LNAT Practice Passage 3: The Triangle on the Ground
Read the passage below and answer the three LNAT-style multiple-choice questions. Full answers and explanations follow.
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The Triangle on the Ground
The distinction between a priori and a posteriori knowledge is most closely associated with Immanuel Kant, who gave it its most influential and systematic modern form in the Critique of Pure Reason, published in 1781. Knowledge is a priori when its justification does not depend upon experience; it is a posteriori when observation is required. This concerns the basis on which a claim is known, not the date on which somebody first learns it. A child may discover arithmetic by counting apples, but the apples do not become evidence for every subsequent calculation.
Why is geometry the favourite exhibit? Euclid proved that the interior angles of a triangle equal two right angles, or 180 degrees, without inspecting a collection of triangular objects. Once his definitions, axioms and propositions are granted, the conclusion follows. It is consequently presented as a model of a priori knowledge. So far, so tidy. The trouble begins when "a triangle" ceases to mean the figure defined within Euclidean geometry and becomes three routes marked upon the Earth.
Take a journey from the North Pole down a meridian to the equator, travel east along the equator for one quarter of its circumference, and then return to the Pole along another meridian. Each part follows what counts on a sphere as the straightest available route. The three corners can each form a right angle, giving a total of 270 degrees. No contradiction has been discovered in Euclid, because his proof concerns a plane governed by Euclidean axioms, not every surface upon which human beings may draw three lines. Yet the schoolroom theorem is often carried into the physical world as though logical necessity came attached to the word "triangle". Whether the ground, the globe or space itself is adequately represented by Euclidean geometry cannot be decided by repeating Euclid's proof. Measurement and physical investigation are required, which makes that further claim a posteriori.
Albert Einstein put the distinction sharply in his 1921 lecture Geometry and Experience: insofar as mathematical laws are certain, they do not refer to reality; insofar as they refer to reality, they are not certain. The remark is easily exaggerated. Einstein was not announcing that mathematics becomes unreliable whenever a surveyor leaves his desk. He was separating deductions within a geometrical system from the empirical question of which system describes physical objects. A builder working across a small, nearly flat site may use Euclidean geometry with excellent results. Its practical accuracy there has nevertheless been learned from the behaviour of the world.
Observing a triangle with angles exceeding 180 degrees therefore does not convert Euclid's proposition into an a posteriori one or prove it false. It shows that two propositions have been carelessly joined: a theorem about Euclidean triangles and a claim that an actual surface is Euclidean. The former is established by proof. The latter requires experience, however obvious it may have appeared before anyone bothered to measure.
Questions
Question 1
According to the passage, what could measurements of a triangle drawn over part of the Earth establish?
- AWhether Euclid's proof contains a contradiction
- BWhether geometrical knowledge can ever be certain without repeated observation
- CWhether the word "triangle" should be confined to figures drawn on a plane
- DWhether Euclidean geometry adequately describes that part of the physical world
Question 2
The child counting apples is used mainly to show that:
- Aexperience can be involved in learning arithmetic
- Blearning and justification are different matters
- Cmathematical knowledge can begin with observation
- Dlater calculations need not use physical objects
Question 3
Which quotation, in its context, most clearly supports the proposition that a mathematically valid theorem need not describe every physical triangle?
- A"Why is geometry the favourite exhibit?"
- B"Once his definitions, axioms and propositions are granted, the conclusion follows."
- C"The three corners can each form a right angle, giving a total of 270 degrees."
- D"A builder working across a small, nearly flat site may use Euclidean geometry with excellent results."
Answers & explanations
Question 1: show answer and explanation
Correct answer: D.
Whether Euclidean geometry adequately describes that part of the physical world
Measurement is required to determine whether Euclidean geometry accurately represents a particular part of the physical world. Trap: Option A confuses testing the physical applicability of a theorem with testing the internal validity of its proof.
- A Measurements of physical objects cannot reveal a contradiction in a proof that follows from stipulated Euclidean axioms.
- B The passage allows geometrical conclusions to be established a priori within a system. Repeated observation is relevant to whether that system describes physical space.
- C The writer distinguishes Euclidean triangles from triangles formed on curved surfaces but does not reserve the word exclusively for plane figures.
- D Measurement is required to determine whether Euclidean geometry accurately represents a particular part of the physical world.
Question 2: show answer and explanation
Correct answer: B.
learning and justification are different matters
The sentence immediately before the example says that the distinction concerns the basis on which a claim is known, not when or how somebody first learns it. The apples illustrate that learning something through experience does not mean experience supplies its ultimate justification. Trap: When a question asks why an example is used, a statement can be true of the example and still be wrong if another option identifies its argumentative purpose more precisely.
- A This is true as a description of the example: the child encounters arithmetic through experience. But it tells us what happens in the example, not the distinction the writer uses the example to establish.
- B Correct. The sentence immediately before the example says that the distinction concerns the basis on which a claim is known, not when or how somebody first learns it. The apples illustrate that learning something through experience does not mean experience supplies its ultimate justification.
- C This is also plausible because the child may first discover arithmetic through counting apples. Again, it concerns how knowledge may first be acquired rather than what ultimately justifies the mathematical claim.
- D The passage says the apples do not become evidence for every subsequent calculation, so this is supported. But it is a consequence of the broader distinction between learning and justification identified in B.
Question 3: show answer and explanation
Correct answer: C.
"The three corners can each form a right angle, giving a total of 270 degrees."
The spherical triangle has three right angles and therefore supplies the concrete case in which a physical triangle does not have the Euclidean total of 180 degrees. Trap: Option B supports the theorem's a priori status and is therefore relevant to one half of the proposition. Option C supplies the missing contrast between that valid theorem and a triangle on a curved physical surface.
- A This introduces geometry as the example but says nothing about the relationship between a theorem and physical triangles.
- B This explains why the Euclidean theorem is valid within its axiomatic system. It does not show that a physical triangle may behave differently.
- C The spherical triangle has three right angles and therefore supplies the concrete case in which a physical triangle does not have the Euclidean total of 180 degrees.
- D The builder's success shows that Euclidean geometry may describe some physical situations accurately. It does not provide the counterexample requested.