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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 5th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded an Idea-section to this (previously stubby) entry --- not meant to be definite, just what came to mind when finding that such a section was still lacking here: *** 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](https://arxiv.org/abs/math/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). *** <a href="https://ncatlab.org/nlab/revision/diff/conjecture/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/conjecture/10">v10</a>, <a href="https://ncatlab.org/nlab/show/conjecture">current</a>

   Added an Idea-section to this (previously stubby) entry — not meant to be definite, just what came to mind when finding that such a section was still lacking here:

   ---

   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).

   ---

   diff, v10, current