Let’s talk a little bit about the axiomatic system

Let's talk a little bit about the axiomatic system

From this week on, I have to go back a bit and look at some of the basics of how human knowledge develops. So, with your permission, I will first go back to Euclid, the great Greek mathematician who lived in Alexandria, Egypt around 300 BC.

Euclid is best known for his 13-volume work Elements. In the first part of these books, Euclid takes mathematics before him, sorts it, and brings the topics together.

Then it establishes a unique system.

What is the system established by Euclid? It starts with 5 suggestions, documents or postulate that do not require proof of authenticity. The first four of these are easy to accept.

Like “only one line passes through two points”;

Such as “a line segment can be stretched indefinitely in both directions”,

Like, “Let’s draw a circle with a center and a point in it.”

Like “all right angles are equal” …

But of course there is a fifth suggestion — axium-postula; Very popular. To translate exactly like this: “If a line falling in two straight lines makes the interior angles on the same side smaller than two right angles, then the two straight lines split at two lower right angles, if extended, to infinity.” This phrase has a simple expression: “Only a parallel can be drawn from a point taken outside a line.”

I really do not like this plain expression, I like Euclid’s original verse, which is confusing. Let me explain if this text confuses you: Draw a parallel and draw another line that separates these two straight lines. If the sum of the angles created by this last line within the same side parallel is less than 180 degrees, you will see the lines split if you stretch the parallelogram as needed. Since the sum of the interior angles on the other side is more than 180 degrees, the rows move away from each other as the length on that side increases.

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Contrary to concept, if the sum of the internal angles is 180 degrees, the parallelogram will remain parallel forever.

Will it survive? This question has plagued almost every mathematician throughout history who has read Euclid’s elements. These include Omar Khayyam and Lobachevsky.

I would like to write the main purpose of this article, replacing a few paragraphs with a discussion of the parallel proposal.

What Euclid did was a great leap of the human mind. Euclid establishes for us a system that is the product of our own mind and of internal stability, in our own language and thought system. Moreover, this system, i.e., geometry, is in harmony with nature, through which we can describe nature.

Such systems are called axiomatic systems. So to begin with, build an entire building under certain basic conditions that no one rejects and no special effort is required to prove it.

I just said that we can describe nature through the system established by Euclid, which is why for centuries mathematicians have been obsessed with the fifth proposition, the parallel proposition.

If we can imagine, if we can, a perfectly flat plane coming from infinity to infinity, there is nothing wrong with this suggestion.

But there is no such thing as a perfect plane in nature.

Although mathematicians are confused and this proposition seems plausible, Euclidean geometry has remained the same geometry for over 2,000 years.

But finally, from the early 1800s, mathematicians began to understand that the problem arose from the “perfect level” that emerged from Euclid’s description, first dreamed up in 1813 by Carl Friedrich Gauss and then in 1818 by Ferdinand Karl Schwartz, a lawyer. Non-Euclidean geometries, but his writings are not published. Subsequently, non-Euclidean geometry was independently developed and published by the Russian mathematician Nikolai Ivanovich Lobachevsky in 1829-30 and the Hungarian mathematician Janos Bolai in 1832.

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My initial explanation, you guessed it from non-Euclidean geometry, is that such “planes” are imagined, the “plane” is not straight, but curved at a positive or negative angle.

For example, imagine the Earth’s crust. There is no way to draw infinite parallels here; Because the earth has a steep surface. If you stretch the lines long enough, you will see that they are no longer parallel.

Non-Euclidean geometry is now commonly referred to as “hyperbolic geometry”. If we did not have such geometry in our hands, that is, if we were still in contact with Euclidean geometry, we would be on our way to the destination by the longest possible route, not by sea or by long distance. In the air.

Without non-Euclidean geometry Albert Einstein could not have written his general theory of relativity, and we would not be able to go into space today, and even then we would not be able to return safely.

May continue next week.

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