a form within. Also In-Formation, like marching soldiers: Discrete units coming together to take a certain form.

If a thing is information (like a number, or picture on your computer) it has the unique trait that it can be in many places at once, while something that is not information, like the physical subject of the picture, is only in one place at once. The word "human" is information, because there are many humans, ("human" appears in many places at once) while a given human, like me or you, tends to be in one place at once.

Information can be objectively measured, because information alters the probably space of what might be to that which is. For instance, say I have a dice you can't see, and I roll it. I tell you that the number is odd, cutting the size of the probability space of it what it might be in half, from 6 possibilities to 3. Putting that 6 over the 3, we get an information measure of 2/1 for that statement. Now if I told you instead that the answer is either a 1 or 3, (reducing the probability space to 1/3 its size) we get an information measure of 3/1. Therefore you can objectively say that telling you the die is either a 1 or 3 gives you more information than telling you its odd (thus 1,3 or 5).

But this is incomplete. Suppose we both have a dice, and we each roll it so the other one can't see it. How much information do we have between us? It should be the information you have (6/1) plus the information I have (6/1) so 12/1. However this clearly isn't true. With both die, we have a 6x6 probability space, with 6 rows for all the possible values of your die and 6 cols for the possible values of mine, so the answer should be 6x6 = 36/1 not 12/1 To fix this so it works, you need a information measuring function I for probabilities p1 (the probability of your answer) and p2 (the probability of mine) such that I(p1) + I(p2) = I(p1*p2) Now a math function that acts this way: log(a) + log(b) = log(a*b), so we can use log(p) as an objective measure of information. Since probabilities are always less than 1, the log is negative and its absolute value is what we are concerned with. So a I(1/2) = 1, while I(1/1000) = 10. (approx for log base 2) which is to say that if you learn I have synaesthesia, as 1 in 1000 do, that gives a great deal more information about me than if you learn I am a male.

You can use this to talk about the total information content in things. The classic example is a binary signal. If a signal comes from the aliens in space consisting of 99,997 zeros and 3 ones, its pretty obvious that whatever they are saying, there is very little information in the message because its almost entirely zeros. IF on the other hand its a rich mix of zeros and ones, than it has the capacity to hold a message. This is Information (Shannon) Entropy. Its the probability of each random variable occuring (in this case the two random variables are zero and one) times the information of that probability, added up. Obviously in a real analysis of the message, more things would come into play like message length, periodicity etc., but these are beyond the purity of this idea.

An intuitive peek into its broader nature: Suppose the economic value of a book is given by its usefulness of its information to the general consumer. I write my book People with Synaesthesia are from Neptune, Everyone else is from Uranus. This book is a real life changer for people with synaesthesia because it focuses in right on them as they relate to the world, a 1 in 1000 minority, and us such has the information value I(1/1000) = 10. which we multiply by, say, $10 for a total economic value of $100 for a person with that condition. However, to the 999 out of 1000 who don't have it, being told that they are from Uranus because they are not part of a a small minority is worthless, having the information value I(999/1000) * $10 = $.01 = one cent. Its a throwaway book. So for a population of 1000, the total economic value of the book is $110. $100 from the person with the condition and a penny from the rest. Now along comes the competition, and writes his book. Men are from Mars Women are from Venus. This book cuts the population in half and speaks in terms of broad generalizations, and thus doesn't offer anything like the information value of the previous book to its target audience. The information content is thus has the measure I(1/2) * $10 = $10 per person. Some good thoughts, nothing life changing. But here's thing: The latter book is worth $10 to everybody in the population, for a total economic value of $10 * 1000, or $10,000, totally eclipsing the $110 made by the previous book.

In more formal terms, the random variable set {Mars, Venus} results in a much higher Shannon Entropy than the random variable set {Neptune, Uranus}, for the population because of the probabilities associated with each of the sets. Which is to say more people get more information from the more general division into halves than the more specialized division into a tiny minority vs everybody else.

And THIS is what lies, I believe, behind the phenomenon of the Dialectic, the tendency to see things in terms of broad dualities: Conservative Liberal, Male Female, Cat people Dog people, etc.

Our senses give us vast amount of data, too much for our brains to handle. So our brains are in the business of sifting through that data for discrete classifications to things. Its the art of chopping the weird, vast, continuous probability space of reality into distinct groupings. The groupings that we seek are those which are the smallest, yet yield the greatest amount of information in the greatest amount of cases. And these, information theory tells us, are broad dialectics, which is why our brain seeks them out.

I think of this a natural part of the human mind. But when I look at the world today in terms of its politics or its stereotypes, I am bothered by it. I think the reason this is such an important idea to me, is that we need to realize, we produce the dialectics through which we view the world... They don't produce us. But we act as though we do. Once we read about Mars and Venus, or our star sign, or anything, it can give us a lense through which to view the world and ourselves, but that very lens can shape us. More on this in Part II.