conjecturing

Conjecturing is the act of formulating conjectures— provisional statements proposed as true on the basis of observation, pattern, or reasoning but not yet proven. In mathematics, a conjecture is a statement believed to be true but lacking a proof; in science and philosophy, conjectures often take the form of hypotheses that guide inquiry and testing.

Conjectures occupy a middle ground between guesswork and established results. They may be supported by partial evidence, special cases, or heuristic arguments, yet remain open to challenge. A conjecture can be overturned by a counterexample or strengthened by a full proof, at which point it becomes a theorem. Some conjectures remain open for decades or centuries, shaping research directions. Notable examples include Goldbach's conjecture, the Collatz conjecture, and the Riemann hypothesis; Fermat's Last Theorem began as a conjecture and later became a theorem.

Practically, conjecturing involves pattern recognition, testing with examples or computations, and exploring generalizations. It relies on informal reasoning, intuition, and peer scrutiny, with the status of a conjecture continually evolving as new evidence appears. In science, conjectures or hypotheses must be testable and falsifiable, whereas in mathematics a proof is the ultimate criterion for establishing truth.