MATH.23:4 - Solution
Local mantra: recover the construction and its use; vary a consequential ingredient; express the suggested law; seek its reason and a separating case; revise the assertion or construction; choose and begin the next useful problem.
MATH.23:4.1 - Recover what makes the starting construction work
Write the inputs and the operations that produce the result. Identify a relation the result satisfies and where that relation is used. Separate what has been proved from what has only occurred in inspected cases.
Look inside the operation. Repeated updates may expose a recurrence. Combining two constructions may expose closure or an order dependence. A successful quotient may reveal which distinctions can be discarded. MATH.17 helps make the operations themselves available for this investigation.
When a program supplies examples, recover the class it actually generates. For instance, a generator that produces only trees cannot investigate a claim about arbitrary graphs unless the missing graph classes are handled elsewhere. The representation and evaluator are part of the available mathematical experiment.
MATH.23:4.2 - Construct a variation with a reason
Choose an ingredient whose change could affect the wanted relation. Useful mathematical moves include:
| Starting opportunity | Construction of the next problem |
|---|---|
| Several parameter values work similarly. | Replace the parameter by a variable, derive the update rule and ask which identity holds over its domain. |
| Two outputs can be combined. | Define their combination and ask whether it remains in the same class and preserves the required relation. |
| A proof uses a special hypothesis. | Remove or weaken it, locate the affected inference and ask for a replacement condition or counterexample. |
| A desired property fails. | Isolate the obstruction and ask whether it completely characterizes failure or only gives one sufficient obstruction. |
| The result survives a change of representation. | Construct the interpretation and ask which assertions, operations and equalities it preserves. |
| An approximation supports a limiting claim. | State the order of its quantifiers and ask which operations or properties survive the limit. |
These are possible operations, not required stages. Choose the one selected by the construction. MATH.8/.11/.13 supplies symmetry or invariant reasoning; MATH.18 supplies interpretations; MATH.22 develops changed assumptions. Use their fuller methods when that mathematical contribution is needed.
Keep the original use visible. Requiring equal-sized groups repairs a mean-of-means formula, but leaves a request about arbitrary groups unanswered. The latter needs another construction, as in :5.2.
MATH.23:4.3 - State the proposed law and its answer form
Give the objects, domains, allowed operations, assumptions and proposed conclusion. In particular, make the quantifiers distinguishable. These statements ask different questions:
- For each input, some construction has the property.
- One construction works for every input.
- Every admitted construction has the property.
A finite pattern of results can suggest a formula. Derive it from the generating operation, a recurrence, an invariant or another mathematical relation when possible. If that derivation is missing, retain the formula as a conjecture and identify the step that would establish it.
Say what would resolve the present question. An explicit construction, a characterization of all admissible cases, a bound, or a counterexample can open different next uses. A conjectured optimal construction needs both an attainable value and an argument excluding better ones when optimality is the required conclusion.
Check whether the proposed law is already a known result under another description. MATH.18 can compare representations. An available theorem can close this question and expose a more useful next one.
MATH.23:4.4 - Alternate a proof attempt with revealing cases
Use MATH.19 to find intermediate claims connecting the construction to the proposed conclusion. Identify which mathematical property each step consumes. This can expose a condition that the conjecture has not yet stated.
At an uncertain step, construct a case that challenges that condition. Prefer one that distinguishes plausible accounts. To test whether a summary supports composition, find inputs with the same summary and place them in the same surrounding construction. If their required outputs differ, the summary has lost information that composition needs.
Inspect the status of a failure. A counterexample satisfying the conjecture’s premises refutes the conjecture. A failure of an auxiliary lemma may leave a different proof possible. A failed program may instead concern its representation or implementation. Recover the mathematical obstruction before revising the claim.
Computational search can propose constructions and counterexamples. Check a returned candidate against the mathematical conditions that its use requires. For a finite explicit object, this may be a small complete calculation. Sampled tests of an infinite family leave its universal claim to be justified.
MATH.23:4.5 - Revise the claim without hiding the lost use
Use the failure to choose the next mathematical change. Three common returns are:
- Repair a hypothesis. State the condition that makes the failed lemma work, prove its sufficiency and check whether the intended cases satisfy it.
- Repair the construction. Retain the original problem and change the objects, representation or operation so that the required property can be obtained.
- Change the conclusion. Obtain a conditional answer, bound, obstruction or weaker property that still serves a useful question.
A sufficient condition can be worth developing before necessity is known. Label that difference: a proof that uniform convergence preserves continuity does not show that every continuous limit was obtained uniformly.
A counterexample can itself become a constructive ingredient. Ask what class it belongs to, which transformations preserve its obstruction, and which related conjectures it can test. The new question can concern the method of generating cases as well as the cases themselves.
Retain the result that survived. Changing the theorem should not erase a valid construction or an earlier unresolved requirement.
MATH.23:4.6 - Choose a next problem and begin its mathematical work
Compare the few continuations whose answers would change further work. A useful result can supply a missing lemma, a reusable combining operation, an explanation of failure, or a method that makes another class of questions approachable. Immediate engineering application is one possible use; developing a new mathematical operation can also enable later inquiry whose destination is not yet known.
Choose a first attempt that the available contributors can perform. For a broad characterization, this might be one direction of the theorem. For an extremal problem, it might be one explicit improved construction or a bound. For a limit question, it might be the estimate needed to justify one passage to the limit.
Return the conjecture, the mathematical basis that suggested it, the most revealing success or obstruction, and the next operation. An expression and a short argument can carry this result.
Use B.5.QD and C.40.CD when several questions and ways must develop together. E.10.INT helps distinguish the interest of further possible work from surprise, fame or a search score. C.36.RP supplies retention of the reconstructible method when continuing capability is at stake. Obtain further observations or checking only when their possible outcomes can change the next move enough to justify the effort.