In the final Analysis, both multiplication-as-substitution and multiplication-as-structure use Substitution to modify structural form.

Figure 14-9 shows the rules of each system that are involved with multiplication.

In the depth-value approach of multiplication-as-substitution, specific forms are identified and substituted after pattern-matching within a target form (the INTO form).

In multiplication-as-structure, a limited number of patterns (the axioms) are identified and substituted after pattern-matching within a target form (the INTO form). If that sounds like the same Process, it’s because it is.

Figure 14-9: Models of multiplication

Figure 14-9

GROUP depth-value arithmetic •..N..• = (•) ☞ 1..N..1 = N x 1 structural algebra o..N..o = N ☞ 1..N..1 = N

MERGE depth-value arithmetic (A)(B) = (A B) ☞ (N x A) + (N x B) = N x (A + B) structural algebra A..N..A = ([A][o..N..o]) ([A][ N ]) ☞ A..N..A = N x A

SUBSTITUTE depth-value arithmetic 〘A • E〙 = 〘E • A〙 ☞ A x E = E x A structural algebra 〘〔([A][M])([A][N])〕 ([A][M N]) E〙 ☞ (A x M) + (A x N) = A x (M + N)

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The difference is that depth-value is essentially an arithmetic technique, it does well with specific numbers.

Structural pattern-matching is essentially an algebraic technique, it does well with abstract variables.

Both can be used to implement either arithmetic or algebra, and with different boundary shapes, both can be mixed within the same form.

Depth-value narrows the use of substitution by making it necessarily commutative. A non-commuting form cannot be substituted.

James Algebra allows the use of substitution in its full generality as a match-and-substitute engine for arbitrary pattern axioms.

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is a kind of substitution system in which multiple states are permitted at any stage. wolfram

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