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-result theorems
proved 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 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
$endgroup$
|
show 1 more 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-result theorems
proved 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 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
$endgroup$
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
4 hours ago
1
$begingroup$
@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
4 hours ago
$begingroup$
Didn't a lot of 19th century focus on quaternions impede the development of dot/cross-product thinking? Also, Newton was right, although his notation... slowed development of calculus on the British Isles?
$endgroup$
– Mark S
1 hour ago
1
$begingroup$
This comment is me thinking out loud about the mechanism by which a theorem might impede or spur progress. I think we carry around beliefs about the likelihood that unproven theorems are true or false, and beliefs about the difficulty of achieving proofs of those theorems. When the truth or falsehood of a theorem becomes known, then one updates one's beliefs about those theorems that remain unproven. So to spur or impede progress, a new theorem should dramatically bias those estimates, thereby causing time/energy to be wasted. (I am not at all certain that I am correct here.)
$endgroup$
– Neal
1 hour ago
$begingroup$
@MarkS, the British Isles in the 18th century had Colin Maclaurin, Thomas Simpson, James Stirling, Brook Taylor, all working in the Newtonian tradition and with mathematical things named after them — so I wouldn’t say that Newtonian notation retarded progress.
$endgroup$
– Matt F.
48 mins ago
|
show 1 more 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-result theorems
proved 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 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
$endgroup$
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-result theorems
proved 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 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
asked 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
edited 1 hour ago
Joseph O'Rourke
asked 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
asked 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
asked 4 hours ago
Joseph O'RourkeJoseph O'Rourke
86.2k16237710
86.2k16237710
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
4 hours ago
1
$begingroup$
@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
4 hours ago
$begingroup$
Didn't a lot of 19th century focus on quaternions impede the development of dot/cross-product thinking? Also, Newton was right, although his notation... slowed development of calculus on the British Isles?
$endgroup$
– Mark S
1 hour ago
1
$begingroup$
This comment is me thinking out loud about the mechanism by which a theorem might impede or spur progress. I think we carry around beliefs about the likelihood that unproven theorems are true or false, and beliefs about the difficulty of achieving proofs of those theorems. When the truth or falsehood of a theorem becomes known, then one updates one's beliefs about those theorems that remain unproven. So to spur or impede progress, a new theorem should dramatically bias those estimates, thereby causing time/energy to be wasted. (I am not at all certain that I am correct here.)
$endgroup$
– Neal
1 hour ago
$begingroup$
@MarkS, the British Isles in the 18th century had Colin Maclaurin, Thomas Simpson, James Stirling, Brook Taylor, all working in the Newtonian tradition and with mathematical things named after them — so I wouldn’t say that Newtonian notation retarded progress.
$endgroup$
– Matt F.
48 mins ago
|
show 1 more 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
4 hours ago
1
$begingroup$
@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
4 hours ago
$begingroup$
Didn't a lot of 19th century focus on quaternions impede the development of dot/cross-product thinking? Also, Newton was right, although his notation... slowed development of calculus on the British Isles?
$endgroup$
– Mark S
1 hour ago
1
$begingroup$
This comment is me thinking out loud about the mechanism by which a theorem might impede or spur progress. I think we carry around beliefs about the likelihood that unproven theorems are true or false, and beliefs about the difficulty of achieving proofs of those theorems. When the truth or falsehood of a theorem becomes known, then one updates one's beliefs about those theorems that remain unproven. So to spur or impede progress, a new theorem should dramatically bias those estimates, thereby causing time/energy to be wasted. (I am not at all certain that I am correct here.)
$endgroup$
– Neal
1 hour ago
$begingroup$
@MarkS, the British Isles in the 18th century had Colin Maclaurin, Thomas Simpson, James Stirling, Brook Taylor, all working in the Newtonian tradition and with mathematical things named after them — so I wouldn’t say that Newtonian notation retarded progress.
$endgroup$
– Matt F.
48 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
4 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
4 hours ago
1
1
$begingroup$
@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
4 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
4 hours ago
$begingroup$
Didn't a lot of 19th century focus on quaternions impede the development of dot/cross-product thinking? Also, Newton was right, although his notation... slowed development of calculus on the British Isles?
$endgroup$
– Mark S
1 hour ago
$begingroup$
Didn't a lot of 19th century focus on quaternions impede the development of dot/cross-product thinking? Also, Newton was right, although his notation... slowed development of calculus on the British Isles?
$endgroup$
– Mark S
1 hour ago
1
1
$begingroup$
This comment is me thinking out loud about the mechanism by which a theorem might impede or spur progress. I think we carry around beliefs about the likelihood that unproven theorems are true or false, and beliefs about the difficulty of achieving proofs of those theorems. When the truth or falsehood of a theorem becomes known, then one updates one's beliefs about those theorems that remain unproven. So to spur or impede progress, a new theorem should dramatically bias those estimates, thereby causing time/energy to be wasted. (I am not at all certain that I am correct here.)
$endgroup$
– Neal
1 hour ago
$begingroup$
This comment is me thinking out loud about the mechanism by which a theorem might impede or spur progress. I think we carry around beliefs about the likelihood that unproven theorems are true or false, and beliefs about the difficulty of achieving proofs of those theorems. When the truth or falsehood of a theorem becomes known, then one updates one's beliefs about those theorems that remain unproven. So to spur or impede progress, a new theorem should dramatically bias those estimates, thereby causing time/energy to be wasted. (I am not at all certain that I am correct here.)
$endgroup$
– Neal
1 hour ago
$begingroup$
@MarkS, the British Isles in the 18th century had Colin Maclaurin, Thomas Simpson, James Stirling, Brook Taylor, all working in the Newtonian tradition and with mathematical things named after them — so I wouldn’t say that Newtonian notation retarded progress.
$endgroup$
– Matt F.
48 mins ago
$begingroup$
@MarkS, the British Isles in the 18th century had Colin Maclaurin, Thomas Simpson, James Stirling, Brook Taylor, all working in the Newtonian tradition and with mathematical things named after them — so I wouldn’t say that Newtonian notation retarded progress.
$endgroup$
– Matt F.
48 mins ago
|
show 1 more comment
3 Answers
3
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 4 hours ago
José Hdz. Stgo.
5,24734877
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
4 hours ago
2
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
4 hours ago
add a comment |
$begingroup$
I have been told that Thurston's work on foliations (for example: Thurston, W. P., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268) essentially ended the subject for some time, even though there was still much work to be done.
Here is a quote from his On Proof and Progress in Mathematics:
"First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn't matter here whether you know what foliations are.) At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog. An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well. I do not think that the evacuation occurred because the territory was intellectually exhausted—there were (and still are) many interesting questions that remain and that are probably approachable."
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
$endgroup$
2
$begingroup$
These mathematicians were afraid to compete with Thurston in foliations, particularly given his trove of unpublished results. Meanwhile the examples in the post show mathematicians confidently drawing the wrong intuitions from others’a results. So I find Thurston an interesting example of a different phenomenon.
$endgroup$
– Matt F.
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 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
$endgroup$
1
$begingroup$
An expemplanary instance 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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3 Answers
3
active
oldest
votes
3 Answers
3
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 4 hours ago
José Hdz. Stgo.
5,24734877
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
4 hours ago
2
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
4 hours 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 4 hours ago
José Hdz. Stgo.
5,24734877
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
$endgroup$
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
4 hours ago
2
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
4 hours 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 4 hours ago
José Hdz. Stgo.
5,24734877
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
$endgroup$
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 4 hours ago
José Hdz. Stgo.
5,24734877
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
edited 4 hours ago
José Hdz. Stgo.
5,24734877
edited 4 hours ago
José Hdz. Stgo.
5,24734877
edited 4 hours ago
José Hdz. Stgo.
5,24734877
5,24734877
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
answered 4 hours ago
Daniel McLauryDaniel McLaury
320217
320217
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
4 hours ago
2
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
4 hours ago
add a comment |
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
4 hours ago
2
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
4 hours ago
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
4 hours ago
$begingroup$
Who realized "that higher homotopy groups are abelian"? Could you provide more details, citations?
$endgroup$
– Joseph O'Rourke
4 hours ago
2
2
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
4 hours ago
$begingroup$
@JosephO'Rourke: see mathoverflow.net/a/13902/25028
$endgroup$
– Sam Hopkins
4 hours ago
add a comment |
$begingroup$
I have been told that Thurston's work on foliations (for example: Thurston, W. P., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268) essentially ended the subject for some time, even though there was still much work to be done.
Here is a quote from his On Proof and Progress in Mathematics:
"First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn't matter here whether you know what foliations are.) At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog. An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well. I do not think that the evacuation occurred because the territory was intellectually exhausted—there were (and still are) many interesting questions that remain and that are probably approachable."
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
$endgroup$
2
$begingroup$
These mathematicians were afraid to compete with Thurston in foliations, particularly given his trove of unpublished results. Meanwhile the examples in the post show mathematicians confidently drawing the wrong intuitions from others’a results. So I find Thurston an interesting example of a different phenomenon.
$endgroup$
– Matt F.
1 hour ago
add a comment |
$begingroup$
I have been told that Thurston's work on foliations (for example: Thurston, W. P., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268) essentially ended the subject for some time, even though there was still much work to be done.
Here is a quote from his On Proof and Progress in Mathematics:
"First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn't matter here whether you know what foliations are.) At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog. An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well. I do not think that the evacuation occurred because the territory was intellectually exhausted—there were (and still are) many interesting questions that remain and that are probably approachable."
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
$endgroup$
2
$begingroup$
These mathematicians were afraid to compete with Thurston in foliations, particularly given his trove of unpublished results. Meanwhile the examples in the post show mathematicians confidently drawing the wrong intuitions from others’a results. So I find Thurston an interesting example of a different phenomenon.
$endgroup$
– Matt F.
1 hour ago
add a comment |
$begingroup$
I have been told that Thurston's work on foliations (for example: Thurston, W. P., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268) essentially ended the subject for some time, even though there was still much work to be done.
Here is a quote from his On Proof and Progress in Mathematics:
"First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn't matter here whether you know what foliations are.) At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog. An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well. I do not think that the evacuation occurred because the territory was intellectually exhausted—there were (and still are) many interesting questions that remain and that are probably approachable."
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
$endgroup$
I have been told that Thurston's work on foliations (for example: Thurston, W. P., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268) essentially ended the subject for some time, even though there was still much work to be done.
Here is a quote from his On Proof and Progress in Mathematics:
"First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn't matter here whether you know what foliations are.) At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog. An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well. I do not think that the evacuation occurred because the territory was intellectually exhausted—there were (and still are) many interesting questions that remain and that are probably approachable."
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
edited 1 hour ago
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
answered 1 hour ago
Sean LawtonSean Lawton
4,02422349
4,02422349
2
$begingroup$
These mathematicians were afraid to compete with Thurston in foliations, particularly given his trove of unpublished results. Meanwhile the examples in the post show mathematicians confidently drawing the wrong intuitions from others’a results. So I find Thurston an interesting example of a different phenomenon.
$endgroup$
– Matt F.
1 hour ago
add a comment |
2
$begingroup$
These mathematicians were afraid to compete with Thurston in foliations, particularly given his trove of unpublished results. Meanwhile the examples in the post show mathematicians confidently drawing the wrong intuitions from others’a results. So I find Thurston an interesting example of a different phenomenon.
$endgroup$
– Matt F.
1 hour ago
2
2
$begingroup$
These mathematicians were afraid to compete with Thurston in foliations, particularly given his trove of unpublished results. Meanwhile the examples in the post show mathematicians confidently drawing the wrong intuitions from others’a results. So I find Thurston an interesting example of a different phenomenon.
$endgroup$
– Matt F.
1 hour ago
$begingroup$
These mathematicians were afraid to compete with Thurston in foliations, particularly given his trove of unpublished results. Meanwhile the examples in the post show mathematicians confidently drawing the wrong intuitions from others’a results. So I find Thurston an interesting example of a different phenomenon.
$endgroup$
– Matt F.
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 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
$endgroup$
1
$begingroup$
An expemplanary instance 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 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
$endgroup$
1
$begingroup$
An expemplanary instance 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 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
$endgroup$
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 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
answered 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
answered 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
answered 3 hours ago
Sam HopkinsSam Hopkins
5,02212557
5,02212557
1
$begingroup$
An expemplanary instance 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 |
1
$begingroup$
An expemplanary instance 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
1
1
$begingroup$
An expemplanary instance 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 expemplanary instance 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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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
4 hours ago
1
$begingroup$
@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
4 hours ago
$begingroup$
Didn't a lot of 19th century focus on quaternions impede the development of dot/cross-product thinking? Also, Newton was right, although his notation... slowed development of calculus on the British Isles?
$endgroup$
– Mark S
1 hour ago
1
$begingroup$
This comment is me thinking out loud about the mechanism by which a theorem might impede or spur progress. I think we carry around beliefs about the likelihood that unproven theorems are true or false, and beliefs about the difficulty of achieving proofs of those theorems. When the truth or falsehood of a theorem becomes known, then one updates one's beliefs about those theorems that remain unproven. So to spur or impede progress, a new theorem should dramatically bias those estimates, thereby causing time/energy to be wasted. (I am not at all certain that I am correct here.)
$endgroup$
– Neal
1 hour ago
$begingroup$
@MarkS, the British Isles in the 18th century had Colin Maclaurin, Thomas Simpson, James Stirling, Brook Taylor, all working in the Newtonian tradition and with mathematical things named after them — so I wouldn’t say that Newtonian notation retarded progress.
$endgroup$
– Matt F.
48 mins ago