A mathematical theorem proved by one mathematician should not be accepted until each step in its proof has been independently verified. Computer-assisted proofs generally proceed by conducting a vast number of calculations—surveying all the possible types of instances in which the theorem could apply and proving that the theorem holds for each type. In most computer-assisted proofs there are astronomically many types of instances to survey, and no human being could review every step in the proof. Hence, computer-assisted proofs involving astronomically many types of instances should not be accepted.

Which one of the following is an assumption on which the argument relies?

(A) The use of the computer to assist in the proof of mathematical theorems has greatly simplified the mathematician's task.
(B) Most attempts to construct proofs of mathematical theorems do not result in demonstrations that the theorems are true.
(C) Computers cannot be used to assist in generating proofs of mathematical theorems that involve only a very limited number of steps.
(D) Any mathematical proof that does not rely on the computer cannot proceed by surveying all possible types of instances to which the candidate theorem might apply.
(E) The use of an independent computer program does not satisfy the requirement for independent verification of each step in a proof that is extended enough to be otherwise unverifiable.

1.Translate/simplify the argument:
- Fact 1: A theorem should be accepted only if each step in its proof has been verified independently
- Fact 2: Computer-assisted proofs work with vast numbers of calculations - (blah blah..surveying instances & proving theorem in each step)
- Fact 3: There are huge number of types of instances, and no one could review all of steps
- Claim: Computer-assisted proofs should not be accepted. (because no one could review all steps!!!)-> Conclusion

2. Quick reasoning (in mind): there is something missed between "independent verification" and "ability of human being". What if there is an alternative way? --> the assumption could eliminate/exclude some possible cases.(which could damage the conclusion)

3. POE:
(A) -> using of computer is not related to the reasoning here ->out of scope
(B) -> attempts to construct proofs blah blah...-> out of scope
(C) -> The argument tell about theorem as a whole, not special case (which involves limited number of steps) -> out of scope
(D) Any mathematical proof that does not rely on the computer cannot proceed by surveying all possible types of instances to which the candidate theorem might apply. ->something has been excluded, not sure it is the correct assumption -> try to negate:
Any proof (non-assisted by computer) can proceed by surveying instances -> so... people still can not review every step in the proof (assisted by computer) --> so... negating this can not damage the conclusion.

(E) The use of an independent computer program does not satisfy the requirement for independent verification of each step in a proof that is extended enough to be otherwise unverifiable. --> oh, it excluded something again -> try to negate:
The use of computer satisfy the requirement for independent verification of each step in a proof (that its extended enough to be otherwise unverifiable) --> yeah, people could not verify it in some cases (or numerous cases), but the fact that computer could satisfy the requirement of independent verification (in that cases) --> definitely destroy the conclusion
--> E is the correct answer

Experts, please correct my reasoning if it is still weak or missing crucial points. Thank you.

Once one realizes that in option E, the pronoun "that" refers to "verification", then the question would be easier to comprehend.

Premise: In most computer-assisted proofs there are astronomically many types of instances to survey, and no human being could review every step in the proof.

Conclusion: computer-assisted proofs involving astronomically many types of instances should not be accepted.

The basic link between the premise and conclusion is that human review (verification) is required in order to accept computer-assisted proofs. Option E states this assumption: it is not possible to verify (review by humans) the "extended verification of each step" (by the computer), i.e. the extended verification (by the computer) of each step is otherwise unverifiable (by humans).

(Note that in the phrase "for independent verification of each step in a proof that is extended enough to be otherwise unverifiable", "verification" refers to verification by computers and "unverifiable" refers to review by humans.)

Please post again if you still have doubts.