Every mathematician will support Kone. We all think that integers and circles are real in some abstract sense, and Plato's view is very attractive. But can it really be supported? If the universe is a one-dimensional space, or even discrete, it is hard to imagine how geometry was conceived and developed in this one-dimensional space. For humans, we seem to be better at integers, and counting is a real primitive concept. But imagine what would happen if civilization did not appear among humans, but among jellyfish hidden in the depths of the Pacific Ocean, living alone and isolated from the world? Jellyfish have no personal experience and only feel the water around them. Exercise, temperature and pressure will provide it with basic sensory experience. In such an environment, there will be no discrete concepts and no technology is needed.

Why is the concept of golden section defined by Euclid considered as an invention? This is because Euclid, with his creative ideas, chose this ratio and made a detailed analysis, which successfully attracted the attention of other mathematicians. However, it is worth noting that the concept of golden ratio was not clearly explained in ancient China, and there was basically no specific description about it in ancient mathematical literature in China. Similarly, ancient India did not invent the concept of golden ratio, but only vaguely mentioned it when studying some theorems of trigonometry.

Many examples can prove that the question of whether mathematics was discovered or invented is actually a false proposition. Mathematics is a combination of invention and discovery! As a concept, the axiom in Euclidean geometry is an invention, just as the rules of chess are human inventions. Axioms are constantly supplemented by various concepts invented by human beings, such as the golden ratio of triangles, parallelograms and ellipses. But in general, Euclid's geometric theorem is discovery, which is a bridge connecting different concepts. In some cases, proof gave birth to theorem mathematicians, who carefully studied what could be proved and summarized and deduced theorems from it. On the other hand, as Archimedes described in Methodology, mathematicians first find out the answer to a question they are interested in, and then look for a proof method.

Generally speaking, the concept was invented. For example, the basic concept of prime numbers was invented by mathematicians, but the related theorems about prime numbers were discovered by people. [5] In ancient Babylon, ancient Egypt and China, mathematicians at that time had developed advanced mathematical theories, but they never put forward the concept of prime numbers. Can we say that they just didn't find the prime number? This is like saying that Britain has not found the only charter compiled into a code. Just as a country can function normally without a constitution, complex mathematics can continue to develop without the concept of prime numbers. Mathematics has indeed developed in this way in history!

What prompted the ancient Greeks to invent the concepts of axiom and prime number? We're not sure. But we can guess that this is due to their unremitting efforts to explore the basic structure of the universe. Prime numbers are the cornerstone of numbers, just as atoms are the foundation of matter. The same axiom is like the source of all geometric truths. Regular dodecahedron is regarded as representing the whole universe, and it is the concept of golden ratio that introduced this symbol.

These discussions reveal another interesting feature of mathematics, which is an important part of human civilization. After the ancient Greeks invented the axiomatic method, all the later mathematical theories in the West followed this method and accepted the same philosophy and practice. Leslie White, an anthropologist, tried to generalize and summarize the human civilization embodied in mathematics. He said that if Newton grew up in a primitive tribe of the Hottentot tribe, his computing power might only be the same as that of hottentots. Many mathematical discoveries, such as knot invariants, and even some important mathematical inventions, such as calculus, are realized by different mathematicians in independent work, which is probably due to the cultural complexity embodied in mathematics.