Finding "what is an important problem or theorem" is often more difficult than solving a known problem or proving a known theorem.When you read any book of ancient Greek geometry, you will be amazed by the simplicity and conciseness of the theorems and proofs that were proposed two thousand years ago, and you can't help but be struck by this style. However, these books often do not provide the reader with clear clues to how these theorems were originally conceived. Archimedes' brilliant treatise "Fang **" fills this gap, revealing how the author himself determined the truth of the theorem before he knew how to prove it. Here is a text taken from a letter written by Archimedes to Eratosthenes, the mathematician of Cyrene. In this letter, Archimedes briefly introduces the main content of his "Fang**":
I will show you how these theorems are proved in this book. As I said, you are a diligent and excellent teacher of philosophy, and you are very interested in any mathematical study, so I think it is necessary to explain to you in detail this particular method that I have adopted in this book. With this special approach, you will be able to understand specific mathematical problems with the help of mechanics. I believe that this will be useful for discovering the proofs of those theorems. Some problems are initially known by physical methods and then proved by geometric methods, because mechanical methods cannot provide real proofs. Because it is much easier to solve problems where some relevant knowledge has been acquired beforehand than to deal with problems that do not have a little background knowledge beforehand. ”
Here, Archimedes touches on one of the most important ideas in scientific research and in the history of mathematicsFinding "what is an important problem or theorem" is often more difficult than solving a known problem or proving a known theorem. So, how did Archimedes discover the new theorem? Using a deep understanding of mechanics, equilibrium theory, and the principle of levers, Archimedes first compared in his mind with the volumes of known objects and the area of the figure, and roughly estimated the volume of the object to be calculated and the area of the figure. In this way, Archimedes found it much easier to prove geometrically the volume of an unknown object and the area of the figure. Subsequently, in "Fang**", Archimedes pointed out the position of the center of gravity of a series of figures and gave a geometric proof.
There are two ways in which the Archimedes' method is different. First of all, in essence, it was Archimedes who introduced the "thought experiment" into the rigorous scientific research. In the 19th century, the German physicist Hans Christian Orsted first named this method of replacing real experiments with fictional experiments "Gedankenexperiment" (which means "thought-guided experiment" in German). In physics, this concept has a high status and value, and thought experiments can be used before real experiments, so that people can understand the experimental process in advance. Or in some cases, where a real experiment is simply not possible in reality due to a lack of necessary conditions, this is where thought experiments come in and can help people understand what the experiment is about. Second, and perhaps more importantly, Archimedes set mathematics free by removing it from the artificial chains built by Euclid, Plato, and others.
For Euclid and Plato, there is one way, and only one way, to do the mathematical work: you have to start from the axioms, use the tools you specify, and prove them in a fixed sequence of logical steps. However, Archimedes, who had a free soul, was not content to be bound by this way, he used all the methods and evidence he could think of, asked new questions, and solved them with his own thinking. He did not hesitate to explore the connection between abstract mathematical objects (Plato's world) and physical reality (real objects), and in the process continued to develop his own mathematical theories.
The last achievement that established and cemented Archimedes' status as a "magician" was his prediction of calculus. Calculus is a branch of mathematics that was formally established and developed by Newton at the end of the 17th century. The German mathematician Leibniz also independently studied and proposed the theory almost at the same time.
The basic idea behind the points is actually quite simple – after being clearly pointed out, of course! For example, if you want to calculate the area of a graph enclosed by an arc on an ellipse and a straight line between two ends of the arc, you can break the graph into many rectangles of equal width. When you add up the areas of these rectangles, you get the area you are looking for (Figures 3 - 5). Obviously, the more rectangles are decomposed, the closer the sum of the areas of these rectangles will be to the real graph area. In other words, when the number of rectangles being decomposed is close to infinity, add up the areas of these rectangles to get the actual area of the graph you want to calculate. This "limit" process is the integral. Using the above method, Archimedes calculated the areas of spheres, cones, ellipses, and parabolas, as well as the volume of the objects formed by them (the object obtained by rotating the ellipse or parabola around its axis).
One of the main goals of differentiation is to calculate the slope of the tangent at a given point on the curve, where the tangent and the curve intersect only at this point. Archimedes gave a method for calculating the tangent slope of a special spiral. Further research on differential calculus was done by Newton and Leibniz. Today, calculus, and the branches of mathematics derived from it, are the basis for the vast majority of mathematical models, with wide applications in physics, engineering, economics, or fluid mechanics.
Archimedes changed the world of mathematics and fundamentally changed people's understanding of the relationship between mathematics and the universe. By demonstrating the shockingly close connection between mathematical theory and practice, Archimedes proposed for the first time that explanations of phenomena in nature appear to have been mathematically engineered on the basis of observation and experimentation, rather than mysticism. It was Archimedes' efforts that gave birth to ideas and understandings that "mathematics is the language of the universe" and "God is the mathematician". Of course, there are some things that Archimedes did not do either. Archimedes, for example, never discussed the possible limitations of the mathematical model he built if it were applied to the actual physical environment.
For example, Archimedes' theory of the principle of the lever does not take into account the weight of the lever itself, and this principle assumes that the hardness of the lever is infinite. It can be said that Archimedes pushed open a door, and after passing through this door, human beings can explain natural phenomena with mathematical models. However, Archimedes pushed the door open to a limited extent, only to the extent of "saving face". This means that mathematical models may only be representative of what humans observe, but they cannot describe the physical world that actually exists. The Greek mathematician Geminus (c. 10 BC – 60 AD) was the first to discuss the differences between mathematical models and physical explanations in detail when studying the motion of celestial bodies. Germinus pointed out the difference between astronomers and physicists, and according to him, the job of an astronomer (or mathematician) was simply to suggest the construction of a model. In fact, this model is a representation of the celestial motion in the sky that they observed, and it is the physicist's job to explain this real motion.
It may be strange to you that Archimedes himself considered his most outstanding achievement to be the discovery of the volume of the cylinder insect sphere (Figs. 3-6). Archimedes was so proud of his discovery that he even asked for it to be engraved on his tombstone as an epitaph. About 137 years after Archimedes' death, the tomb of the great mathematician was discovered by the famous Roman orator Marcus Tullius Cicero (c. 106–43 BC).
"When I was a treasurer in Sicily, I searched for Archimedes' grave. The people of Syracuse knew nothing about this and refused to acknowledge the existence of the Archimedes necropolis. But this small area, completely covered in thorns and surrounded by shrubs, is indeed the burial place of the great Archimedes.I once heard a few verses, supposedly short verses engraved on his tombstone, which referred to cylinder and sphere models. To do this, I visited all the cemeteries near the Agrigentine Gate, looking at the tombstones erected on each of them. Finally I noticed that after a tombstone had been cleaned, a small cylinder could be vaguely inscribed on it, and above it was a sphere and a column, and I immediately realized and told the Syracuse people around me that this was what I was looking for. We had some hands cleared the weeds around us with scythes and carved out a path that led directly to the tombstone. The verses on the stele are faintly recognizable, but the second half of each sentence has been blurred by the erosion of time. The city was one of the most famous cities in the ancient Greek world and a great center of learning in bygone years, but it was ignorant of the burial place of the most glorious citizens it had ever conceived. Thankfully, I, the man from Arpinum, showed up and recognized it! ”
This article **Turing New Knowledge, excerpted from "The Last Math Problem", [Meet Math] has been licensed**.Recommended Reading:
**Wan Fan Incentive Plan Author: Mario Livio (Mario Livio) Translator: Huang Zheng.
Is mathematics a human invention or a discovery? Where does the ubiquitous and omnipotent power of mathematics come from?
The world's best-selling classic in the history of mathematical philosophy, showing the extraordinary power of mathematics to be ubiquitous and omnipotent.
The legends of the great masters of science and philosophy, the magnificent history of mathematics, physics, astronomy and philosophy.