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

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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 |
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mathematical concepts |
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