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[math-learn] Accelerating universe etc. via converting division to subtraction - Doctorow
Posted:
Jan 27, 2001 11:51 AM
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From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sat. Jan. 27, 2001 8:32AM
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.
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