Historian: Newton developed mathematical concepts and techniques that are fundamental to modern calculus. Leibniz developed closely analogous concepts...

GMAT Critical Reasoning : (CR) Questions

Source: Official Guide

Critical Reasoning

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Historian: Newton developed mathematical concepts and techniques that are fundamental to modern calculus. Leibniz developed closely analogous concepts and techniques. It has traditionally been thought that these discoveries were independent. Researchers have, however, recently discovered notes of Leibniz's that discuss one of Newton's books on mathematics. Several scholars have argued that since the book includes a presentation of Newton's calculus concepts and techniques, and since the notes were written before Leibniz' own development of calculus concepts and techniques, it is virtually certain that the traditional view is false. A more cautious conclusion than this is called for, however. Leibniz' notes are limited to early sections of Newton's book, sections that precede the ones in which Newton's calculus concepts and techniques are presented.

In the historian's reasoning, the two portions in boldface play which of the following roles?

The first provides evidence in support of the overall position that the historian defends; the second is evidence that has been used to support an opposing position.

The first provides evidence in support of the overall position that the historian defends; the second is that position.

The first provides evidence in support of an intermediate conclusion that is drawn to provide support for the overall position that the historian defends; the second provides evidence against that intermediate conclusion.

The first is evidence that has been used to support a conclusion that the historian criticizes; the second is evidence offered in support of the historian's own position.

The first is evidence that has been used to support a conclusion that the historian criticizes; the second is further information that substantiates that evidence.

Solution

Understanding the Passage

Text from Passage	Analysis
"Newton developed mathematical concepts and techniques that are fundamental to modern calculus."	What it says: Newton created key mathematical ideas that form the foundation of calculus as we know it today. Visualization: Newton's work in 1600s → Core calculus concepts → Modern math textbooks in 2024 What it does: Establishes Newton as a key figure in calculus development Source: Historian's factual statement
"Leibniz developed closely analogous concepts and techniques."	What it says: Leibniz independently created very similar mathematical concepts and methods to Newton's. Visualization: Newton's calculus concepts ≈ Leibniz's calculus concepts (parallel development) What it does: Establishes Leibniz as another key figure who developed similar ideas Source: Historian's factual statement
"It has traditionally been thought that these discoveries were independent."	What it says: The conventional belief has been that Newton and Leibniz developed their calculus ideas separately, without one copying from the other. Visualization: Traditional view: Newton (1600s) → calculus ideas, Leibniz (1600s) → similar calculus ideas (no connection between them) What it does: Presents the established historical viewpoint that will be challenged Source: Traditional scholarly view
"Researchers have, however, recently discovered notes of Leibniz's that discuss one of Newton's books on mathematics."	What it says: New evidence has emerged - scientists found Leibniz's personal notes that reference and discuss Newton's mathematical work. Visualization: Recent discovery: Leibniz's notebook → contains discussion of Newton's math book → challenges independence theory What it does: Introduces new evidence that contradicts the traditional view Source: Recent researchers' findings
"Several scholars have argued that since the book includes a presentation of Newton's calculus concepts and techniques, and since" (Boldface 1) "the notes were written before Leibniz' own development of calculus concepts and techniques"	What it says: Some academics claim that because Newton's book contained his calculus ideas, and because Leibniz wrote his notes about Newton's book before Leibniz developed his own calculus, this proves Leibniz copied from Newton. Visualization: Timeline: Leibniz reads Newton's book (1670) → Leibniz writes notes (1671) → Leibniz develops his calculus (1675) → Suggests copying What it does: Presents scholars' argument against independence using timing evidence Source: Several scholars' argument
"it is virtually certain that the traditional view is false."	What it says: These scholars conclude that the old belief about independent discovery is almost definitely wrong. Visualization: Scholars' confidence level: 95% certain that traditional independence view = false What it does: States the strong conclusion these scholars draw from the evidence Source: Several scholars' conclusion
"A more cautious conclusion than this is called for, however."	What it says: The historian believes we should be more careful and not jump to such a strong conclusion. Visualization: Scholars say 95% certain → Historian says we should be maybe 30% certain (more cautious) What it does: Signals the historian's disagreement and introduces their counter-argument Source: Historian's position
(Boldface 2) "Leibniz' notes are limited to early sections of Newton's book, sections that precede the ones in which Newton's calculus concepts and techniques are presented"	What it says: The historian points out that Leibniz's notes only discuss the beginning chapters of Newton's book, and Newton's calculus ideas appeared later in the book that Leibniz apparently didn't read. Visualization: Newton's book structure: Chapters 1-3 (basic math, Leibniz read these) → Chapters 7-10 (calculus concepts, Leibniz didn't read these) What it does: Provides counter-evidence that weakens the scholars' argument about copying Source: Historian's counter-evidence

Overall Structure

The historian presents a debate about whether Leibniz copied from Newton, first showing why some scholars think he did, then providing evidence for why that conclusion is premature.

Main Conclusion: A more cautious conclusion is needed rather than declaring the traditional independence view false.

Boldface Segments

Boldface 1: the notes were written before Leibniz' own development of calculus concepts and techniques
Boldface 2: Leibniz' notes are limited to early sections of Newton's book, sections that precede the ones in which Newton's calculus concepts and techniques are presented

Boldface Understanding

Boldface 1:

Function: Serves as key evidence supporting the scholars' argument that Leibniz copied from Newton
Direction: Opposite to author's conclusion (the historian disagrees with using this to make strong claims)

Boldface 2:

Function: Provides counter-evidence that undermines the scholars' copying argument
Direction: Same as author's conclusion (supports the historian's call for caution)

Structural Classification

Boldface 1:

Structural Role: Evidence for a position the author ultimately opposes
Predicted Answer Patterns: "evidence for a view the author challenges" or "support for a conclusion the author finds premature"

Boldface 2:

Structural Role: Counter-evidence supporting the author's more cautious position
Predicted Answer Patterns: "evidence supporting the author's position" or "counter-evidence that supports a more cautious conclusion"

Answer Choices Explained

The first provides evidence in support of the overall position that the historian defends; the second is evidence that has been used to support an opposing position.

'The first provides evidence in support of the overall position that the historian defends' - ✗ WRONG - The first boldface actually supports the scholars' position that the historian opposes, not the historian's own position
'the second is evidence that has been used to support an opposing position' - ✗ WRONG - The second boldface is the historian's own counter-evidence, not evidence supporting the opposing scholars' view

The first provides evidence in support of the overall position that the historian defends; the second is that position.

'The first provides evidence in support of the overall position that the historian defends' - ✗ WRONG - Again, the first boldface supports the scholars' copying argument, which the historian disagrees with
'the second is that position' - ✗ WRONG - The second boldface is evidence, not the historian's main conclusion itself

'The first provides evidence in support of an intermediate conclusion that is drawn to provide support for the overall position that the historian defends' - ✗ WRONG - The first boldface supports the scholars' conclusion that the historian criticizes, not any conclusion supporting the historian's position
'the second provides evidence against that intermediate conclusion' - ✗ WRONG - While the second does provide counter-evidence, the setup is wrong because there's no intermediate conclusion supporting the historian's view

The first is evidence that has been used to support a conclusion that the historian criticizes; the second is evidence offered in support of the historian's own position.

'The first is evidence that has been used to support a conclusion that the historian criticizes' - ✓ CORRECT - The timing evidence (boldface 1) is used by scholars to conclude copying occurred, which the historian then criticizes as too strong
'the second is evidence offered in support of the historian's own position' - ✓ CORRECT - The information about limited note coverage (boldface 2) directly supports the historian's call for a more cautious conclusion

The first is evidence that has been used to support a conclusion that the historian criticizes; the second is further information that substantiates that evidence.

'The first is evidence that has been used to support a conclusion that the historian criticizes' - ✓ CORRECT - This part accurately describes boldface 1's role
'the second is further information that substantiates that evidence' - ✗ WRONG - The second boldface actually undermines the first evidence rather than substantiating it

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