Chapter 1 has developed index notation and has given a certain amount of practice in the use of this notation. We now consider in greater detail the algebra of holors.

In the algebra of real numbers, the following relations apply:

1. Addition is commutative: a + b = b + a.

2. Addition is associative: a + (b + c) = (a + b) + c.

3. Multiplication is commutative: ab = ba.

4. Multiplication is associative: a(bc) = (ab)c.

5. Multiplication is distributive with respect to addition: a(b + c) = ab + ac.

In the eighteenth and nineteenth centuries, the naive belief was prevalent that mathematics was an absolute: something “present in the Divine Mind before Creation.” The modern idea, on the other hand, is much more sophisticated: Mathematics is merely a man-made game whose rules can be changed at will. Thus, for holor manipulation, the rules of real algebra may be accepted or not, depending entirely on whether the results are what we need for a particular application.

The structure of the world being what it is, there seems to be little or no advantage in modifying the commutative and associative rules of addition. The distributive rule also seems to be universally accepted. With multiplication, however, considerable flexibility in holor algebra is required to meet the needs of modern applications. This chapter is devoted to a study of holor algebra and how it differs from the algebra of real numbers.