Consider that you are the United States president and that you
have a very specific social-political agenda. Consider that (due
to some tragedy) that you have the opportunity to appoint all
nine members of the US supreme court. Consider that (at the
moment) there are enough like-minded people in the US senate
so that your appointments will be ratified. Now, since you
have a specific social-political agenda, what you want is for the
new supreme court to always make the decisions that you want
them to make. That is, at this particular moment in time, you
have a procedure to make decisions (that satisfies your agenda)
and you want the supreme court justicies to always follow that
same decision procedure. Consider that this is what you want.
Now, if you could pull this off (somehow get the new justices
to make decisions in this way), the new justices would be
making decisions in a path-independent way. (Although the
appointment of the new justicies was VERY path dependent).
The new judges would not be thinking for themselves. They
would just be mindlessly following a pre-written formula. It
would be completely arbitrary what order the cases were
brought to the court. Any particular decision would provide
no new information (as for how other cases would later be
decided). Everything is already entailed by the decision
formula that you provided at the time of the initial
appointments.
Now, consider a different situation. Consider that the justices
are free to think for themselves. Consider that they aren't just
implementing a pre-written decision procedure. Consider that
it isn't guaranteed that they will come to the same decisions
that you want. In such a case, each decision provides new
information and can affect future decisions. In this situation, it
would be impossible (over time) to guarantee that your
particular social-political agenda would always be followed.
The order of the cases brought to the court matters. There is
now path dependence.
Path independence (in this example) is associated with the
mindless following of rules or formulas. No new information
emerges along the way. Everything is entailed from the
beginning. The end-point is certain. It doesn't matter what
order the cases come in.
Path dependence (in this example) is associated with the
justicies being free to make real decisions. New information is
being produced (emerging) along the way as a result of these
decisions. This new information then partly influences future
decisions. It becomes impossible to predict (from the start)
how the justicies will decide in future cases. Everything is not
entailed from the beginning. It does matter what order the
cases come in.
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I hope this helps. Please comment if anybody thinks that
aspects of the above explanation misses the point of the
distinction between path-dependence and path-independence.
Jeff Prideaux
P.S. I kind of patterned my explanation after the types of things
Kampis said in his book "Self-modifying systems in biology and cognitive
science".