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Fermi Paradox in Mathematics
This question sums up the core problem of the so-called Fermi Paradox: in 1950, physicist Enrico Fermi assumed that the evolution on the earth is no exception in the universe. Considering the age of our universe (138 billion years), there should be many other civilizations, many of which may be more advanced than us in science and technology. Some civilizations may even be so old that they are likely to explore other planets and colonize them. But so far, we have not found any evidence of the existence of extraterrestrial intelligent life.

Before discussing why we can't detect any intelligent alien species so far, we need to know the number of alien civilizations we should be able to communicate with in the galaxy. Drake equation provides a rough estimate (with many uncertainties). The equation is probabilistic, so it can't provide accurate results-the possible change between the upper limit and the lower limit depends on the assumption of its factors. We can apply each factor in the equation as a filter-by adding additional filters, the total will be smaller and smaller.

These factors are:

1. The average rate of star formation in our galaxy;

2. The ratio of stars to planets;

3. The average number of planets that may support the existence of life (one planet per star);

4. The proportion of planets that can support life and develop life at a certain time;

5. The proportion of living planets that continue to evolve into intelligent life;

6. Proportion of civilizations that have developed technologies to release detectable signs of existence into space.

7. The length of time that these civilizations release detectable signals into space.

According to the assumptions of various factors, it can be determined that the number of civilizations we should be able to reach may be very wide: from 1 to about15.6 million (Wikidrake equation). In June 2020, astronomers from the University of Nottingham reported that according to the estimation of astrophysics, there may be more than 30 intelligent alien civilizations in our galaxy.

Based on Drake equation and its many factors, Fermi paradox has many possible explanations. Just a few examples:

Factor 5 may be much lower than previously estimated, so the formation of intelligent life is almost impossible, so at least in our galaxy, we are lonely;

No other civilization in our galaxy has developed the technology of releasing detectable signals into space, because all civilizations will destroy themselves at some point in history (for example, through weapons of mass destruction);

We are located in an area of the Milky Way, too far away from other civilizations, and have not yet reached the technical level, so we cannot release detectable signals in time.

Among the many possible explanations of Fermi paradox, one is very popular in recent years-the so-called dark forest theory. It is popular in Liu's trilogy of "three-body" novels, and the title of the second one is "Dark Forest". This hypothesis explains the Fermi paradox in factor 6, which comes down to the following: Although there may be a large number of advanced civilizations in our galaxy that can release detectable signals into space, they choose not to do so. Why? Because exposing their existence may have potentially destructive consequences. A civilization with more advanced technology may strike first and prevent other civilizations from striking first.

The dark forest hypothesis is based on game theory: because alien species can't communicate effectively (considering their long distance, different languages, different communication methods and cultural concepts), it is assumed that the worst case scenario is that other civilizations will destroy us at the first encounter. But we are not sure whether other civilizations are stronger or weaker than ourselves militarily, so the best strategy is to keep silent first and not to expose our existence to other civilizations.

In fact, if the technology of attacking civilization is advanced enough, it may not even be necessary to conquer the solar system of another civilization-it is enough to send a military probe that may or may not arrive in time (even if it takes hundreds of earth years or more), and other civilizations can reach the advanced technical level enough to effectively defend against the incoming probe.

If human history has any guiding significance, then the more dominant civilization did conquer or destroy the inferior civilization-whether voluntarily or involuntarily, whether or not there was a prior communication attempt.

From the perspective of the dark forest hypothesis, all attempts to communicate voluntarily or involuntarily with other civilizations are foolish, because exposing our existence may lead to our destruction. What seems impossible on a planetary scale like the earth-avoiding the conqueror-is likely to become possible in the vast space.

In fact, even among some famous scientists such as Stephen Hawking, the dark forest hypothesis has gained some support. They pointed out that considering the age of our galaxy, there may be an alien civilization with a history of at least 654.38 billion years, which may have the technical ability to reach and destroy us-after all, in their view, we are to them what primitive people are to them and ants are to us.

But it's too late now, because the first radio communication signal was actually sent out in the late 9th century, so about 120 years ago, our radio communication activities really started to break out from the second world war. Because of the directionality and weakness of these signals, terrestrial radio signals are unlikely to make alien civilizations aware of our existence and location, but there have been many attempts (since 1970' s) to send directional high-energy signals to other star systems. This information usually contains mathematical and other scientific information, such as atomic number, human genome, description of our solar system and earth home, and description of human body. So according to the assumption of the dark forest, this is a very bad idea. Maybe the deadly detector destined to destroy us is on its way ... or?

In order to explain why the dark forest hypothesis is not the most likely explanation of Fermi paradox, several hypotheses will be made. The main argument is based on:

1. All life forms, including alien life forms (intelligent or primitive), are composed of individuals as part of a group. The relationship between group members can be anything from very primitive (bacteria) to highly complex (such as human society).

2. Intelligent life can only appear after a long development process called evolution. Evolution may not follow the same rules in the whole universe, but it can be assumed that a more advanced species takes longer to develop than the original species.

3. All intelligent life forms existing in the form of individuals and organized in the form of groups will be organized into factions once the total population growth exceeds a certain threshold. Different groups began to form and enter different geographical areas. In a complex society, individuals can belong to many factions at different levels.

4. Intelligent species can be distinguished from non-intelligent species by the following characteristics, all of which must be satisfied: complex language (not necessarily phonetic language), the ability to plan and execute coordinated actions (based on considering several outcomes and scenarios of future events), and the ability to accumulate and archive knowledge and pass it on to future generations.

5. Smart species can create technology, improve technology and pass it on from generation to generation.

Even if some of these assumptions are too inapplicable to all possible intelligent alien species, I firmly believe that if intelligent alien civilizations exist, these characteristics will be applicable to most.

On earth, the most important faction mentioned in hypothesis 3 is now called the country. If our history tells us one thing, it is very difficult to form a world government that all countries obey. Why do countries cede their military power to the world government? The international system is an anarchic structure rather than a hierarchical structure-the organizational principle that best describes the internal organization of a country is actually a hierarchical structure. The United Nations is not a world government, but a forum for coordinating the actions of all countries in order to bring better results to all and prevent crises and wars. So far, the success of the United Nations has been very limited at best. This is because all countries in the international system are sovereign countries.

Therefore, if these assumptions are applied to alien civilizations, it turns out that it is impossible to decide to remain silent for a long time and not to send any signals to outer space (despite the existence of technology). Just because there will always be individuals and groups with different ideas. Apply to our planet: suppose there is a country that prohibits all its people from sending signals into space, then why do other countries do so?

The fundamental defect of the dark forest hypothesis is actually a very naive simplification that is very familiar to cosmologists: the simplification of "one culture, one ruler". Whenever a starship visits a planet (first contact or otherwise), there is usually some kind of planet government, even a ruler who can make decisions for the whole planet.

The dark forest hypothesis implicitly assumes that alien civilization, as an entity, has consistent behavior for a long time. This is simply impossible-even if it applies to one species, it definitely does not apply to all species. Yes, one day there may be the possibility of a world government that all factions or countries can abide by. But on the road to the future, there will be more attempts to communicate with alien civilizations.

So where did everyone go? If the dark forest hypothesis is not the most likely explanation of Fermi paradox, what is it? It is believed that there may be thousands of worlds with alien life forms in our galaxy. However, there may be only a few-perhaps no-intelligent alien life forms except the earth. Even if there are other civilizations-perhaps their life and technological evolution will take longer, or it will take longer to transmit and receive signals. Some alien civilizations may not constantly send out strong enough signals to outer space for us to receive after hundreds or thousands of earth years. In addition, they may send some signals in the wrong direction (from our point of view, there are actually many wrong directions).

Let's look at this behavior objectively: technically, we can only receive signals from outer space for decades-the first radio telescope was built only in 1932. Compared with the whole time span of human existence, 89 years is a very short period of time. Even if the time of human existence is very short compared with the age of the earth (about 4.5 billion years) ... Considering the vastness of space, it takes a long time for signals to reach us through space. Can you really draw a conclusion by using the radio? Telescopes have only been used for decades. Is it because we didn't receive their signal that alien civilization doesn't exist (or they are hiding)? Maybe we just need to wait a few more centuries to make sure they don't destroy the earth.