Beware of mathematicians!
Sep. 21st, 2017 12:16 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
vit_ryesterday part
Now please take a pen and draw a straight line that goes through points "1" and "2". Measure the size between your dots and draw the point "0" before "1". It would be nice, if the lengths from "0" to "1" and from "1" to "2" are equal, however this is not important.
Imagine two small mathematical demons standing at the point "0". They are big enough to go forward on top of your line. The black demon holds a flag with "{i : i ∈ N}" on it. He counts natural numbers. The red demon holds a flag with "{2*i : i ∈ N}". He counts only even natural numbers.
They start to go forward and they both make one move each second. The black "i" steps on the point "1", the red "2*i" jumps on the point "2". Then the "i" steps on the point "2". The "2*i" jumps again two times longer.
A mathematics teacher would draw a big circle and label it with "N" then a smaller one within it and label the smaller with something like "N2". The sentence you would hear during this may sound like "The set of even natural numbers is included in the set of natural numbers".
The representation is obvious, the definition is clear. Does this mean that there are two times more natural numbers than even natural numbers? (We use the definition of "N" that excludes the number "0".)
Before answering this question look at your demons. They have gone further away but they are still moving on top of the mathematical straight line. They will never stop. This is the same "never" that leads the first demon who is digging perfection within your first dot.
The red "i*2" demon is two times further than the black "i" but he has made the same number of his longer steps. Note, the numbers of steps will always be equal. The "i" demon is behind but he steps on odd and on even points. This means in the future he will reach the point the red "i*2" demon stands now. In 1000 seconds the "i*2" demon will be on the point "2000" and the "i" on the point "1000". However after 1000 additional seconds the "i" demon will step on "2000" too.
In an hour, in a day, in hundred days, in hundred years, in thousand years, in billions of years... the black "i" demon and the red "i*2" demon will have the same number of steps behind them and the "i" demon could never reach the "i*2" but after some time he will step on every dot the "i*2" demon had stepped before. Ever, forever, indefinite.
Mathematics uses the sign containing two loops to represent the concept of infinity "∞".
We could write the sentence "There are two times more natural numbers than even natural numbers" as "2 * ∞ = ∞".
Does this mean that we can answer the question "Are there two times more natural numbers than even natural numbers?"
We write "∞ / ∞ = ? "... and the mathematics says "Stop! You are not allowed to do this!"
We have found the next word: "rejection".
By the way, this is the most important mathematical word.
Despite of common understanding, the equation "1 + 1 = 2" is not the essence of mathematics. This is the obvious everyday knowledge. We can count dots, apples or fingers and say "Two!"
Mathematics states that "1 + 1" cannot be "1" and cannot be "3", and cannot be "2.00000001", and cannot be something else then "2". However we may sometimes get such results when we try to prove mathematics with our physical world measurements.
Mathematical equations do not change depending on our experience. However they strictly depend on context.
If we switch from decimal to binary system, mathematics would say "1 + 1 = 10".
Where is "2"? It does not exist in the binary representation.
This means you also cannot write "1 + 2", if you use the binary system. The number "2" is outside the domain of definition.
A side note especially for mathematicians: I do know about limits. We could use "2 times more" during the time demons are moving into infinity but the teacher's representation is fixed.
There is the indefinite number of even natural numbers within the smaller circle "N2" and the indefinite number of natural numbers within the circle "N". The teacher could draw the area of the circle "N" two times bigger than the area of the circle "N2" but this representation is misleading.
The visually obvious division and comparison operations are rejected.
Friday part
Now please take a pen and draw a straight line that goes through points "1" and "2". Measure the size between your dots and draw the point "0" before "1". It would be nice, if the lengths from "0" to "1" and from "1" to "2" are equal, however this is not important.
Imagine two small mathematical demons standing at the point "0". They are big enough to go forward on top of your line. The black demon holds a flag with "{i : i ∈ N}" on it. He counts natural numbers. The red demon holds a flag with "{2*i : i ∈ N}". He counts only even natural numbers.
They start to go forward and they both make one move each second. The black "i" steps on the point "1", the red "2*i" jumps on the point "2". Then the "i" steps on the point "2". The "2*i" jumps again two times longer.
A mathematics teacher would draw a big circle and label it with "N" then a smaller one within it and label the smaller with something like "N2". The sentence you would hear during this may sound like "The set of even natural numbers is included in the set of natural numbers".
The representation is obvious, the definition is clear. Does this mean that there are two times more natural numbers than even natural numbers? (We use the definition of "N" that excludes the number "0".)
Before answering this question look at your demons. They have gone further away but they are still moving on top of the mathematical straight line. They will never stop. This is the same "never" that leads the first demon who is digging perfection within your first dot.
The red "i*2" demon is two times further than the black "i" but he has made the same number of his longer steps. Note, the numbers of steps will always be equal. The "i" demon is behind but he steps on odd and on even points. This means in the future he will reach the point the red "i*2" demon stands now. In 1000 seconds the "i*2" demon will be on the point "2000" and the "i" on the point "1000". However after 1000 additional seconds the "i" demon will step on "2000" too.
In an hour, in a day, in hundred days, in hundred years, in thousand years, in billions of years... the black "i" demon and the red "i*2" demon will have the same number of steps behind them and the "i" demon could never reach the "i*2" but after some time he will step on every dot the "i*2" demon had stepped before. Ever, forever, indefinite.
Mathematics uses the sign containing two loops to represent the concept of infinity "∞".
We could write the sentence "There are two times more natural numbers than even natural numbers" as "2 * ∞ = ∞".
Does this mean that we can answer the question "Are there two times more natural numbers than even natural numbers?"
We write "∞ / ∞ = ? "... and the mathematics says "Stop! You are not allowed to do this!"
We have found the next word: "rejection".
By the way, this is the most important mathematical word.
Despite of common understanding, the equation "1 + 1 = 2" is not the essence of mathematics. This is the obvious everyday knowledge. We can count dots, apples or fingers and say "Two!"
Mathematics states that "1 + 1" cannot be "1" and cannot be "3", and cannot be "2.00000001", and cannot be something else then "2". However we may sometimes get such results when we try to prove mathematics with our physical world measurements.
Mathematical equations do not change depending on our experience. However they strictly depend on context.
If we switch from decimal to binary system, mathematics would say "1 + 1 = 10".
Where is "2"? It does not exist in the binary representation.
This means you also cannot write "1 + 2", if you use the binary system. The number "2" is outside the domain of definition.
A side note especially for mathematicians: I do know about limits. We could use "2 times more" during the time demons are moving into infinity but the teacher's representation is fixed.
There is the indefinite number of even natural numbers within the smaller circle "N2" and the indefinite number of natural numbers within the circle "N". The teacher could draw the area of the circle "N" two times bigger than the area of the circle "N2" but this representation is misleading.
The visually obvious division and comparison operations are rejected.
Friday part
Tags:
no subject
Date: 2017-09-21 03:50 am (UTC)no subject
Date: 2017-09-21 05:11 am (UTC)Не-а. Много - это много, бесконечность - это бесконечность. У них разная природа. И делить их нельзя, потому что нет "конкретной бесконечности", а про lim при x -> ∞ я уже написал.
no subject
Date: 2017-09-22 10:12 pm (UTC)То есть, еще раз: формула - вполне конкретная, бесконечно только ее значение для некоего аргумента. Поэтому такие значения делить нельзя. А формулы - можно. И иногда оказывается, что после деления формул получается такая формула, значение которой для того же аргумента уже не бесконечно.
no subject
Date: 2017-09-22 10:17 pm (UTC)no subject
Date: 2017-09-21 05:00 am (UTC)I dunno, my demon bumped into a wall after number 17.
no subject
Date: 2017-09-21 06:00 am (UTC)I dunno, my demon bumped into a wall after number 17.
I have a window on the right side of my desk. Maybe it is good to mention that demons do not respect physical objects and physical laws to some extent. If we do not restrict physics, the collisions will start from the coordinate system because the Earth is rotating and the Solar System also does not fly straight.
no subject
Date: 2017-09-21 08:51 am (UTC)no subject
Date: 2017-09-21 11:30 am (UTC)Вот как раз на русском писать что-то дельное нет никакого смысла. Английский же можно переработать и использовать для чего-то полезного. Это всё-таки записная книжка, а не трибуна для выступлений. Если хочешь ублажить читателей, писать надо не про математику, а про сиськи.
no subject
Date: 2017-09-21 12:44 pm (UTC)Почему? Читают вас в основном по русски, насколько я могу судить по комментариям.
no subject
Date: 2017-09-21 12:56 pm (UTC)Вот, в данный момент я слушаю на немецком вебинар про повышение продаж он-лайн магазинов. На русском это появится лет через пять и криво.
Блог ведётся для развлечения и философствований, а не для работы. Плюс есть вещи, которые не должны попадаться клиентам в поиске.
С полезными текстами ситуация другая. Хоть кириллическая раскладка и позволяет печатать быстрее, всё равно при составлении сложных текстов усилий на правку и формулировки уходит примерно столько же, сколько на других языках, так что сначала обтёсывать на русском, а потом переводить тоже смысла не имеет. Слишком мал выигрыш времени.