MATH.16:5 - Archetypal Grounding

MATH.16:5.1 - Two classifications, then one shared integer

Two separately chosen integers have been classified. The first report gives its remainder modulo 2; the second gives its remainder modulo 4. The task is to combine the reports so that both can be recovered, and to add combined reports by adding their corresponding remainders. The two reports should determine the complete combined result.

Let A={0,1}, with addition reduced modulo 2, and B={0,1,2,3}, with addition reduced modulo 4. Adding the two reported numbers into one number loses recovery: reports (0,1) and (1,0) both give 1. Keeping a pair supplies both projections and requires no additional choice. All eight pairs are possible because the original integers may be chosen separately.

Now both reports must describe the same integer. The pair (0,1) fails: an integer with remainder 1 modulo 4 is odd. The modulo-4 report determines parity through t(b)=b modulo 2. Set s to the identity on A and construct the compatible pairs:

Q={(0,0),(1,1),(0,2),(1,3)}.

Every member comes from an integer. Addition stays within Q: (1,1)+(1,3)=(0,0). A report determines the original integer only modulo 4.

Additional structure – groups. With the stated modular additions, A and B are groups. The identity on A and the parity map t preserve addition, so Q with componentwise addition is also their pullback in groups.

The projection (a,b) -> b has inverse b -> (t(b),b). If the next calculation is easier with one modulo-4 value, MATH.7 transports it through these maps. If the next question is which integers may be identified while retaining both reports, MATH.2 instead constructs their quotient modulo 4.

Returning to functions on the underlying sets, a proposed update exposes another choice. Incrementing the modulo-4 component alone sends (0,0) to (0,1), outside Q. Incrementing both components gives (1,1) and preserves agreement for every pair in Q. The intended change to the underlying integer determines which component updates belong together.

Additional structure – groups. The joint increment is not a homomorphism: it takes the identity (0,0) to (1,1). It is suitable for updating the represented integer, but fails a requirement to preserve the group operation.

MATH.16:5.2 - Process both results or either result

One calculation returns an integer count; another returns a text label. A report containing both results needs a product. Its projections recover the count and the label, and the two values determine the report.

A different interface accepts either an integer count or a text label. It must display a count numerically and leave a text label as supplied. A tagged union, the coproduct of these sets, allows both handlers to determine one display function. An input tagged as a count follows the numeric handler; a text-tagged input follows the text handler.

The word “combine” did not decide which object to build. The question about formation and use did: recover two components from one report, or process either input through its own rule. These constructions describe values and functions. Whether executing the calculations reads or changes shared state is a further question about their execution.

MATH.16:5.3 - Two views of one quantity

A model has candidate states A and B for two component descriptions. Each description specifies the value of a shared quantity in C. For example, the two ends of an ideal connection may be required to have equal potential.

The product contains arbitrary state pairs. Requiring agreement selects the pullback of the two quantity maps. Its projections retain both component states, so a later calculation can still use their other quantities.

The physical account must justify the ideal connection and the meaning of that potential. If the connection has a relevant drop, the equality premise changes. A relation involving the drop and other quantities must be modeled before constructing its compatible states. The mathematical construction supplies the combination once that relation has been formulated; it does not choose the physical interaction law.

MATH.16:5.4 - Construct an object that can itself be applied

A function can be prepared by supplying a setting, then used with different inputs. We want a mathematical object representing the prepared function. Fix the input set A and output set B. A possible set of settings X supplies behavior eX:X x A -> B: it returns an output for a setting x and input a. Different X and eX describe different ways to prepare such behavior.

Seek a set E of prepared functions and an evaluation rule ev:E x A -> B for applying them. Preparing from a setting should give a map h:X -> E. To retain the original behavior, require:

ev(h(x),a)=eX(x,a) for every x and a.

Here prepared functions are equal when they give the same answer for every input. Thus the behavior specified by eX should determine h completely. Require a unique such h for every X and eX.

Construct E=B^A, the set of all functions from A to B. Define ev(k,a)=k(a), and let h(x) be the function sending a to eX(x,a). The required equation follows by evaluation. Any other proposed value for h(x) must give that same answer at every a, so it is the same function. This proves uniqueness.

For example, take A, B and X to be the integers and eX(n,a)=n+a. Then h(3) is the function that adds 3; ev(h(3),6)=9. The construction allows us to pass, apply and compare that function as an object. The conversion from a two-input function to a function returning a function is called currying. Distinguishing procedures with the same answers but different costs requires a further computational description of those procedures.