Feynman defined "science" as a bunch of rules for solving
problems and avoiding mistakes. The first step toward
avoiding mistakes is to keep in mind that mistakes are
possible. Be careful not to fool yourself, and keep in
mind that the one person you can most easily fool is
yourself.
In other words ... check your work! This is something that
everybody was supposed to learn in third grade, or maybe
even in kindergarten. This is the cornerstone of critical
thinking. It's not the whole story, but it's a good start.
At a slightly more detailed level:
-- When checking things, check *all* of the plausible
hypotheses. The existence of a semi-OK model does not
preclude the existence of better models.
-- On the other side of the same coin, check each of the
hypotheses against *all* of the available data.
-- Furthermore, if a theory makes multiple predictions,
check each of the predictions. It is common to find a
theory that makes five predictions, four of which are
completely wrong. If there is no good reason for this,
i.e. if the logic behind all five predictions is the
same, it tells you that the one correct prediction was
fortuitous. Nothing about the theory can be trusted.
========
In my experience, whenever I state a bunch of general
principles like that, everybody nods in agreement. They
say I am once again demonstrating my keen grasp of the
obvious.
However...
Q: Do you know the difference between "theory" and "practice"?
A: In theory there is no difference, but in practice there is.
Specifically, there seem to be a lot of situations where
people had an opportunity to check their work, but dropped
the ball. Here's a situation that appears to be an example:
B) The idea was elaborated in this paper:
Massimiliano Esposito, Ryoichi Kawai, Katja Lindenberg, and Christian Van den Broeck,
“Efficiency at Maximum Power of Low-Dissipation Carnot Engines”
Phys. Rev. Lett. 105, 150603 (2010) http://prl.aps.org/abstract/PRL/v105/i15/e150603
C) The idea is covered at length, in detail, in the textbook:
Chabay & Sherwood
_Matter and Interactions_
We are talking about a model heat engine that has some thermal
losses in /series/ with the main heat reservoirs. This was
discussed in this forum back in March 2013, in the "efficiency
versus Carnot efficiency" thread.
One can identify five predictions that the Curzon/Ahlborn model
makes, four of which are categorically wrong. The fifth prediction
is more-or-less in the right ballpark, but I consider this to
be completely fortuitous and devoid of physical significance.
You don't even need to know much about real-world engines to
figure out that the C/A model is seriously flawed. Virtually
everybody has operated a car engine. Consider idling in
neutral. The thermodynamic efficiency in this situation is
zero. Energy is going into the engine, while the shaft is
delivering zero useful mechanical work.
In contrast, the C/A model predicts that at the lowest
throttle settings, the efficiency is highest. Furthermore,
the model assumes that fuel is cheap and predicts that
nobody cares about efficiency. Gaaack!
==================================
What is going on here? Why are not the problems with this
model more widely known?
The Curzon/Ahlborn article came out in 1975, and has been
cited more than 1000 times. It may be that somewhere in
those 1000 citations somebody pointed out the flaws, but
still there remain lots and lots of people who think the
model is perfectly OK. As recently as 2010 Esposito et
al. were able to get a PRL published, with no hint that
there was anything wrong with the model.
I emphasize that it does not take any esoteric knowledge of
real-world engines to see what the problem is. Just the
Gedankenexperiment of idling an engine suffices.
Similarly AFAICT it does not take much dedication to checking
your work to find the bugs. The three documents cited above
check the model at high power settings and get a plausible
number for the efficiency ... but if they had checked the
model at any other power setting they would have exposed
the bugs. Perhaps this was a case of circular reasoning:
You convince yourself that the engine ought to run at a
high power setting, so you check the high-power case and
then go home. This shows that the model is consistent with
/some/ of the data; however, due diligence requires checking
the model against /all/ of the data, including the low-power
data, including idling.
Another possibility is excessive compartmentalization. If
you decide that thermodynamics is esoteric and cars are
prosaic, you might not connect your knowledge of one to
the other.
I tell students: In terms of critical thinking, the next
step beyond "check your work" is "check the other guy's
work". That includes checking what you're told in class,
and checking what you read in the textbook. There aren't
enough hours in the day to double-check everything you're
told, but you ought to be /somewhat/ skeptical. Pick some
random subset of what you are told, and give it a careful
check. Note that if you are told N things, checking one
of them carefully is better than checking all of them
uncarefully.
Many thousands of students have been exposed to the C/A
engine model. So, how many of them have raised objections?
Possibly none. There is a blog where this sort of issue
could be raised, and there is no record of any objections
or even questions. http://ipv6.google.com/search?q=engine+site%3Amatterandinteractions.org
This suggests that there is something seriously wrong with
our educational system.
-- I don't care whether students know that the efficiency
of a real-world power plant is XX percent of the Carnot
efficiency.
-- I care a great deal whether the students know how to
learn and to think.
All too often, I see students being told stuff that cannot
possibly be true. They have long-since learned to shrug
and learn it by rote. This is the opposite and the enemy
of critical thinking.
Constructive suggestion #1: Charity begins at home. That
is, practice what you preach. In addition to telling students
to check their work, it helps if you, the instructor, check
your own work, and check the stuff in the textbook. It's not
hard to find dozens and dozens of errors in a typical book
... and I'm not talking about misspellings and misplaced
commas; I'm talking about wrong physics.
Complacency is an occupational hazard for professors. After
20 years of being the smartest guy in the room, it's easy to
become less-than-fastidious about checking your own work.
Suggestion #2: Set up a reward system that encourages
students to report errors in the book and (gasp!) errors
in what you say in class.
When you say you want corrections, you have to actually mean
it. When folks find bugs in stuff I've said or written, it
used to hurt my feelings, but after the first few thousand
bugs I got used to it. I'd really much rather have the bug
reports than not have them.
Suggestion #3: On homework and on exams, the rules should be:
-- show your work
-- check your work
-- show the checks
-- design the work from the outset to be easily checkable
This includes taking off points if the checks are not
shown, even if the result is numerically correct. This
includes covering fewer topics on the exam, so that there
is time to check the items that are covered. I'm serious
about this. I really don't care very much whether the
students know XYZ physics fact. I care a great deal
whether they know how to learn and to think.
Suggestion #4: Discourage compartmentalization. Look for
connections between things ... real things, as opposed to
monkey-shooting. Choose exercises where a_priori knowledge
of the real world can be applied to physics, and where physics
can be used to build up a_posteriori intuition about the real
world.
Suggestion #5: Remember that this is not something you can
teach as a topic unto itself, once a year on National Critical
Reasoning Day. It is something that needs to be baked into
every lesson, all day every day. It needs to be baked in,
like the oatmeal in oatmeal cookies. It's not something
you can sprinkle on afterwards.
Sometimes people tell me that although critical thinking is
important, it's impossible to teach. I disagree. It's just
not that tricky.
============================================
Just to keep things in perspective: I'm picking on
_Matter and Interactions_ even though -- and indeed because --
it's a relatively enlightened book. Certain other widely-
used texts are so much worse that picking on them would be
unsporting.