When does 1 + 1 not equal 2?

Is a bag of potato chips one thing? Many? Both? Neither? Other? Depends on how you look at it.


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 nonconsciously, 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.

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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 fit.

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.

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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 an subjective and arbitrary rule any less subjective and arbitrary.