Paradox                         魏澤人                     國立東華大學應用數學學系                       2013-04-13魏澤人 (國立東華大學應用數學學系)   ...
1Proof.                       1        −1                           =                      −1        1                    ...
2Proof.                               6           3    3            −1 = (−1)3 = (−1) 2 = ((−1)2 ) 2 = 1 2 = 1魏澤人 (國立東華大學應...
3    任何一個人的團體,所有人頭髮數量一定一樣多。    假設所有 n 個人的團體,都必然每個人有相同數量頭髮。    考慮一個 n + 1 人團體,編號為 1, 2, . . . , n + 1    1, 2, . . . , n 每個...
4證明 a = b for every a, b ∈ N對 max (a, b) 做數學歸納法case max (a, b) = 0: a = 0 = b, done.設                         max (a, b) =...
5    1   設 BC 中點 D    2   設 ∠A 平分線及 BC 中垂線交於O (否則 AB = AC )    3   做 OR⊥AB, OQ⊥AC    4   由 AAS, RAO ∼ QAO                 ...
Braess’s ParadoxWithout ——————-Total 4000 cars.                         ARoute Start->A->END: 100 + 45                    ...
Braess’s ParadoxWithout ——————-Total 4000 cars.                         ARoute Start->A->END: 100 + 45                    ...
Braess’s ParadoxWith —————–Total 4000 cars.                   XRoute Start->A: 100 < 45                   XRoute B->END: 1...
Zeno’s paradoxes   Achilles and the tortoise   The dichotomy paradox   The arrow paradox魏澤人 (國立東華大學應用數學學系)             Par...
Zeno’s paradoxes   Achilles and the tortoise   The dichotomy paradox   The arrow paradox魏澤人 (國立東華大學應用數學學系)             Par...
Zeno’s paradoxes   Achilles and the tortoise   The dichotomy paradox   The arrow paradox魏澤人 (國立東華大學應用數學學系)             Par...
Zeno’s paradoxes   Achilles and the tortoise   The dichotomy paradox   The arrow paradox魏澤人 (國立東華大學應用數學學系)             Par...
Thomson’s Lamp  time    Status  2:00      on  3:00      off  3:30      on  3:45      off 3:52.5     on    ...  4:00      ?魏澤...
Infinite    Hilbert’s paradox of the Grand Hotel    Squares and Natural numbers    Balls and vase problem魏澤人 (國立東華大學應用數學學系)...
Infinite    Hilbert’s paradox of the Grand Hotel    Squares and Natural numbers    Balls and vase problem魏澤人 (國立東華大學應用數學學系)...
Infinite    Hilbert’s paradox of the Grand Hotel    Squares and Natural numbers    Balls and vase problem魏澤人 (國立東華大學應用數學學系)...
玫瑰小木中午前 1 分鐘,放 10 個球到花瓶中,然後拿出一個。中午前1/2分鐘,放 10 個球到花瓶中,然後拿出一個。中午前1/4分鐘,放 10 個球到花瓶中,然後拿出一個。...中午時,花瓶有幾顆球?魏澤人 (國立東華大學應用數學學系)  ...
玫瑰小木球編號 1, 2, 3, ...中午前 1 分鐘,放 1-10號球到花瓶中,然後拿出 1 號。中午前1/2分鐘,放 11-20 個球到花瓶中,然後拿出 2 號。中午前1/4分鐘,放 21-30 個球到花瓶中,然後拿出 3 號。...中午...
Banach–Tarski paradox魏澤人 (國立東華大學應用數學學系)      Paradox   2013-04-13   14 / 30
Voter’s Paradox2                  23   think A > B,   3   think B > C , then A > C ?      Voter 1: A > B > C      Voter 2:...
Voter’s Paradox2                  23   think A > B,   3   think B > C , then A > C ?      Voter 1: A > B > C      Voter 2:...
Arrow’s ParadoxIt is impossible to satisfies     Universal Domain : Contain all possible choices.     Irrelevant Alternativ...
Arrow’s ParadoxIt is impossible to satisfies     Universal Domain : Contain all possible choices.     Irrelevant Alternativ...
Arrow’s ParadoxIt is impossible to satisfies     Universal Domain : Contain all possible choices.     Irrelevant Alternativ...
Arrow’s ParadoxIt is impossible to satisfies     Universal Domain : Contain all possible choices.     Irrelevant Alternativ...
Arrow’s ParadoxIt is impossible to satisfies     Universal Domain : Contain all possible choices.     Irrelevant Alternativ...
錢包A和B兩人進行一場賭博。賭法是:由第三者計算A、B二君錢包裡面的錢,錢少者可以贏走錢多者的錢。A對於這場賭博的想法為:若B君的錢比我少,我可能輸掉我現有的錢。但若B君的錢比我多,我贏了,就會得到多於我現有的錢。我能夠贏的錢比輸的錢多,所以這...
信封兩個外觀一樣的信封,都放了錢,一個裡面的金額是另外一個的兩倍。你隨機的拿起了一個,然後給你一個機會來交換,你是否要交換?魏澤人 (國立東華大學應用數學學系)   Paradox   2013-04-13   18 / 30
Barber Paradox   There is a town with just one male barber.   Every man in the town keeps himself clean-shaven       some ...
Barber Paradox   There is a town with just one male barber.   Every man in the town keeps himself clean-shaven       some ...
Barber Paradox   There is a town with just one male barber.   Every man in the town keeps himself clean-shaven       some ...
Barber Paradox   There is a town with just one male barber.   Every man in the town keeps himself clean-shaven       some ...
Barber Paradox   The situation presented is in fact impossible:       If the he does not shave himself, he must shave hims...
Barber Paradox   The situation presented is in fact impossible:       If the he does not shave himself, he must shave hims...
Liar’s Paradox    Epimenides paradox: A Cretan says: "All Cretans are liars".    Liar paradox: "This sentence is false." o...
Liar’s Paradox    Epimenides paradox: A Cretan says: "All Cretans are liars".    Liar paradox: "This sentence is false." o...
Self Reference Paradoxes   Berry paradox: "the first number not nameable in under ten words".       computability and compl...
Self Reference Paradoxes   Berry paradox: "the first number not nameable in under ten words".       computability and compl...
Self Reference Paradoxes   Berry paradox: "the first number not nameable in under ten words".       computability and compl...
Self Reference Paradoxes   Berry paradox: "the first number not nameable in under ten words".       computability and compl...
More Self Reference Paradoxes   Opposite Day: "It is opposite day today."   Paradox of the Court: A law student agrees to ...
聖彼得堡擲硬幣,若第一次擲出正面,你就賺1元。若第一次擲出反面,那就要再擲一次,若第二次擲的是正面,你便賺2元。若第二次擲出反面,那就要擲第三次,若第三次擲的是正面,你便賺2*2元. . . . . . 如此類推,即可能擲一次遊戲便結束,也可能...
Penney Ante   HHH         TTT   HHT         TTH   HTH         THT   HTT         THH魏澤人 (國立東華大學應用數學學系)   Paradox   2013-04-...
Simpson’s Paradox                   Men                       Women Dept A   250/500, 50% accepted        9/10, 90% accept...
Simpson’s Paradox                 Treatment A     Treatment B     Both       78% (273/350)   83% (289/350) Small Stones   ...
UC Berkeley 1973         Applicants   % admitted  Men      8442          44% Women     4321          35%魏澤人 (國立東華大學應用數學學系)...
UC Berkeley 1973 Major   Men    %    Women    %  A      825   62%    108    82%  B      560   63%    25     68%  C      32...
Related Paradox   Low birth weight paradox   Batting average魏澤人 (國立東華大學應用數學學系)            Paradox   2013-04-13   30 / 30
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Imo talk2

  1. 1. Paradox 魏澤人 國立東華大學應用數學學系 2013-04-13魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 1 / 30
  2. 2. 1Proof. 1 −1 = −1 1 1 −1 = −1 1 √ √ 1 −1 √ = √ −1 1 √ √ √ √ 1 1 = −1 −1 1 = −1魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 2 / 30
  3. 3. 2Proof. 6 3 3 −1 = (−1)3 = (−1) 2 = ((−1)2 ) 2 = 1 2 = 1魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 3 / 30
  4. 4. 3 任何一個人的團體,所有人頭髮數量一定一樣多。 假設所有 n 個人的團體,都必然每個人有相同數量頭髮。 考慮一個 n + 1 人團體,編號為 1, 2, . . . , n + 1 1, 2, . . . , n 每個人頭髮數量相同。 2, . . . , n, n + 1 每個人頭髮數量相同 所以 1, 2, . . . , n, n + 1 每個人的頭髮都相同魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 4 / 30
  5. 5. 4證明 a = b for every a, b ∈ N對 max (a, b) 做數學歸納法case max (a, b) = 0: a = 0 = b, done.設 max (a, b) = n ⇒ a = b成立。當 max (a, b) = n + 1 時, max (a, b) = n + 1 ⇒ max (a − 1, b − 1) = n ⇒ a−1=b−1 ⇒ a=b魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 5 / 30
  6. 6. 5 1 設 BC 中點 D 2 設 ∠A 平分線及 BC 中垂線交於O (否則 AB = AC ) 3 做 OR⊥AB, OQ⊥AC 4 由 AAS, RAO ∼ QAO = 5 由 SAS, ODB ∼ ODC = 6 所以 ∼ QOC ROB = 7 得 AB = AC魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 6 / 30
  7. 7. Braess’s ParadoxWithout ——————-Total 4000 cars. ARoute Start->A->END: 100 + 45 BRoute Start->B->END: 100 + 45Best solution is A = B = 2000, Time = 2000 + 45 = 65. 100魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 7 / 30
  8. 8. Braess’s ParadoxWithout ——————-Total 4000 cars. ARoute Start->A->END: 100 + 45 BRoute Start->B->END: 100 + 45Best solution is A = B = 2000, Time = 2000 + 45 = 65. 100魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 7 / 30
  9. 9. Braess’s ParadoxWith —————–Total 4000 cars. XRoute Start->A: 100 < 45 XRoute B->END: 100 < 45Best solution is 100 + 4000 = 80. 4000 100魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 8 / 30
  10. 10. Zeno’s paradoxes Achilles and the tortoise The dichotomy paradox The arrow paradox魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
  11. 11. Zeno’s paradoxes Achilles and the tortoise The dichotomy paradox The arrow paradox魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
  12. 12. Zeno’s paradoxes Achilles and the tortoise The dichotomy paradox The arrow paradox魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
  13. 13. Zeno’s paradoxes Achilles and the tortoise The dichotomy paradox The arrow paradox魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
  14. 14. Thomson’s Lamp time Status 2:00 on 3:00 off 3:30 on 3:45 off 3:52.5 on ... 4:00 ?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 10 / 30
  15. 15. Infinite Hilbert’s paradox of the Grand Hotel Squares and Natural numbers Balls and vase problem魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 11 / 30
  16. 16. Infinite Hilbert’s paradox of the Grand Hotel Squares and Natural numbers Balls and vase problem魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 11 / 30
  17. 17. Infinite Hilbert’s paradox of the Grand Hotel Squares and Natural numbers Balls and vase problem魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 11 / 30
  18. 18. 玫瑰小木中午前 1 分鐘,放 10 個球到花瓶中,然後拿出一個。中午前1/2分鐘,放 10 個球到花瓶中,然後拿出一個。中午前1/4分鐘,放 10 個球到花瓶中,然後拿出一個。...中午時,花瓶有幾顆球?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 12 / 30
  19. 19. 玫瑰小木球編號 1, 2, 3, ...中午前 1 分鐘,放 1-10號球到花瓶中,然後拿出 1 號。中午前1/2分鐘,放 11-20 個球到花瓶中,然後拿出 2 號。中午前1/4分鐘,放 21-30 個球到花瓶中,然後拿出 3 號。...中午時,花瓶有幾顆球?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 13 / 30
  20. 20. Banach–Tarski paradox魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 14 / 30
  21. 21. Voter’s Paradox2 23 think A > B, 3 think B > C , then A > C ? Voter 1: A > B > C Voter 2: B > C > A Voter 3: C > A > B魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 15 / 30
  22. 22. Voter’s Paradox2 23 think A > B, 3 think B > C , then A > C ? Voter 1: A > B > C Voter 2: B > C > A Voter 3: C > A > B魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 15 / 30
  23. 23. Arrow’s ParadoxIt is impossible to satisfies Universal Domain : Contain all possible choices. Irrelevant Alternatives - An individual preference stated in binary terms shouldn’t affect his ordering of other preferences Pareto Optimality : If every one votes the same way, the society is decisive. Monotonicity+Non-imposition Nondictatorship : More than one person’s vote should be used.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
  24. 24. Arrow’s ParadoxIt is impossible to satisfies Universal Domain : Contain all possible choices. Irrelevant Alternatives - An individual preference stated in binary terms shouldn’t affect his ordering of other preferences Pareto Optimality : If every one votes the same way, the society is decisive. Monotonicity+Non-imposition Nondictatorship : More than one person’s vote should be used.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
  25. 25. Arrow’s ParadoxIt is impossible to satisfies Universal Domain : Contain all possible choices. Irrelevant Alternatives - An individual preference stated in binary terms shouldn’t affect his ordering of other preferences Pareto Optimality : If every one votes the same way, the society is decisive. Monotonicity+Non-imposition Nondictatorship : More than one person’s vote should be used.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
  26. 26. Arrow’s ParadoxIt is impossible to satisfies Universal Domain : Contain all possible choices. Irrelevant Alternatives - An individual preference stated in binary terms shouldn’t affect his ordering of other preferences Pareto Optimality : If every one votes the same way, the society is decisive. Monotonicity+Non-imposition Nondictatorship : More than one person’s vote should be used.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
  27. 27. Arrow’s ParadoxIt is impossible to satisfies Universal Domain : Contain all possible choices. Irrelevant Alternatives - An individual preference stated in binary terms shouldn’t affect his ordering of other preferences Pareto Optimality : If every one votes the same way, the society is decisive. Monotonicity+Non-imposition Nondictatorship : More than one person’s vote should be used.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
  28. 28. 錢包A和B兩人進行一場賭博。賭法是:由第三者計算A、B二君錢包裡面的錢,錢少者可以贏走錢多者的錢。A對於這場賭博的想法為:若B君的錢比我少,我可能輸掉我現有的錢。但若B君的錢比我多,我贏了,就會得到多於我現有的錢。我能夠贏的錢比輸的錢多,所以這場賭博對我有利。而B的想法也是如此。二人想法的邏輯都正確,但若認為二人的想法都正確,又將做出這場賭博對A、B二人都有利的錯誤結論。這顯然是一個悖論。魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 17 / 30
  29. 29. 信封兩個外觀一樣的信封,都放了錢,一個裡面的金額是另外一個的兩倍。你隨機的拿起了一個,然後給你一個機會來交換,你是否要交換?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 18 / 30
  30. 30. Barber Paradox There is a town with just one male barber. Every man in the town keeps himself clean-shaven some by shaving themselves some by attending the barber. The barber shaves all and only those men who do not shave themselves. seems reasonable. Does the barber shave himself?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
  31. 31. Barber Paradox There is a town with just one male barber. Every man in the town keeps himself clean-shaven some by shaving themselves some by attending the barber. The barber shaves all and only those men who do not shave themselves. seems reasonable. Does the barber shave himself?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
  32. 32. Barber Paradox There is a town with just one male barber. Every man in the town keeps himself clean-shaven some by shaving themselves some by attending the barber. The barber shaves all and only those men who do not shave themselves. seems reasonable. Does the barber shave himself?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
  33. 33. Barber Paradox There is a town with just one male barber. Every man in the town keeps himself clean-shaven some by shaving themselves some by attending the barber. The barber shaves all and only those men who do not shave themselves. seems reasonable. Does the barber shave himself?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
  34. 34. Barber Paradox The situation presented is in fact impossible: If the he does not shave himself, he must shave himself. If he does shave himself, he should not shave himself. Barber Paradox = Russell’s Paradox {x |x ∈ x }. Solution: modern set theory, ZFC. ZFC is the foundation of modern mathematics.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 20 / 30
  35. 35. Barber Paradox The situation presented is in fact impossible: If the he does not shave himself, he must shave himself. If he does shave himself, he should not shave himself. Barber Paradox = Russell’s Paradox {x |x ∈ x }. Solution: modern set theory, ZFC. ZFC is the foundation of modern mathematics.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 20 / 30
  36. 36. Liar’s Paradox Epimenides paradox: A Cretan says: "All Cretans are liars". Liar paradox: "This sentence is false." or "Is the answer to this question no?" Godel’s incomplete theorem: “This sentence can not be proved”. The sentence is true and can not be proved. No formal system is complete. Huge impact.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 21 / 30
  37. 37. Liar’s Paradox Epimenides paradox: A Cretan says: "All Cretans are liars". Liar paradox: "This sentence is false." or "Is the answer to this question no?" Godel’s incomplete theorem: “This sentence can not be proved”. The sentence is true and can not be proved. No formal system is complete. Huge impact.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 21 / 30
  38. 38. Self Reference Paradoxes Berry paradox: "the first number not nameable in under ten words". computability and complexity implication. Curry’s paradox: "If this sentence is true, the world will end in a week." logic and philosophy interests. Grelling-Nelson paradox: "heterological" := "not applicable to itself". Yablo’s paradox. A self reference paradoxes without self reference.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
  39. 39. Self Reference Paradoxes Berry paradox: "the first number not nameable in under ten words". computability and complexity implication. Curry’s paradox: "If this sentence is true, the world will end in a week." logic and philosophy interests. Grelling-Nelson paradox: "heterological" := "not applicable to itself". Yablo’s paradox. A self reference paradoxes without self reference.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
  40. 40. Self Reference Paradoxes Berry paradox: "the first number not nameable in under ten words". computability and complexity implication. Curry’s paradox: "If this sentence is true, the world will end in a week." logic and philosophy interests. Grelling-Nelson paradox: "heterological" := "not applicable to itself". Yablo’s paradox. A self reference paradoxes without self reference.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
  41. 41. Self Reference Paradoxes Berry paradox: "the first number not nameable in under ten words". computability and complexity implication. Curry’s paradox: "If this sentence is true, the world will end in a week." logic and philosophy interests. Grelling-Nelson paradox: "heterological" := "not applicable to itself". Yablo’s paradox. A self reference paradoxes without self reference.魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
  42. 42. More Self Reference Paradoxes Opposite Day: "It is opposite day today." Paradox of the Court: A law student agrees to pay his teacher after winning his first case. Quine’s paradox: "Yields a falsehood when appended to its own quotation" yields a falsehood when appended to its own quotation. Richard’s paradox:魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 23 / 30
  43. 43. 聖彼得堡擲硬幣,若第一次擲出正面,你就賺1元。若第一次擲出反面,那就要再擲一次,若第二次擲的是正面,你便賺2元。若第二次擲出反面,那就要擲第三次,若第三次擲的是正面,你便賺2*2元. . . . . . 如此類推,即可能擲一次遊戲便結束,也可能反覆擲沒完沒了。問題是,你最多肯付多少錢參加這個遊戲?魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 24 / 30
  44. 44. Penney Ante HHH TTT HHT TTH HTH THT HTT THH魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 25 / 30
  45. 45. Simpson’s Paradox Men Women Dept A 250/500, 50% accepted 9/10, 90% accepted Dept B 1/10, 10% accepted 100/500, 20% accepted Totals 251/510 accepted 109/510 accepted魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 26 / 30
  46. 46. Simpson’s Paradox Treatment A Treatment B Both 78% (273/350) 83% (289/350) Small Stones 93% (81/87) 87% (234/270) Large Stones 73% (192/263) 69% (55/80)魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 27 / 30
  47. 47. UC Berkeley 1973 Applicants % admitted Men 8442 44% Women 4321 35%魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 28 / 30
  48. 48. UC Berkeley 1973 Major Men % Women % A 825 62% 108 82% B 560 63% 25 68% C 325 37% 593 34% D 417 33% 375 35% E 191 28% 393 24% F 272 6% 341 7%魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 29 / 30
  49. 49. Related Paradox Low birth weight paradox Batting average魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 30 / 30

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