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A mathematical pattern can be persuasive long before it has been proved: a relationship that holds for many examined cases naturally invites the expectation that further cases will behave in the same way. That expectation may guide investigation, but a statement about every member of an unlimited collection requires more than agreement among the particular members already inspected. The distinction concerns the scope of the claim: the evidence covers a specified selection, whereas the claim extends to cases that have not been individually checked.

A computer can enlarge that selection dramatically and can reveal regularities that would be difficult to notice by hand, making computation valuable in the discovery of possible relationships. Nevertheless, the mere enlargement of a finite list does not remove the difference between checking that list and establishing a rule for all cases to which a general statement refers. A counterexample, by contrast, can have a decisive role: a single case that satisfies a universal statement’s conditions but contradicts its conclusion is enough to refute that statement.

This asymmetry helps explain why mathematical exploration often moves between generating conjectures, searching for exceptions, and developing arguments that account for the entire intended domain. A proof may draw on a structure common to the cases rather than repeat the calculation for each one, thereby showing why the result follows under the stated assumptions. Computation and proof can consequently support one another without being interchangeable, since suggesting what may be true and establishing why it must be true are different intellectual tasks.