On Jun 11, 1:12 pm, Iain hotmail.com> wrote:

Very good. I remember that McCrone spoke about how there are three
moments or something. How there is the immediate past the present and
the next moment and how subjectivity is always in transition through
them. The past becoming the present while the present becomes the
future, all at the same time. The elemental processes are of shorter
duration than the length of a present moment and these elementals
overlap. Strange trying to create and anology with MSUs when MSUs are
necessary for moments to procede anyway. I think you were talking
about "transitivity" when you spoke about order and have some text
further down this page and below is a snippit from the moments but
more is on that page.

Benjamin Libet famously suggested it takes about half a second for the
brain to get through all the processing steps needed to settle our
view of the moment just past. But this immediately raises the question
of why don't we notice a lag? How does anyone ever manage to hit a
tennis ball or drive a car? The answer is that we anticipate. We also
have a level of preconscious habit which "intercepts" stuff before it
reaches a conscious level of awareness. And yet it really does take
something like half a second to develop a fully conscious experience
of life. You can read about the cycle of processing story and its
controversies in the following....

see: the cycle of processing
http://www.dichotomistic.com/readings_intro.html

Here's an obvious fact: if x > y and y > z, then x > z, right?

Or to give an example in plain english, lets say I have more money
than Herbert. Also, Bill Gates has more money than I do. Therefore,
Bill Gates also has more money than Herbert.

In each of these cases, the conclusion follows because the greater-
than relation ">" is transitive.

Now lets look at a nontransitive example from real life. Let's say
Lee, Danny, and Chris decide to have a chess tournament. Danny wins
against Lee, Lee wins against Chris, and Chris wins against Danny. Who
is the best chess player? There is no clear victor because the method
of measuring the best player (i.e. the "best-player" relation) is not
transitive.

So that brings us to the main topic: nontransitive dice. Lets say we
have four dice. Each die has six sides, and each die is numbered as
follows:

http://www.geocities.com/CapeCanaveral/Hangar/7773/dice.html
http://www.sciencenews.org/20020413/mathtrek.asp

- Nontransitive Dice, Paradoxical Possibility Sets, A>B, B>C, C>A, 1
System X Outcomes

There are many problems with the phrase "survival of the fittest" as a
description of evolution. It suggests that fitness is a static
concept, one that lives on a simple numerical scale. On the contrary,
the fitness of an organism depends on what it is competing for—the
game that it is playing. A simple illustration of how counterintuitive
the concept of fitness can be is given by so-called nontransitive
dice, whose name we explain in a moment. Imagine three three-sided
dice whose faces are numbered as follows:

A: 3 4 8
B: 1 5 9
C: 2 6 7

Each player chooses one die, and rolls it; whoever gets the highest
number wins.

You can calculate the probabilities of winning by drawing up tables of
the combinations, like this:

(Fig 10 - 01.jpg)

For example, if die A throws a 3 and die B throws 1, then A wins, so
we put an A in column '3' and row T of the first table. We see that B
beats A with probability 5/9, because out of the nine combinations of
throws, B wins on five occasions. Similarly C beats B with probability
5/9. So B is better than A, and C is better than B. Therefore C is
better than A? No! Die A beats die C with probability 5/9 as well.

We're used to this in real life. In football games it is not unusual
to find that the Giants beat the Eagles, the Eagles beat the Jets, and
the Jets beat the Giants. This rather paradoxical kind of behavior,
which mathematicians call nontransitive, may even happen fairly
consistently because of the way the respective teams' styles of play
interact. Effectively, each pairing is a different game, even though
on a meta-level they are all football. This is what makes the dice
games behave so strangely: Each choice of two dice effectively changes
the competition to a different game. We may therefore expect
nontransitive behavior in evolutionary games. This is just one of many
reasons why it is impossible to label a particular mutation as bad or
good, independent of context. Each of the dice A, B, C is good in the
context of one game and bad in the context of another.

Because the status of a mutation is context-dependent, the argument
behind Muller's ratchet—that bad mutations accumulate faster than back-
mutations to good ones—needs more careful examination. It still seems
likely that rather a lot of bad mutations will usually occur before
any good ones do, but the position is no longer so clear-cut. In the
absence of detailed knowledge of the meaning of DNA for development,
it's not really possible to decide what the proportion of good
mutations to bad ones is likely to be. It remains true, however, that
sexual reproduction is far superior at preserving the good and
surviving the bad.

--------------------------------------------------

- What is a Species, Breeding Criteria & Transitivity, Speciation at
Edge of Population

Let's take one step backward, before we launch into the unknown.
Ecologies are large-scale systems, and their main bits and pieces are
not so much individual organisms as species. A woodland ecology
consists not just of this rabbit and tbattree, for over quite short
periods of time both will die. Other rabbits and other trees will
appear instead, but not in exactly the same place. An ecosystem is a
process, much of which happens on the species level. "You never step
into the same river twice," said Heraclitus— meaning that the water is
constantly changing. For similar reasons, you never step into the same
wood twice, and you never step into the same ecosystem twice. Yet
river, wood, and ecosystem possess a definite continuity of existence;
they are recognizably the same process, going on just as before. Human
beings simplify their world, attaching short labels such as "river,"
"wood," "species," or "ecology" to hugely complicated processes. As a
warm-up exercise before we think about ecologies, we'll try to sort
out just what is meant by the term "species."

There is a conventional, textbook answer: Organisms belong to the same
species if they can interbreed to produce fertile offspring. A horse
and a donkey can make a mule, but mules are sterile, so horses and
donkeys are distinct species. A horse and a hummingbird are incapable
of cross-breeding a pegasus, so horses and hummingbirds are distinct
species. A poodle and a German shepherd can produce mongrel puppies
that in turn can breed with other dogs: dogkind is a single species.
This approach to species is essentially reductionist in character: It
looks inside the members of a given species to see what it is that
makes them similar. Not literally—but the emphasis is on "able to
breed," looking within a given species, rather than on "unable to
breed," which looks outside in the following sense: You don't find
zoologists trying to breed a horse with a hummingbird to prove they
really are different species. And as it stands there are problems with
this definition. It doesn't apply to amoebas, or to any asexual
species, but we can exclude them from consideration for the moment. If
taken literally, the definition implies that any sterile animal is the
unique member of its own exceptional species—so worker bees are a
different species from queen bees. In order to prove that all horses
belong to the same species, it looks as if we have to breed them in
all possible pairings, and then test the fertility of the offspring—
rather a tall order. Of course we don't do that at all. We use various
extraneous clues to assign organisms to relatively well-defined groups—
how many legs, what color coat, with or without wings—and then use the
interbreeding test to resolve any fine points. You don't need to try
to breed a hummingbird with a stallion to prove that horses aren't
birds; but if two types of bird look extremely similar, interbreeding
might resolve the matter.

Unfortunately, interbreeding is not a totally satisfactory answer. We
expect species to be rather like exclusive clubs, with each organism
belonging to one and only one species. Mathematicians would say that
belonging to the same species is a transitive relation, by which they
mean that if animal A belongs to the same species as B, and B belongs
to the same species as C, then A necessarily belongs to the same
species as C. This may look pretty obvious, but recall our
nontransitive dice: A beat B, which beat C, which beat A. Let's
interpret transitivity in terms of interbreeding: If A can interbreed
with B, and B can interbreed with C, then A can interbreed with C. Now
it doesn't look so obvious. We could imagine a string of dogs, each
larger than the last, so that each can breed with its neighbors; but
the two on the ends are incompatible, if only for physical reasons. If
you try hard you may be able to envisage, purely as a theoretical
possibility, a continuum of animals stretching from hummingbird to
horse. Neighboring animals could differ so little that they could
interbreed; but the two extremes? Never!

In fact, we don't need such thought experiments: A continuum of this
kind can occur in nature. There is a more or less continuous chain of
gulls, which starts in Britain, goes right around the world, and ends
up back near where it started. The gulls change slowly as you go
around the chain. At one end you find black-backed gulls; at the other
herring gulls. The gulls of these two types can't interbreed, so by
that criterion they belong to different species. However, in between
there is a continuous range of gulls, and all the gulls (of the type
being discussed) on any particular local section of the chain can
interbreed with those on nearby sections, so each set of gulls is by
that very same criterion the same species as its neighbors are. So Ms.
Herring Gull has a next-door neighbor of the same species, who has a
neighbor of the same species, who . . . has a neighbor of the same
species; but this ultimate neighbor "of the same species" is not Mr.
Herring Gull, but Mr. Black-backed Gull, with whom she cannot
interbreed at all.

This is a spatial analogue of the problem that so puzzled Darwin's
detractors: How can an ape have a child that is an ape that has a
child that is an ape that . . . has a child that is a man that has a
child that is a man that has a child that is a man?

This lack of transitivity is a structural feature of our concept of
species. It can't be repaired by taking an even more reductionist
stance, by looking for genetic similarities or differences in the DNA
of all the gulls. It is emphatically not a matter of finding ever more
esoteric genetic criteria for drawing the line between black-backed
gulls and herring gulls. Drawing fine lines is a human tendency, an
attempt to make our simplified mental labeling system match a
differently structured world; but in that world boundaries may be
fuzzy, or fractal, or may not exist at all.

-------------------------

As a mathematics paper, Lorenz's climate work would have been a failure
—he proved nothing in the axiomatic sense. As a physics paper, too, it
was seriously flawed, because he could not justify using such a simple
equation to draw conclusions about the earth's climate. Lorenz knew
what he was saying, though. "The writer feels that this resemblance is
no mere accident, but that the difference equation captures much of
the mathematics, even if not the physics, of the transitions from one
regime of flow to another, and, indeed, of the whole phenomenon of
instability." Even twenty years later, no one could understand what
intuition justified such a bold claim, published in Tellus, a Swedish
meteorology journal. ("Tellus! Nobody reads Tellus," a physicist
exclaimed bitterly.) Lorenz was coming to understand ever more deeply
the peculiar possibilities of chaotic systems—more deeply than he
could express in the language of meteorology.

As he continued to explore the changing masks of dynamical systems,
Lorenz realized that systems slightly more complicated than the
quadratic map could produce other kinds of unexpected patterns. Hiding
within a particular system could be more than one stable solution. An
observer might see one kind of behavior over a very long time, yet a
completely different kind of behavior could be just as natural for the
system. Such a system is called intransitive. It can stay in one
equilibrium or the other, but not both. Only a kick from outside can
force it to change states. In a trivial way, a standard pendulum clock
is an intransitive system. A steady flow of energy comes in from a
wind-up spring or a battery through an escapement mechanism. A steady
flow of energy is drained out by friction. The obvious equilibrium
state is a regular swinging motion. If a passerby bumps the clock, the
pendulum might speed up or slow down from the momentary jolt but will
quickly return to its equilibrium. But the clock has a second
equilibrium as well—a second valid solution to its equations of motion—
and that is the state in which the pendulum is hanging straight down
and not moving. A less trivial intransitive system—perhaps with
several distinct regions of utterly different behavior—could be
climate itself.

Climatologists who use global computer models to simulate the long-
term behavior of the earth's atmosphere and oceans have known for
several years that their models allow at least one dramatically
different equilibrium. During the entire geological past, this
alternative climate has never existed, but it could be an equally
valid solution to the system of equations governing the earth. It is
what some climatologists call the White Earth climate: an earth whose
continents are covered by snow and whose oceans are covered by ice. A
glaciated earth would reflect seventy percent of the incoming solar
radiation and so would stay extremely cold. The lowest layer of the
atmosphere, the troposphere, would be much thinner. The storms that
would blow across the frozen surface would be much smaller than the
storms we know. In general, the climate would be less hospitable to
life as we know it. Computer models have such a strong tendency to
fall into the White Earth equilibrium that climatologists find
themselves wondering why it has never come about. It may simply be a
matter of chance.

To push the earth's climate into the glaciated state would require a
huge kick from some external source. But Lorenz described yet another
plausible kind of behavior called "almost-intransitivity." An almost-
intransitive system displays one sort of average behavior for a very
long time, fluctuating within certain bounds. Then, for no reason
whatsoever, it shifts into a different sort of behavior, still
fluctuating but producing a different average. The people who design
computer models are aware of Lorenz's discovery, but they try at all
costs to avoid almost-intransitivity. It is too unpredictable. Their
natural bias is to make models with a strong tendency to return to the
equilibrium we measure every day on the real planet. Then, to explain
large changes in climate, they look for external causes—changes in the
earth's orbit around the sun, for example. Yet it takes no great
imagination for a cli-matologist to see that almost-intransitivity
might well explain why the earth's climate has drifted in and out of
long Ice Ages at mysterious, irregular intervals. If so, no physical
cause need be found for the timing. The Ice Ages may simply be a
byproduct of chaos.

The Collapse of Chaos: Discovering Simplicity in a Complex World
Jack Cohen and Ian Stewart / 1994
http://www.amazon.com/exec/obidos/tg/detail/-/0140178740/