Theorems that impeded progressWhat are some famous rejections of correct mathematics?How to unify various...
Theorems that impeded progress
What are some famous rejections of correct mathematics?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
$begingroup$
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 2 hours 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 2 hours 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
2 hours 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
2 hours 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 2 hours 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 2 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 2 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 2 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 2 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237709
asked 2 hours 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
2 hours 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
2 hours 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
2 hours 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
2 hours 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
2 hours 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
2 hours 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
2 hours 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
2 hours ago
add a comment |
2 Answers
2
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 2 hours ago
José Hdz. Stgo.
5,24734877
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
1 hour ago
1
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
1 hour ago
add a comment |
$begingroup$
Here I quote from the introduction to "Shelah’s pcf theory and its applications" by Burke and Magidor (https://core.ac.uk/download/pdf/82500424.pdf):
Cardinal arithmetic seems to be one of the central topics of set theory. (We
mean mainly cardinal exponentiation, the other operations being trivial.)
However, the independence results obtained by Cohen’s forcing technique
(especially Easton’s theorem: see below) showed that many of the open problems
in cardinal arithmetic are independent of the axioms of ZFC (Zermelo-Fraenkel
set theory with the axiom of choice). It appeared, in the late sixties, that cardinal arithmetic had become trivial in the sense that any potential theorem seemed to be refutable by the construction of a model of set theory which violated it.
In particular, Easton’s theorem showed that essentially any cardinal
arithmetic ‘behavior’ satisfying some obvious requirements can be realized as the
behavior of the power function at regular cardinals. [...]
The general consensus among set theorists was that the restriction to regular cardinals was due to a weakness in the proof and that a slight improvement in the methods for constructing models would show that, even for powers of singular cardinals, there are no deep theorems provable in ZFC.
They go on to explain how Shelah's pcf theory (and its precursors) in fact show that there are many nontrivial theorems about inequalities of cardinals provable in ZFC.
So arguably the earlier independence results impeded the discovery of these provable inequalities, although I don't know how strongly anyone would argue that.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
$endgroup$
$begingroup$
An explanar example of my query. Possibly Cohen's forcing was the "culprit" in jumping so far that there was a natural retraction?
$endgroup$
– Joseph O'Rourke
1 hour ago
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
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 2 hours ago
José Hdz. Stgo.
5,24734877
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
1 hour ago
1
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
1 hour 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 2 hours ago
José Hdz. Stgo.
5,24734877
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
1 hour ago
1
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
1 hour 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 2 hours ago
José Hdz. Stgo.
5,24734877
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
$endgroup$
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 2 hours ago
José Hdz. Stgo.
5,24734877
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
edited 2 hours ago
José Hdz. Stgo.
5,24734877
edited 2 hours ago
José Hdz. Stgo.
5,24734877
edited 2 hours ago
José Hdz. Stgo.
5,24734877
5,24734877
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
answered 2 hours ago
Daniel McLauryDaniel McLaury
290217
290217
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
1 hour ago
1
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
1 hour ago
add a comment |
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
1 hour ago
1
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
1 hour ago
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
1 hour ago
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
1 hour ago
1
1
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
1 hour ago
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
1 hour ago
add a comment |
$begingroup$
Here I quote from the introduction to "Shelah’s pcf theory and its applications" by Burke and Magidor (https://core.ac.uk/download/pdf/82500424.pdf):
Cardinal arithmetic seems to be one of the central topics of set theory. (We
mean mainly cardinal exponentiation, the other operations being trivial.)
However, the independence results obtained by Cohen’s forcing technique
(especially Easton’s theorem: see below) showed that many of the open problems
in cardinal arithmetic are independent of the axioms of ZFC (Zermelo-Fraenkel
set theory with the axiom of choice). It appeared, in the late sixties, that cardinal arithmetic had become trivial in the sense that any potential theorem seemed to be refutable by the construction of a model of set theory which violated it.
In particular, Easton’s theorem showed that essentially any cardinal
arithmetic ‘behavior’ satisfying some obvious requirements can be realized as the
behavior of the power function at regular cardinals. [...]
The general consensus among set theorists was that the restriction to regular cardinals was due to a weakness in the proof and that a slight improvement in the methods for constructing models would show that, even for powers of singular cardinals, there are no deep theorems provable in ZFC.
They go on to explain how Shelah's pcf theory (and its precursors) in fact show that there are many nontrivial theorems about inequalities of cardinals provable in ZFC.
So arguably the earlier independence results impeded the discovery of these provable inequalities, although I don't know how strongly anyone would argue that.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
$endgroup$
$begingroup$
An explanar example of my query. Possibly Cohen's forcing was the "culprit" in jumping so far that there was a natural retraction?
$endgroup$
– Joseph O'Rourke
1 hour ago
add a comment |
$begingroup$
Here I quote from the introduction to "Shelah’s pcf theory and its applications" by Burke and Magidor (https://core.ac.uk/download/pdf/82500424.pdf):
Cardinal arithmetic seems to be one of the central topics of set theory. (We
mean mainly cardinal exponentiation, the other operations being trivial.)
However, the independence results obtained by Cohen’s forcing technique
(especially Easton’s theorem: see below) showed that many of the open problems
in cardinal arithmetic are independent of the axioms of ZFC (Zermelo-Fraenkel
set theory with the axiom of choice). It appeared, in the late sixties, that cardinal arithmetic had become trivial in the sense that any potential theorem seemed to be refutable by the construction of a model of set theory which violated it.
In particular, Easton’s theorem showed that essentially any cardinal
arithmetic ‘behavior’ satisfying some obvious requirements can be realized as the
behavior of the power function at regular cardinals. [...]
The general consensus among set theorists was that the restriction to regular cardinals was due to a weakness in the proof and that a slight improvement in the methods for constructing models would show that, even for powers of singular cardinals, there are no deep theorems provable in ZFC.
They go on to explain how Shelah's pcf theory (and its precursors) in fact show that there are many nontrivial theorems about inequalities of cardinals provable in ZFC.
So arguably the earlier independence results impeded the discovery of these provable inequalities, although I don't know how strongly anyone would argue that.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
$endgroup$
$begingroup$
An explanar example of my query. Possibly Cohen's forcing was the "culprit" in jumping so far that there was a natural retraction?
$endgroup$
– Joseph O'Rourke
1 hour ago
add a comment |
$begingroup$
Here I quote from the introduction to "Shelah’s pcf theory and its applications" by Burke and Magidor (https://core.ac.uk/download/pdf/82500424.pdf):
Cardinal arithmetic seems to be one of the central topics of set theory. (We
mean mainly cardinal exponentiation, the other operations being trivial.)
However, the independence results obtained by Cohen’s forcing technique
(especially Easton’s theorem: see below) showed that many of the open problems
in cardinal arithmetic are independent of the axioms of ZFC (Zermelo-Fraenkel
set theory with the axiom of choice). It appeared, in the late sixties, that cardinal arithmetic had become trivial in the sense that any potential theorem seemed to be refutable by the construction of a model of set theory which violated it.
In particular, Easton’s theorem showed that essentially any cardinal
arithmetic ‘behavior’ satisfying some obvious requirements can be realized as the
behavior of the power function at regular cardinals. [...]
The general consensus among set theorists was that the restriction to regular cardinals was due to a weakness in the proof and that a slight improvement in the methods for constructing models would show that, even for powers of singular cardinals, there are no deep theorems provable in ZFC.
They go on to explain how Shelah's pcf theory (and its precursors) in fact show that there are many nontrivial theorems about inequalities of cardinals provable in ZFC.
So arguably the earlier independence results impeded the discovery of these provable inequalities, although I don't know how strongly anyone would argue that.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
$endgroup$
Here I quote from the introduction to "Shelah’s pcf theory and its applications" by Burke and Magidor (https://core.ac.uk/download/pdf/82500424.pdf):
Cardinal arithmetic seems to be one of the central topics of set theory. (We
mean mainly cardinal exponentiation, the other operations being trivial.)
However, the independence results obtained by Cohen’s forcing technique
(especially Easton’s theorem: see below) showed that many of the open problems
in cardinal arithmetic are independent of the axioms of ZFC (Zermelo-Fraenkel
set theory with the axiom of choice). It appeared, in the late sixties, that cardinal arithmetic had become trivial in the sense that any potential theorem seemed to be refutable by the construction of a model of set theory which violated it.
In particular, Easton’s theorem showed that essentially any cardinal
arithmetic ‘behavior’ satisfying some obvious requirements can be realized as the
behavior of the power function at regular cardinals. [...]
The general consensus among set theorists was that the restriction to regular cardinals was due to a weakness in the proof and that a slight improvement in the methods for constructing models would show that, even for powers of singular cardinals, there are no deep theorems provable in ZFC.
They go on to explain how Shelah's pcf theory (and its precursors) in fact show that there are many nontrivial theorems about inequalities of cardinals provable in ZFC.
So arguably the earlier independence results impeded the discovery of these provable inequalities, although I don't know how strongly anyone would argue that.
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
answered 1 hour ago
Sam HopkinsSam Hopkins
5,00212557
5,00212557
$begingroup$
An explanar example of my query. Possibly Cohen's forcing was the "culprit" in jumping so far that there was a natural retraction?
$endgroup$
– Joseph O'Rourke
1 hour ago
add a comment |
$begingroup$
An explanar example of my query. Possibly Cohen's forcing was the "culprit" in jumping so far that there was a natural retraction?
$endgroup$
– Joseph O'Rourke
1 hour ago
$begingroup$
An explanar example of my query. Possibly Cohen's forcing was the "culprit" in jumping so far that there was a natural retraction?
$endgroup$
– Joseph O'Rourke
1 hour ago
$begingroup$
An explanar example of my query. Possibly Cohen's forcing was the "culprit" in jumping so far that there was a natural retraction?
$endgroup$
– Joseph O'Rourke
1 hour 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
2 hours 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
2 hours ago