This one is due to Errett Bishop: "all meaningful mathematics is reducible to finite calculations with strings of $0$s and $1$s" (imho Bishop formulated this not as a conjecture but as an article of faith but that doesn't necessarily affect the truth or falsity thereof).
A reference for Bishop's claim is his article "Crisis in contemporary mathematics" where it is formalized in terms of a principle called LPO ("limited principle of omniscience"). The LPO is discussed starting on page 511 of the article.