Thinking is hard. One of the books I’ve recently read,
Daniel Kahneman’s Thinking Fast and Slow
(2011), has convinced me that it’s harder than I thought. Aristotle and other
ancient writers tried to help by making rules for logical argument, but the
thinking we do most of the time doesn’t fit into the deductive logic scheme of
those books. In 1620 Francis Bacon came up with a helpful list of how our
thinking goes wrong. His book is called Novum
Organum (Aristotle’s book on logic had been called Organon—the instrument; Bacon thought he’d come up with a new
instrument). It detailed a lot of mental habits that get in the way of clear
thinking. We imagine more order in the world than is actually there, for
example. We cling to ideas and see only the evidence that confirms them. Our
observations tend to be skewed toward what we want to be true. The sort of
problems Bacon warned against are the reasons drug testing has to be double
blind; that is, neither patient nor doctor knows whether the patient’s pills
are actually the medicine being tested or blanks.
Kahneman
discovered that narratives often override our sense of the way chance and
probability work. The way a story is told can trump our knowledge of numbers. A
simple example: we are more likely to go under the knife if the surgeon
emphasizes the 90% of his patients who survive rather than the 10% mortality
rate. We are likely to ignore the phenomenon known as regression to the mean in any series of events or numbers. Kahneman
quotes an Israeli pilot instructor who says “when I praise someone for an
exceptionally good performance, he almost always does worse the next time, but
when I chew out someone for a bad job, he always does better the next time.” He
thinks it works better to criticize than to reward, but the results in each
case are merely reversion toward the mean from extremes of good or bad performance
of the student pilots.
We tend
to classify according to a narrative story rather than our knowledge of the real probability that object or person A
belongs in category B. An outstanding case is the Linda Problem, in which
respondents go for plausibility when asked about probability (the tendency to
answer a less-difficult question than the one asked is one of the drawbacks of
thinking fast). Linda is single, 32, outspoken and bright. She majored in
philosophy and as a student was deeply concerned with issues of discrimination
and social justice. Subjects are given this information and told to rank the
following scenarios in order of probability:
A Linda is a
teacher in elementary school
B Linda is a bank
teller
C Linda is an
insurance salesman
D Linda is a bank
teller who is active in the feminist movement,
Most respondents rank D as more likely than B, defying
logic, since there are more bank tellers than there are bank tellers with a
limiting characteristic.
We also
tend to misconceive of the way chance works in particular ways. Given a
six-sided die with 4 green and 2 red faces, rank in order of probability these
sequences coming up:
A RGRRR
B GRGRRR
C GRRRRR
As in the Linda Problem, most respondents will say B is
more likely than A, even though there are two possibilities for a previous
throw in A and thus it is more likely.
Kahneman
lists a number of other pitfalls to our thinking. He believes armed with his
information, we can be wary of quick decision-making, slow down our thinking, and
eliminate some of these “cognitive biases” as he calls them. But after reading
through four hundred pages of his examples, I’m fearful that we can’t easily
avoid such problems thinking fast or
slow.
A number
of Kahneman’s thinking errors come about through our failure to think through
the way numbers work. A general ignorance of simple mathematical rules and
probabilities is what the mathematician John Allen Paulos has complained about
for many years in books such as Innumeracy
(1988) and A Mathematician Reads the
Newspaper (1995). Paulos believes that such ignorance is dangerous for the
welfare of individuals and for public policy making. But he wasn’t the first to
try to show people how looking more carefully at numbers can keep us from
exaggerated or misleading advertising claims, political demagoguery, or overt
attempts to defraud. Such attention is also the message of Darrell Huff’s
little 1954 classic, How to Lie with
Statistics. Huff is especially good on the way graphs and pictorial
representations of statistics can distort the real meaning of the numbers, but
he gives general advice about how to confront and question a statistic by
asking who it’s coming from, how that person or organization knows, what might
be missing, whether it makes sense, and so on. Huff’s tone is light and his
book is illustrated throughout.
Some
books can help with the rigors of thinking, and I don’t mean yesterday’s
throwaway self-help scribbles but books that have stood the test of time and
usefulness. One of my favorites is Henning Nelms’s Thinking with a Pencil (1957). Nelms argues that drawing can help
one think through a problem as well as communicate information easily to
others. We can use a drawing to collect information (as we’re listening to directions,
for example), and Nelms illustrates that even a mathematical formula can be
depicted through a simple geometric drawing. Though it is not one of his 700
illustrations in the book, the often-used drawing illustrating Pythagoras’s
square of the hypotenuse theorem comes to mind here. In a chapter called
“Visualizing Numerical Data,” Nelms points out that any quantity can be
expressed as a number and any number can be graphically represented by a length
or an angle or a set of images. He provides much technical detail about
handling graphs, in the process giving us the same warnings about the mishandling of graphs as Huff does. Nelm
says even if you don’t think you can draw, you are capable of using a pencil to
help you think through many problems with doodles or simple shapes.
Probably
the most thorough book on thinking through a problem is George Pólya’s How to Solve It (1945), which I think
has been continuously in print since he wrote it. Pólya’s book presents a
heuristic, a method for solving problems that is first laid out in schematic
form at the end of his table of contents and then shown in operation with
several examples. The method is illustrated primarily though not exclusively
with examples from mathematics and geometry; Pólya clearly believes it has a
wider application. He invites us to divide the task of solution into understanding the problem, devising a plan,
carrying out the plan, and checking the result. Most of the attention goes
to the first and second of these steps. In understanding
the problem, we need to ask what is the unknown, what are the data, and
what is the condition. Can we draw a figure to illustrate the problem? (Nelms
would say “yes!”) If the problem is abstract, try looking at a concrete
example. In devising a plan, Pólya suggests
we ask ourselves whether we have seen this problem or an analogous problem
before, or whether we have seen a problem with this unknown before. He suggests
we try to restate the problem. Can we perhaps solve a related problem, a more
general problem, or part of the
problem? Have we used all the data? Can we imagine a solution and work
backward? In carrying out the plan,
he advises us to check each step and attempt to prove its correctness. In checking the whole solution, he asks
whether we could derive the result in a different way, and whether our solution
or its method could be used to solve another problem.
In his
first chapter Pólya imagines a teacher prompting a student through these steps
in the solution of several problems. Another chapter is arranged as a
dictionary explaining and exemplifying key terms of the heuristic such as analogy and condition. This glossary is somewhat idiosyncratic and intended to
be read through rather than merely consulted; Pólya introduces people into the
glossary such as Descartes and Leibniz who were important in the history of
heuristics, and some of the terms are phrases or questions such as Did you use all the data? Though some of
the examples may be beyond the reach of non-technical readers, Pólya’s style is
clear and precise and his book very readable. Where Kahneman’s book may stagger
your confidence in human thought processes, Pólya will reassure you that a
patient, organized approach will get you to clear thinking.
