Mathematics is often about finding patterns. For example, if you look at the squared numbers 1, 4, 9, 16, 25, 36, and 49, you will notice that the differences between two squares follow an odd number sequence: 3, 5, 7, 9, 11, 13 and so on. Collecting examples of numbers and how they appear in the world really isn’t any different than searching for patterns amongst the squared numbers. When collecting examples, try to find what the number itself might mean.
Not all examples are created equally, though. The strength or quality of an example depends on what kind it is. Here are some guidelines to get things started.
Changing Versus Permanent Examples
Most things in life change around us every day. Count the number of clouds in the sky, the number of coins in your pocket, or the number of cars outside your window and these numbers change constantly. You might count eight swans on a lake, but tomorrow there might only be two. What would be far more interesting is if there were always eight swans on a that lake. How weird would that be? Examples that are easily changeable generally do not help establish what the personality of a number really is.
Permanent examples are those that essentially never change, or those that have VERY few exceptions. For example, there are always 60 minutes in an hour. Or with very few exceptions, all humans are born with five fingers on each hand. And that number doesn’t change (except through accidents). We don’t for example, grow more fingers as we age. So, why five? What is it about five-ness that goes with fingers? Why are they different lengths? And why did the ancient Egyptians sometimes depict people with fingers all the same length? Because permanent examples basically never change, they can give us insight into various numbers.
Natural Versus Human-Made Examples
Nature is fact. Nature is free of personal opinions and extraneous emotions. Nature just is. How things turn up in nature is according to patterns that have emerged after thousands of years. For example, white pine trees have 5 needles in a cluster, whereas red pines have 2. Snowflakes are almost always formed based on the number six. Examples from nature are clear and true.
Human-made things, though, can be based in opinion, ego or invention. Just because an architect designs a eight-sided building does not mean that it truly reflects the personality of the number eight. For example, the design of the building could have been influenced by politics, zoning laws, taxes, and so on, so that the eight sides was merely a convenience, rather than an archetypal example of the number eight.