Intuitionism in mathematics, developed by Brouwer, rejects completed infinities and non-constructive proofs. For intuitionists, proving 'there exists an x such that P(x)' requires constructing such an x, not merely showing non-existence leads to contradiction. This rejects the law of excluded middle for infinite domains—'either every even number is the sum of two primes or there's one that isn't' isn't necessarily true until proven or disproven. Classical mathematicians resist these restrictions on powerful proof techniques.
The passage suggests that
all mathematicians accept intuitionism
foundational positions in mathematics may have implications for which proof methods are considered valid
the law of excluded middle is accepted in intuitionistic logic for infinite domains
constructing objects is never required for existence proofs in any mathematical framework
Correct Answer: B
Choice B is the best answer. Intuitionism rejects non-constructive proofs; classicism accepts them.
- Context clues: Intuitionists require construction; classical mathematicians use "powerful proof techniques" intuitionists reject.
- Meaning: Different foundations validate different methods.
- Verify: The rejection of classical excluded middle shows proof acceptability varies by foundation.
đź’ˇ Strategy: When foundational positions disagree about valid methods, infer foundation affects method validity.
Choice A is incorrect because classical mathematicians "resist these restrictions." Choice C is incorrect because intuitionism rejects excluded middle for infinite domains. Choice D is incorrect because intuitionism precisely requires construction.