PureInsight | December 30, 2002
Many mathematicians, among them Davis S. Ross, Ph.D. (mathematician at Eastman Kodak Research labs) and Greg Chaitin (IBM's Thomas J. Watson Research Lab), are asserting that the field of mathematics has passed through a series of logical crises during the past century (1902 to 2002) that have done it severe damage. Greg Chaitin stated that, normally, pure mathematics is static, unchanging, perfect, yet, in this century and in the past century, there has been a lot of controversy over the foundations of mathematics and what constitutes a valid proof. Nonetheless the majority of scientists are not too concerned about the crisis in the mathematical sciences, as with time all crises have been successfully resolved by new theories. The new theories, though, evolved from existing theories, i.e., they were built upon prior mathematical theories, and did not claim to be new discoveries. In this way, the possible flaws in the foundation of mathematical theory have been satisfactorily resolved for the time being. Consequently, science's foundation in existing mathematical theory has not been not challenged. There is no need for a new Math 101.
The present day alleged mathematical crisis is bound to be different from past crises with respect to mathematical theory, if we look at the rate of scientific development and its influence on human society. We have to admit that science has developed at an unprecedented rate during the past century, compared to other historical periods. Especially during the past fifty years, science has become the core of virtually everything in human society. Furthermore, physics, mechanics, economics, and other disciplines are built on mathematical foundations. Also, Prof. E. Colman stated in 1931, "mathematics at the bottom determines the development of production, technology and economy." Isn't it true that the rate of scientific development was so rapid that the term "scientific explosion" does more justice to describe the phenomenon? It is puzzling that contrary to the influence and incredible rate of mathematical development, there has been no visible outcry in resolving the alleged crisis with respect to mathematical development. It is worrisome when science is heading towards a danger zone, when it disregards its flaws and continues to be systematically repetitive.
Why is this happening? Perhaps everyone has become too fascinated with today's technology, given the tangible benefits of the computer driven technologies. Everyone is oblivious to the negative effect on the human psyche, given the physical comforts because of scientific development. Who would expect a drug addict to be seriously concerned about the danger of narcotics?
In this article I propose to review the overall mathematical development in layman's terms. I do not intend to use any mathematical equations or formulas, because that would simply prevent people from understanding my arguments. I believe that a genuine scientist must be able to describe science in layman's terms, easily understood by the majority of people, regardless of their level of education.
I first wish to explain the relationship between mathematics and other disciplines as they evolved over an extended period in history. Basically, mathematics, physics, philosophy and a number of other disciplines were by and large lumped together under the catchall term of science, as they are closely related. Today, although inconsequential in my mind, science is divided into numerous disciplines. Yet, each discipline has a close relationship to the other disciplines. Therefore, I can't see the value in discussing any of them individually. The sole reason for accentuating mathematics is that mathematics has laid the foundation for and is the core of science. In addition, all of the most fundamental concepts in science, regardless of forms and content, lie in the mathematical concepts, calculations, and proofs. We humans value some of the hidden implications in these mathematical concepts, calculations, and formulas implicitly. Yet, most of these values simply escape our conventional conscious analytical thinking.
That leads me to digress slightly from my original intent and discuss the origin of science. Human society of two thousand years ago was undeniably dissimilar to today's human society in many ways, since society evolves continuously. They may have been quite different with respect to their ideologies and concepts. Even human society of the not so remote past, for example society of four hundred years ago, was very different from that of today. Yet, from what I can discern, all scientists, from the earliest up to Newton and Copernicus, conducted scientific research for one purpose only, to prove the existence of God. In fact, I emphatically state that they have been successful. Now, let me get back to my original thought.
Let us ask ourselves, what is mathematics and what is the value of mathematical concepts? How can I best explain this? Well, building mathematical models is an important concept in modern mathematics. Researchers express their theories based on scientific observations after stating a hypothesis. Therefore, what we call mathematics today should be more accurately referred to as mathematical models. They may or may not be based on demonstrable truth, or facts. Mathematical models represent nothing more than personal concepts or ideas. The idea of using models to express concepts and theories has been used in all scientific disciplines. Scientists build models of the universe, celestial dynamics, molecular and nuclear dynamics, as well as models in economics, and languages. If I may be candid, in most scientific fields, except quantum mechanics or economics, effective models are rarely found. It appears that in most scientific fields it is difficult to derive a model from ordinary observations.
Generally speaking, a model, or an expression of mathematical or logical representation, has to be quantified. One may develop a model through logical application of known quantities. The model then allows one to develop theories to explain the model. However, to test a model in the real world, one must use numbers. On the other hand, if one cannot express his idea by building a model, numbers will be useless, although the scientist understands mathematics. Mathematical concepts, such as integers, real numbers, imaginary numbers, addition, subtraction, multiplication and division are actually models, or assemblies of models. They are the concepts that form the basic elements of mathematics. It requires the ability to perform abstract analysis to build models. It is through abstract analysis that one develops ones own concepts. One looks for phenomena that match the concepts in the experiments. I hope I have described briefly the basics of mathematics and the relationship with human thinking.
I hope to have made it clear that humans express scientific ideas through mathematical models. A closer look at the logic we employ in mathematics may help us see the mathematical boundary delimited by our own ignorance. Mathematics, simply, is the theory of numbers and operations of numbers, while numbers and operations of numbers correspond to quantity and logic.
The simplest example of a mathematical model is the aggregation of natural numbers from 1, 2, and 3 to infinity. How many natural numbers are there? This question seems rather simple. There are an infinite number of natural numbers! Infinite means "endless", so we will not be able to find out the total number of natural numbers. Regarding this question, two viewpoints have existed since ancient times: actual infinity (there are no natural numbers beyond infinity) and potential infinity (there exist natural numbers beyond infinity). There is one shared assumption in both theories: a known natural number increasing progressively (as by addition) in a known order becomes another natural number, while "natural number" means numbers that exist naturally. The concept of "infinity" is, therefore, built upon the linear concept.
Is such a numerical concept correct? Can such a numerical concept reflect the attributes of a substance? Is addition a representation of a natural phenomenon? Does N and N + 1 describe more than just a change in quantity? The answers to all these questions are yes, if we choose to verify them within the scope of what we can see and hear. This is how the most basic concepts of modern science are verified, and why no one seems to dispute the foundation of modern science. However, if we search carefully within the scope of what we can see and hear, we can find a lot of things that will lead us to answer 'No' to all the above questions.
The root cause for our flaws lie in: entity, range and space. The knowledge of range and space-time based on our instincts is extremely confined. The maximum in the range of numbers humans encounter is 1024 light-years, and the minimum, 10-36 meter, in Superstring Theory (Superstring Theory is an attempt to describe strong nuclear forces. This theory modifies the understanding of space-time and gravitational forces.). The range of numbers in most scientific research does not go beyond the range of the Earth. In our imagination the numbers may increase to as large as we want them to, but that is not how numbers work in the real world. In the real world, when a number increases to a certain limit, it will enter another space-time dimension, and become another independent entity. An entity, in our imagination, is superimpositions of countless entities of different space-time dimensions. Actually, the word "superimposition" can hardly describe the phenomenon when a number advances to another space-time dimension. In addition, the concept of addition does not correspond to natural phenomena.
Given this, I will expound on the concept of aggregation, the concept itself, as well as many rules inferred from it. It is essentially a problem of discrepancy between an inferred concept and the base concept. Now that we have an expanded understanding of space-time dimension, we can look at some mathematical models and find out the flaws in today's basic concepts. I will use the mathematical formulas to prove my arguments later when such opportunities arise.
What comes after natural numbers is the aggregation of integers. This concept is a natural result brought by the introduction of subtraction operations. It is an inevitable expansion of induction and deduction. Remembering that mathematical concepts influence all facets in society, positive and negative can be seen to be related to yin and yang, while addition and subtraction are related to profit and loss, gain and loss, and variations on these.
Next the concept of rational and irrational numbers was developed. Each step in the progression of mathematical thinking was a successful effort at re-packaging of old thinking and concepts. The concepts of the continuum, expansion and separation continue to promote the development of wrong theories built upon older wrong theories, sealing the sciences into a tight spot, with nowhere to go to. It was not until the introduction of the concept of real numbers that the continuous development of wrong theories built upon older wrong theories finally came to an end. All the complex expressions of operations and logics in modern science are the product of linear thinking. In fact, the so-called dialectic is a direct product of linear thinking. Linear thinking is the inevitable result of direct observation, because all human observations, and experiences must be transformed to human sensory experiences. The superficial sensory organs of a human being cannot surpass the threshold of phenomena that occur at the molecular level. They are merely confined within the narrow and small scope of nerve impulses conducted at a low speed.
No one who could fathom that the concept of real numbers actually has denied the existence of other space-times dimensions at both the macroscopic and the microscopic levels. When a number becomes small enough, it will enter another space. But the concept of real numbers stipulates that all real numbers exist within our space-time dimension. In today's systematic science environment, scientists believe that in using more parameters and conditions in their models, they will make new discoveries. As the aforementioned rule implies, when a number becomes large and complex enough, it will, too, advance to another space-time dimension. Introducing more parameters and conditions in the experiments just won't do, unless if the scientists were to conduct their research from multiple space-time dimensions. In other words, the research will need to be done beyond our space-time dimension.
All in this universe lives. So we could actually discuss the flaw of human science from the angle of life. All human ideas are under some type of manipulation, but I have not yet mentioned this point. I have not proposed a brand new theory either. What I have done here is merely to discuss mathematics from a fresh perspective. I have pointed out the root cause for the boundary in today's science. When the time comes, the entire world will surely begin to admire the Theory of the Five Elements, expounded in ancient China. Moreover, everyone will be astounded by the knowledge presented in the broad and profound Fa.
"A dream of ten thousand years is over, eventually the coast." (From "Arduously Saving" in Hong Yin.) When the time comes, the eternal sorrow of humanity will come to an end.
Translated from: http://www.zhengjian.org/zj/articles/2002/11/13/19196.html
