I have been developing on geometry-research various applications of logic-based probability (LBP) which converts division of probabilities to subtraction of probabilities (plus the constant 1). The generalized method can be used not only to enable students to explore and actually research Non-Euclidean analogues in algebra, geometry, probability/statistics, logic, etc., but to research remarkably simple models for the accelerating versus decelerating universe. The conversion of division to subtraction is represented as y/x --> y - x + 1 with the stipulation that y < = x, although in some branches of mathematics the direction of this inequality is somewhat arbitrary as long as the same direction is used consistently. It is not a function or transformation (although with restrictions of domain it can be made into a function, and its "inverse" appears to be a function), and I refer to it as a conversion. It has already been established that the conversion changes Bayesian conditional probability (BCP) to logic-based probability (LBP) and that it changes Product/Goguen fuzzy multivalued logical implication to Lukaciewicz fuzzy multivalued logical implication, and these two types of logic or alternatively Godel logic substituted for one of them generate generalized Boolean logic. To proceed to the accelerating universe scenario, which has been generally accepted now for the expansion of the universe at present, we use a fact which I established recently, namely, that the proximity function p(x,y) = 1 + y - x for y < = x is roughly speaking an inverse of the metric or distance-function with the scale [0,1] replacing the usual distance scale [0, infinity). I refer to it as a semi-inverse distance (SID). It satisfies the triangle inequality and is nonnegative (between 0 and 1, in fact), and while it does not satisfy reflexivity p(x,y) = p(y,z) non-trivially, it does technically satisfy it since y is never greater than x from the condition y < = x. Intuitively, it is a sort of one-sided metric or inverse metric. p(x,y) is maximum when x = y and minimum when x and y differ maximally on [0,1], which means that x = 1 and y = 0 since y < = x. This is essentially the opposite of the metric or distance-functions as usually defined. There is a similar proximity function for three variables, p(x,y,z) = 1 + (y-x) + (y-z) with the constraints y < = x and y = z, and it satisfies the triangle inequality and nonnegativity, although instead of going from 0 to 1 it goes from 0 to 2. The core of the accelerating universe argument in the quintessence formulation is that there is an attractive force (gravitation) versus a repulsive force (quintessence). We draw upon recent research in continuum physics to the effect that forces can be generalized to be the fundamental things (Springer's recent mathematics volumes have some excellent treatises on this - I will cite references another time), or alternatively we mimic special relativity's idea of contraction along the direction of motion (x) and expansion along the orthogonal direction (y) but interpret it as a force argument so that x and y can represent either coordinates or (opposite type) forces. For the p(x,y) case, differentiating twice with respect to time yields p" = y" - x", where prime denotes differentiation with respect to time, and we have p" > 0 iff y" > x", so the universe is decelerating iff y" > x" and accelerating iff y" < x" (an accelerating proximity would correspond to a decelerating distance). Likewise, p' > 0 iff y' > x'. The student or the teacher or researcher can then explore the reasons for this behavior.
Osher Doctorow Ventura College, West Los Angeles College, etc.