Is a bag of potato chips
one thing? Many? Both? Neither? Other? Depends on how you look
A basic part of mathematics, physics, chemistry, engineering,
economics and daily life is counting. Counting is popularly considered
to be an objective activity. In the field, however, it involves
subjectivity. Not over whether 1 + 1 = 2, but over what is 1.
Both scientists and non-scientists have personal and varying
views of what is 1 and what is 2, 3, 4 and 5.
Humans mentally, even subconsciously, individualize things,
isolate, group and count things- whether or not the things were
designed to be individualized, isolated, grouped and counted.
To humans a dog is one thing. A cat is one. A dog and a cat are
two. This numbering is not just intellectual, but often psychological,
aesthetic, moral, religious, political and philosophical. A human
being is popularly regarded as a single thing, a proverbial island
unto itself. Some will be morally offended if you count a human
differently. Two humans, even if physically connected by holding
hands, are not considered one human, but two.
A distant snow capped mountain of one billion stones is commonly
referred to as one thing, not one billion things. Yet three of
the stones removed and held in one's hand will be labeled as
three. This shift is a reflection of the counter's mind and eyesight
more than the counted. Mountains and stones existed fine before
humans were around to count and individualize them. How or whether
or why we count them makes no difference to mountains and what
they are. The counting is a human exercise.
When a long cloud briefly separates in the middle many call
it two things, two clouds. What is the legitimacy of this representation?
Could it just as well be called one? Is either number an arbitrary
choice, a definition of terms?
A lake and connected creek that share the same water and fish
are commonly considered two things. Is this the correct representation?
Could they instead be considered one? Is 1, 2 or any number a
true representation of the body of water, or merely a convenient
representation for humans?
Perpendicularly intersecting roads are often considered two
things, while a wooden cross is commonly considered one. What
is interesting about this example is that the roads are more
physically one than two boards nailed or glued together. If you
stand at the middle of the intersection, the two roads at that
point are physically the same. It is not one road or the other
road, it is both roads simultaneously. A piece of asphalt belongs
to both. The two cannot be separated or distinguished from each
other. At the intersection of the cross, on the other hand, the
two pieces of wood are easily distinguished and can be separated.
Physically at least, the roads could be considered more one thing
than the cross.
My sandbox of stones
Say I have in my front yard a sand box filled deep with an unchanging
amount of stones. Just as with a sandbox of sand, no matter how
I fiddle or play or scoop or make stone castles there is never
a gap with no stones.
In this ever unbroken sea of stones, I make two tall mounds
of stones on the surface. If I pull someone off the sidewalk,
point to the box of stones and say, "How many things do
you see?," she likely will say two. She may even point out
that the two things she sees are the mounds. If I had instead
made three mounds, it's likely she will say there are three things.
If there was one mound, it's fair to assume she would have said
one. If the surface was flat (no mounds), she may say there is
one thing. Even if her answers aren't as I just said, they likely
would change depending on the number of mounds.
Duly note that my question was 'How many things do you see?'
I didn't ask how many stones or how many shapes or how many mounds.
I let the woman define what was a thing and count as she see
There are two interesting aspects about her counting of things
in the box. First, it is not clear that the number of things
in the box ever changed. There was always a body containing an
identical amount of stones. The body was constant, other than
the changing surface shape. No one I know counts lakes by counting
the number of surface waves. To most people, a strong wind doesn't
create more lakes. People don't count triangles as objects differently
than squares, or two humped camels differently than one hump
camels ("Guess what, Mom. I saw two camels at the zoo today.
One one-humped camel and one two-humped camel.") There was
never any separation that created isolated islands of stones.
It was the changing surface shape that caused different number
answers. Her counting was personal. A different person looking
at the same stones might come up with different numbers, as he
defined things differently.
The second interesting thing was that, even if accepting her
definition of surface mounds as the things, the woman's math
was goofy. When there were one, two, three mounds, the woman
counted things by the number of mounds. But when there were no
mounds, she didn't say there was nothing. She likely would have
said there was one thing (the body of stones) or been confused
as to what she was supposed to count or perhaps said "There
are a lot of stones. I can't count them all." Her definition
of what is a thing and her method of counting was inconsistent.
In her math, removing 1 thing (mound) from 1 thing did not equal
0, and in fact may have equaled more than 1.
The act of counting the box of stones, or land or clouds or
a herd of wildebeest, has at least as much to do with the counter,
her biases and perceptions and idiosyncrasies and choices, as
with the subject being counted. That the woman's definitions
changed and different people off the sidewalk may count the box
of stones differently demonstrates this. Many people believe
that the individualizing and counting of things is intrinsic
to the things being individualized and counted, but there is
no evidence this is true. The human counting of a mountain may
have nothing to do with what it is. Is a cross 1 or 2? Why
does it have to be either?
Many will point out that counting is essential for humans,
an important tool for functioning. This is correct, but again
demonstrates that counting is about humans. Having a practical
use doesn't make a conceit any less of a conceit.
(c) david rudd cycleback, cycleback.com
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