Logical Reasoning

PT122 · S2 · Q22 A mathematical theorem proved

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A mathematical theorem proved by one mathematician should not be accepted until each step in its proof has been independently verified.

Conclusion (Hence)

Computer-assisted proofs involving astronomically many types of instances should not be accepted.

Evidence

A math theorem proved by one mathematician should not be accepted until each step in its proof has been independently verified.

In most computer-assisted proofs there are astronomically many types of instances in which the theorem could apply, and no human being could review every step in the proof.

Evaluate

The conclusion is attempting to prove that a certain type of proof "should not be accepted".

Our first question should be, ? The argument provided that rule. It says:

If each step in the then proof proof has not been → should not be independently verified accepted

Did the author establish that in the case of "computer-assisted proofs involving astronomically many types of instances", each step in the proof has not been independently verified?

Close, but not explicitly. The author just said, .

Goal

So the author seems to be assuming, .

Or by contrapositive, .

22.

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

  1. The use of the computer

    Out of Scope: greatly simplified

    Nothing in this argument cares about whether computers have greatly simplified or only moderately simplified or not simplified at all the task of a mathematician. Anyone of those possibilities would be fine with the author's argument. However much computers have / haven't simplified the mathematician's task, the author is still going to argue that we shouldn't accept computer-assisted proofs.

    1% picked this

  2. Most attempts to construct proofs

    Too Strong

    Too Strong: most Unrelated to Goal: not about computer proofs The word "most" kills 99% or more of the Necessary Assumption answers it's in. (It can generally only be correct if the conclusion uses the word "most" or uses an equivalent concept like "usually / probably / generally / likely"). When you negate a Most claim, you're basically arguing over whether the frequency of something is at least 51% of the time or at most 49% of the time. Who cares? This argument won't change one iota, whether 51% of attempts to construct proofs result in demonstrations that the theorems are true or whether it's only 49% of them that do. This answer should look unappealing since it's referring to ALL attempts to construct proofs, and we only care about computer-assisted proofs involving huge numbers of instances.

    3% picked this

  3. Computers cannot be used to

    Too Strong: cannot be used

    The author is only claiming that we can't accept computer based proofs that involve an astronomical number of instances to independently verify. She hasn't commented on whether we can accept computer-assisted proofs for things that have a very limited number of steps. Presumably, for those types of proofs, a human being COULD review every step in the proof and so the author's objection would no longer apply.

    4% picked this

  4. Any mathematical proof that does

    Too Strong: any

    This answer is talking about EVERY SINGLE math proof that doesn't rely on computers. Did the author say anything that tells us what she would believe is true about every single non-computer proof? Of course not. We don't really need to read the second half of this answer. We know the author did not commit to any universal idea that applies to all non-computer proofs. We should be unnerved that this answer is even talking about non-computer proofs, since the conclusion is all about computer proofs.

    7% picked this

  5. Correct

    The use of an independent

    Why this is right

    This has that lovable feature, the ruling-out "not", that is in so many correct Necessary Assumption answers. These "nots" are ripe for the Negation Test. If we remove the 'not', does this become an objection? The use of an independent computer program does satisfy the requirement for independent verification of each step in a proof that is extended enough (i.e. that has astronomically many types of instances) to be otherwise unverifiable. Yes! We were looking for this move the author was making: "If no human being can review every step in the proof, then it's not the case that each step in the proof has been independently verified". This negation is saying, "That's not true. Even if no human can review every step in the proof, you can still independently verify each step in a proof with astronomically many instances by using an independent computer program to verify each step."

    Skill tested: Necessary Assumption · how this choice captures the argument's function is the move to repeat next time.

    85% picked this

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