A Proof Checked by Computer
Written by Studio AM.
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The four-color theorem concerns maps drawn on a plane. If each region is connected, four colors suffice to distinguish regions sharing a boundary segment. Regions meeting only at a point do not count as neighbors. The statement is simple; establishing it required a much more elaborate argument.
In 1976, Kenneth Appel and Wolfgang Haken announced a proof that relied on extensive computer calculations. Mathematical reasoning reduced the problem to a finite collection of configurations, and programs checked crucial cases. The scale of that checking made ordinary line-by-line human inspection impractical.
This raised a question about surveyability: how far a proof should be open to checking in full by a human reader. If acceptance depended partly on a program running correctly, did mathematical knowledge begin to resemble knowledge from an experiment? The comparison concerned the basis of confidence, not whether colored maps were being physically measured.
In 2005, Georges Gonthier reported a formal proof checked with the Coq proof assistant. Formal statements and deductions could be verified according to explicit logical rules, including the computational portions. This moved more of the checking into a systematic framework while retaining questions about the formal statement and trusted software.
The development broadened the discussion of proof. Human understanding, independent review, and formal verification offer different kinds of assurance; examining how they work together is more useful than treating either humans or computers as incapable of error.
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Questions
Choose an answer. The explanation appears after you answer.
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Question 1 of 4
What is the passage mainly about?
The answer is C: Computer-assisted proofs made mathematicians reconsider how a proof earns trust.
The passage follows how extensive computation and later formal verification changed discussion of proof and confidence.
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Question 2 of 4
Why does the passage compare accepting a computer-assisted proof to accepting an experimental result?
The answer is A: Because belief in it rested on trusting equipment to have worked correctly, not on following the reasoning oneself.
The analogy concerns relying on a correctly functioning apparatus for part of the warrant, rather than following all computation manually.
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Question 3 of 4
What does “surveyability” mean in this passage?
The answer is D: the property of being checkable in full by a human reader
The third paragraph defines surveyability as how far a human reader can check a proof in full.
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Question 4 of 4
What did the 2005 proof assistant do, according to the passage?
The answer is B: It checked formal deductions according to explicit logical rules.
The fourth paragraph identifies rule-based checking of the formal proof, including its computational portions.
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