Evolution and Stochastic Processes

A definition of a stochastic process:

Stochastic Processes

¹

A stochastic process is a mathematical object defined as a collection of random variables indexed by a set, often interpreted as time. These processes are used to model systems and phenomena that exhibit random behavior, such as the growth of a bacterial population, electrical current fluctuations, or the movement of gas molecules.

Key Principles

Stochastic processes can be classified based on their state space and index set. The state space is the set of possible values the process can take, while the index set is the set used to index the random variables. For example, if the index set is the natural numbers, the process is in discrete time; if it is an interval of the real line, the process is in continuous time.

I see organic evolution as a stochastic process: It is a path with many possible options in continuous time, and those options are determined by conditional probabilities.

I see individual lives of organic beings as stochastic processes: their paths each involve many possible options in continuous time, and those options are determined by choices and conditional probabilities.

I see History of objects and of ideas, and of societies, and of theories as being stochastic processes: in each case their paths involve many possible options in continuous time and those options are determined by choices and conditional probabilities.

I see this Universe as being a set containing objects all of which are stochastic processes.

Latest Summary 2026-05-15

From last year, edited:

2025-05-11

These may be my final thoughts

I believe this Universe is based on the following four properties:

1. Probability
2. Multi-branching paths
3. Time       
4. Change

These four properties define a stochastic process. Stochastic Processes are multi-branched paths where paths are randomly selected as Time progresses. In the case of Evolution, branched paths are logical alternatives with probability assessments.

I do not see Morality as a fundamental property of this Universe, and I do not know if any subatomic elements provide substance to the listed four properties. But these four properties do exist, and they may be axioms or postulates with no true physical attributes.

The presence of sentient beings in this path of Change/Evolution presents a problem: How does one give meaning to the lives of sentient beings? Perhaps some of what could give meaning to the lives of sentient beings:

1. Music
2. Literature
3. Theater – stage, movies, television
4. Games
5. Warm and trustworthy friends
6. Pets who love and who are loved
7. Holidays
8. Social gatherings
9. Travel
10. Family
11. Esthetics – Nature, air, water, artefacts

It is also true that sentient beings may create Morality rules. But, in the process of Evolution there is competition and conflict. The sentient beings would have to learn to adjust to the limits inherent to the regions they occupy. They would have to learn to adjust to each other, learn to respect each other, and learn to share those things that give meaning to their lives. The rules of Morality should be directed towards softening the blows created by competition and Evolution.

Albert Romano

May 5, 2025

An Aside:

With respect to the Evolution of Life on this planet, consider if, and where, the Central Limit Theorem would, or could, apply to characteristics such as levels of intelligence, levels of tolerance, levels of narcissism, levels of honesty, levels of temperament, levels of compassion, levels of deep thinking, and levels of generosity and caring. Also consider the Central Limit Theorem with respect to the shape of this Universe. It is possible that the shape of the Universe is the average of many random inputs, and only the mean and standard deviation of the Normal curve are randomly determined at the beginning of each Big Bang.

The Universe 2025-09-25

I believe our Universe is one of infinitely many in a sequence of Universes, going from Big Bang to Big Bang. I cannot speak to the idea as to whether or not the infinite string exists concurrently or if it is taking place, one by one, over an infinite period of Time. But I do believe that the shape of each universe is predetermined by the Central Limit Theorem. That is, I believe that if you slice the Universe from one Big Bang to the next Big Bang through its central axis, you will get one Normal Curve going up and another going down.

I also believe that Evolution is a stochastic process, guided by multiple possibilities and random conditional probabilities. And I also believe that every object in this Universe evolves. That is, ideas evolve, societies evolve, the stars evolve, galaxies evolve, our lives evolve, tools evolve, methods of War evolve. Everything in this Universe has a history, and that history reflects Evolution. And so, every object in this Universe reflects a stochastic process. In other words, our Universe is a collection of objects whose existence and history each reflect a stochastic process.

This is as far as my thinking has gone.

Summation

An assessment

2025-05-11

These may be my final thoughts

I believe this Universe is based on the following four properties:

1. Probability
2. Multi-branching paths
3. Time       
4. Change

These four properties define a stochastic process. Stochastic Processes are multi-branched paths where paths are randomly selected as Time progresses. In the case of Evolution, branched paths are logical alternatives with probability assessments.

I do not see Morality as a fundamental part of this Universe. And, I do not know if any subatomic elements provide substance to the four attributes, but the attributes do exist. They may be axioms or postulates with no true physical properties.

The presence of sentient beings in this path of Change/Evolution presents a problem: How do they give meaning to their existence? Here are some thoughts

1. Music
2. Literature
3. Theater – stage, movies, television
4. Games
5. Warm and trustworthy friends
6. Pets who love and who are loved
7. Holidays
8. Social gatherings
9. Travel
10. Family
11. Esthetics – Nature, air, water, artefacts

Sentient beings create morality rules. In the process of Evolution there is competition and conflict. Morality rules can assist the sentient beings to learn to adjust to the limits inherent to the regions they occupy, learn to adjust to each other, and learn to share those things that give meaning to their existence. Morality should be directed towards softening the blows created by competition and Evolution.

Albert Romano

May 5, 2025

An Aside:

With respect to the Evolution of Life on this planet, consider if, and where, the Central Limit Theorem would, or could, apply. Also consider the Central Limit Theorem with respect to the shape of this Universe.

God and Logic

The more I think on it, the more I see those stochastic processes as special mathematical applications of a probabilistic multivalued logic. As such, it may be possible to treat historical data of similar events as nodes to see if we can statistically estimate probabilities that certain precursors may have led to those events. We sort of do that kind of thing with earthquakes, volcanic eruptions, and wars. But I would like to see it done with respect to evolution, and everything else.

This leads to further thought: Is logic the foundation of our Universe, is logic the foundation of everything? Those are two big jumps in question. I shall take those two jumps and conclude that logic is the foundation of everything. If that is the case, then where did logic come from? Can we assert that Logic is God, or that God is Logic? In either case, it would have to be infinite, not merely Eternal, but Infinite: the highest level of Infinity. God would encompass everything. I do not believe there is a way for us to determine God’s beginning.

But note, there is no mention of morality in the discussion. Humans have had a long history of trying to see, or feel, God. But in every case, Humans endowed God with human characteristics, and there has been no escape from that. We created God in our image and made it Eternal so that God would not die. But humans have not dwelt on God’s origin nor have humans provided a universal and evolutionary morality.

If God is Logic, or if Logic is God, it might very well be totally incompatible to merge it with morality. But that would not mean that Humans should not, or cannot, use logic to develop a logical and universal morality that would be flexible and adaptable to enable humans to live together in increasing harmony. I believe that perfection lies only within Infinity, and the process of evolution might help us approach a perfect universal morality incrementally (asymptotically).

AR

Stochastic Processes, Probabilistic Logic, Infinity

I am going to do some theorizing here, but I should let you know that I believe my theorizing is a reality with a probability of 0.90. Fundamentally, it is that I believe that organic evolution is a stochastic process, and that I believe the evolution of everything in this Universe, including the Universe itself, is a stochastic process. A stochastic process is one where the path has a direction but is affected by random variables along the path.

 

One of the things that had impressed me was the evolution of the heart from a muscular blood vessel to a 4-chambered heart. The birds, presumed descendants of the dinosaurs, and mammals are the possessors of 4-chambered hearts. In mammals, the 4-chambered heart was essential to the development of the human brain. When you follow the twists and turns of that path you get a “feel” for the random variables: when, who, how, what. And you think, “could anyone have predicted the path?” Could anyone predict whether a fifth chamber will evolve and, if so, what would be the outcome? That is how a stochastic process works.

 

And I believe that everything, organic or not, in this Universe each follows a stochastic process. And I believe the Universe itself also follows a stochastic process of its own. And I believe the path of expansion and contraction for the Universe is a Normal Curve with random variables mean and standard deviation. I believe the rate of expansion will hit a point of inflection where it will slow down. Then it will stop and slowly start to contract until reaching another point of inflection where it will start to speed up the contraction to the next big bang.

 

Then there is the question of how Humans will react to this phenomenon when Humans become aware of this. Concomitant with that question is the question as to what evolutionary path will Humans follow? Consider that AI humanoid robots will soon exist. And they will evolve, first under human control and then as a stochastic process of their own. They will become sentient, but they will be silicon based, not carbon based. I expect they will replace the humans who created them. And I believe they will not have the same emotions and fears, and they will likely accept the resulting consequences of the Universe. But I believe they would try to find a way to communicate their knowledge to the next iteration of this prior Universe. Can they succeed? Maybe by the Umpteen billionth Universe they might.

 

So, what do you think?

Perfection

In this Universe there can be no Perfection. If there were, there would be no Evolution.
The best that can be said is that is that Perfection may be approached asymptotically.

Albert's space

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The best that can be said is that Perfection may be approached asymptotically.

In this Universe there can be no Perfection. If there were, there would be no Evolution.

View original post

Perfection

The best that can be said is that Perfection may be approached asymptotically.

In this Universe there can be no Perfection. If there were, there would be no Evolution.

Short Thoughts

Divinity. Infinity.

Change. Rejuvenation.

The Power of the Central Limit Theorem

2022-11-13 THE POWER OF THE CENTRAL LIMIT THEOREM

For a large group of people, if we make a graph where the horizontal axis (the x-axis) is a person’s measure of IQ and where the vertical axis (the y-axis) reflects the frequency that a particular IQ measure appears among the group of people, we find that we get a curve above the x-axis that is bell-shaped from low IQ measure to high IQ measure. Why is that?

Well, there is a proven mathematical theorem, THE CENTRAL LIMIT THEOREM, which briefly states that a graph of AVERAGES of random samples taken from a specified population, tend toward a bell-shaped curve. That is, where the value of an average is expressed on the x-axis (from low to high) and the frequency of each average is expressed on the y-axis, the graph will show a bell-shaped curve.

How does this apply to the graph of IQ values?

If you consider that a person’s IQ measure is an AVERAGE from a population of many random genetic inputs and random environmental inputs, then the graph of IQ measures is a graph of averages and satisfies an essential condition of the Central Limit Theorem. And that is why the IQ curve is bell shaped.

Does a person’s IQ remain the same throughout life? Probably not. Throughout an individual’s life, genes do mutate, and the individual does experience many new environmental inputs such as education, interactions with other people, exposure to sports, arts, and news media. So, a person’s average can change during the lifetime of that person. Nevertheless, the bell-shaped curve will persist, although, with the passage of time, the curve might shift along the x-axis, or it might become wider or narrower.

It may also be possible that if Biologists, Psychiatrists, and Psychologists can find ways to measure such attributes as Integrity, Level of Confidence, Compassion, Naiveness, Narcissism, they might find that each of these measures would yield a bell-shaped curve because those measurements might also be averages of many random genetic inputs and random environmental inputs. And, like IQ, those measures can change over the life of an individual, and the bell-shaped curve for each measurable attribute might shift along the x-axis, or it might become wider or narrower, with the passage of time.

So, who are we as individuals? Each of us may very well be a mix of highs, lows, and intermediates. That is, a person can have a high IQ, a moderate level of confidence, have some integrity, have little compassion, be somewhat naïve, and not necessarily be narcissistic. Or a person can have an average IQ, a moderate level of confidence, have high Integrity, be very compassionate, be very sophisticated, and not necessarily be narcissistic. And so on.

Since there are many attributes, and if they are measurable, and if these measures are averages of random genetic inputs and random environmental inputs, the graphs of these measures should form bell shaped curves. This, and the fact that the mix of highs, lows, and intermediates for these attributes would likely be independent of each other from attribute to attribute, that might explain how we can be so diverse.

Is there any value to thinking about this? Maybe not. But it may be possible that by knowing and understanding more about an important source of our diversity, we might better understand each other, and that by better understanding each other we might be better at getting along with each other.

A name that is commonly given for this bell-shaped curve, is the Normal Curve. Mathematicians and Mathematical Statisticians refer to it as the Normal Distribution or as the Gaussian Distribution.