This is a kind of magic. Representing several things as though it was a modified version of one thing, frees up the mind to do so much. It allows us to store large amounts of information *outside* of our bodies.

The simple human mind can only really conceive of about 3-6 things at once. If a person without counting is asked which group is larger and is shown two groups, one with 33 apples, and another with 31, is extremely difficult to tell. But with numbers a person can count. They can set aside the reality of the apples and use several kinds of abstract representation to tell how many there are. They can arrange the apples into groups of three – which can be easily identified – and use their fingers outstretched to represent their place in counting each group. This is storing information outside of oneself.

This is a* *profound transformation. It can be shown that numbers are a kind of representative logic. Adding the ability to store information outside the human body transforms humans from just an animal into t*uring complete*. Turing machines can solve any problem that is computable given enough time.

To the extent that we are right that one thing is like another thing, abstraction and counting save us a lot of brainpower. It’s a kind of compression. When we use numbers to represent things, we discover that there are certain logical properties that can rearrange these groups (numbers) in ways that are more understandable without affecting their accuracy or changing the number at all. For instance, three groups of 10 apples is the same as 30 apples. Multiplying doesn’t do anything to the groups but it does make a simpler token to represent it in our memory (30 as opposed to 3 sets of 10).

These conceptual simplifications let us represent other relationships we discover. Like the fact that planets (from the Greek for *wanderer*) seem to look like stars that moves throughout the sky. By putting numbers on how much they move we can compare this that are hard to directly observe – just like the large groups of apples. And we can store that information outside of our minds so we can compare it over long periods of time.

Comparing these numbers lets us discover patterns that describe how the planets behave like Newton’s equations of motion and gravitation. What’s more, they let us predict how they will behave.

The most interesting and readable author on the subject is Bertrand Russell in his book Introduction to Mathematical Philosophy.