The presen t s tudy examines the s eman t i c problems involved in comput ing the meaning of the non s tandard uses of if. The cen t r a l ques t ion is whe the r or not it is necessary to in t roduce d i f f e r en t meanings of if. Austin proposed two non s tandard meanings for if. We show that these can be accoun ted for by the s t andard meaning t oge the r with shi f ts in the posi t ion o...

Noises: a nonstandard analysis. Thanks to the nonstandard formalization of fast oscillating functions, due to P. Cartier and Y. Perrin, an appropriate mathematical framework is derived for new non-asymptotic estimation techniques, which do not necessitate any statistical analysis of the noises corrupting any sensor. Various applications are deduced for multiplicative noises, for the length of t...

A large variety of supervised or unsupervised learning algorithms is based on a metric or similarity measure of the patterns in input space. Often, the standard euclidean metric is not sufficient and much more efficient and powerful approximators can be constructed based on more complex similarity calculations such as kernels or learning metrics. This procedure is benefitial for data in euclide...

Stability arises as the consistency criterion in a betting interpretation for hyperreal imprecise previsions, that is imprecise previsions (and probabilities) which may take infinitesimal values. The purpose of this work is to extend the notion of stable coherence introduced in [8] to conditional hyperreal imprecise probabilities. Our investigation extends the de Finetti-Walley operational char...

The paper applies some very elementary model theory to explain how Kenneth Arrow’s classical theory of collective choice sheds some light on the nature of mathematical structures arising in physics.

The chapter represents a comparison of ultrapower fields (loosely surreals, hyper-reals, long line) and number fields generated by infinite primes having a physical interpretation in Topological Geometrodynamics. Ultrapower fields are discussed in very physicist friendly manner in the articles of Elemer Rosinger and these articles are taken as a convenient starting point. The physical interpret...

Non-standard interactions (NSI) arise naturally when considering physics beyond the Standard Model. In particular, they can have an important influence in the solar neutrino analysis. The current status of NSI constraints will be discussed and the perspectives of future neutrino experiments will be briefly outlined.

A new construction is given of non-standard uniserial modules over certain valuation domains; the construction resembles that of a special Aronszajn tree in set theory. A consequence is the proof of a sufficient condition for the existence of non-standard uniserial modules; this is a theorem of ZFC which complements an earlier independence result.

In the classical theory of probability we start by a pair (Ω, ), with Ω set of elementary events and σ-algebra of subsets of Ω, called events. A probability on is defined as a countable additive and positive function p such that p(Ω)=1. If A and B are two events, the conditional probability p(A/B) is defined if and only if p(B) is not null, by considering the quotient p(AB)/p(B). In the theory ...