When a lunch guest at your home asks for directions to the
bathroom, you likely give the shortest, simplest route ("Go
straight down that hallway and it's the second door on the left.
You can't miss it."). Unless you can't stand the person
or are playing a kid's game you don't give an unnecessarily long,
circuitous route ("Go outside, circle the house twice, come
back in through the cellar window …").
When giving driving directions to someone new in town, it
is usually best to give the shortest, easiest route to the destination.
If you give a bumblebee route the visitor will spend unneeded
travel time and is more likely to get lost. There will be occasions
where a more roundabout route is desirable. Perhaps the shortest
route goes through a bad neighborhood or the visitor asks for
the scenic route or has to pick up groceries along the way. Without
these extra requirements, however, you will give the simplest,
shortest route to the destination.
In science, mathematics and many other academic areas, simplicity
is similarly desired. It is a standard preference that a scientific
theory or model be as simple as is possible without sacrificing
accuracy or necessary meaning. Another way to put it is that
it is considered undesirable for a theory to have unnecessary
parts, difficultly or complexity. If something is tacked onto
a model because it is needed to make the theory work or work
better, that is fine. If something unneeded is tacked on, it
should be removed. If two models are otherwise equivalent and
produce the same results, it is considered best to pick the model
that is the simplest.
In modern science and mathematics the common reasons for simplicity
are similar to for when giving simple road and bathroom directions.
A scientific model that is needlessly complicated is harder to
read, understand and communicate. In the classroom and the laboratory,
it may confuse students and the teacher alike. The simpler model
is easier to test. It's easier to identify which parts work and
which parts don't. With a plethora of needless parts, it's harder
and sometimes impossible to know what's working and what isn't.
Even in our normal lives, we simplify things in a scientific
way when trying to figure out a mystery. If you hear a strange
nighttime noise outside, you turn off the television volume and
have the people in the room with you stop talking. The inside
noises can only serve to mask or muddle the auditory information
you're trying to identify. You want to reduce the auditory information
to one source: the noise outside. The reason you don't want the
other auditory sources is the same reason a laboratory scientist
doesn't want extra, useless parts on his test model.
Ockham's razor
This principle of simplicity in scientific models and theories
is commonly called Ockham's razor, or Occham's razor. It is popularly
attributed to 1400s English friar and philosopher William of
Ockham, also known as William of Occham. The razor alludes to
the shaving away of unneeded detail. A common paraphrase of
Ockham's principle, originally written in Latin, is "All
things being equal, the simplest solution tends to be the best
one."
Misunderstanding and misuse of Ockham's razor
The famous simplicity principle has been deftly used by famous
scientists from Albert Einstein to Max Plank. It has also been
misapplied, misunderstood, misinterpreted and misused by the
uninformed. The following looks at common misuses and misunderstandings
of the principle.
One common error in applying Ockham's razor is to use it as
the absolute arbiter of good and bad, right and wrong. Ockham
wrote "All other things being equal, the simplest solution
tends to be the best one." He didn't say "the simplest
is always right" or "the more complicated automatically
is wrong." His "tends to be" implies that sometimes
the simplest isn't the best solution. Look at the below equations
for an example where simpler is not better:
a) 1 + 1 = 5
b) 1 + 0 + 0 + 0 + 1 = 2 + 0
Equation a is simpler and wrong. Equation b
is long winded but correct. The relative quality of the two equations
isn't judged by simplicity alone. In math, simplicity may be
desired, but accuracy trumps simplicity. Most mathematicians
would say that ' 1 + 1 = 2' trumps both a and b,
as it is both simple and correct. In other words, they'd take
equation b and simplify it.
In the first paragraph's directions to the bathroom, the circuitous
route wasn't directionally incorrect. They would have lead the
visitor to the same destination as the short directions: the
bathroom. The simpler directions, however, were deemed better
in the situation, including the social aspect of treating a guest
politely.
Another common misuse of Ockham's razor is to apply it to
two disparate models. The principle is not for arbitrating apples
versus oranges. Remember, the principle starts with, "All
(other) things being equal …." If model d and
e are otherwise the same, use the simpler one. If model
d and e are different animals, Ockham doesn't apply.
As this all shows, Ockham's razor is a useful tool but not
the sole tool in making and judging models and equations. A scientist
who uses Ockham's razor in the making a nuclear physics model,
will also test the model for accuracy. A simple model that gives
bad results sends the scientist back to the drawing board.
It is important to realize that scientists deal in often complex,
vast and obscure areas, where the answers are not known. It's
not like equation a and b, where anyone with a third grade education
can pick out and the correct equation in three seconds. The scientist
makes rough models to test theories and study complex ideas.
The razor is often used to make the testing easier and more reliable.
The rule is a practical rule of thumb for making a good model,
not an arbiter of truth and accuracy.
Ockham's razor as value and aesthetic judgment
Whether good or bad, right or wrong, humans tend to use simplicity
as a tool to judge aesthetics, morals, truth, good and bad. People
often say things like good is simple, messy is incorrect,
simple truths, the beauty is in the simplicity,
disordered mind. In some form and degree these sentiments
show up in the application of Ockham's razor, often producing
dubious results when the sentiments are used too seriously. For
some, Ockham's guide is not only used to identify correctness
or convenience, but sometimes aesthetics, beauty and morals.
This happened centuries ago, when simplicity was considered tied
to religion, nature and the supernatural.
Below are two interesting quotes from Aristotle. Notice how
the philosopher equates simplicity with truth, nature and perfection.
Aristotle: "Nature operates in the shortest way possible."
This of course is untrue. Nature can be complicated.
It's a moral or aesthetic judgment that is factually wrong.
Aristotle: "The more perfect nature is the fewer means
it requires for its operation."
This is a statement of personal aesthetic beliefs,
and I would say dubious. There is no proof that simpler is better
or worse than complex, and in fact complexity is required for
somethings to work. A car with one wheels falls over. Duplication
of wheels is required.
If a scientist likes beauty and simplicity in his models for
beauty's and simplicity's sake, that is reasonable personal taste
so long as this principle doesn't override accuracy … By
the same token a mathematician wanting his equations to have
neat margins and handwriting is reasonable, but this neatness
doesn't make the equation right or wrong. The neatness may be
important for his students' and colleagues' easy reading. This
is important, but, again, doesn't make the equation more or less
right or wrong.
When judging someone else's model, first understand what
is the model's purpose
A common error in judging models is not knowing what the model
is trying to represent and/or do. If you don't know what the
model is trying to do, how can you judge if it is a good or bad
model? Some models are trying to represent the essential meaning
of the universe, while others are merely trying to give directions
to the nearest 7-11 on the back of a napkin. No doubt the napkin
sketch is a terrible model of the universe, but you'd look like
a fool pointing out this obvious fact to the barroom crowd. You
might as well also point out to them that the sketch looks nothing
like the Taj Mahal.
In short, much dubious criticism of models is caused by the
criticizer not understand the purpose of the model. His criticism
is a product of his dubious assumptions and ignorance.
Once you understand the purpose, you can criticize the purpose
itself. "Sure, that'll get you to the 7-11, but my recommendation
is you go the Piggly Wiggly instead. It's about the same distance
and the price are cheaper."
The problem of defining what is simple
The problem in simplifying models and theories and equations,
is that what is simple differs from person to simple. Simplicity
is subjective. Same goes for aesthetics when aesthetics is used
to make models.
The following is a brief look at
simplicity
Same subject models can look different from scientist to scientist
due to the scientists' personal views of simplicity, aesthetics
and order. There are often different but equally legitimate ways
to order information. Due to personal views, one scientist may
group the information by colors, another by shape, another by
size.
Organize the following into two groups of related objects
Scientists, and non-scientists, often find it convenient and
practical to group information. I asked different people, including
a university applied science professor, to group the above objects
into two groups of like objects. One person grouped by color
(black objects and white objects), another by size, another by
letters (he saw the objects as E's and C's. Interesting, as I
drew the Cs as moons!), another by direction left or right (problematic
as one doesn't know if a moon is faced left or right). Their
reasons for pairing were equally legitimate, but produced different
pairings. This should show you how one scientist's model can
look different than another's, not due to scientific theory or
knowledge but different views of aesthetics, simplicity and association.
When making a simple cosmic model, the human must realize
that humans' concept of simplicity may not match the cosmos'
concept of simplicity. Any scientific model is a representation
of the human's way of seeing.
Some philosophers and scientists have rebelled against Ockham's
razor, in particular when simplicity is used to define nature.
They said that if nature is complex, a simple model can't be
a true representation of nature. This view has merit when the
simple models are used to depict the truth of nature. However,
as we've already seen, some Ockham models are practical devises
for specific purposes. They aren't pretending to represents the
whole of nature.
Positivism
Positivism is a practical philosophy related to Ockham's razor
that says "sense perceptions are the only admissible basis
of human knowledge and precise thought." It dismisses the
subjective and that which can't be empirically measured and tested.
It has been applied in science, and is much a definition of the
scientific method.
Positivist models will be derived from verifiable and experienced
facts, which of course is a good thing. However, they will also
be products of our limited senses, excluding real but immeasurable
things. Artists and the religious often say science can't explain
everything, which is true.
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