From the Introduction to Objectivist Epistemology: "The element of similarity is crucially involved in the formation of every concept..." (p.13)
To get the element of similarity right, make some comparisons. Compare the names "Potter" and "Peter." They are different. Now compare them with "Petrograd." Measured against "Petrograd," "Potter" and "Peter" do not look so different. They look similar. By making a rough measurement of difference, you came up with similarity.
Now imagine looking at a landscape with the eyes of a painter. You see various shapes and colors. They are all different, but some differences are not very great, while others are enormous. You can group smooth shapes together and contrast them to jagged shapes. You can see many shades of green, many shades of blue, many shades of brown.
That is why similarity is so crucial: the ability to make all those measurements of difference makes it possible to classify things into groups. You don't even have to keep track of the measurements, as long as you keep in mind that the groups are not arbitrary. To be similar, things must have something in common that can be measured, such as color or shape.
Your senses see differences, but your mind can transform some of those differences into similarities. By measuring in different ways, you can see different similarities, depending on what you want to do. If you want to drive down the street without a crash, you measure motion, and look for movement headed your way. If you want to count red cars, you measure shades of color, and choose a range to call red. If you want to decide what new car to buy, you measure all cars against your personal standard, and form a mental group of cars that come close. By seeing common characteristics among things, you begin the process of forming mental integrations.
People get similarity wrong by ignoring essentials, and by ignoring differences. Instead of the essential of friendship, mutual esteem, they might think of "friends" as including people they like, people they can use, and people they can dominate. They spend their lives wondering who their real friends are. An arbitrary similarity is a pretend similarity, which gives no help in sorting things out.
Instead of measuring difference to get similarity, some wipe out difference to get similarity. Then all members of a group they like are equally good, and all members of a group they don't like are equally bad. They forget that to sense reality is to sense differences. Pretending that differences don't exist is trying to handle reality by throwing it away.
Ayn Rand's phrase "measurement omission," is a good way to remember the method. It means that you know about differences all the time, but you don't use that knowledge all the time. You see similarity, and use it to make classifications. That works unless the similarity is false.
A "similarity" produced by whim is of no use in handling reality. Neither is a "similarity" that pretends to be identity. Similarity is not produced by wiping out difference, but by quantifying difference.
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