Self-study
Support A mathematical theorem proved by one mathematician should not be accepted until each step in its proof has been independently verified. █████████████████ ██████ █████████ ███████ ██ ██████████ █ ████ ██████ ██ ████████████████████████ ███ ███ ████████ █████ ██ █████████ ██ █████ ███ ███████ █████ █████ ███ ███████ ████ ███ ███████ █████ ███ ████ █████ ██ ████ █████████████████ ██████ █████ ███ ██████████████ ████ █████ ██ █████████ ██ ███████ ███ ██ █████ █████ █████ ██████ █████ ████ ██ ███ ██████ ██████ █████████████████ ██████ █████████ ██████████████ ████ █████ ██ █████████ ██████ ███ ██ █████████
Summary
The author concludes that computer-assisted proofs involving astronomically many types of instances should not be accepted. His reasoning is that each step of a math proof must be independently verified, and no human being could verify every step of the proof.
Notable Assumptions
The author’s conclusion is about independent verification of math proofs, but his support is only about human verification. What if other computers were capable of independent verification? The author’s argument would provide no grounds for rejecting computer-assisted proofs in that case.
Therefore, the author must assume that, for the proofs in question, computer verification is not possible.
22.
Which one of the following ██ ██ ██████████ ██ █████ ███ ████████ ███████
a
The use of ███ ████████ ██ ██████ ██ ███ █████ ██ ████████████ ████████ ███ ███████ ██████████ ███ ███████████████ █████
The author is trying to partially discredit the use of computers to prove mathematical theorems, so making positive assumptions about them is not necessary to his argument.
b
Most attempts to █████████ ██████ ██ ████████████ ████████ ██ ███ ██████ ██ ██████████████ ████ ███ ████████ ███ █████
The proportion of successful attempts is irrelevant: the author is concerned with the process for verifying attempts.
c
Computers cannot be ████ ██ ██████ ██ ██████████ ██████ ██ ████████████ ████████ ████ ███████ ████ █ ████ ███████ ██████ ██ ██████
The author’s argument is about verifying theorems that have many steps. So he doesn’t need to make any assumptions about proofs with only a few steps.
d
Any mathematical proof ████ ████ ███ ████ ██ ███ ████████ ██████ ███████ ██ █████████ ███ ████████ █████ ██ █████████ ██ █████ ███ █████████ ███████ █████ ██████
This would suggest a limitation of non-computer math proofs. Since the author’s aim is to partially discredit computer-assisted math proofs, this assumption is not necessary to his argument.
e
The use of ██ ███████████ ████████ ███████ ████ ███ ███████ ███ ███████████ ███ ███████████ ████████████ ██ ████ ████ ██ █ █████ ████ ██ ████████ ██████ ██ ██ █████████ █████████████
Negated, this is: the use of an independent computer program does satisfy the requirement for independent verification. If so, the author’s argument that computer-assisted proofs shouldn’t be accepted because they can’t be verified collapses. Consequently, he must assume that (E) is true.