Critical Reasoning

Historian: Leibniz, the seventeenth- century philosopher, published his version of calculus before Newton did. But then Newton revealed his private notebooks, which showed he had been using these ideas for at least a decade before Leibniz's publication. Newton also claimed that he had disclosed these ideas to Leibniz in a letter shortly before Leibniz's publication. Yet close examination of the letter shows that Newton's few cryptic remarks did not reveal anything important about calculus. Thus, Leibniz and Newton each independently discovered calculus.

Which one of the following is an assumption required by the historian's argument?

A. Leibniz did not tell anyone about calculus prior to publishing his version of it.
B. No third person independently discovered calculus prior to Newton and Leibniz.
C. Newton believed that Leibniz was able to learn something important about calculus from his letter to him.
D. Neither Newton nor Leibniz knew that the other had developed a version of calculus prior to Leibniz publication.
E. Neither Newton nor Leibniz learned crucial details about calculus from some third source.

In a way this is a cool question because there is all kinds of stuff going on with the letter and the timing and things. So finding the assumption requires sorting through what is going on and determining what is really relevant.

With that in mind, eliminating the answers is pretty straightforward once one sees what the conclusion is.

The conclusion is that they each discovered calculus independently. So we need to see what is assumed in order for that to be true.

A) This is irrelevant. Whom he told does not affect whether he discovered it.
B) This is irrelevant. Even if someone else discovered it at the same time, the two mentioned could still have each discovered it independently.
C) In a way this is tempting. One could almost get the impression that this is relevant, but actually if Newton were to have believed this and if further it were somehow true, it would weaken the conclusion, because then Leibniz would maybe have learned from Newton rather than making an independent discovery. So that's out.
D) This makes no difference. They could have found out about each other's work before his publication and still discovered calculus independently before that. So this assumption is not necessary.
E) This is necessary for the conclusion to be correct. In order for them to have discovered it independently they can't have learned it from someone else, obviously.

Choose E.

Historian: Leibniz, the seventeenth- century philosopher, published his version of calculus before Newton did. But then Newton revealed his private notebooks, which showed he had been using these ideas for at least a decade before Leibniz's publication. Newton also claimed that he had disclosed these ideas to Leibniz in a letter shortly before Leibniz's publication. Yet close examination of the letter shows that Newton's few cryptic remarks did not reveal anything important about calculus. Thus, Leibniz and Newton each independently discovered calculus.

Which one of the following is an assumption required by the historian's argument?

Conclusion Leibniz and Newton each independently discovered calculus.

Premises: We know Leibniz published first. We know Newton had written his ideas in a notebook before Leibniz published. We know that Newton sent Leibniz a letter that doesn't seem to have been important.

A) Leibniz did not tell anyone about calculus prior to publishing his version of it.
He may have told lots of people. (I certainly would have!) That wouldn't effect the conclusion that he and Newton discovered calculus independently. (Remember, Newton was writing his ideas in his notebooks before Leibniz published.) A is out.

B) No third person independently discovered calculus prior to Newton and Leibniz.
The conclusion never says that Newton and Leibniz were the only
ones to have discovered calculus independently. It's possible that three people discovered calculus independently. B is out.

C) Newton believed that Leibniz was able to learn something important about calculus from his letter to him.
Tempting, but we don't know what Newton believed about the importance of what he wrote in his letter. It's possible that Newton thought he was revealing ideas that Leibniz already knew about.

D) Neither Newton nor Leibniz knew that the other had developed a version of calculus prior to Leibniz publication.

It's possible that Leibniz knew that an eccentric Englishman had already discovered a version of calculus before he did. But he could still have worked out his own version independently.

E) Neither Newton nor Leibniz learned crucial details about calculus from some third source.

This looks good. The correct answer, when negated, would destroy the conclusion. Negated, the answer choice would be: Both Newton and Leibniz learned crucial details about calculus from some third source.
Well, if they both learned about calculus from a third source, suddenly it becomes dubious to claim that they each discovered calculus.


E is the answer.

Hi David,

Isn't the conclusion talking about each independently discovering calculus? If this happens, wouldn't option D be correct? If they learnt calculus from a third source, its true that the duo would not have discovered calculus. However, how does this affect the conclusion that they did not discover calculus independently of each other?

Regards,
Amit