Glimpses beyond the Two Millennia Old Bondage of the Archimedean Axiom: Part 1

Glimpses beyond the Two Millennia Old Bondage of the Archimedean Axiom (by Dainis Zeps): Abstract: This is my Editorial Note as Editor-at-Large on this Focus Issue featuring Elemér E. Rosinger’s recent work.

How Far Should the Principle of Relativity Go? (by Elemer E. Rosinger): Abstract: The Principle of Relativity has so far been understood as the covariance of laws of Physics with respect to a general class of reference frame transformations. That relativity, however, has only been expressed with the help of one single type of mathematical entities, namely, the scalars given by the usual continuum of the field R of real numbers, or by the usual mathematical structures built upon R, such as the scalars given by the complex numbers C, or the vectors in finite dimensional Euclidean spaces Rn, infinite dimensional Hilbert spaces, etc. This paper argues for progressing deeper and wider in furthering the Principle of Relativity, not by mere covariance with respect to reference frames, but by studying the possible covariance with respect to a large variety of algebras of scalars which extend significantly R or C, variety of scalars in terms of which various theories of Physics can equally well be formulated. First directions in this regard can be found naturally in the simple Mathematics of Special Relativity, the Bell Inequalities in Quantum Mechanics, or in the considerable amount of elementary Mathematics in finite dimensional vector spaces which occurs in Quantum Computation. The large classes of algebras of scalars suggested, which contain R and C as particular cases, have the important feature of typically no longer being Archimedean, see Appendix, a feature which can prove to be valuable when dealing with the so called "infinities" in Physics. The paper has a Comment on the so called "end of time".

George Boole and the Bell Inequalities (by Elemer E. Rosinger): Abstract: As shown by Pitowsky, the Bell inequalities are related to certain classes of probabilistic inequalities dealt with by George Boole, back in the 1850s. Here a short presentation of this relationship is given. Consequently, the Bell inequalities can be obtained without any assumptions of physical nature, and merely through mathematical argument.

Which Are the Maximal Ideals? (by Elemer E. Rosinger): Abstract: Ideals of continuous functions which satisfy an off diagonality condition proved to be important connected with the solution of large classes of nonlinear PDEs, and more recently, in General Relativity and Quantum Gravity. Maximal ideals within those which satisfy that off diagonality condition are important since they lead to differential algebras of generalized functions which can handle the largest classes of singularities. The problem of finding such maximal ideals satisfying the off diagonality condition is formulated within some background detail, and commented upon.

Heisenberg Uncertainty in Reduced Power Algebras (by Elemer E. Rosinger): Abstract: The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely,within one or the other of the infinitely many reduced power algebras which can replace the usual real numbers R, or complex numbers C. A major advantage of such a re-formulation is, among others, the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics". The use of reduced power algebras also opens up a foundational question about the role, and in fact, about the very meaning and existence, of fundamental constants in Physics, such as Planck’s constant h. A role, meaning, and existence which may, or on the contrary, may not be so objective as to be independent of the scalars used, be they the usual real numbers R, complex numbers C, or scalars given by any of the infinitely many reduced power algebras, algebras which can so easily be constructed and used.