For the first post in this series, see *here*.

(1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that have dual approaches. Let us begin with the standard approaches to these three topics, and then define duals to each of them. In some ways, the original and the dual may be used together.

(2) Number and algebra

The concept of counting and number is as universal as language, though the full definition of number did not occur until the 19th century. Algebra came to the West from India and Arabia in the Middle Ages but its formal definition did not occur until the 19th century. Abstract algebra also began in the 19th century.

The basic rules of algebra are as follows: addition and multiplication are commutative and associative; multiplication distributes over addition; addition and multiplication have identities and inverses with one exception: there is no multiplicative inverse for zero.

An idea of infinity comes from taking the limit of a number as its value approaches zero: ∞ ∼ 1/x as x → 0. Infinity can be partially incorporated via limits.

Dual: harmonic numbers

An additive dual can be defined by negating every number. A more interesting dual comes from taking the multiplicative dual of every number. This latter case can be called *harmonic numbers* and *harmonic algebra* because of its relation to the harmonic mean.

The harmonic isomorphism relates every number x to its harmonic dual by H(*x*) := 1/*y*. The dual of zero is ∞.

For harmonic algebra: see *here*.

Harmonic algebra is the multiplicative inverse of ordinary algebra. There is a sense in which harmonic algebra counts down rather than up. Zero in harmonic numbers is like infinity in ordinary numbers. Larger harmonic numbers correspond to smaller ordinary numbers. Smaller harmonic numbers correspond to larger ordinary numbers.