#22 - A mathematical theorem proved by one mathematician

[Administrator]

Complete Question Explanation

Assumption. The correct answer choice is (E)

The conclusion of this stimulus is presented at the end of this stimulus: computer assisted proofs should not be accepted. This is based on the following premises:

• Premise 1: a theorem should not be accepted until each step is independently verified.
  
  Premise 2: computer assisted proofs run numerous scenarios.

To draw from these two premises the conclusion that computer assisted proofs should not be accepted, the author must assume that the computer's approach of running scenarios is not sufficient to provide independent verification of each step.

Answer choice (A): This choice does not provide the link needed to draw the author's conclusion, and the discussion does not reference simplification.

Answer choice (B): The proportion of attempts that successfully prove theorems is irrelevant to the author's conclusion, so this answer choice is incorrect.

Answer choice (C): The fact that computers can consider all possible types of instances does not mean that they cannot do calculations with few steps, and this is certainly not an assumption on which the conclusion relies.

Answer choice (D): This answer choice does not provide the link prephrased above, so this answer choice is incorrect.

Answer choice (E): This is the correct answer choice, exactly reflecting the prephrased answer discussed above. The author must assume that the computer's approach does not satisfy the criterion presented.

[andriana.caban]

Hey!

Can you explain why answer choice D is wrong? I don't understand based on the prephrase explanation?

[Luke Haqq]

Hi Andriana!

I can do my best to help illuminate why (D) is incorrect, though I can only guess in the explanation without knowing the specific reasons you had for choosing that one.

The argument on this question is that (1) mathematical theorems should only be accepted if they can be independently verified, (2) some recent computer-based theorems cannot be independently verified by any human being; (3) therefore, these recent computer-based theorems should not be accepted.

The problem--or the "gap," since this is an assumption question--is with (2). For example, my prephrase when reading the stimulus was that the fact that "any human being" cannot verify these theorems left open the possibility that many human beings might be able to verify them and thus render them acceptable. That's one possible assumption. The right answer, (E), is another assumption--it raises the possibility that verification could come by way of another computer program. In order for the stimulus's conclusion to follow from the premises, (E) would have to be true--that such verification is not possible.

Answer (D) doesn't get to acceptability, which is what the conclusion of the stimulus is about. (D) seems similar to the prephrase I had come up with, however, the wording of that answer choice seems too strong to be correct: "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." The stimulus isn't saying that humans can never independently verify a mathematical theorem, as I take (D) to be saying. Rather, the argument is specifically about those theorems that are generated by computers in the first place.

Hope that helps!

[Pragmatism]

[quote="Luke Haqq"]Hi Andriana!

Code: Select all

I can do my best to help illuminate why (D) is incorrect, though I can only guess in the explanation without knowing the specific reasons you had for choosing that one.

The argument on this question is that (1) mathematical theorems should only be accepted if they can be independently verified, (2) some recent computer-based theorems cannot be independently verified by any human being; (3) therefore, these recent computer-based theorems should not be accepted. 

The problem--or the "gap," since this is an assumption question--is with (2). For example, my prephrase when reading the stimulus was that the fact that "any human being" cannot verify these theorems left open the possibility that [i]many[/i] human beings might be able to verify them and thus render them acceptable. That's one possible assumption. The right answer, (E), is another assumption--it raises the possibility that verification could come by way of another computer program. In order for the stimulus's conclusion to follow from the premises, (E) would have to be true--that such verification is not possible. 

Answer (D) doesn't get to acceptability, which is what the conclusion of the stimulus is about. (D) seems similar to the prephrase I had come up with, however, the wording of that answer choice seems too strong to be correct: "[i]Any[/i] mathematical proof that does not rely on the computer [i]cannot proceed[/i] by surveying all possible types of instances to which the candidate theorem might apply." The stimulus isn't saying that humans can never independently verify a mathematical theorem, as I take (D) to be saying. Rather,[b] the argument is specifically about those theorems that are generated by computers in the first place.[/b]

Furthermore, I would add, this answer choice talks about "mathematical proof that does not rely on the computer," and all we possibly know about proofs that do not rely on the computer is that they "should not be accepted until each step in its proof has been independently verified."

Also, the part where the answer choice state, "by surveying all possible types of instances to which the candidate theorem might apply," I believe, is confusing it for, "In most computer-assisted proofs there are astronomically... every step in the proof." So, while this answer choice may seem tempting, it is a mere mirage of what the right answer choice should look like.