Theorems that impeded progressHow to unify various reconstruction theorems (Gabriel-Rosenberg,...
Theorems that impeded progress
How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)Theorems first published in textbooks?Theorems that are 'obvious' but hard to proveAn undergraduate's guide to the foundational theorems of logicProofs that inspire and teachExamples of major theorems with very hard proofs that have NOT dramatically improved over timeHistory of preservation theorems in forcing theoryAre there any Algebraic Geometry Theorems that were proved using Combinatorics?Did Euler prove theorems by example?Theorems demoted back to conjectures
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It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:
- Faber's Theorem on polynomial interpolation
- Squire's Theorem on hydrodynamic instability
Trefethen, Lloyd N. "Inverse Yogiisms." Notices of the American Mathematical Society 63, no. 11 (2016).
Also: The Best Writing on Mathematics 2017 6 (2017): 28.
Google books link.
In my own experience, I have witnessed the several negative-results theorems in
Marvin Minsky and Seymour A. Papert.
Perceptrons: An Introduction to Computational Geometry , 1969.
MIT Press.
impede progress in neural-net research for more than a decade.1
Q. What are other examples of theorems whose (correct) proofs (possibly temporarily)
suppressed research advancement in mathematical subfields?
1
Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." Social Studies of Science 26, no. 3 (1996): 611-659.
Abstract: "[...]I devote particular attention to the proofs and arguments of Minsky and Papert, which were interpreted as showing that further progress in neural nets was not possible, and that this approach to AI had to be abandoned.[...]"
RG link.
ho.history-overview big-picture
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
$endgroup$
add a comment |
$begingroup$
It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:
- Faber's Theorem on polynomial interpolation
- Squire's Theorem on hydrodynamic instability
Trefethen, Lloyd N. "Inverse Yogiisms." Notices of the American Mathematical Society 63, no. 11 (2016).
Also: The Best Writing on Mathematics 2017 6 (2017): 28.
Google books link.
In my own experience, I have witnessed the several negative-results theorems in
Marvin Minsky and Seymour A. Papert.
Perceptrons: An Introduction to Computational Geometry , 1969.
MIT Press.
impede progress in neural-net research for more than a decade.1
Q. What are other examples of theorems whose (correct) proofs (possibly temporarily)
suppressed research advancement in mathematical subfields?
1
Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." Social Studies of Science 26, no. 3 (1996): 611-659.
Abstract: "[...]I devote particular attention to the proofs and arguments of Minsky and Papert, which were interpreted as showing that further progress in neural nets was not possible, and that this approach to AI had to be abandoned.[...]"
RG link.
ho.history-overview big-picture
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
$endgroup$
1
$begingroup$
I remember reading, I believe in some other MO post, about how whereas Donaldson's work on smooth 4 manifolds launched a vibrant program of research with invariants coming from physics, Freedman's contemporaneous work on topological 4 manifolds essentially ended the study of topological 4 manifolds. But maybe that's not what you mean by "impeded progress"
$endgroup$
– Sam Hopkins
19 mins ago
$begingroup$
@SamHopkins: I am seeking more misleading impeding, as opposed to closing off a line of investigation. Certainly when a line has terminated, that's it. But there are also misleading endings, which are not terminations afterall.
$endgroup$
– Joseph O'Rourke
6 mins ago
add a comment |
$begingroup$
It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:
- Faber's Theorem on polynomial interpolation
- Squire's Theorem on hydrodynamic instability
Trefethen, Lloyd N. "Inverse Yogiisms." Notices of the American Mathematical Society 63, no. 11 (2016).
Also: The Best Writing on Mathematics 2017 6 (2017): 28.
Google books link.
In my own experience, I have witnessed the several negative-results theorems in
Marvin Minsky and Seymour A. Papert.
Perceptrons: An Introduction to Computational Geometry , 1969.
MIT Press.
impede progress in neural-net research for more than a decade.1
Q. What are other examples of theorems whose (correct) proofs (possibly temporarily)
suppressed research advancement in mathematical subfields?
1
Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." Social Studies of Science 26, no. 3 (1996): 611-659.
Abstract: "[...]I devote particular attention to the proofs and arguments of Minsky and Papert, which were interpreted as showing that further progress in neural nets was not possible, and that this approach to AI had to be abandoned.[...]"
RG link.
ho.history-overview big-picture
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
$endgroup$
It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:
- Faber's Theorem on polynomial interpolation
- Squire's Theorem on hydrodynamic instability
Trefethen, Lloyd N. "Inverse Yogiisms." Notices of the American Mathematical Society 63, no. 11 (2016).
Also: The Best Writing on Mathematics 2017 6 (2017): 28.
Google books link.
In my own experience, I have witnessed the several negative-results theorems in
Marvin Minsky and Seymour A. Papert.
Perceptrons: An Introduction to Computational Geometry , 1969.
MIT Press.
impede progress in neural-net research for more than a decade.1
Q. What are other examples of theorems whose (correct) proofs (possibly temporarily)
suppressed research advancement in mathematical subfields?
1
Olazaran, Mikel. "A sociological study of the official history of the perceptrons controversy." Social Studies of Science 26, no. 3 (1996): 611-659.
Abstract: "[...]I devote particular attention to the proofs and arguments of Minsky and Papert, which were interpreted as showing that further progress in neural nets was not possible, and that this approach to AI had to be abandoned.[...]"
RG link.
ho.history-overview big-picture
ho.history-overview big-picture
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 26 mins ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
86.2k16237709
1
$begingroup$
I remember reading, I believe in some other MO post, about how whereas Donaldson's work on smooth 4 manifolds launched a vibrant program of research with invariants coming from physics, Freedman's contemporaneous work on topological 4 manifolds essentially ended the study of topological 4 manifolds. But maybe that's not what you mean by "impeded progress"
$endgroup$
– Sam Hopkins
19 mins ago
$begingroup$
@SamHopkins: I am seeking more misleading impeding, as opposed to closing off a line of investigation. Certainly when a line has terminated, that's it. But there are also misleading endings, which are not terminations afterall.
$endgroup$
– Joseph O'Rourke
6 mins ago
add a comment |
1
$begingroup$
I remember reading, I believe in some other MO post, about how whereas Donaldson's work on smooth 4 manifolds launched a vibrant program of research with invariants coming from physics, Freedman's contemporaneous work on topological 4 manifolds essentially ended the study of topological 4 manifolds. But maybe that's not what you mean by "impeded progress"
$endgroup$
– Sam Hopkins
19 mins ago
$begingroup$
@SamHopkins: I am seeking more misleading impeding, as opposed to closing off a line of investigation. Certainly when a line has terminated, that's it. But there are also misleading endings, which are not terminations afterall.
$endgroup$
– Joseph O'Rourke
6 mins ago
1
1
$begingroup$
I remember reading, I believe in some other MO post, about how whereas Donaldson's work on smooth 4 manifolds launched a vibrant program of research with invariants coming from physics, Freedman's contemporaneous work on topological 4 manifolds essentially ended the study of topological 4 manifolds. But maybe that's not what you mean by "impeded progress"
$endgroup$
– Sam Hopkins
19 mins ago
$begingroup$
I remember reading, I believe in some other MO post, about how whereas Donaldson's work on smooth 4 manifolds launched a vibrant program of research with invariants coming from physics, Freedman's contemporaneous work on topological 4 manifolds essentially ended the study of topological 4 manifolds. But maybe that's not what you mean by "impeded progress"
$endgroup$
– Sam Hopkins
19 mins ago
$begingroup$
@SamHopkins: I am seeking more misleading impeding, as opposed to closing off a line of investigation. Certainly when a line has terminated, that's it. But there are also misleading endings, which are not terminations afterall.
$endgroup$
– Joseph O'Rourke
6 mins ago
$begingroup$
@SamHopkins: I am seeking more misleading impeding, as opposed to closing off a line of investigation. Certainly when a line has terminated, that's it. But there are also misleading endings, which are not terminations afterall.
$endgroup$
– Joseph O'Rourke
6 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.
edited 13 mins ago
José Hdz. Stgo.
5,24734877
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
2 mins ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.
edited 13 mins ago
José Hdz. Stgo.
5,24734877
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
2 mins ago
add a comment |
$begingroup$
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.
edited 13 mins ago
José Hdz. Stgo.
5,24734877
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
2 mins ago
add a comment |
$begingroup$
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.
edited 13 mins ago
José Hdz. Stgo.
5,24734877
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
$endgroup$
I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.
edited 13 mins ago
José Hdz. Stgo.
5,24734877
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
edited 13 mins ago
José Hdz. Stgo.
5,24734877
edited 13 mins ago
José Hdz. Stgo.
5,24734877
edited 13 mins ago
José Hdz. Stgo.
5,24734877
5,24734877
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
answered 17 mins ago
Daniel McLauryDaniel McLaury
280217
280217
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
2 mins ago
add a comment |
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
2 mins ago
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
2 mins ago
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
2 mins ago
add a comment |
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$begingroup$
I remember reading, I believe in some other MO post, about how whereas Donaldson's work on smooth 4 manifolds launched a vibrant program of research with invariants coming from physics, Freedman's contemporaneous work on topological 4 manifolds essentially ended the study of topological 4 manifolds. But maybe that's not what you mean by "impeded progress"
$endgroup$
– Sam Hopkins
19 mins ago
$begingroup$
@SamHopkins: I am seeking more misleading impeding, as opposed to closing off a line of investigation. Certainly when a line has terminated, that's it. But there are also misleading endings, which are not terminations afterall.
$endgroup$
– Joseph O'Rourke
6 mins ago