Mazur, B. (1997). Conjecture. Synthese, 111(2), 197–210. https://doi.org/10.1023/A:1004934806305
Atomic notes
Reading notes
Architectural conjectures
- ”Plausible reasoning” is distinct from general conjectures, which include guessed answers to open problems: “Some manner of conjecturing has always been present as a vital, but informal, penchant of mathematicians: how else does any mathematics progress except by questions whose answers are guessed at, except by waves of surmises? … There have also always been open problems which are in every sense conjectures.” (198)
- Example. The Beilinson Conjectures may be unresolved, but their formulation “unifies a research program” since their set-up requires developing a lot of mathematics—the statement of the Conjectures requires “convincing a reader that there is internal, at least, consistency and coherence to this constellation of conjectures.” (199)
- Conjectures can be fleshed out—becoming theorems, propositions, etc.—by demonstrations.
Cautions about conjecture
- Conjecture may be a way for researchers to impose agendas that the overall development of mathematics: “David Kazhdan, responding to the statement that “conjecture” is a way for present-day researchers to imprint their intent, their goals, on future researchers, cautioned that if the practice achieved too much prominence, it, like the “five-year plans” of the former Soviet Union might eventually have a stultifying effect.” (200)
Uncertainty and rigor
- Uncertainty of conjecture is tolerated by their appeal: “Or is it that we are simply more at home with large theories, Gaudiesque cathedrals, that are incomplete, and whose eventual dimensions are uncertain, by virtue of their very grandeur. And yet we must deal with these theories in some orderly way. If so, instead of thinking of “conjecture” as temporary scaffolding, to be removed upon being replaced by theorems, we might expect that as our experience widens, we will replace proved constellations of conjectures with ever bolder ones.” (208)
- Conjecture is distinguished from rigor as long as there are shared standards of proof: “To distinguish conjectures from proofs you need generally accepted standards. These need not be “right” in any given sense, certainly need not be formal. But they need to be generally accepted to give a meaningful distinction. … Shared standards of rigor can only arise where there are large shared projects, and vice versa. I come to think of “cathedrals” and “rigor” as the social and theoretical sides (respectively) of one development”.” (209)