Central to the establishment concept of continuity is that there can be no "next-to." It is integral to set theory and modern mathematics. Bertrand Russell noticed that it is in conflict with the common sense understanding of differential equations. Nonetheless, he accepted it. This led to a bizzare notion of a "physical object." Is there is an alternative to this concept of continuity? Yes, there is.
This is a review of George de Bothezat's account of the spatial infinitesimal in his work, Back To Newton: A Challenge to Einstein's Theory of Relativity.
The veritable number system is more versatile with respect to division than the real number system.
This paper reviews the author's soon-to-be-published book, The Nature of Negative Numbers. In this book, the veritable number system is presented much more fully than it was in the earlier book, Absolute Space, Absolute Time & Absolute Motion, or elsewhere, including the 2005 NPA Proceedings. It is compared and contrasted to imaginary numbers and, especially, to the received real number system. The argument is that it is superior to the other two.
This number system, already discussed in a 2005 NPA paper and in the published book, has important applications. Its characterization of paths, whether straight or curvilinear, is far superior to that of the better known real number system.
Quaternions were discovered (or invented) early in the 19th century, when Sir William Rowan Hamilton set out to duplicate a significant part of the imaginary number system. His efforts were not entirely right; however, the later attempts by Josiah Willard Gibbs and others to replace the quaternions with vectors were not entirely right either. But much can be done with the newly discovered veritable number system.
Space is three-dimensional only, and consists of infinitesimals. Infinitesimals are points of location, without area, shapeless, indivisible, continuous in all directions. Nothing inconsistent with them, such a square circle, can exist. There is no microscopic infinity. Knowing that they exist provides the answer to the mystery of irrational numbers, asymptotes, "infinite series," and much else.
This paper aka "The Nature of Space".
Time does not move or flow. Neither is it a fourth dimension. It is completely independent of space, though both necessarily exist. It consists of infinitesimals, the instants. Each instant is discrete; it is here, then no more. Yet, it is continuous; there is no time that is not a time. Our consciousness of time does not come from the senses, but from memory. It is an innate idea.
This lecture is taken from the author's book, Absolute Space, Absolute Time, & Absolute Motion. Common sense teaches that space is different from the objects within it. This is also the conclusion of sustained reason. The key to understanding space is its smallest division, the infinitesimal. The opposing thesis, namely that space is infinitely divisible can be refuted. By understanding the nature of the spacial infinitesimal, we can answer such questions as: What is the nature of an irrational number? Under what conditions is division by zero valid? Is space curved?
This lecture is taken from the author's book, Absolute Space, Absolute Time, & Absolute Motion. The idea of the limit when applied to the derivative in the infinitesimal calculus is wrong. It does not solve the problem that the derivative is usually different from dy/dx.. Instead, it conceals this problem. As a result of applying the limit idea, certain products of the process of derivation are commonly rejected, leaving only the derivative. Yet, inspection shows that they must still exist. Since the derivative is not an approximation, but an exact product, the commonly rejected extra terms must be recognized as present. As such, they may provide an avenue for the future advance of physical science.
Absolute Space, Absolute Time, and Absolute Motion exist. These are shown to be facts through an investigation of the nature of infinitesimals. Knowledge of that nature also makes the irrational magnitudes within the unit comprehensible. The number line is shown to be cognitively superior to set theory; furthermore, non-Euclidean geometry is shown to be a mere manipulation of symbols and not an expression of a ?parallel universe?. Inside, the reader will also learn about a hitherto unknown number system locked within the square root of negative one. He will also discover in the infinitesimal calculus a hidden key to a level of reality beneath that of nano-technology. The foundation of science is not some vague generality, but the exercise of reason as originating from the human sensorium. There is no difference between mathematical and ordinary inductive reasoning. The paper is the first chapter of the book, Absolute Space, Absolute Time, & Absolute Motion.
This is an excerpt from a forthcoming book on the nature of the infinitesimal. This square root of negative one has been in use for centuries. Most consider it to be either something transcendental or a mere direction to perform a certain operation. Einstein used it to help make plausible his idea of the fourth dimension. Actually, neither the idealists nor the logical positivists are correct. The -1 inside the radical sign is from a number system discovered by the author. It is called the "Veritable Number System." It?s special properties are shown herein. It stands with the real number system and absolute numbers as one of the three ways to handle the concept of direction. The author has found some applications to physical science not discussed in this paper. Students of nature and others are welcome to explore the implications of what has been disclosed, and use it to increase man?s knowledge of reality.