On model theory, non-commutative geometry and physics

1.1 Our motivation for working on the subject presented below comes from the realisation of the rather paradoxical situation with the mathematics used by physicists in the last 70 or so years. Physicists have always been ahead of mathematicians in introducing and testing new methods of calculations, leaving to mathematicians the task of putting the new methods and ideas on a solid and rigorous foundation. But this time, with developments in quantum field theory huge progress achieved by physicists in dealing with singularities and non-convergent sums and integrals (famous Feynman path integrals) has not been matched so far, after all these years, with an adequate mathematical theory. A nice account of some of these methods with a demonstration of challenging calculations can be found in [6], and more detailed account in [7] (Chapter 2. The basic strategy of exctracting finite information from infinities). One may suggest that the success in developing this calculus in the absence of a rigorous mathematical theory is due to the fact that the physicist, in fact, uses an implicit or explicit knowledge of the structure of his model which is not yet available in mathematical terms. A beautiful and honest account by a mathematician attemting to translate physicists’ vision into a mathematical concept provides the introductory section of [10]. In particular, many formulas which to the mathematician’s eye are defined in terms of a metric or a measure are not what they look. Typically, it is crucial that discrete approximations to the continuous models have the same type