Self-study
No mathematician today would flatly refuse to accept the results of an enormous computation as an adequate demonstration of the truth of a theorem. ██ █████ ████████ ████ ███ ███ ███ █████ ████ ██████████████ ██ ████ ████ ███████ ██ ██████ ███ ███████ ██ █ ███████ ████████ █████████████ ██ █ ████ ██████ ███████ ████████ ████████ ████ ██████████████ █████ ████ █ ██████ ██████ ████ █ ██████ ███████ █████ ██ ████ █ ██████ ██████ ██████ ██ █████ ████ ██████ ████████ ████ ████████ ████████ ███████
Stimulus Breakdown
The stimulus discusses mathematicians and how they approach proving theorems. Since we'll be looking for an answer that must be true, it's helpful to get all the facts laid out clearly. Even a small misunderstanding could lead to choosing a wrong answer.
(1) currently, no mathematician would flatly refuse an "enormous computation" as a means of proving a theorem;
(2) in 1976, some mathematicians did refuse such a computation to prove a simple theorem;
(3) currently, some mathematicians believe that simple theorems should have simple proofs;
(4) some simple theorems require complex proofs.
Analysis: Intersecting Sets
Notice all of the quantifier language in this stimulus, and the various statements about mathematicians: it looks like we're dealing with intersecting sets. Using that framework, we can start making some up-front inferences.
(1) + (2) if any mathematicians who rejected enormous computational proofs in 1976 are still mathematicians, they must have become more open to such proofs;
(1) + (3) some people who wouldn't flatly refuse computational proofs nonetheless believe that simple theorems should have simple proofs;
(3) + (4) some simple theorems require proofs that run counter to the beliefs of some mathematicians.
There's no guarantee that one of these inferences will be the correct answer, but we can still do a quick scan to check if any of them shows up. Otherwise, if the correct answer is one we overlooked, we may have to revert to process of elimination.
12.
If all of the statements ██ ███ ███████ ███ █████ █████ ███ ██ ███ █████████ ████ ████ ██ █████
a
Today, some mathematicians ███ ███████ ████ █ ██████ ███████ █████ ██ ████ █ ██████ █████ █████ ████████ █████████ ███ ███████ ██ ██ ████████ ███████████ ██ █ █████████████ ██ ███ █████ ██ █ ████████
(A) reflects a synthesis of claims (1) and (3) from the stimulus. We know
It would also be true to say that all current mathematicians who believe in simple proofs would consider accepting computational proofs. But since "some" includes "all", (A) must still be true as is.
b
Some individuals who ███████ ████ █ ██████ ███████ █████ ██ ████ █ ██████ █████ ███ ███ ███████████████
The stimulus doesn't say anything about nonmathematicians. The stimulus only tells us what some mathematicians believe about simple theorems and simple proofs. That isn't a basis to include or exclude anyone from being a mathematician, so (B) is out.
c
Today, some individuals ███ ██████ ██ ██████ ███ ███████ ██ ██ ████████ ███████████ ██ █ █████████████ ██ ███ █████ ██ █ ███████ ███████ ████ █ ██████ ███████ █████ ██ ████ █ ██████ ██████
The stimulus says that
d
Some individuals who ██ ███ ███████ ████ █ ██████ ███████ █████ ██ ████ █ ██████ █████ █████ ███ ██ ███████ ██ ██████ ███ ███████ ██ ██ ████████ ███████████ ██ █████ ██ █ ███████ ████████
The stimulus only tells us that some mathematicians believe simple theorems should have simple proofs. That doesn't let us infer anything about individuals who don't hold that belief—they could be mathematicians or nonmathematicians, we don't know. And there's equally no basis to say that such individuals would reject computational proofs.
e
Some nonmathematicians do ███ ███████ ████ █ ██████ ███████ █████ ██ ████ █ ██████ ██████
The stimulus doesn't say anything about nonmathematicians. The stimulus only tells us what some mathematicians believe about simple theorems and simple proofs. That doesn't let us determine what nonmathematicians believe, so (E) is out.