Natural Information

I want to compare the number of combinations of n bits:

2ⁿ

with growing processes, like with accruing of capital with annual compounding

(1 + (i)/(100))ⁿ

or the especially interesting natural growing

e^x = lim_{n −  > ∞}(1 + (1)/(n))^nx = lim_{m −  > ∞}(1 + (x)/(m))^m

Mathematics describes reality. So the similar formulas should have similar, though abstracted, realities: This similarity can be found in the context of information.

Note

I often use the variable - value pair. Value, though, has the connotation of additiveness. This is not implied in my usage, but also not excluded. Variable - state, or variable - element would be alternatives.

The key to compare them is to understand information in the shape of bits as a growing process.

The bit values are identifiable, have a real existence in the computer. Every bit variable in a set of bits is also identifiable. Let's not name the bit variables and bit values, though, but instead only look at the size of distinguishable bit value patterns. Every bit increases this size by 1 times what is there already. Let's denote this aspect of the bit by (1 + 1) to emphasize that an additional 1 is added to the one there already. The parentheses make this an operator, an element of the number set Q, the set where the number also includes the multiplication operation, thus making it an operator. The operator's effect is to duplicate the distinguishable multitude. n repeated applications of (1 + 1) produces a multitude of size

(1 + 1)ⁿ = 2ⁿ

Every new bit is compounded to the existing combinations.

The entry in this multitude is used to represent a real value in a real variable of size C < 2ⁿ. The information measure for a real variable, when dealing with bits, is the number of bits n = log₂C needed to create the multitude of size C. This measure arises from the wish to create a multitude, using bit variables, that can be mapped to the values of the real variable. The real information is the size of the actual variable we map to.

Note

If we start from a number of variables, the exponential function gives the number of value combinations. If we start from a number of values, the logarithm gives the number of variables needed to represent it.

For interest calculation we look at an amount of money (the 1), which is deposited in the bank with interest i. After n years the 1 has grown to

(1 + i ⁄ 100)ⁿ = qⁿ

q is not 2, normally just a little above 1. The corresponding "information" measure in a financial context of interest i would be the number of years, or whatever unit of compounding period one chooses to use.

The essential difference with respect to bit information is that what is added is a fraction of what is there. But then, fraction is actually just a matter of units.

The units of living organisms are cells and the ultimate units in the real world are the quantum particles. Both of them are small compared to the things around us. And with such small units one can also compound arbitrarily (infinitely) often:

lim_{m −  > ∞}(1 + (x)/(m))^m = lim_{n −  > ∞}(1 + (1)/(n))^nx = e^x

In the first equality we see that, given a certain growth, varying the compounding steps amounts to varying the growth rate.

Note

Actually in the financial world the real compounding takes place in very small steps, just that the bank forwards them to the customer in larger units of time for several reasons.

x is the information in the natural information unit nat. Basically we split up the size of the variable to infinitely many infinitely small fractional variables, whose size are just a very little bit larger than 1.

What is special about e^x is that its change is also e^x. It is a consequence of the infinite amount of compounding steps and that ∞ − 1 = ∞:

(de^x)/(dx) = ∞(1 + (x)/(∞))^{∞ − 1}(1)/(∞) = (1 + (x)/(∞))^∞ = e^x

So to summarize:

• The actual information is the size of the real variable.
• The logarithm x = log_bC arises from the view of this size by combinations of constituent or representative variables like bits (b = 2), years (b = q), trits (b = 3), nats (b = e), digits (b = 10); cells, atoms, quantum particles, ...

This change of perspective is done very often in physics. We can think of it whenever we see an exponential function or a logarithm in a formula.

In programming the size of a program can also be measured

• by the number of variables (assumed independent by a good design, and only those, which actually contain information, i.e. assume more than one value)

  This is a relevant measure for design and reviewing of source code.

• the number of value combinations throughout the lifetime of the program

  This is a relevant measure when testing a program.

Note

Following these thoughts helps to feel at ease with the two ways to view information:

• the number of representational variables (information measure)
• the number of values (states) of the actual variable

Relate health issues first to your diet

The human species started out of Africa and spread over the whole planet. At those times the human world population was small compared to today. The roaming and scattering was done in small groups. As with all species, once separated through distance, independent breeding and environmental differences in selection our development forked. But we all did quite well. The populations grew and some groups became peoples, the first civilizations burgeoned. This was facilitated by domesticating animals and farming the land. From there on we talk about history. But the time before, the prehistory, was a lot longer. It is there that genetic differences developed.

In the long prehistory time humans adapted genetically to their environment, especially in what we ate. The tribes following the herds to the north fed on meat mostly, especially during the long winters. Those in the warm regions ate more plants and fruits because they were available either throughout the year or at least during summer.

When the populations grew, the peoples spread and met again: They waged war, fought for resources, mixed genetically again. These are time periods we learned to survey in our history education, periods of 2, 3 thousand years. That is no time compared to the hundreds of thousand of years before that. That is no time for genetic adaption, which counts in generations. The history time accounts for a hundred generations, the prehistory time for 10000 generations.

The history time brought us in today's small world for the 7 billions of us. We now can reach every spot of the world within a day. We mixed up and we mix up more than ever today, genetically, culturally and, in regard to the topic, especially in food culture.

We may be born into a "homogeneous" culture, but actually might not be so much in a genetic sense, because the mixing went on for some time already. So we start to live in a food culture that is good for some of us, but is bad for others. Some of us can drink milk and eat sweets throughout their life, others get intolerant to the lactose in the milk and can't eat fruits when they become adults. Others report issues when eating meat.

Health issues were not very much linked to food throughout history, because there was not the awareness brought by through the scientific method. Surely some individuals found out by observing themselves and couldn't explain why everybody else seemed to have no problems at all. Still today diets are advertised as the good diet for all. But it is not the case. We are different in what we can eat and in what we can't, even if we live side by side.

Were I was born people have a saying: "You eat what is on the table". I was born on a little mountain farm and I know of grandfathers who had stomach issues and died early. These people lived before the modern world had reached the village. They lived of the farming, and a lot of it was grain. The land was quite split already and rather small. So one had to be parsimonious to get along though the winter. One could not slaughter too often and so one ate grain products mostly. Their health issues were not associated to what they ate. They themselves were not aware of it and the people around them would also not have accepted special food for one of them.

All people have a long history of natural genetic selection behind them and are or were well adapted to their environment. This means that they also felt well, felt lively and full of energy. The fast change, compared to evolutionary times, in the food culture though cultural mixing and systematic creation of new food products, made some of us feel tired most of the times.

If you have health issues, mentally scan your diet first and do trial and error with food until you feel better. Observe your digestion.

A doctor will not see how you feel day in, day out. In general one cannot give away responsibility to others, also not to a doctor. Doctors often don't search carefully, because they don't have time, because they are in a routine to walk people through their practice. Don't become a congestion of the health system and don't waste money for doctors or health gurus.

But what is a health issue in the first place? If you feel tired and can't concentrate although you have slept enough, you might think, that's the way you are. But think again, think of what you ate, Tiredness is the first symptom and can last for decades, possibly accompanied by red eyes or light skin issues. The next symptoms show a lot later, maybe when you are 40. You get swollen eyes, more visible skin issues. But then your inner organs and all the rest of your cells have a long time of suffering behind them and have spent their very resourceful ability to cope with the poisonous milieu you have kept them for too a long time.

That is a big problem about eating related issues. It takes a long time until the symptoms become serious. So people tend to neglect them.

Take the first signs serious. I've heard that from an old professor who was shaking like hell while lecturing. Ultimately the nerves are the most precious part that will cease functioning: shaking, dementia an the like. Of course everything will cease functioning finally at one time. It is just that eating related issues tend to be a slow decline.