full transcript

From the Ted Talk by Jeff Dekofsky: Is math discovered or invented?

Unscramble the Blue Letters

Would mathematics esixt if people didn't? Since ancient times, mankind has hotly debated whether mathematics was discovered or ieetvnnd. Did we create mathematical concepts to help us understand the universe around us, or is math the naitve laungage of the universe itself, existing whether we find its trtuhs or not? Are numbers, polygons and equations truly real, or merely ethereal representations of some theoretical ideal? The independent reality of math has some ancient advocates. The Pythagoreans of 5th Century Greece believed numbers were both lnviig entities and universal principles. They called the number one, "the monad," the generator of all other numbers and source of all creation. Numbers were active atengs in nature. Plato argued mathematical concepts were coctrene and as real as the uisnerve itself, regardless of our knowledge of them. Euclid, the father of geometry, believed nature itself was the physical mentsifaoaitn of meathiacmtal laws. Others argue that while numbers may or may not exist physically, mathematical statements definitely don't. Their trtuh values are based on rleus that hamuns created. Mathematics is thus an invented logic exercise, with no existence outside mankind's conscious thought, a language of abstract risoltianhpes based on patterns discerned by brains, bilut to use those patterns to invent useful but artificial order from chaos. One proponent of this sort of idea was Leopold Kronecker, a professor of mathematics in 19th century Germany. His belief is summed up in his famous statement: "God created the natural numbers, all else is the work of man." During maitthimeacan David Hilbert's lifetime, there was a push to establish mathematics as a logical construct. Hilbert apmteettd to axiomatize all of mathematics, as Euclid had done with geometry. He and others who attempted this saw mathematics as a deeply philosophical game but a game nonetheless. Henri poacrnié, one of the father's of non-Euclidean geometry, beleevid that the existence of non-Euclidean geometry, daienlg with the non-flat surfaces of hyperbolic and elliptical curvatures, proved that Euclidean geometry, the long standing geometry of flat surfaces, was not a universal truth, but rather one outcome of using one particular set of game rules. But in 1960, Nobel Physics laureate euegne Wigner coined the phsare, "the unreasonable effectiveness of mathematics," pushing slgtrony for the idea that mictamtehas is real and deivsorecd by people. winger ptoenid out that many pleruy mathematical theories developed in a vacuum, often with no view towards describing any pyaihcsl pmeneonha, have proven decades or even centuries later, to be the framework necessary to explain how the universe has been wionrkg all along. For instance, the number theory of British mathematician Gottfried Hardy, who had boasted that none of his work would ever be found useful in describing any phenomena in the real world, helped eabilstsh cryptography. Another piece of his purely theoretical work became known as the Hardy-Weinberg law in genetics, and won a Nobel pzrie. And Fibonacci stumbled upon his famous sequence while looking at the growth of an idealized rabbit population. Mankind later found the sequence everywhere in nature, from sunflower seeds and flower petal arrangements, to the structure of a pineapple, even the branching of bronchi in the lungs. Or there's the non-Euclidean work of Bernhard Riemann in the 1850s, which Einstein used in the model for general riievtatly a century later. Here's an even bigger jump: mathematical knot theory, first developed around 1771 to describe the geometry of position, was used in the late 20th century to explain how DNA unravels itself during the replication pcresos. It may even provide key elxpinatnaos for string treohy. Some of the most influential mathematicians and scientists of all of human history have chimed in on the issue as well, often in surprising ways. So, is mathematics an invention or a discovery? arfaiciitl construct or universal truth? Human pdrcuot or natural, possibly dinvie, creation? These questions are so deep the dabtee often becomes spiritual in nature. The answer might depend on the specific concept being looked at, but it can all feel like a distorted zen koan. If there's a number of trees in a forest, but no one's there to count them, does that number exist?

Open Cloze

Would mathematics _____ if people didn't? Since ancient times, mankind has hotly debated whether mathematics was discovered or ________. Did we create mathematical concepts to help us understand the universe around us, or is math the ______ ________ of the universe itself, existing whether we find its ______ or not? Are numbers, polygons and equations truly real, or merely ethereal representations of some theoretical ideal? The independent reality of math has some ancient advocates. The Pythagoreans of 5th Century Greece believed numbers were both ______ entities and universal principles. They called the number one, "the monad," the generator of all other numbers and source of all creation. Numbers were active ______ in nature. Plato argued mathematical concepts were ________ and as real as the ________ itself, regardless of our knowledge of them. Euclid, the father of geometry, believed nature itself was the physical _____________ of ____________ laws. Others argue that while numbers may or may not exist physically, mathematical statements definitely don't. Their _____ values are based on _____ that ______ created. Mathematics is thus an invented logic exercise, with no existence outside mankind's conscious thought, a language of abstract _____________ based on patterns discerned by brains, _____ to use those patterns to invent useful but artificial order from chaos. One proponent of this sort of idea was Leopold Kronecker, a professor of mathematics in 19th century Germany. His belief is summed up in his famous statement: "God created the natural numbers, all else is the work of man." During _____________ David Hilbert's lifetime, there was a push to establish mathematics as a logical construct. Hilbert _________ to axiomatize all of mathematics, as Euclid had done with geometry. He and others who attempted this saw mathematics as a deeply philosophical game but a game nonetheless. Henri ________, one of the father's of non-Euclidean geometry, ________ that the existence of non-Euclidean geometry, _______ with the non-flat surfaces of hyperbolic and elliptical curvatures, proved that Euclidean geometry, the long standing geometry of flat surfaces, was not a universal truth, but rather one outcome of using one particular set of game rules. But in 1960, Nobel Physics laureate ______ Wigner coined the ______, "the unreasonable effectiveness of mathematics," pushing ________ for the idea that ___________ is real and __________ by people. ______ _______ out that many ______ mathematical theories developed in a vacuum, often with no view towards describing any ________ _________, have proven decades or even centuries later, to be the framework necessary to explain how the universe has been _______ all along. For instance, the number theory of British mathematician Gottfried Hardy, who had boasted that none of his work would ever be found useful in describing any phenomena in the real world, helped _________ cryptography. Another piece of his purely theoretical work became known as the Hardy-Weinberg law in genetics, and won a Nobel _____. And Fibonacci stumbled upon his famous sequence while looking at the growth of an idealized rabbit population. Mankind later found the sequence everywhere in nature, from sunflower seeds and flower petal arrangements, to the structure of a pineapple, even the branching of bronchi in the lungs. Or there's the non-Euclidean work of Bernhard Riemann in the 1850s, which Einstein used in the model for general __________ a century later. Here's an even bigger jump: mathematical knot theory, first developed around 1771 to describe the geometry of position, was used in the late 20th century to explain how DNA unravels itself during the replication _______. It may even provide key ____________ for string ______. Some of the most influential mathematicians and scientists of all of human history have chimed in on the issue as well, often in surprising ways. So, is mathematics an invention or a discovery? __________ construct or universal truth? Human _______ or natural, possibly ______, creation? These questions are so deep the ______ often becomes spiritual in nature. The answer might depend on the specific concept being looked at, but it can all feel like a distorted zen koan. If there's a number of trees in a forest, but no one's there to count them, does that number exist?

Solution

  1. explanations
  2. universe
  3. phrase
  4. purely
  5. language
  6. debate
  7. truth
  8. humans
  9. truths
  10. agents
  11. native
  12. strongly
  13. built
  14. discovered
  15. working
  16. artificial
  17. establish
  18. poincaré
  19. wigner
  20. manifestation
  21. mathematical
  22. living
  23. relativity
  24. pointed
  25. believed
  26. physical
  27. eugene
  28. relationships
  29. rules
  30. mathematics
  31. prize
  32. divine
  33. process
  34. theory
  35. product
  36. mathematician
  37. phenomena
  38. dealing
  39. attempted
  40. concrete
  41. exist
  42. invented

Original Text

Would mathematics exist if people didn't? Since ancient times, mankind has hotly debated whether mathematics was discovered or invented. Did we create mathematical concepts to help us understand the universe around us, or is math the native language of the universe itself, existing whether we find its truths or not? Are numbers, polygons and equations truly real, or merely ethereal representations of some theoretical ideal? The independent reality of math has some ancient advocates. The Pythagoreans of 5th Century Greece believed numbers were both living entities and universal principles. They called the number one, "the monad," the generator of all other numbers and source of all creation. Numbers were active agents in nature. Plato argued mathematical concepts were concrete and as real as the universe itself, regardless of our knowledge of them. Euclid, the father of geometry, believed nature itself was the physical manifestation of mathematical laws. Others argue that while numbers may or may not exist physically, mathematical statements definitely don't. Their truth values are based on rules that humans created. Mathematics is thus an invented logic exercise, with no existence outside mankind's conscious thought, a language of abstract relationships based on patterns discerned by brains, built to use those patterns to invent useful but artificial order from chaos. One proponent of this sort of idea was Leopold Kronecker, a professor of mathematics in 19th century Germany. His belief is summed up in his famous statement: "God created the natural numbers, all else is the work of man." During mathematician David Hilbert's lifetime, there was a push to establish mathematics as a logical construct. Hilbert attempted to axiomatize all of mathematics, as Euclid had done with geometry. He and others who attempted this saw mathematics as a deeply philosophical game but a game nonetheless. Henri Poincaré, one of the father's of non-Euclidean geometry, believed that the existence of non-Euclidean geometry, dealing with the non-flat surfaces of hyperbolic and elliptical curvatures, proved that Euclidean geometry, the long standing geometry of flat surfaces, was not a universal truth, but rather one outcome of using one particular set of game rules. But in 1960, Nobel Physics laureate Eugene Wigner coined the phrase, "the unreasonable effectiveness of mathematics," pushing strongly for the idea that mathematics is real and discovered by people. Wigner pointed out that many purely mathematical theories developed in a vacuum, often with no view towards describing any physical phenomena, have proven decades or even centuries later, to be the framework necessary to explain how the universe has been working all along. For instance, the number theory of British mathematician Gottfried Hardy, who had boasted that none of his work would ever be found useful in describing any phenomena in the real world, helped establish cryptography. Another piece of his purely theoretical work became known as the Hardy-Weinberg law in genetics, and won a Nobel prize. And Fibonacci stumbled upon his famous sequence while looking at the growth of an idealized rabbit population. Mankind later found the sequence everywhere in nature, from sunflower seeds and flower petal arrangements, to the structure of a pineapple, even the branching of bronchi in the lungs. Or there's the non-Euclidean work of Bernhard Riemann in the 1850s, which Einstein used in the model for general relativity a century later. Here's an even bigger jump: mathematical knot theory, first developed around 1771 to describe the geometry of position, was used in the late 20th century to explain how DNA unravels itself during the replication process. It may even provide key explanations for string theory. Some of the most influential mathematicians and scientists of all of human history have chimed in on the issue as well, often in surprising ways. So, is mathematics an invention or a discovery? Artificial construct or universal truth? Human product or natural, possibly divine, creation? These questions are so deep the debate often becomes spiritual in nature. The answer might depend on the specific concept being looked at, but it can all feel like a distorted zen koan. If there's a number of trees in a forest, but no one's there to count them, does that number exist?

Frequently Occurring Word Combinations

ngrams of length 2

collocation frequency
mathematical concepts 2

Important Words

  1. abstract
  2. active
  3. advocates
  4. agents
  5. ancient
  6. answer
  7. argue
  8. argued
  9. arrangements
  10. artificial
  11. attempted
  12. axiomatize
  13. based
  14. belief
  15. believed
  16. bernhard
  17. bigger
  18. boasted
  19. brains
  20. branching
  21. british
  22. bronchi
  23. built
  24. called
  25. centuries
  26. century
  27. chaos
  28. chimed
  29. coined
  30. concept
  31. concepts
  32. concrete
  33. conscious
  34. construct
  35. count
  36. create
  37. created
  38. creation
  39. cryptography
  40. curvatures
  41. david
  42. dealing
  43. debate
  44. debated
  45. decades
  46. deep
  47. deeply
  48. depend
  49. describe
  50. describing
  51. developed
  52. discerned
  53. discovered
  54. discovery
  55. distorted
  56. divine
  57. dna
  58. effectiveness
  59. einstein
  60. elliptical
  61. entities
  62. equations
  63. establish
  64. ethereal
  65. euclid
  66. euclidean
  67. eugene
  68. exercise
  69. exist
  70. existence
  71. existing
  72. explain
  73. explanations
  74. famous
  75. father
  76. feel
  77. fibonacci
  78. find
  79. flat
  80. flower
  81. forest
  82. framework
  83. game
  84. general
  85. generator
  86. genetics
  87. geometry
  88. germany
  89. gottfried
  90. greece
  91. growth
  92. hardy
  93. helped
  94. henri
  95. hilbert
  96. history
  97. hotly
  98. human
  99. humans
  100. hyperbolic
  101. idea
  102. ideal
  103. idealized
  104. independent
  105. influential
  106. instance
  107. invent
  108. invented
  109. invention
  110. issue
  111. key
  112. knot
  113. knowledge
  114. koan
  115. kronecker
  116. language
  117. late
  118. laureate
  119. law
  120. laws
  121. leopold
  122. lifetime
  123. living
  124. logic
  125. logical
  126. long
  127. looked
  128. lungs
  129. man
  130. manifestation
  131. mankind
  132. math
  133. mathematical
  134. mathematician
  135. mathematicians
  136. mathematics
  137. model
  138. monad
  139. native
  140. natural
  141. nature
  142. nobel
  143. number
  144. numbers
  145. order
  146. outcome
  147. patterns
  148. people
  149. petal
  150. phenomena
  151. philosophical
  152. phrase
  153. physical
  154. physically
  155. physics
  156. piece
  157. pineapple
  158. plato
  159. poincaré
  160. pointed
  161. polygons
  162. population
  163. position
  164. possibly
  165. principles
  166. prize
  167. process
  168. product
  169. professor
  170. proponent
  171. proved
  172. proven
  173. provide
  174. purely
  175. push
  176. pushing
  177. pythagoreans
  178. questions
  179. rabbit
  180. real
  181. reality
  182. relationships
  183. relativity
  184. replication
  185. representations
  186. riemann
  187. rules
  188. scientists
  189. seeds
  190. sequence
  191. set
  192. sort
  193. source
  194. specific
  195. spiritual
  196. standing
  197. statements
  198. string
  199. strongly
  200. structure
  201. stumbled
  202. summed
  203. sunflower
  204. surfaces
  205. surprising
  206. theoretical
  207. theories
  208. theory
  209. thought
  210. times
  211. trees
  212. truth
  213. truths
  214. understand
  215. universal
  216. universe
  217. unravels
  218. unreasonable
  219. vacuum
  220. values
  221. view
  222. ways
  223. wigner
  224. won
  225. work
  226. working
  227. world
  228. zen