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For anyone claiming to write a history of a science of which
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01 Feb 2026, 15:29
01 Feb 2026, 15:29
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For anyone claiming to write a history of a science of which reasoning forms the very essence, the question of the logic is of paramount importance. For example, a modern western account of any historical period in mathematics would, as a matter of course, show a detailed proof justifying each and every mathematical result discussed. Despite this obvious fact, general histories of Chinese mathematics rarely show concern for this issue. They insist above all on presenting only the mathematical results, the logical underpinnings of which are unclear, and rarely do they provide the reader with any semblance of a proof. While this approach to the history of mathematics is naturally a result of various causes, one which probably plays an essential role is the fact that most Chinese mathematical works themselves contain no logical justifications: according to this worldview, apparently it was enough to state authoritatively that something was true --- it was completely superfluous to demonstrate why it was true.
There is one major exception to this general pattern, namely a set of Chinese argumentative discourses which has been handed down to us from the first millennium A.D. We are referring to the commentaries and sub-commentaries on the Jiuzhang Suanshu ["The Nine Chapters on the Mathematical Art"], the key work which inaugurated Chinese mathematics and served as a reference for it over a long period of its history. This fact, which was long unrecognized, means that we are now in a position to know a lot more about the logical construction of mathematics in China than, for example, in Egypt, Mesopotamia, or India.
There is one major exception to this general pattern, namely a set of Chinese argumentative discourses which has been handed down to us from the first millennium A.D. We are referring to the commentaries and sub-commentaries on the Jiuzhang Suanshu ["The Nine Chapters on the Mathematical Art"], the key work which inaugurated Chinese mathematics and served as a reference for it over a long period of its history. This fact, which was long unrecognized, means that we are now in a position to know a lot more about the logical construction of mathematics in China than, for example, in Egypt, Mesopotamia, or India.
1. What distinguishes the commentaries on the Jiuzhang Suanshu from almost all other works of Chinese mathematical history is that the authors of the former
A. made clear exactly what theorems are true
B. spent time justifying their qualifications as authorities
C. did not fully disclose all the results in the fields discussed
D. provided explicit proofs for the mathematical results presented
E. had influence over a large portion of Chinese history
2. The author implies all of the following EXCEPT:
A. The ancient mathematical texts of Mesopotamia do not provide explicit proofs for all their results.
B. The first Western scholars studying the history of Chinese mathematics were unaware of the proofs available in the commentaries and subcommentaries on the Jiuzhang Suanshu
C. Proofs are a method of demonstrating the logical arguments underlying a mathematical result.
D. The majority of important Chinese mathematicians between 1000 and 1500 would have known of the Jiuzhang Suanshu
E. The authors of the Jiuzhang Suanshu do not make any claim justifying their own authority.
3. The author is primarily concerned with:
A discussing the most effective way to elucidate the history of science
B explaining how one ancient work conforms to our modern expectations for logical transparency
C arguing for a change in the methodology used in studying the history of mathematics
D demonstrating that ancient Chinese mathematics was far more advanced than that of other ancient civilizations
E providing an example of how authors making clear their own qualifications enhances the respectability of the work as a whole
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Posts: 2932
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Re: For anyone claiming to write a history of a science of which
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02 Feb 2026, 08:04
02 Feb 2026, 08:04
2
Expert Reply
Official Explanation
1. What distinguishes the commentaries on the Jiuzhang Suanshu from almost all other works of Chinese mathematical history is that the authors of the former
Difficulty Level: Medium
Explanation
Most Chinese mathematical works stated results only, without the proofs justifying the results. The commentaries on the Jiuzhang Suanshu are different in that they contain both the results and the proofs. In other words, they provide "explicit proofs for the mathematical results presented", which is precisely what choice (D), the credited answer, says.
The majority of Chinese mathematical works did not include proofs, but they all said what was true. Both the commentaries on the Jiuzhang Suanshu and the other works "made clear exactly what theorems are true", so this does not distinguish them. Choice (A) is not correct.
The passage talks about "justification" in terms of the logical justification a mathematical proof implies. Nowhere does it discuss people justifying themselves. Choice (B) is not correct.
Choice (C) sounds awfully fancy, but it's not 100% clear what it even means. The authors of the commentaries on the Jiuzhang Suanshu "did not fully disclose all the results in the fields discussed"(??) Well, the only "field discussed" is mathematics. The verb "disclosed" is particularly odd, because it implies that someone was keeping secrets by withholding information, something not mentioned in the passage at all. As stated in answer choice (A), all Chinese mathematical works stated "results", i.e. what was true; only the commentaries on the Jiuzhang Suanshu also showed the proofs, i.e. why things were true. In that sense, the commentaries on the Jiuzhang Suanshu showed (disclosed?) more than the other works. Insofar as this choice says anything meaningful, it contradicts the passage. Don't be fooled by fancy wording. Choice (C) is not correct.
We know the commentaries on the Jiuzhang Suanshu "had influence over a large portion of Chinese history" because they "served as a reference for it over a long period of [Chinese mathematical] history." Does this make the commentaries on the Jiuzhang Suanshu different from other works of Chinese mathematics? Do we have the grounds, based on the passage, to conclude that absolutely no other work of Chinese mathematics had any influence over the course of China's long history? Hmm. That seems like a stretch. The passage only talks about the influence of the commentaries on the Jiuzhang Suanshu, and says zilch about whether other works had a long influence. We just don't know. Choice (E) is not correct.
Answer: D
Difficulty Level: Medium
Explanation
Most Chinese mathematical works stated results only, without the proofs justifying the results. The commentaries on the Jiuzhang Suanshu are different in that they contain both the results and the proofs. In other words, they provide "explicit proofs for the mathematical results presented", which is precisely what choice (D), the credited answer, says.
The majority of Chinese mathematical works did not include proofs, but they all said what was true. Both the commentaries on the Jiuzhang Suanshu and the other works "made clear exactly what theorems are true", so this does not distinguish them. Choice (A) is not correct.
The passage talks about "justification" in terms of the logical justification a mathematical proof implies. Nowhere does it discuss people justifying themselves. Choice (B) is not correct.
Choice (C) sounds awfully fancy, but it's not 100% clear what it even means. The authors of the commentaries on the Jiuzhang Suanshu "did not fully disclose all the results in the fields discussed"(??) Well, the only "field discussed" is mathematics. The verb "disclosed" is particularly odd, because it implies that someone was keeping secrets by withholding information, something not mentioned in the passage at all. As stated in answer choice (A), all Chinese mathematical works stated "results", i.e. what was true; only the commentaries on the Jiuzhang Suanshu also showed the proofs, i.e. why things were true. In that sense, the commentaries on the Jiuzhang Suanshu showed (disclosed?) more than the other works. Insofar as this choice says anything meaningful, it contradicts the passage. Don't be fooled by fancy wording. Choice (C) is not correct.
We know the commentaries on the Jiuzhang Suanshu "had influence over a large portion of Chinese history" because they "served as a reference for it over a long period of [Chinese mathematical] history." Does this make the commentaries on the Jiuzhang Suanshu different from other works of Chinese mathematics? Do we have the grounds, based on the passage, to conclude that absolutely no other work of Chinese mathematics had any influence over the course of China's long history? Hmm. That seems like a stretch. The passage only talks about the influence of the commentaries on the Jiuzhang Suanshu, and says zilch about whether other works had a long influence. We just don't know. Choice (E) is not correct.
Answer: D
GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2932
Given Kudos: 1259
GPA: 3.39
Re: For anyone claiming to write a history of a science of which
[#permalink]
02 Feb 2026, 08:08
02 Feb 2026, 08:08
1
Expert Reply
Official Explanation
2. The author implies all of the following EXCEPT:
Difficulty Level: Very Hard
Explanation
This is an "EXCEPT" question. We expect that the passage clearly implies four of the answers, and for the last, it either implies something else or simply is not clear on that point. This last one, the one not clearly implied, is the right answer.
In the last sentence, the passage clearly says we know more about the logical reasoning in Chinese math because of this one work that shows proofs. If we know less about the logical reasoning in ancient Mesopotamian math, it's because those works don't show proofs. Choice (A) is clearly implied, so it is not the answer to the EXCEPT question.
The passage clearly says "This fact, which was long unrecognized, ..." --- in other word, the first folks studying this did not realize that the commentaries and sub-commentaries on the Jiuzhang Suanshu provided proofs. Choice (B) is clearly implied, so it is not the answer to the EXCEPT question.
The passage clearly says that proofs "justify" mathematical results, and explain why it is true. In other words, proofs "demonstrate the logical argument underlying a mathematical result". Choice (C) is clearly implied, so it is not the answer to the EXCEPT question.
The passage clearly says that the Jiuzhang Suanshu "served as a reference for [Chinese mathematics] over a long period of its history." Therefore, it is quite reasonable to infer that most Chinese mathematicians between 1000 and 1500 would have known about it. Choice (D) is clearly implied, so it is not the answer to the EXCEPT question.
The passage talks about "justification" in terms of the logical justification a mathematical proof implies. Nowhere does it discuss people justifying themselves. Choice (E) is not implied, so this is the correct answer to the EXCEPT question.
Answer: E
Difficulty Level: Very Hard
Explanation
This is an "EXCEPT" question. We expect that the passage clearly implies four of the answers, and for the last, it either implies something else or simply is not clear on that point. This last one, the one not clearly implied, is the right answer.
In the last sentence, the passage clearly says we know more about the logical reasoning in Chinese math because of this one work that shows proofs. If we know less about the logical reasoning in ancient Mesopotamian math, it's because those works don't show proofs. Choice (A) is clearly implied, so it is not the answer to the EXCEPT question.
The passage clearly says "This fact, which was long unrecognized, ..." --- in other word, the first folks studying this did not realize that the commentaries and sub-commentaries on the Jiuzhang Suanshu provided proofs. Choice (B) is clearly implied, so it is not the answer to the EXCEPT question.
The passage clearly says that proofs "justify" mathematical results, and explain why it is true. In other words, proofs "demonstrate the logical argument underlying a mathematical result". Choice (C) is clearly implied, so it is not the answer to the EXCEPT question.
The passage clearly says that the Jiuzhang Suanshu "served as a reference for [Chinese mathematics] over a long period of its history." Therefore, it is quite reasonable to infer that most Chinese mathematicians between 1000 and 1500 would have known about it. Choice (D) is clearly implied, so it is not the answer to the EXCEPT question.
The passage talks about "justification" in terms of the logical justification a mathematical proof implies. Nowhere does it discuss people justifying themselves. Choice (E) is not implied, so this is the correct answer to the EXCEPT question.
Answer: E
