MATH.1:5 - Archetypal Grounding

MATH.1:5.1 - A continuation that needs a permission

Two routes p and q start at S and reach location V; their costs are 1 and 4. A final step r costs 2. Under the first rule, r is available after either route, so p;r and q;r are paths and their costs are 3 and 6.

Now change the rule: only q grants the permission needed for r. Keeping a single endpoint V would still make p;r appear composable.

Construct two intermediate objects, V0=(V,0) and V1=(V,1). Set:

The path q;r exists and costs 6. The expression p;r fails the endpoint test. Selecting the cheaper prefix first would therefore lose the available completion. The useful result is the permitted path and its cost; the refined endpoint explains why it is permitted.

If a further generator a:V0→V1 grants permission at cost 1, a new path p;a;r becomes available at cost 4. The former failure has opened a construction question: what additional arrow would connect the available prefix to the required continuation?

MATH.1:5.2 - Natural numbers from repetition

Take one object X and one generating loop a:X→X. The paths are the empty word, a, a;a, and longer repetitions. Write a^n for the list containing n copies, with a^0=id_X.

Concatenating a^m and a^n gives a^(m+n). Thus lengths supply an arithmetic account of this structure: the identity corresponds to 0 and composition corresponds to addition. Every path is determined by its length in this one-generator case.

Adding a second loop b changes the situation. The paths a;b and b;a both have length 2 but are different lists. Counting generators now loses their order. It still answers a length question; a question about which generator acts first requires the path.

MATH.1:5.3 - Order matters under interpretation

On integers let f(x)=x+1 and g(x)=2*x. Both generators start and end in the integer type, so both orders are composable. Starting from 0, f;g returns 2, while g;f returns 1.

The associativity argument allows regrouping a longer list. It does not authorize exchanging f and g. The differing results make that boundary consequential.

MATH.1:5.4 - Keep the intermediate states of interacting updates

Two updates to a counter each read its current value, retain that reading and later write the reading plus one. From zero, finishing one update before the other gives two. If both read zero before either writes, the final value is one.

To represent the difference, a state retains the counter, each update’s saved reading and whether its read and write have occurred. A read copies the counter into that update’s saved value; its later write replaces the counter by that saved value plus one. Form paths in which each read precedes its own write. The paths read-A, write-A, read-B, write-B and read-A, read-B, write-A, write-B return two and one respectively.

Treating each update as one indivisible arrow would lose the second path. If the work can require one complete update to finish before the other, the restricted paths preserve both increments. FPF C.29 establishes how these transitions describe the implemented work; Method Engineering ME.7 helps change its composition. Choosing an implementation also depends on how it handles waiting, interruption and failure.