Consequences of lack of rigour
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The standards of rigour in mathematics have increased several times during history. In the process some statements, previously considered correct where refuted. I wonder if these wrong statements were "applied" anywhere before (or after) refutation to some harmful effect. For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?
Sorry if this is a silly question.
Edit: Let me clarify. I am looking for examples of bad things happening, which fall in the following scheme:
Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics.
That wrong statement was correctly applied to some other field, or somehow used in calculations etc.
The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering.
This led to some real-world harm or danger.
It is crucial that there was no mistakes or omissions in 2,3,4, and hypothetical being omniscient on the level of 1 would approve the project.
Sorry if I made it even sillier. Feel free to close the question.
soft-question ho.history-overview
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|
show 5 more comments
$begingroup$
The standards of rigour in mathematics have increased several times during history. In the process some statements, previously considered correct where refuted. I wonder if these wrong statements were "applied" anywhere before (or after) refutation to some harmful effect. For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?
Sorry if this is a silly question.
Edit: Let me clarify. I am looking for examples of bad things happening, which fall in the following scheme:
Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics.
That wrong statement was correctly applied to some other field, or somehow used in calculations etc.
The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering.
This led to some real-world harm or danger.
It is crucial that there was no mistakes or omissions in 2,3,4, and hypothetical being omniscient on the level of 1 would approve the project.
Sorry if I made it even sillier. Feel free to close the question.
soft-question ho.history-overview
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12
$begingroup$
Structures have fallen apart because of inadequate attention to numerical analysis (see www-users.math.umn.edu/~arnold//disasters), but not because of exotica in pure math.
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– KConrad
14 hours ago
2
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
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– J.J. Green
14 hours ago
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Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
14 hours ago
3
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
13 hours ago
6
$begingroup$
The standards of rigor have not just increased. They have evolved. They are different according to subfields, countries, communities, education and personal taste. They might have been consider to "decrease" at some point, or to become fussy and cumbersome at other points. There are many opposite extreme points of view about this.
$endgroup$
– YCor
13 hours ago
|
show 5 more comments
$begingroup$
The standards of rigour in mathematics have increased several times during history. In the process some statements, previously considered correct where refuted. I wonder if these wrong statements were "applied" anywhere before (or after) refutation to some harmful effect. For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?
Sorry if this is a silly question.
Edit: Let me clarify. I am looking for examples of bad things happening, which fall in the following scheme:
Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics.
That wrong statement was correctly applied to some other field, or somehow used in calculations etc.
The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering.
This led to some real-world harm or danger.
It is crucial that there was no mistakes or omissions in 2,3,4, and hypothetical being omniscient on the level of 1 would approve the project.
Sorry if I made it even sillier. Feel free to close the question.
soft-question ho.history-overview
$endgroup$
The standards of rigour in mathematics have increased several times during history. In the process some statements, previously considered correct where refuted. I wonder if these wrong statements were "applied" anywhere before (or after) refutation to some harmful effect. For example, has any bridge fallen because every continuous function was thought to be differentiable except on a set of isolated points?
Sorry if this is a silly question.
Edit: Let me clarify. I am looking for examples of bad things happening, which fall in the following scheme:
Lack of rigour led to a wrong statement (by "today's standard") in pure mathematics.
That wrong statement was correctly applied to some other field, or somehow used in calculations etc.
The conclusion from item 2 was then applied to a real-world situation, perhaps in construction or engineering.
This led to some real-world harm or danger.
It is crucial that there was no mistakes or omissions in 2,3,4, and hypothetical being omniscient on the level of 1 would approve the project.
Sorry if I made it even sillier. Feel free to close the question.
soft-question ho.history-overview
soft-question ho.history-overview
edited 12 hours ago
community wiki
erz
12
$begingroup$
Structures have fallen apart because of inadequate attention to numerical analysis (see www-users.math.umn.edu/~arnold//disasters), but not because of exotica in pure math.
$endgroup$
– KConrad
14 hours ago
2
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
14 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
14 hours ago
3
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
13 hours ago
6
$begingroup$
The standards of rigor have not just increased. They have evolved. They are different according to subfields, countries, communities, education and personal taste. They might have been consider to "decrease" at some point, or to become fussy and cumbersome at other points. There are many opposite extreme points of view about this.
$endgroup$
– YCor
13 hours ago
|
show 5 more comments
12
$begingroup$
Structures have fallen apart because of inadequate attention to numerical analysis (see www-users.math.umn.edu/~arnold//disasters), but not because of exotica in pure math.
$endgroup$
– KConrad
14 hours ago
2
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
14 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
14 hours ago
3
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
13 hours ago
6
$begingroup$
The standards of rigor have not just increased. They have evolved. They are different according to subfields, countries, communities, education and personal taste. They might have been consider to "decrease" at some point, or to become fussy and cumbersome at other points. There are many opposite extreme points of view about this.
$endgroup$
– YCor
13 hours ago
12
12
$begingroup$
Structures have fallen apart because of inadequate attention to numerical analysis (see www-users.math.umn.edu/~arnold//disasters), but not because of exotica in pure math.
$endgroup$
– KConrad
14 hours ago
$begingroup$
Structures have fallen apart because of inadequate attention to numerical analysis (see www-users.math.umn.edu/~arnold//disasters), but not because of exotica in pure math.
$endgroup$
– KConrad
14 hours ago
2
2
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
14 hours ago
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
14 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
14 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
14 hours ago
3
3
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
13 hours ago
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
13 hours ago
6
6
$begingroup$
The standards of rigor have not just increased. They have evolved. They are different according to subfields, countries, communities, education and personal taste. They might have been consider to "decrease" at some point, or to become fussy and cumbersome at other points. There are many opposite extreme points of view about this.
$endgroup$
– YCor
13 hours ago
$begingroup$
The standards of rigor have not just increased. They have evolved. They are different according to subfields, countries, communities, education and personal taste. They might have been consider to "decrease" at some point, or to become fussy and cumbersome at other points. There are many opposite extreme points of view about this.
$endgroup$
– YCor
13 hours ago
|
show 5 more comments
2 Answers
2
active
oldest
votes
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Quoting from
Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes, Scientific American Vol. 208, No. 2 (February 1963), pp. 148-158:
Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry. To give an example, it might be taught that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.
$endgroup$
4
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
11 hours ago
2
$begingroup$
This "might be taught". Do we know if it actually was taught this way at some point?
$endgroup$
– Michael Lugo
9 hours ago
2
$begingroup$
"Most people would say yes:" one can indeed detect a certain lack of rigor in this method of proof (especially when the question is one that "most people" will never have thought about).
$endgroup$
– Christian Remling
7 hours ago
1
$begingroup$
Most people do not build fuel booster tanks for space shuttles.
$endgroup$
– Théophile
4 hours ago
1
$begingroup$
The original question is not one about space shuttles, but Is the circle the only closed curve of constant width?, i.e. Is the circle the only possible shape for wheels? This is a very natural question, I think...
$endgroup$
– Francesco Polizzi
3 hours ago
|
show 4 more comments
$begingroup$
In classical mechanics, dissipative forces are typically regarded as having a stabilizing effect. However, this is not always the case as the folks behind the first satellite launched by the United States Explorer I found out. To quote from the linked wikipedia article
Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years—to address this kind of momentum-preserving energy dissipation.
In short, the satellite ended up rotating like a windmill blade because of a counterintuitive phenomenon known as dissipation-induced instabilities; for a review article on this see
Krechetnikov, R.; Marsden, J. E., Dissipation-induced instabilities in finite dimensions, Rev. Mod. Phys. 79, No. 2, 519-553 (2007). ZBL1205.70002.
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Quoting from
Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes, Scientific American Vol. 208, No. 2 (February 1963), pp. 148-158:
Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry. To give an example, it might be taught that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.
$endgroup$
4
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
11 hours ago
2
$begingroup$
This "might be taught". Do we know if it actually was taught this way at some point?
$endgroup$
– Michael Lugo
9 hours ago
2
$begingroup$
"Most people would say yes:" one can indeed detect a certain lack of rigor in this method of proof (especially when the question is one that "most people" will never have thought about).
$endgroup$
– Christian Remling
7 hours ago
1
$begingroup$
Most people do not build fuel booster tanks for space shuttles.
$endgroup$
– Théophile
4 hours ago
1
$begingroup$
The original question is not one about space shuttles, but Is the circle the only closed curve of constant width?, i.e. Is the circle the only possible shape for wheels? This is a very natural question, I think...
$endgroup$
– Francesco Polizzi
3 hours ago
|
show 4 more comments
$begingroup$
Quoting from
Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes, Scientific American Vol. 208, No. 2 (February 1963), pp. 148-158:
Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry. To give an example, it might be taught that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.
$endgroup$
4
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
11 hours ago
2
$begingroup$
This "might be taught". Do we know if it actually was taught this way at some point?
$endgroup$
– Michael Lugo
9 hours ago
2
$begingroup$
"Most people would say yes:" one can indeed detect a certain lack of rigor in this method of proof (especially when the question is one that "most people" will never have thought about).
$endgroup$
– Christian Remling
7 hours ago
1
$begingroup$
Most people do not build fuel booster tanks for space shuttles.
$endgroup$
– Théophile
4 hours ago
1
$begingroup$
The original question is not one about space shuttles, but Is the circle the only closed curve of constant width?, i.e. Is the circle the only possible shape for wheels? This is a very natural question, I think...
$endgroup$
– Francesco Polizzi
3 hours ago
|
show 4 more comments
$begingroup$
Quoting from
Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes, Scientific American Vol. 208, No. 2 (February 1963), pp. 148-158:
Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry. To give an example, it might be taught that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.
$endgroup$
Quoting from
Martin Gardner: Curves of constant width, one of which makes it possible to drill square holes, Scientific American Vol. 208, No. 2 (February 1963), pp. 148-158:
Is the circle the only closed curve of constant width? Most people would say yes, thus providing a sterling example of how far one's mathematical intuition can go astray. Actually there is an infinity of such curves. Any of them can be cross section of a roller that will roll a platform as smoothly as a circular cylinder! The failure to recognize such curves can have and has had disastrous consequences in industry. To give an example, it might be taught that the cylindrical hull of a half-built submarine could be tested for circularity by just measuring maximum widths in all directions. As will soon be made clear, such a hull can be monstruously lopsided and still pass such a test. It is precisely for this reason that the circularity of a submarine hull is always tested by applying curved templates.
edited 5 hours ago
community wiki
3 revs
Francesco Polizzi
4
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
11 hours ago
2
$begingroup$
This "might be taught". Do we know if it actually was taught this way at some point?
$endgroup$
– Michael Lugo
9 hours ago
2
$begingroup$
"Most people would say yes:" one can indeed detect a certain lack of rigor in this method of proof (especially when the question is one that "most people" will never have thought about).
$endgroup$
– Christian Remling
7 hours ago
1
$begingroup$
Most people do not build fuel booster tanks for space shuttles.
$endgroup$
– Théophile
4 hours ago
1
$begingroup$
The original question is not one about space shuttles, but Is the circle the only closed curve of constant width?, i.e. Is the circle the only possible shape for wheels? This is a very natural question, I think...
$endgroup$
– Francesco Polizzi
3 hours ago
|
show 4 more comments
4
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
11 hours ago
2
$begingroup$
This "might be taught". Do we know if it actually was taught this way at some point?
$endgroup$
– Michael Lugo
9 hours ago
2
$begingroup$
"Most people would say yes:" one can indeed detect a certain lack of rigor in this method of proof (especially when the question is one that "most people" will never have thought about).
$endgroup$
– Christian Remling
7 hours ago
1
$begingroup$
Most people do not build fuel booster tanks for space shuttles.
$endgroup$
– Théophile
4 hours ago
1
$begingroup$
The original question is not one about space shuttles, but Is the circle the only closed curve of constant width?, i.e. Is the circle the only possible shape for wheels? This is a very natural question, I think...
$endgroup$
– Francesco Polizzi
3 hours ago
4
4
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
11 hours ago
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
11 hours ago
2
2
$begingroup$
This "might be taught". Do we know if it actually was taught this way at some point?
$endgroup$
– Michael Lugo
9 hours ago
$begingroup$
This "might be taught". Do we know if it actually was taught this way at some point?
$endgroup$
– Michael Lugo
9 hours ago
2
2
$begingroup$
"Most people would say yes:" one can indeed detect a certain lack of rigor in this method of proof (especially when the question is one that "most people" will never have thought about).
$endgroup$
– Christian Remling
7 hours ago
$begingroup$
"Most people would say yes:" one can indeed detect a certain lack of rigor in this method of proof (especially when the question is one that "most people" will never have thought about).
$endgroup$
– Christian Remling
7 hours ago
1
1
$begingroup$
Most people do not build fuel booster tanks for space shuttles.
$endgroup$
– Théophile
4 hours ago
$begingroup$
Most people do not build fuel booster tanks for space shuttles.
$endgroup$
– Théophile
4 hours ago
1
1
$begingroup$
The original question is not one about space shuttles, but Is the circle the only closed curve of constant width?, i.e. Is the circle the only possible shape for wheels? This is a very natural question, I think...
$endgroup$
– Francesco Polizzi
3 hours ago
$begingroup$
The original question is not one about space shuttles, but Is the circle the only closed curve of constant width?, i.e. Is the circle the only possible shape for wheels? This is a very natural question, I think...
$endgroup$
– Francesco Polizzi
3 hours ago
|
show 4 more comments
$begingroup$
In classical mechanics, dissipative forces are typically regarded as having a stabilizing effect. However, this is not always the case as the folks behind the first satellite launched by the United States Explorer I found out. To quote from the linked wikipedia article
Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years—to address this kind of momentum-preserving energy dissipation.
In short, the satellite ended up rotating like a windmill blade because of a counterintuitive phenomenon known as dissipation-induced instabilities; for a review article on this see
Krechetnikov, R.; Marsden, J. E., Dissipation-induced instabilities in finite dimensions, Rev. Mod. Phys. 79, No. 2, 519-553 (2007). ZBL1205.70002.
$endgroup$
add a comment |
$begingroup$
In classical mechanics, dissipative forces are typically regarded as having a stabilizing effect. However, this is not always the case as the folks behind the first satellite launched by the United States Explorer I found out. To quote from the linked wikipedia article
Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years—to address this kind of momentum-preserving energy dissipation.
In short, the satellite ended up rotating like a windmill blade because of a counterintuitive phenomenon known as dissipation-induced instabilities; for a review article on this see
Krechetnikov, R.; Marsden, J. E., Dissipation-induced instabilities in finite dimensions, Rev. Mod. Phys. 79, No. 2, 519-553 (2007). ZBL1205.70002.
$endgroup$
add a comment |
$begingroup$
In classical mechanics, dissipative forces are typically regarded as having a stabilizing effect. However, this is not always the case as the folks behind the first satellite launched by the United States Explorer I found out. To quote from the linked wikipedia article
Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years—to address this kind of momentum-preserving energy dissipation.
In short, the satellite ended up rotating like a windmill blade because of a counterintuitive phenomenon known as dissipation-induced instabilities; for a review article on this see
Krechetnikov, R.; Marsden, J. E., Dissipation-induced instabilities in finite dimensions, Rev. Mod. Phys. 79, No. 2, 519-553 (2007). ZBL1205.70002.
$endgroup$
In classical mechanics, dissipative forces are typically regarded as having a stabilizing effect. However, this is not always the case as the folks behind the first satellite launched by the United States Explorer I found out. To quote from the linked wikipedia article
Explorer 1 changed rotation axis after launch. The elongated body of the spacecraft had been designed to spin about its long (least-inertia) axis but refused to do so, and instead started precessing due to energy dissipation from flexible structural elements. Later it was understood that on general grounds, the body ends up in the spin state that minimizes the kinetic rotational energy for a fixed angular momentum (this being the maximal-inertia axis). This motivated the first further development of the Eulerian theory of rigid body dynamics after nearly 200 years—to address this kind of momentum-preserving energy dissipation.
In short, the satellite ended up rotating like a windmill blade because of a counterintuitive phenomenon known as dissipation-induced instabilities; for a review article on this see
Krechetnikov, R.; Marsden, J. E., Dissipation-induced instabilities in finite dimensions, Rev. Mod. Phys. 79, No. 2, 519-553 (2007). ZBL1205.70002.
edited 1 hour ago
community wiki
2 revs, 2 users 95%
Nawaf Bou-Rabee
add a comment |
add a comment |
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12
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Structures have fallen apart because of inadequate attention to numerical analysis (see www-users.math.umn.edu/~arnold//disasters), but not because of exotica in pure math.
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– KConrad
14 hours ago
2
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
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– J.J. Green
14 hours ago
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Are you asking about "applications" outside math, or applications within maths?
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– YCor
14 hours ago
3
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You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
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– KConrad
13 hours ago
6
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The standards of rigor have not just increased. They have evolved. They are different according to subfields, countries, communities, education and personal taste. They might have been consider to "decrease" at some point, or to become fussy and cumbersome at other points. There are many opposite extreme points of view about this.
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– YCor
13 hours ago