Re: multi-step reasoning

["John S. Denker" <jsd@MONMOUTH.COM>]

Herbert H Gottlieb wrote:
> I was quite interested in your PV question below and
> wondered if I was able to arrive at the correct answer myself..
> I reasoned that if V is doubled, (other things being equal)
> then P would immediately be reduced to half its value and the
> nRt would remain the same.

OTBE.....  Other Things Being Equal.....   Dangerous words!

*) If a fireplace bellows expands, when V is doubled, N doubles
at constant pressure and temperature.

*) If an ordinary (albeit rather loud) sound wave expands, it
happens isentropically and the pressure goes down by a factor
of 2^1.4 at non-constant temperature, while the N of a given
parcel remains constant.

*) Herb's scenario arguably cannot happen exactly as stated,
depending on what one means by "immediately", but if the gas
expands slowly in a piston that is held at constant temperature
by a massive heat bath, then yes, P will go down by a factor of
two, at constant N and T.

*) Other possibilities, including intermediate cases, abound.

For a simple general-purpose illustration of OTBE pitfalls, see
  http://www.monmouth.com/~jsd/physics/causation.htm#fig-lever

====================

 -- Since we have no context for the question "what happens if V
doubles" we have no hope of knowing what OTBE means.
 -- In other cases, we might be able to figure out from context
what OTBE means, but this will be always be super-sensitive to
details.

The Subject: line speaks of multi-step reasoning, and a
related issue might be best described as _multi-viewpoint_
reasoning.  It is common, indeed unavoidable to get lured
into blind alleys.  For example, one might start by assuming
the expansion is isothermal, only to discover later that
it is not, and other hypotheses (such as isentropic expansion)
must be considered.  It is a crucial skill to be able to
backtrack out of the blind alley without making a shambles
of the rest of the analysis, so that one can continue with a
systematic exploration of the other hypotheses.  To summarize:
 *) one needs enough power of imagination to imagine all the
relevant hypotheses, and
 *) one needs a method for systematically searching the space
of hypotheses.  (This requires, among other things, the
ability to keep track of what is a solidly-known fact and
what is merely a hypothesis.)

In computer science, this topic goes by the name of
"combinatorial search" since in complicated cases there are
combinatorially many possibilities.  There are sophisticated
ways of organizing the search to make it more efficient...
tree diagrams and linked lists and all that;  I won't go into
details right now.

Many of the exercises suggested in my previous note can be
attacked using combinatorial search techniques.

A multi-step reasoning process, with one or two hypotheses
to be considered at each step, has a total number of
possibilities that grows combinatorially as a function
of the number of steps.  For this reason multi-step reasoning
can be spectacularly hard when the number of steps is even
moderately large.

==========

Pedagogical point:  I've seen lots of bogus "explanations"
in classrooms and in textbooks that conceal the combinatorial-
search aspect of problem solving.  That is, for each problem
they use 20/20 hindsight to steer a course through the
branching tree of possibilities, unerringly taking the
luckiest branch at every node.  Such an "explanation" does
not permit any sort of learning beyond rote learning.  It
sheds no light on how one actually solves problems (i.e.
problems where the solution is not already known).

That is a trap unwary teachers can fall into.  The teacher
knows the solution, and even the method of solution.  But the
students know neither.  The students need to search.  Search
skills must be taught!  People are not born with these skills!