TedTest

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

Unscramble the Blue Letters

Would mathematics exist if people didn't? Since ancient times, mannkid has hotly debated whether mathematics was discovered or invented. Did we create mathematical cnpteocs to help us understand the universe around us, or is math the navtie language of the universe itself, enisxitg whether we find its truths or not? Are nbumres, polygons and equations truly real, or merely ethereal representations of some theoretical ideal? The independent reality of math has some ancient ateadovcs. The Pythagoreans of 5th Century gceere believed numbers were both lnivig entities and universal principles. They called the nmeubr one, "the monad," the generator of all other numbers and source of all craoietn. Numbers were active agents in nature. Plato argued mathematical concepts were concrete and as real as the uervsine 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 thhguot, a language of abstract raeilipsnhots 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 llooped kocnekrer, a professor of mathematics in 19th crteuny Germany. His belief is semmud up in his fuoams statement: "God cretaed the nautarl numbers, all else is the work of man." During mathematician David Hilbert's ltifeime, there was a push to establish mtcmeatiahs as a liogacl construct. Hilbert attempted to axiomatize all of mathematics, as Euclid had done with geometry. He and others who attempted this saw mathematics as a dleepy philosophical game but a game nonetheless. hneri Poincaré, one of the father's of non-Euclidean geometry, beiveeld that the existence of non-Euclidean gmetroey, dealing with the non-flat surfaces of hyperbolic and epatililcl curvatures, proved that elcauiedn geometry, the long snnitdag 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 sgtonrly 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 disrcneibg any physical phenomena, have proven decades or even ceineurts later, to be the framework necessary to explain how the universe has been working all along. For itascnne, 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 peice 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 fewolr petal arrangements, to the structure of a pineapple, even the branching of bronchi in the lungs. Or there's the non-Euclidean work of benrrahd Riemann in the 1850s, which Einstein used in the model for general relativity a century later. Here's an even begigr jump: mathematical knot tohrey, 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 sitciensts of all of human hstoriy have cehmid in on the issue as well, often in surprising ways. So, is mathematics an invention or a discovery? Artificial construct or universal trtuh? 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 spicefic concept being looked at, but it can all feel like a distorted zen koan. If there's a number of trees in a foerst, but no one's there to count them, does that number exist?

Open Cloze

Would mathematics exist if people didn't? Since ancient times, _______ has hotly debated whether mathematics was discovered or invented. Did we create mathematical ________ to help us understand the universe around us, or is math the ______ language of the universe itself, ________ whether we find its truths or not? Are _______, polygons and equations truly real, or merely ethereal representations of some theoretical ideal? The independent reality of math has some ancient _________. The Pythagoreans of 5th Century ______ believed numbers were both ______ entities and universal principles. They called the ______ one, "the monad," the generator of all other numbers and source of all ________. Numbers were active agents in nature. Plato argued mathematical concepts were concrete and as real as the ________ 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 _______, a language of abstract _____________ 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 _______ _________, a professor of mathematics in 19th _______ Germany. His belief is ______ up in his ______ statement: "God _______ the _______ numbers, all else is the work of man." During mathematician David Hilbert's ________, there was a push to establish ___________ as a _______ construct. Hilbert attempted to axiomatize all of mathematics, as Euclid had done with geometry. He and others who attempted this saw mathematics as a ______ philosophical game but a game nonetheless. _____ Poincaré, one of the father's of non-Euclidean geometry, ________ that the existence of non-Euclidean ________, dealing with the non-flat surfaces of hyperbolic and __________ curvatures, proved that _________ geometry, the long ________ 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 ________ 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 __________ any physical phenomena, have proven decades or even _________ later, to be the framework necessary to explain how the universe has been working all along. For ________, 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 _____ 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 ______ petal arrangements, to the structure of a pineapple, even the branching of bronchi in the lungs. Or there's the non-Euclidean work of ________ Riemann in the 1850s, which Einstein used in the model for general relativity a century later. Here's an even ______ jump: mathematical knot ______, 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 __________ of all of human _______ have ______ in on the issue as well, often in surprising ways. So, is mathematics an invention or a discovery? Artificial construct or universal _____? 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 ________ concept being looked at, but it can all feel like a distorted zen koan. If there's a number of trees in a ______, but no one's there to count them, does that number exist?

Solution

centuries
specific
natural
piece
relationships
euclidean
bigger
living
bernhard
strongly
concepts
flower
mathematics
forest
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logical
history
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kronecker
created
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henri
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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

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