Consequences of lack of rigourIs rigour just a ritual that most mathematicians wish to get rid of if they...
Consequences of lack of rigour
Is rigour just a ritual that most mathematicians wish to get rid of if they could? Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?Are there any (interesting) consequences of the irrationality of π?Consequences of the Langlands programShuffle (co-)multiplication and generalized Leibniz formula in tensor calculusWhat are the 'wonderful consequences' following from the existence of a minimal dense subspace?
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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 2 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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7
$begingroup$
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
4 hours ago
1
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@KConrad: Italian differential geometry is an example of the latter :-)
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– J.J. Green
4 hours ago
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Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
4 hours ago
1
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
4 hours ago
3
$begingroup$
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
4 hours ago
|
show 2 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 3 hours ago
community wiki
erz
7
$begingroup$
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
4 hours ago
1
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
4 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
4 hours ago
1
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
4 hours ago
3
$begingroup$
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
4 hours ago
|
show 2 more comments
7
$begingroup$
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
4 hours ago
1
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
4 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
4 hours ago
1
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
4 hours ago
3
$begingroup$
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
4 hours ago
7
7
$begingroup$
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
4 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
4 hours ago
1
1
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
4 hours ago
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
4 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
4 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
4 hours ago
1
1
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
4 hours ago
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
4 hours ago
3
3
$begingroup$
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
4 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
4 hours ago
|
show 2 more comments
1 Answer
1
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.
$endgroup$
2
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
1 hour ago
add a comment |
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$begingroup$
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.
$endgroup$
2
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
1 hour ago
add a comment |
$begingroup$
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.
$endgroup$
2
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
1 hour ago
add a comment |
$begingroup$
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 2 hours ago
community wiki
Francesco Polizzi
2
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
1 hour ago
add a comment |
2
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
1 hour ago
2
2
$begingroup$
In one of his books, Feynman mentions the same issue regarding reusable fuel booster tanks from the space shuttle
$endgroup$
– Rodrigo A. Pérez
1 hour 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
1 hour ago
add a comment |
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7
$begingroup$
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
4 hours ago
1
$begingroup$
@KConrad: Italian differential geometry is an example of the latter :-)
$endgroup$
– J.J. Green
4 hours ago
$begingroup$
Are you asking about "applications" outside math, or applications within maths?
$endgroup$
– YCor
4 hours ago
1
$begingroup$
You might find the answers at math.stackexchange.com/questions/1220875/… to be of interest.
$endgroup$
– KConrad
4 hours ago
3
$begingroup$
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
4 hours ago