Glimpses beyond the Two Millennia Old Bondage of the Archimedean Axiom: Part 2

Mathematics and "The Trouble with Physics", How Deep We Have to Go? (by Elemer E. Rosinger): Abstract: The parts contributed by the author in recent discussions with several physicists and mathematicians are reviewed, as they have been occasioned by the 2006 book "The Trouble with Physics", of Lee Smolin. Some of the issues addressed are the possible and not yet sufficiently explored relationship between modern Mathematics and theoretical Physics, as well as the way physicists may benefit from becoming more aware of what at present appear to be certain less than fortunate yet essential differences between modern Mathematics and theoretical Physics, as far as the significant freedom of introducing new fundamental concepts, structures and theories in the former is concerned. A number of modern mathematical concepts and structures are suggested for consideration by physicists, when dealing with foundational issues in present day theoretical Physics. Since here discussions with several persons are reviewed, certain issues may be brought up more than one time. For such repetitions the author ask for the kind understanding of the reader.

String Theory: A Mere Prelude to Non-Archimedean Space-Time (by Elemer E. Rosinger): Abstract: It took two millennia after Euclid and until in the early 1880s, when we went beyond the ancient axiom of parallels, and inaugurated geometries of curved spaces. In less than one more century, General Relativity followed. At present, physical thinking is still beheld by the yet deeper and equally ancient Archimedean assumption which entraps us into the limited view of "only one walkable world". In view of that, it is argued with some rather easily accessible mathematical support that Theoretical Physics may at last venture into the Non-Archimedean realms.

Cosmic Contact to Be, or Not to Be Archimedean (by Elemer E. Rosinger): Abstract: This is a two part paper which discusses various issues of cosmic contact related to what so far appears to be a self-imposed censorship implied by the customary acceptance of the Archimedean assumption on space-time.

From Reference Frame Relativity to Relativity of MathematicalModels: Relativity Formulas in a Variety of Non-Archimedean Setups (by Elemer E. Rosinger): Abstract: Galilean Relativity and Einstein’s Special and General Relativity showed that the Laws of Physics go deeper than their representations in any given reference frame. Thus covariance or independence of Laws of Physics with respect to changes of reference frames became a fundamental principle. So far, all of that has only been expressed within one single mathematical model, namely, the traditional one built upon the usual continuum of the field R of real numbers, since complex numbers, finite dimensional Euclidean spaces, or infinite dimensional Hilbert spaces, etc., are built upon the real numbers. Here, following [55], we give one example of how one can go beyond that situation and study what stays the same and what changes in the Laws of Physics, when one models them within an infinitely large variety of algebras of scalars constructed rather naturally. Specifically, it is shown that the Special Relativistic addition of velocities can naturally be considered in any of infinitely many reduced power algebras, each of them containing the usual field of real numbers and which, unlike the latter, are non-Archimedean. The nonstandard reals are but one case of such reduced power algebras, and are as well non-Archimedean. Two surprising and strange effects of such a study of the Special Relativistic addition of velocities are that one can easily go beyond the velocity of light, and rather dually, one can as easily end up frozen in immobility, with zero velocity. Both of these situations, together with many other ones, are as naturally available, as the usual one within real numbers.