Mathematics often makes me feel incredible.
Primary school began to learn π. A compass draws a semicircle with a radius of 1 and a circumference of π, a strange number with an endless tail.
This number is endless, irregular, untraceable and can never be fully expressed. In mathematics with simple beauty as its beauty, this number seems so out of place that I feel confused, bored and even uneasy.
What is the significance of the existence of such a wonderful thing as irrational numbers? Is it to break the illusion of human pursuit of beauty?
Why can't even describe the circumference accurately?
Why is the circle, which is full of symmetry and looks so harmonious and perfect, tied to irrational numbers, in a mess and without beauty?
π, was God created to mock human ignorance?
What is even more puzzling is that π, as a geometric concept, can even be related to arithmetic. For example, the following formula:
I stared at this formula for two minutes.
This formula is really amazing. One is a number representing geometry, and the other is an odd-numbered column and a square-numbered column in arithmetic. Geometry and arithmetic, two branches of mathematics, come together in the form of infinite analysis. On the left is the square of irrational number, and on the right is the rational number of infinite series. The form of the sequence is so beautiful that an equal sign can be marked in the middle. ...
In the silent universe, π is commented as follows:
What's more. The formula that makes me speechless most is Euler formula, which is familiar to math lovers.
The ancients lamented the beauty of music-"this song should only exist in the sky, but it can be heard on earth"; Similarly, the beauty of Euler's formula does not seem to exist in this world.
These five important mathematical elements are composed of three basic operations: addition, multiplication and exponent. Formally, it is extremely simple; In this sense, I think very carefully-
Multiplication of π and I? What kind of weird number should this be? And then this number is used as the index of e? The result will be-1? ……
Euler's formula, which is called God's formula, is also called "the most beautiful equation ever". The wordless universe said, "This formula is one of the most contradictory propositions in mathematics." In the eyes of math lovers, this formula shows all the beauty of mathematics.
The magic of mathematics is also reflected in the interlacing of geometry and arithmetic, time and space.
For example, if you go back to the imaginary unit I and take-1 as the square root, it's like "dividing by zero", and you just want to answer "How is it possible". But this number used by scientists as an arithmetic tool has geometric significance. ...
Algebraic operation "multiply I" is equivalent to geometric operation: "turn 90 degrees counterclockwise"-this is the square root of-1, how can it be equivalent to turn 90 degrees counterclockwise?
Another example is Hamilton, who created quaternions. In his theory, time and space merge into a single "time and space" (as for how to merge, I don't understand).
In Hamilton's quaternion (the expression is a+bi+cj+dk), where I, J and K representing three-dimensional space are imaginary numbers and time (a) is real number. This is a wonderful irony-
Space is imaginary, time is real, and the positions of the real world and the virtual world are just reversed.
These incredible connections shown by mathematics are like miracles, and each item has a compelling force. I can't imagine, and I can't imagine, what is the meaning behind all this.
Mathematics is special. Empirical science, including physics, chemistry and biology, is the language to describe the world and express the knowledge of the universe. They exist to interpret the world. The particularity of mathematics lies in that it is an independent knowledge system besides having the same function as empirical science. In other words, the purpose of mathematics can also be mathematics itself.
Mathematics is rigorous. The facts described in mathematics cannot be proved by experience, rationality, statistics or tests, but only by mathematics itself. This is a major principle that distinguishes mathematics from other disciplines, including empirical science.
Therefore, it is mentioned in the Silent Universe that physics is a matter within the scope of our universe, because physical theories ultimately need to be tested by experiments. But mathematics is outside our universe, and its universality of laws may describe all possible universes.
Judging from the mathematical miracles I have seen, perhaps only mathematics can undertake the impossible task of "describing all possible universes".
However, human beings who have mastered mathematical tools are far from reaching the realm of omniscient and omnipotent God.
The most important achievement of mankind in the 20th century, in my opinion, is not the discovery of relativity, the mastery of quantum mechanics and nuclear energy, nor the proof of Fermat's last theorem, but a clear understanding of the limitations of human cognition.
One limitation comes from quantum mechanics. Heisenberg's uncertainty principle delimits the boundary of human cognition from a microscopic point of view.
Another limitation comes from mathematics. Proof of godel's incomplete theorem;
In other words, man can never prove an axiomatic system. This is just a starting point. Maybe one day, a species can provide a completely justified proof of 1+ 1=3, and build a new axiom system on this basis.
From this perspective, the science of quantum mechanics, the mathematics behind Godel's theorem and the philosophy represented by Camus' absurd theory all point in the same direction:
Humans can never fully and accurately understand the world.
On the road to absolute truth, philosophy can't go, and neither can science and mathematics.
The sentence that touched me the most in Three-body is: "Weakness and ignorance are not obstacles to survival, but arrogance is."
So I recommend Silent Universe: Let's be more awed by the unknown and less arrogant by ignorance.