It is a rather universal tacit and unquestioned belief—and even more so among physicists—that there is one and only one real line, namely, given by the coodinatisation of Descartes through the usual field ℝ of real numbers. Such a dramatically limiting and thus harmful belief comes, unknown to equally many, from the similarly tacit acceptance of the ancient Archimedean Axiom in Euclid’s Geometry. The consequence of that belief is a similar belief in the uniqueness of the coordinatization of the plane by the usual field ℂ of complex numbers, and therefore, of the various spaces, manifolds, etc., be they finite or infinite dimensional, constructed upon the real or complex numbers, including the Hilbert spaces used in Quantum Mechanics. A near total lack of awareness follows therefore about the rich self‐similar structure of other possible coordinatisations of the real line, possibilities given by various linearly ordered scalar fields obtained through the ultrapower construction. Such fields contain as a rather small subset the usual field ℝ of real numbers. The concept of walkable world, which has highly intuitive and pragmatic algebraic and geometric meaning, illustrates the mentioned rich self‐similar structure.

This content is only available via PDF.
You do not currently have access to this content.