Searching of decisions
To illustrate how difficult it is to find a solution to the equation n
=
a
3 +
b
3 +
c
3, let's look at what happens for
n
= 1 and
n
= 2. For
n
= 1, the solution is obvious: 13 + 13 + (–1)3 = 1. Are there others?
Yes, there is: 93 + (–6)3 + (–8)3 = 1. This calculation is not the only other solution. In 1936, the German mathematician Kurt Mahler proposed an infinite number of them. For any integer p
:
(9p
4)3 + (3
p
– 9
p
4)3 + (1 – 9
p
3)3 = 1
This can be proven using the sum cube identity. An infinite number of solutions are also known for n
= 2. It was discovered in 1908 by mathematician A. S. Verebrusov.
For any integer p
:
(6p
3 + 1)3 + (1 – 6
p
3)3 + (–6
p
2)3 = 2
By multiplying each term of these equations by the cube of the integer r3, we come to the conclusion that there are also infinitely many solutions for both the cube and the double cube of any integer. Let's look at example 16, that is (23)*2. For p = 1 we get: 143 + (–10)3 + (–12)3 = 16. Note that for n
= 3 As of August 2020, only two solutions were known: 13 + 13 + 13 = 3; 43 + 43 + (–5)3 = 3. Naturally, the question arises: is there at least one solution for each non-forbidden value?
Nihilism
Most often, the word “nihilism” is associated with the predecessors of the Russian revolutionaries of the early 20th century, but this term is much more complex. Nihilism—from the Latin hihil (“nothing”)—believes that such things as “value” or “meaning” do not exist in nature, and therefore human existence has no meaning.
Nietzsche believed that the spread of nihilistic beliefs would eventually lead to people ceasing to do anything at all. This, as we see, did not happen, but nihilism as indifference to what is happening still remains popular.
Tibetan philosophy
These teachings are common in Tibet and other parts of the Himalayas. Very similar to classical Buddhism, Tibetan philosophy believes that the meaning of life is the end of earthly suffering. The first step to this is understanding the world. By understanding the world, you can come to the knowledge necessary to end suffering.
Philosophy provides the opportunity to choose the “Path of Small Opportunities,” on which a person is primarily concerned with his own salvation from the world, or the “Path of Greater Opportunities,” on which a person helps others. The true meaning of life is found in practice. Tibetan philosophy is also memorable because it offers its followers precise instructions on behavior.
Albert Einstein
Einstein was one of the most outstanding representatives of humanity. In 1951, a young woman asked him in a letter what the meaning of life was. The answer was short: “To create satisfaction for yourself and for others.”
In a letter to his son Eduard, Einstein was more specific. He wrote to him that he believed in “a higher stage of consciousness as the highest ideal,” and that the human ability to create new things out of nothing was more than we might think. It is the act of creation that allows us to experience happiness. He also reminded that you need to create not out of a desire to be remembered, but out of love for the thing you create.
Aztec philosophy
The highest meaning of life for the Aztecs was to live in harmony with nature. Such a life allows one to continue energy and form new generations. This energy was called "teotl" and was not a deity, but something like the Jedi Force. Teotl fills the world, all our knowledge and extends beyond knowledge.
In teotl there are polar opposites that fight each other and thereby maintain balance in the universe. Neither life nor death is bad - they are just part of a cycle. The Aztecs believed that it was best to stay in the middle, not striving for wealth and using what you already had wisely. This was a guarantee that the children would receive the world in the same condition as their fathers.
Cyrenaica
The Cyrenacians were supporters of one of the branches of the teachings of Socrates. This group was founded around 400 BC in North Africa and was led by Aristippus, one of Socrates' students. Their teaching contained the proposition that the experience and knowledge available to an individual are always subjective. Therefore, no one person will be able to see the world the way another sees it. They also believed that we do not know anything definite about the world, and the only knowledge available is sensory experience.
They taught that the only purpose of life is to experience pleasure in the present, instead of making plans for the future. Physical pleasures are paramount and a person should take all measures to maximize their quantity. Overall, this was a very selfish point of view, placing the pleasure of the individual above the well-being of the community, city or country.
The Cyrenaics ignored not only foreign philosophy, but also traditional social norms. Thus, Aristippus taught that there is nothing wrong with incest - in his opinion, only social convention led to the taboo of consanguineous marriages.
Mohism
Mohism was developed by Chinese philosophers around the same time that the Cyrenaics appeared in the Hellenistic world. This teaching was created by Mo Di, who was one of the first in China to raise the question of the meaning of life. He outlined 10 principles that people should follow in everyday life, the central one of which was impartiality.
According to this teaching, the meaning of life will be achieved when each person pays equal attention to everyone else, without putting any person above others. This meant, of course, a renunciation of luxury, wealth and pleasure. The Mohists saw equality as the ideal of human relationships and believed that they would be rewarded for this with the same equality in the afterlife.
Epicureans
Epicurean philosophy is often oversimplified. According to Epicurus, everything consists of tiny particles, including the human body, which is made up of particles of the soul. Without soul particles, the body is dead, and without the body, the soul is unable to perceive the outside world. Thus, after death, neither the soul nor the body can continue to exist. After death there is no punishment, no reward - nothing. This means that a person needs to focus on earthly affairs.
Particles of the soul are capable of experiencing both pleasure and pain. Therefore, you need to avoid pain and enjoy. What we can't control (unexpected death) we just have to accept.
This doesn't mean you can do whatever you want. Even if robbing a bank brings some pleasant experiences, a true epicure remembers that feelings of guilt and anxiety can lead to greater discomfort later. Epicureans are also committed to friendship, the most pleasant, safe and reliable feeling that can be available to a person.
Darwinism
Charles Darwin had a complex relationship with religion and the religious meaning of life. Initially he adhered to Christian beliefs, but later his views noticeably wavered.
Some of his heirs began to practically deify evolution - after all, it was it that ensured the emergence of man. They see in this the highest meaning of evolution and believe that it inevitably had to lead to modern people. Some, on the contrary, emphasize that evolution is a combination of a chain of chance and the ability to survive. But both agree that the meaning of life is to pass on part of your DNA to future generations.
Cynics
The Cynics were another group close to Socrates. They found the meaning of life in living in obedience to the natural order of things rather than to ethics and traditions. Cynics believed that social conventions such as wealth or hypocrisy prevented people from achieving virtue.
They did not abandon public institutions entirely, but believed that each person develops his own personal ideas about good and evil and has the right to go against society by following his own guidelines. This is where the principle of “paresia” arose - the principle of telling the truth.
Another important principle of cynicism was self-sufficiency. Cynics believed that a person can maintain freedom only if he is ready at any moment to refuse communication with other people and the benefits of civilization.
