PrepTest 112, Section 4, Question 14
Joseph: My encyclopedia says that the mathematician Pierre(89 more words not shown)
Which one of the following(9 more words not shown)
- AIt purports to(15 more words not shown)
- BIt mistakenly assumes(22 more words not shown)
- CIt mistakes something(16 more words not shown)
- DIt uses the(5 more words not shown)
- EIt fails to(25 more words not shown)
Reading the argument
Someone else proved the theorem. Did Fermat prove it? Those are different questions. Joseph goes too far in treating the earlier lack of a proof as a reason to think the theorem cannot be proved. Laura corrects that part of his reasoning with a recent success. But a later mathematician's achievement doesn't automatically vindicate Fermat's own claim. He could have been mistaken while someone else eventually found a valid proof.
Imagine claiming that I designed a complicated machine. Showing that someone has built a working version of that machine wouldn't establish that I designed one. It would show that the task can be done. We would still need evidence that I did it.
What we’re looking for
The question asks for Laura's reasoning error. I'd look for that jump from an achievable task to a particular person's achievement. She has challenged Joseph's basis for doubt, but she hasn't shown that Fermat himself had a proof.
Working the answers
- A
The later proof doesn't contradict Laura's conclusion. It could coexist with Fermat having found his own proof earlier. The trouble is that it could also coexist with his never having found one. Evidence that leaves both possibilities open fails to establish the conclusion; it doesn't necessarily contradict it.
- B
Laura doesn't describe Fermat as an honest or admirable person and use that description to defend his claim. Her evidence is that someone recently proved the theorem. The argument concerns what is possible and who accomplished it, not a guarantee based on character.
- CCorrect
For Fermat to have truthfully claimed a valid proof, the theorem must be provable. That is a requirement his claim would have to meet. But meeting the requirement doesn't establish that he actually produced the proof.
Laura treats provability as enough to clear him of lying or error. Our machine example has the same gap: the machine must be possible for my design claim to be true, but its possibility doesn't establish my accomplishment. C gives an abstract description of the concrete mistake we noticed.
- D
We can understand 'provable' as capable of being established by a proof. Defining the term more explicitly wouldn't resolve the gap between someone being able to prove the theorem and Fermat actually having done so. Laura's problem remains even with perfectly clear terminology.
- E
This describes a mix-up about whether a claim is true and how people have regarded it. The problem here is more specific: a later proof establishes that the theorem can be proved, but not that Fermat's personal claim was accurate. Joseph and Laura disagree about that claim; disagreement alone doesn't show that Laura has confused the two kinds of mistaken belief E describes.
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