Symbol is the essence of mathematical thought. The linguistic turn of philosophy in the 20th century shows once again that when human beings try to achieve "objective communication between subjects" and realize * * * knowledge based on different cognitive backgrounds, they will inevitably encounter systematic errors brought by language tools (here, daily language).
As mentioned in the previous book review of Six Lectures on Loneliness, each of us is lonely because our thoughts and consciousness are bound in our own bodies. In order to realize the exchange of different ideas, language tools are needed, and the grammatical system of this daily language is not strict. Everyone will have a different understanding of the same word.
Language is indispensable to eliminate misunderstandings and achieve near-perfect mutual understanding. As a supplement to everyday language, mathematical language is bound to be symbolic and rigorous, although at the beginning of mathematical development, the understanding of many symbols still stays on the basis of intuition.
But even simple mathematical concepts have helped us to establish a set of "seemingly complete" and "strict" mathematical symbol system-Euclidean geometry at first. At least, it is in line with our intuition and seems to be "self-evident".
In the eyes of Plato and his successors, the rational world is full of metaphors of the real world. Although we have never seen a strict straight line or circle in our life, we still say that the sun is round and the ruler is straight. This is the real world we live in-a world with different observation scales, different observation tools and different observation angles.
Even our eyes are biased tools: although the molecules and atoms that make up our bodies are so small and the gaps between them are so large, our eyes can magically "ignore" the gaps between particles that occupy 99.9% of the space and only see the aggregate of solid particles of 0. 1%!
On the contrary, high-frequency electromagnetic waves can always penetrate our body easily, as if the space occupied by that body is "empty", and these particles can't see our human existence in this world.
There are countless such "reality paradoxes", which fully shows that it is often difficult for us to observe the real world and understand it based on observation. On the contrary, most of the concepts we use are "approximations" of the real world to some extent. Yes, different observation scales require different "approximation" tools.
On the way to know the world, mathematics, as an abstract product of human thinking, has the characteristics of rigor and symbolism. The rational world casts a shadow, and we follow the shadow to find the light of the hole. Is that God's domain? At least, in the face of the previous axioms, human beings think so, because they cannot be falsified intuitively, and only "God created mathematics" can explain this mystery.
Unfortunately, non-Euclidean geometry has broken this long-standing discipline belief. If mathematics is only a game rule that can be self-modified, and if the rational world is only an indicator with "positional deviation" in human eyes, what else do we need to seek along this road?
This is the anxiety we have to face when we are lucky enough to be struck by lightning of wisdom. Wisdom always drives us to seek the "objective world" beyond the scope of experience, to explore the roots of this world, and to reflect on the meaning of "who am I" and "why am I here".
Wisdom is always looking for answers and never stops. As for the end point, I just want to say that the end point of science is faith.
The process of thinking about mathematics is actually to locate the coordinates of our existence in the world.