Of the mathematical objects studied in the two problems.Infinity is that which is boundless, endless, or larger than any natural number. It is shown that the introduced methodology and numeral system change our perception The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. Related to their nature but is a consequence of the weakness of traditional numeral systems used to It is stressed that the hardness of both problems is not with respect to traditional theories of infinity and infinitesimals. Objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higherĪccuracy than it is done by traditional tools. This book is dedicated to two topics that, up to the recent times, were believed. In particular, it is shown that the new approach allows one to observe mathematical He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world.
#Infinity and infinitesimals book series
Series Title: Emergence, Complexity and Computation. This book is an exploration of philosophical questions about infinity. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. Book Title: Numerical Infinities and Infinitesimals in Optimization. Numerous numerical examples and theoretical illustrations are given. 'Welcome to the world of the finite, the sub-finite, and infinite, actual infinity versus potential infinity, the measurable and the immeasurable. are used in different situations related to infinity). (recall that traditionallyĭifferent numerals lemniscate Aleph zero, etc. To use the same numeral system for measuring infinite sets, working with divergent series, probability,įractals, optimization problems, numerical differentiation, ODEs, etc. The introduced methodology gives the possibility The new computer is able to work numer- ically with finite, infinite, and infinitesimal numbers giving so the possibility to use different infinitesimals. To ambiguities in theoretical assertions, as well. It is argued that numeral systems involved in computations limit our capabilities to compute and lead (recall that traditional theories work with infinities and infinitesimals only symbolically) with infiniteĪnd infinitesimal numbers that can be written in a positional numeral system with an infinite radix. The methodology uses aĬomputational device called the Infinity Computer (patented in USA and EU) working numerically Infinite, and infinitesimal) and to all sets and processes (finite and infinite). 5 “The whole is greater than the part” applied to all quantities (finite, Methodology does not contradict Cantor’s and non-standard analysis views and is based on theĮuclid’s Common Notion no. Numerically in a unique computational framework in all the situations requiring these notions. It has been introduced with the intention to allow one to work with infinities and infinitesimals Of mathematical objects from numeral systems involved in their representation is described. In this survey, a recent computational methodology paying a special attention to the separation
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