Susan D'Agostino
- Published in print:
- 2020
- Published Online:
- April 2020
- ISBN:
- 9780198843597
- eISBN:
- 9780191879388
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198843597.003.0029
- Subject:
- Mathematics, Educational Mathematics, Applied Mathematics
“Be contradictory, because of the infinitude of primes” offers encouragement and practice with the “proof-by-contradiction” method of mathematical proof. Any mathematician will tell you that the ...
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“Be contradictory, because of the infinitude of primes” offers encouragement and practice with the “proof-by-contradiction” method of mathematical proof. Any mathematician will tell you that the collection of prime numbers is infinite. However, readers are guided in proving this statement by contradicting it. The activity pushes readers to engage deeply with the reasons supporting the fact that there are an infinite number of primes. Mathematics students and enthusiasts are encouraged to debate with enthusiasm in their mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.Less
“Be contradictory, because of the infinitude of primes” offers encouragement and practice with the “proof-by-contradiction” method of mathematical proof. Any mathematician will tell you that the collection of prime numbers is infinite. However, readers are guided in proving this statement by contradicting it. The activity pushes readers to engage deeply with the reasons supporting the fact that there are an infinite number of primes. Mathematics students and enthusiasts are encouraged to debate with enthusiasm in their mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.
Jody Azzouni
- Published in print:
- 2020
- Published Online:
- October 2020
- ISBN:
- 9780197508817
- eISBN:
- 9780197508848
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780197508817.003.0013
- Subject:
- Philosophy, Metaphysics/Epistemology
The hangman/surprise-examination/prediction paradox is solved. It is not solved by denying knowledge closure (although knowledge closure is false). It is not solved by denying KK or denying that ...
More
The hangman/surprise-examination/prediction paradox is solved. It is not solved by denying knowledge closure (although knowledge closure is false). It is not solved by denying KK or denying that knowing p implies other iterated knowing attitudes (although these are false). It is not solved by misleading evidence causing the students to lose knowledge because students cannot lose knowledge this way. It is solved by showing that a tacit assumption (what is being said to the students/prisoner is informative) is overlooked and that inferences by contradiction are invalid if assumptions are left out. The phenomenology of the surprise-exam paradox is explored to explain why this solution has been missed. Crucial is that in many cases the students/prisoner know(s) there will be a surprise exam/execution because of an inference from what the teacher/judge meant to say, and not directly by the literal application of what he did say.Less
The hangman/surprise-examination/prediction paradox is solved. It is not solved by denying knowledge closure (although knowledge closure is false). It is not solved by denying KK or denying that knowing p implies other iterated knowing attitudes (although these are false). It is not solved by misleading evidence causing the students to lose knowledge because students cannot lose knowledge this way. It is solved by showing that a tacit assumption (what is being said to the students/prisoner is informative) is overlooked and that inferences by contradiction are invalid if assumptions are left out. The phenomenology of the surprise-exam paradox is explored to explain why this solution has been missed. Crucial is that in many cases the students/prisoner know(s) there will be a surprise exam/execution because of an inference from what the teacher/judge meant to say, and not directly by the literal application of what he did say.
