Desiderata Content The notion of number Desiderata to the extension of the numbers Summary Abstract Although the numbers are yet considered as the simplest abstractions of mathematics, it is increasingly taking shape is a picture in which the numbers will the most complex and exciting concept in mathematics. The geometric algebra is a probable candidate for the new number system, but desiderata of the author is unlike the criteria of the geometric algebra in three points: The number line of real numbers is orthogonal to the space-dimensions according to the definition of scalar product. As the fundamental operation of arithmetic the product is commutative. The space-dimensions evolve from time-dimension which is represented by number line of real numbers.   After…

Two-element numbers as space-time elements of probability theory Abstract Introductory The probability theory and the space-time Different interpretations and problems of the concept of probability The Bertrand paradox and his explanation Relative frequency concept and the paradoxes Kolmogorov’s probability theory Epilog   After registration, you may ask for a translation of the full text by email.

The role of the concept of computability in physics “It would require infinite precision for the coordinates of a phase-space point- i.e. all the decimal places!- in order for it to make sense to say that the point is non-computable. (A number described by a finite decimal is always computable.) A finite portion of .a decimal expansion of a number tells us nothing about the computability of the entire expansion of that number. But all physical measurements have a definite limitation on how accurately they can be performed, and can only give information about a finite number of decimal places. Does this nullify the whole concept of 'computable number' as applied to physical measurements?” (Roger Penrose, The Emperor’s New Mind – Concerning Computers, Minds, and The…

Content The infinity and the two-element numbers Infinity differently – a heuristic approach Infinity described with homogeneous coordinates The infinity and the geometric algebra The complex and hyperbolic numbers as two-dimensional Clifford algebras - examples from the literature The hiding infinity AbstractTypes of geometric algebra generated by vectors of 1-dimensional vector space are not at all exhibit poor structure. These geometric algebras have very exciting basic structure as they are represented by two-element-numbers which are the complex, the parabolic (dual) and the hyperbolic numbers. On the other hand, special types of infinities – related to the continuum hypothesis – are modeled by the two-element numbers. So it can be said that vectors of 1-dimensional vector space, which generates the above…

In the geometric algebra1 there are some problems, which not found in the literature, and I would like to share my thoughts about these now. I restrict my examination to those geometric algebras, where the starting point is a finite dimensional Euclidean vector space and the real numbers. In the Euclidean vector space an internal product is defined with an outer product generalized by Grassmann. After that, a so-called geometric product, which will be the basic operation in the geometric algebra, is defined. ___________________________________________________ 1 See for example the book of Chris Doran - Anthony Lasenby titled Geometric Algebra for Physicists;http://www.cambridge.org/nl/academic/subjects/physics/theoretical-physics-and-mathematical-physics/geometric-algebra-physicists?format=PB or an article titled „Imaginary Numbers are not Real — the Geometric Algebra of Spacetime” by Gull-Lasenby-Doranhttp://www.researchgate.net/publication/226188504_Imaginary_numbers_are_not_realThe_geometric_algebra_of_spacetime After registration, you may ask for a translation of the full text…