🚀 go-pugleaf

In article <jaeastman-011192163128@jmac.msd.anl.gov>
jaeastman@anl.gov (Jeff Eastman) writes:

> [a] The North Pole ...
>
> [b] ... a "circle" of points near the South Pole such that when you walk
>     south 1 km, a circle drawn through your position and parallel to the
>     equator has a circumference equal to 1km (so when you walk 1 km west
>     you go around the circumference once...
> 
> [c] You can continue drawing these circles closer to the South Pole such
>     that your westward walk takes you around the circle 1, 2, 3, ... times.

Especially for [c], I tip my hat to you.  The usual sequence of solution,
which often takes mathematically conversant folks many hours or even days
to arrive at, is:

  [a] almost instantaneously (except for one peculiar AI/mathematical type
      who got [b] and [c] almost instantly, and had to be *told* about the
      North pole.  Don't ask me...)

  [b] after a few hours, or until they abandon the problem in an abject
      state of frustration.

  [c] essentially never *unless* they are told explicitly that further
      solutions beyond [b] do in fact exist.  Even then it may again take
      anywhere from hours to days to quitting in frustration.

There are a lot of intriguing implication about problems solving strategies
in this little problem.  In particular, most people seem to have all sorts
of little "heuristics" attached to words such as "east" and "west", ideas
that normally make navigation problems easier (rather than harder) to
solve.

But for a problem like this, those heuristics definitely get in the way --
for the problem is actually quite straightforward if you can confine your
thoughts to simply tracing a line from the north pole to the south pole
and examining for any type of topological transitions -- which begin
happening rapidly as you approach the south pole!

But our intuitions keep distracting us.  Some folks become absolutely
absolutely fixated on the north pole, perhaps assuming that because one
solution was there the others "must" be nearby.

The [c] set is hidden not by logical complexity -- it is an almost trivial
extension of the [b] solutions -- but by our resolute refusal to think of
"going east or west on km" as meaning going *around the earth*.  After all,
we all "know" that the earth is "very big" and thus cannot be circum
navigated with such a "short" travel distance.


My point in all this?

No one thinks in a vacuum, particularly in a highly complex discipline such
as physics.  Whether you know it or not, each one of you reading this has
his own set of distinctive "heuristics" that you use to navigate possible
solutions that have been presented to you.  Many of the heuristics you use
are uniquely yours, but others are an aspect of the training you have had
in school -- e.g., many mathematical models are loaded with implicit ideas
about what is important and what is not.  Professors often promulgate such
implicit problem solving without even realizing that they are doing so --
which is a bit risky, since an undeclared premise is for many, many folks
an *invisible* premise that is difficult to even recognize, let alone
challenge.


Examples?  Well, there's the classic of special relativity.  In his own
writings, Dr. Einstein stated that the real insight that made special
relativity possible happened while he was simply describing the problem
to a friend.  The insight?  Very simple: there is no *single* flow of
time.  After that he wrapped up the whole thing in about two weeks, since
compared to that simple insight the task of mathematically quantifying the
issue was (by his standards) almost trivial.

Nowadays we are so familiar and comfortable with the idea of relative time
flow that it doesn't seem like that big of an insight.  That is very, very
wrong.  In Einstein's time the assumption of universal time flow was so
terribly, trivially obvious that it needed no examination, no explanation,
and indeed no recognition at all.  It simply was.  The huge transition made
by Einstein, the revolution that made him a truly great thinker, was not
his equations per se, but that *all* of humanity was laboring under a single
false assumption that had to be recognized and swept away before the results
of new sets of experiments could be resolved in a consistent fashion.

So...  Do you wish to be an exceptional physicist, or a nice, mediocre one?

If the former, then you might at least consider the possibility that you
have a number of hidden assumptions that are keeping you from making really
interesting observations or insights about whatever your may be working on.

				Cheers,
				Terry Bollinger


P.S. -- Shucks, did you have to solve is so *quickly*?!  Sigh.  It's much
        easier to keep folks guessing for a few hours or days when you don't
        broadcast the problem around the world!

P.S.S. -- Did you have any particular training that helped solve it?  I had
        one fellow who was greatly assisted by his familiarity with non-
        Euclidian geometry.  (On the other hand, he was the same guy who
        was the *only* person who never recognized the North Pole solution!)