W. B. Wu University of Chicago

Introduction

Being the guiding ideology of modern science, empiricism, the view that experience is the only source of knowledge, is regarded as the sole philosophy for exploring the laws of nature by almost all scientists. The fact that empiricism has led to significant developments in every scientific field and brought unprecedented comfort to human beings' material life has made people believe that it was the only way to explore the laws of nature. In mathematical science, first of all, empiricism established an axiomatic system, followed by logical deduction, from which a variety of theorems and corollaries were derived. In this paper, I will briefly discuss the limitations of empiricism through studies of some famous discoveries in the history of mathematics, the understanding of statistics, and the path of genuine exploration in science.

Some Discoveries in the History of Mathematics

S. Ramanujan (Dec. 12, 1887 -- April 26, 1920), a mathematical genius, was born in India. He made many important achievements in the area of mathematics, such as the analytical number theory, elliptic function theory, continued fractions and infinite series, and others. Obtaining his B.S. degree at Cambridge University in 1916, he collaborated extensively in many fields of mathematics with G. H. Hardy, who was a famous mathematician at that time. The research style of Ramanujan was completely different from other mathematicians. Having not much mathematical training, Ramanujan knew very little about rigorous deduction in contemporary mathematics. However, he wrote down 4,000 theorems, even though he did not know how to prove many of them. What he did was only to write down the complicated, yet delicate principles that appeared in his mind. Later mathematicians have been able to prove many of his theorems. (For readers who have more interest, please refer to Ramanujan's manuscripts and Ramanujan's notebooks).

Ramanujan's discoveries are amazing to people. However, the fact that he just knew these principles and simply wrote them down was even more incomprehensible. Those who have studied modern mathematics know that mathematics is based on strict logic. Therefore it is very hard to imagine how those sophisticated and delicate principles could simply appear in one's head. To prove these existing principles with logic is another issue. But Ramanujan could just "dream up" these theorems. Hardy once said, "His primitive knowledge was as astonishing as his profound thoughts." So how did he discover these theorems and principles?

Obviously, Ramanujan had a different way of thinking, which is similar to that of ancient Chinese scholars. At the time of the ancient Chinese scholars, the atmosphere of cultivation was intense in society. Many people, including many scholars, practiced sitting meditation and believed in emptying the mind and regulating the breath. Many famous scholars were also lay Buddhists. It was easy for them to experience things in other dimensions during their meditation. Ramanujan also practiced cultivation. If these things, these experiences, were recorded, they would be beyond comprehension for modern people because empiricism could not explain them.

An example from the history of ancient Chinese science and technology is the calculation of Pi by Zu Chongzi (429 BC-500 BC). He found an approximate value of Pi, 355/113 by splitting 113355 into two. This was a glorious achievement . The way Zu Chongzi discovered this value, however, remains a mystery. In ancient times there were no symbols or logical systems that were commonly used in modern mathematics, not to mention numeric approximation theory. Current mathematical historians have no idea how this value was computed.

These phenomena are well explained in Zhuan Falun. In Chapter Nine, it reads, "First of all, let us talk about the origin of one's thoughts. In ancient China, there was the phrase 'heart-thinking.' Why was it called heart thinking? Ancient Chinese science was very advanced since its research directly targeted things such as the human body, life, and the universe." "In fact, the ancient science of China was completely different from the science introduced from the present-day Europe. In ancient China, people made direct researches on the human body, life, and universe. The ancients dared to touch the intangible and invisible things and were able to verify their existence. The sensation one feels when he sits in meditation will become more and more keen, until finally one not only has a very keen sensation, but also can touch or see it, which has raised something invisible to something tangible. The ancients took another way to probe the secrets of life and the relationship between the human body and the universe. That is completely different from the way taken by the present-day science of authentic proof." (Zhuan Falun, Volume II. Preaching the Law at Lantau Island.)

Mathematical Statistics and Prediction

Because practicing cultivation is such an extraordinary thing, should it manifest even a little bit, people would feel it to be fathomless. People who were practicing cultivation or who had some supernatural abilities wrote many prophecies that have been passed down from ancient times. These prophecies documented major events that would happen in several hundred years. Many of them have been borne out. The level of precision was astonishing.

Time series theory in modern mathematical statistics also explores the issue of making future forecasts. In order to build a time series model, researchers usually adopt various assumptions such as parametric, nonparametric, linear or nonlinear, and so forth. They then use historical data to determine if the time series model fits. That is how various statistical models are developed to make future forecasts. This way of researching is a typical method of empiricism and it can indeed provide some useful information. However, the model assumptions are only an approximation and simplification of the real situation. In addition, reality involves much randomness. Therefore, the model is unable to accurately describe the real situation. Short-term forecasts may have some precision, but a forecast spanning several hundred years is almost impossible, since the fundamental underlying mechanism is often beyond recognition by empirical methods. Specifically, it can only describe superficial connections within the surface dimension. Statisticians themselves also recognize this. George Box once said that all models were wrong but some of them were useful.

How could the prophets predict things that happened several hundred years later? They could do this because they actually saw all things that would happen during their cultivation practice or in some other situations. In the section, "The Supernormal Ability of Precognition and Retrocognition" in Chapter Two of Zhuan Falun, Teacher states, "One can see very accurately a person's future and past. No matter how well the fortune-telling works, it cannot tell the minor events and their details. This person, however, is able to see things very clearly, including the time of year. The details of changes are all visible since what one sees is the actual reflection of people and things from different dimensions." Certainly, different people are at different levels of practicing cultivation; therefore, the things they can see are not necessarily the same.

Conclusion

In the previous paragraphs, I mainly discussed the limitations of empiricism as well as some extraordinary phenomena manifested through practicing cultivation. To genuinely understand the laws of nature, we should make a breakthrough in our traditional ways of thinking, and thus bring about a new science and new insights. Practicing cultivation provides a true path for exploring nature and understanding material existence; thus, future science and culture can be enlightened through practicing cultivation.

Reprinted from PureInsight Net

Posting date: 8/31/2002
Original article date: 8/30/2002
Category: Open Forum

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