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.