In mathematics, a conjecture is a proposition which is expected to be true, hence expected to have a proof, but for which no proof is (currently) known.
Hence being a conjecture is a sociological aspect of a proposition, not a mathematical aspect: Once a proof (or else a counterexample) is found, the conjecture ceases to be a conjecture and instead becomes a theorem.
It happens that conjectures remain unproven while being perceived as trustworthy enough that further theorems are proven assuming the conjectures – in this case the conjecture plays the role of a hypothesis in the sense of formal logic.
For example, the “standard conjectures” in algebraic geometry serve as hypotheses in a wealth of theorems which are all proven (only) “assuming the standard conjectures” (cf. e.g. arXiv:9804123).
In other cases the term “hypothesis” is used synonymously with “conjecture” – e.g. for the homotopy hypothesis (key cases of which have long become theorems) or the cobordism hypothesis (on which a proof has famously been claimed but not universally accepted).
geometric Langlands correspondence-conjecture
mathematical statements
Last revised on March 5, 2023 at 13:35:10. See the history of this page for a list of all contributions to it.