PrepTest 122, Section 2, Question 22
A mathematical theorem proved by one mathematician should(82 more words not shown)
Which one of the following(8 more words not shown)
- AThe use of(16 more words not shown)
- BMost attempts to(15 more words not shown)
- CComputers cannot be(18 more words not shown)
- DAny mathematical proof(23 more words not shown)
- EThe use of(26 more words not shown)
Reading the argument
A proof can be too long for a person to check. Does that automatically make it unacceptable?
The author begins with a standard for a theorem proved by one mathematician: each step must be independently verified before we accept it. That is a demand to check the work, not merely trust the person who produced it. Computer-assisted proofs can involve so many cases that no human could inspect every step.
My first question is whether the author has justified applying the same standard to these computer-assisted proofs. Could their method of producing results matter to what verification they need? We shouldn't assume computers are reliable just because they're computers, but the shift from one mathematician's work to computer-assisted work is worth examining.
What we’re looking for
For the necessary assumption, we need something the author must accept to reject these very long proofs. I'd initially look for a claim defending the verification requirement in this setting. But we should also check exactly what the author needs to rule out before saying the proof cannot be accepted.
Working the answers
- A
Making a mathematician's job easier doesn't establish whether the resulting proof meets the verification standard. Computer assistance could simplify the work, complicate some of it, or do both. The argument's objection concerns checking every step, not how much effort the original mathematician saves.
- B
The success rate of attempts to construct proofs is not needed. Even if most attempts succeeded, the author could still insist on independent verification before accepting them. B adds a claim about how often proofs are successfully constructed rather than what is required to accept this kind.
- C
The argument targets computer-assisted proofs with astronomically many cases. Computers could also assist with short proofs that a person could check. Allowing that possibility doesn't undermine the author's objection to the long ones, so the claim that computers cannot help with short proofs isn't required.
- D
A person might survey every possible case when there are only a few. The author needn't deny that method to every proof produced without a computer. The problem identified here is the enormous number of cases, not the mere act of checking cases one by one.
- ECorrect
This exposes another gap: “no human can check it” doesn't mean “it cannot be independently checked.” What if a separate computer program could verify each step and satisfy the stated requirement? Then the absence of a human reviewer wouldn't justify rejecting the proof.
The author therefore needs this alternative route to fail. E says an independent program does not satisfy the requirement for these otherwise unverifiable proofs. That doesn't claim every computer result is untrustworthy. It blocks a specific way the proof might meet the author's standard despite being too long for a human to check.
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