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I did a Ph.D. in number theory, published a few dozen research papers, and have programmed a lot and this post sounds about right to me. I did CS as an undergr

I did a Ph.D. in number theory, published a few dozen research papers, and have programmed a lot and this post sounds about right to me. I did CS as an undergrad, before doing a math Ph.D., and remember being very surprised that math papers weren't a lot more wrong than they actually are (since computer software is so often full of bugs, and all it takes is one single bug to completely invalidate an entire paper). When I was a grad student at Berkeley, I once asked Serge Lang (who I think wrote the most advanced math textbooks of anybody who ever lived) why mathematics research papers don't have a lot more incorrect results in them. He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof, and it was unlikely that multiple very different reasons for something being true would all be wrong. Of course, there are serious mistakes and gaps in the literature, though what one person considers a gap, somebody else who knows more might not consider a gap. For example, I remember once that Ernst Kani was complaining to me about a big gap in a paper by Ribet; I didn't believe it, so I read the relevant section and was able to fill in the gap in a few hours sufficiently to satisfy him. In any case, I've spent decades both programming and doing number theory, and despite some similarities, they are very different things. Another related way these threads come together is with Kevin Buzzard's big project to formalize mathematics using LEAN; he's another number theorist, and he told me that a few years ago he started really worrying about the correctness of a lot of number theory papers, and that strongly motivated his interest in LEAN.

shuckles 5y ago
During an undergraduate research program, I discovered a gap in a theory in PDEs which I’m assuming nobody else did because I was the only one who tried to follow each step (just getting that far was the entire purpose of the effort!). My advisor said it would be best to fix the gap before publishing about it because multiple students entering the job market depended on the result being true. That was an eye-opening social experience.
- gameshot911 5y ago
  So were you able to fix it??
- redtexture 5y ago
  Partial Differential Equations for the rest of us.
- lcq92 5y ago
  Can you share what the gap was?
  - ummwhat 5y ago
    I could tell you, but it might cost some people their job.
CJefferson 5y ago
I find the further off the "main path" you go, the easier it is to find incorrect papers. In particular, I found a few papers which claim to provide algorithms which solve some problem. If there is no evidence the algorithm was ever implemented, there is a distribingly good chance it doesn't work -- I used to sometimes give such papers as projects for students, but now I always start on day 1 saying "Be prepared for this to be wrong".
hyperbovine 5y ago
> He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof. I don't do pure math, but I write the occasional theory paper, and this resonates. So much ends up on the cutting room floor--usually you proved the key result three or four different ways before finding a proof that is actually incisive/aesthetically pleasing/whatever to justify signing your name to it sending it out the door. Also most mathematicians are extremely averse to even saying something that's incorrect, let alone putting in print. There's ton of sanity-checking and trying to break your own result that, again, is rarely if ever mentioned in the final publication.
- caddemon 5y ago
  As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere. Is it left out because it's not worth the effort to include compared to the number of people that would actually read it? Or is it something that runs the risk of negative perception by those established in the field?
  - andai 5y ago
    It does seem like including a few alternative proofs would greatly aid in understanding.
  - concreteblock 5y ago
    I think it's fundamentally a question of incentives. A mathematician's career prospects and status depends heavily on being able to prove things that others can't. Why spend extra time helping your 'competitors'? Not saying this is ideal, but it seems to explain many behaviours in the community, and it seems unavoidable given the ever-increasing competition for jobs.
  - thehappypm 5y ago
    It's actually pretty tough to write out a proof, and going from a whiteboard where you're 99.9% sure it's right to something in LaTeX in a paper in your field is actually a long number of steps to take.
jkhdigital 5y ago
Are you familiar with homotopy type theory? One of its proponents, Fields medalist Vladimir Voevodsky, has stated that his contributions are part of a “personal mission” to bring mathematics into a new age of formal verification [1]. I’d be curious to know how this compares to LEAN. [1] https://www.ias.edu/ideas/2014/voevodsky-origins https://www.ias.edu/ideas/2014/voevodsky-origins
- deleted 5y ago
  [deleted]
- mhh__ 5y ago
  I am kind of sceptical of these verified-mathematics projects in the sense that my litmus is whether I could easily verify my workings rather than the proofs I rely upon, but nonetheless I have enormous respect for those doing this work and really hope it comes to fruition. I really hope this work trickles down to our programming - state machine problems are basically "solved" (for my needs at least), but more complex programs are really hard to prove.
  - pramodbiligiri 5y ago
    When you say that state machine problems are basically solved, are you referring to any tools that you're using to reason about them?
    - mhh__ 5y ago
      By solved I really mean "in commercial use", there's probably a lot of research let to be done, but people are already verifying TLA+ specs all day everyday.
  - jkhdigital 5y ago
    I'm a PhD student in computer science, and after spending some time on this subject I'm pretty well convinced that parts of mathematics and CS are converging towards a more unified foundational framework that will benefit both fields tremendously. In a nutshell, the key challenge is taking the core of complexity theory (which is easy to formalize in terms of state machines) and migrating it into the language(s) of category/type theory (which is much more modular/compositional). A couple interesting recent works in this area: Categorical Complexity: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FB2CAC644C697FEF1092C285D784CAB4/S2050509420000262a.pdf/categorical-complexity.pdf https://www.cambridge.org/core/services/aop-cambridge-core/c... Dusko Pavlovic's Monoidal computer series: 1. https://arxiv.org/pdf/1208.5205 https://arxiv.org/pdf/1208.5205 2. https://arxiv.org/pdf/1402.5687 https://arxiv.org/pdf/1402.5687 3. https://arxiv.org/pdf/1704.04882 https://arxiv.org/pdf/1704.04882
nicoburns 5y ago
> He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof, and it was unlikely that multiple very different reasons for something being true would all be wrong. This was the most frustrating thing for me when I was studying undergraduate maths. Often a result was just stated, with the reasoning behind how it was generated completely omitted. Good professors would go through all of that, but those seemed to be few and far between. The rest just expected you to work it out (despite not ever being exposed to it or pointed to a good source), or rote memorise the result.
- ska 5y ago
  There is a fundamental tension here because a) doing this stuff properly takes a long time and b) lots of undergraduate math teaching is "service" teaching, meant to get students up to a level where they can do the manipulations needed in other courses. This is why there is typically separate Calculus and Real Analysis streams in undergraduate - the former is needed for all the hard sciences and engineering, the latter for math students and fellow travelers who really want to understand what is going on and be able to prove things properly. Typically the analysis stream will start with a pretty in-depth look at what real numbers actually are (spoiler: weirder than you thought) and the implications of that. You get a much better understanding this way but you don't cover ground nearly as quickly.
  - smallnamespace 5y ago
    Knowing and using a thing and proving it are (usually) different skills.
    - ska 5y ago
      True; but so are understanding a thing well and being able to use it. I wasn't really talking about proof-based vs. not though - it's true that an analysis course will be more proof based (than Calculus) because that is also a skill you are expected to be developing. However most calculus graduates, even the ones at the top of their class, still have at best a somewhat superficial understanding of the set of real numbers and functions on it, regardless of whether or not they can regurgitate an epsilon-delta proof. This isn't a knock on calculus courses, there is an opportunity cost to the time. By the time a continuous math student hits measure theory and understands why they need a(nother) different definition of an integral, most calculus students have been happily chugging along calculating things they need, blissfully ignorant of the "problems" with the integrals they barely remember being defined.
hannob 5y ago
Given that you seem to know this space, I have been wondering: Was there ever any major flaw in an important mathematical proof that was taken as a given, and subsequently plenty of papers building on that turned out to be wrong as well? Like I don't mean "this proof has a gap in its justification, but the thing claimed likely is still true". I mean "we thought someone had a proof for X, but actually later someone else showed X isn't true".
- Penyngton 5y ago
  I'm not sure if this quite matches your request (I suppose much hinges on the word "important"), but here's a story of a fairly famous and long-standing error: https://www.jamesrmeyer.com/ffgit/godel-error-2.html https://www.jamesrmeyer.com/ffgit/godel-error-2.html Personally, I'd say the fact that someone bothered to discover it was false fifty years after it was published shows its importance. The short version is that Gödel claimed in a paper that he had an algorithm to determine satisfiability for a certain logic, but the algorithm actually worked for a slightly different logic instead. People basically took Gödel's word for it for fifty years until a logician called Goldfarb found the mistake. EDIT: As a disclaimer, I don't have any association with or endorsement of the website I've linked to. I found it just now because I needed a source for the story that I've known for many years (I think I first heard it in person, but it's definitely contained in the book `The Classical Decision Problem' by Börger, Grädel and Gurevich). I skimmed the post and it seemed to cover the relevant details, but that's the extent of my knowledge of the site and its contents.
- dynrzk 5y ago
  Somewhat famously, symplectic geometry had its fair bit of foundational issues.
- Sniffnoy 5y ago
  This[0] may be a good place to start, but I'm not sure if it contains proper examples of what you say (of serious collapse). E.g., I expect a lot of people probably relied on the false result mentioned by Matheus, but most of them could probably be repaired with the corrected statement. So it seems like this probably hasn't happened? OTOH, there was the Italian school of algebraic geometry, but that was more than one single flaw... [0] https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong https://mathoverflow.net/questions/35468/widely-accepted-mat...
oolonthegreat 5y ago
wanted to drop a link to buzzard's very interesting talk https://youtu.be/Dp-mQ3HxgDE https://youtu.be/Dp-mQ3HxgDE starting at 13:00 he talks about that frailness. also loved the bit about "elders" at 40:37
deleted 5y ago
[deleted]

shuckles 5y ago

During an undergraduate research program, I discovered a gap in a theory in PDEs which I’m assuming nobody else did because I was the only one who tried to follow each step (just getting that far was the entire purpose of the effort!). My advisor said it would be best to fix the gap before publishing about it because multiple students entering the job market depended on the result being true. That was an eye-opening social experience.

gameshot911 5y ago
So were you able to fix it??
redtexture 5y ago
Partial Differential Equations for the rest of us.
lcq92 5y ago
Can you share what the gap was?
- ummwhat 5y ago
  I could tell you, but it might cost some people their job.

CJefferson 5y ago

I find the further off the "main path" you go, the easier it is to find incorrect papers. In particular, I found a few papers which claim to provide algorithms which solve some problem. If there is no evidence the algorithm was ever implemented, there is a distribingly good chance it doesn't work -- I used to sometimes give such papers as projects for students, but now I always start on day 1 saying "Be prepared for this to be wrong".

hyperbovine 5y ago

> He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof. I don't do pure math, but I write the occasional theory paper, and this resonates. So much ends up on the cutting room floor--usually you proved the key result three or four different ways before finding a proof that is actually incisive/aesthetically pleasing/whatever to justify signing your name to it sending it out the door. Also most mathematicians are extremely averse to even saying something that's incorrect, let alone putting in print. There's ton of sanity-checking and trying to break your own result that, again, is rarely if ever mentioned in the final publication.

caddemon 5y ago
As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere. Is it left out because it's not worth the effort to include compared to the number of people that would actually read it? Or is it something that runs the risk of negative perception by those established in the field?
- andai 5y ago
  It does seem like including a few alternative proofs would greatly aid in understanding.
- concreteblock 5y ago
  I think it's fundamentally a question of incentives. A mathematician's career prospects and status depends heavily on being able to prove things that others can't. Why spend extra time helping your 'competitors'? Not saying this is ideal, but it seems to explain many behaviours in the community, and it seems unavoidable given the ever-increasing competition for jobs.
- thehappypm 5y ago
  It's actually pretty tough to write out a proof, and going from a whiteboard where you're 99.9% sure it's right to something in LaTeX in a paper in your field is actually a long number of steps to take.

jkhdigital 5y ago

Are you familiar with homotopy type theory? One of its proponents, Fields medalist Vladimir Voevodsky, has stated that his contributions are part of a “personal mission” to bring mathematics into a new age of formal verification [1]. I’d be curious to know how this compares to LEAN. [1] https://www.ias.edu/ideas/2014/voevodsky-origins https://www.ias.edu/ideas/2014/voevodsky-origins

deleted 5y ago
[deleted]
mhh__ 5y ago
I am kind of sceptical of these verified-mathematics projects in the sense that my litmus is whether I could easily verify my workings rather than the proofs I rely upon, but nonetheless I have enormous respect for those doing this work and really hope it comes to fruition. I really hope this work trickles down to our programming - state machine problems are basically "solved" (for my needs at least), but more complex programs are really hard to prove.
- pramodbiligiri 5y ago
  When you say that state machine problems are basically solved, are you referring to any tools that you're using to reason about them?
  - mhh__ 5y ago
    By solved I really mean "in commercial use", there's probably a lot of research let to be done, but people are already verifying TLA+ specs all day everyday.
- jkhdigital 5y ago
  I'm a PhD student in computer science, and after spending some time on this subject I'm pretty well convinced that parts of mathematics and CS are converging towards a more unified foundational framework that will benefit both fields tremendously. In a nutshell, the key challenge is taking the core of complexity theory (which is easy to formalize in terms of state machines) and migrating it into the language(s) of category/type theory (which is much more modular/compositional). A couple interesting recent works in this area: Categorical Complexity: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/FB2CAC644C697FEF1092C285D784CAB4/S2050509420000262a.pdf/categorical-complexity.pdf https://www.cambridge.org/core/services/aop-cambridge-core/c... Dusko Pavlovic's Monoidal computer series: 1. https://arxiv.org/pdf/1208.5205 https://arxiv.org/pdf/1208.5205 2. https://arxiv.org/pdf/1402.5687 https://arxiv.org/pdf/1402.5687 3. https://arxiv.org/pdf/1704.04882 https://arxiv.org/pdf/1704.04882

nicoburns 5y ago

> He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof, and it was unlikely that multiple very different reasons for something being true would all be wrong. This was the most frustrating thing for me when I was studying undergraduate maths. Often a result was just stated, with the reasoning behind how it was generated completely omitted. Good professors would go through all of that, but those seemed to be few and far between. The rest just expected you to work it out (despite not ever being exposed to it or pointed to a good source), or rote memorise the result.

ska 5y ago
There is a fundamental tension here because a) doing this stuff properly takes a long time and b) lots of undergraduate math teaching is "service" teaching, meant to get students up to a level where they can do the manipulations needed in other courses. This is why there is typically separate Calculus and Real Analysis streams in undergraduate - the former is needed for all the hard sciences and engineering, the latter for math students and fellow travelers who really want to understand what is going on and be able to prove things properly. Typically the analysis stream will start with a pretty in-depth look at what real numbers actually are (spoiler: weirder than you thought) and the implications of that. You get a much better understanding this way but you don't cover ground nearly as quickly.
- smallnamespace 5y ago
  Knowing and using a thing and proving it are (usually) different skills.
  - ska 5y ago
    True; but so are understanding a thing well and being able to use it. I wasn't really talking about proof-based vs. not though - it's true that an analysis course will be more proof based (than Calculus) because that is also a skill you are expected to be developing. However most calculus graduates, even the ones at the top of their class, still have at best a somewhat superficial understanding of the set of real numbers and functions on it, regardless of whether or not they can regurgitate an epsilon-delta proof. This isn't a knock on calculus courses, there is an opportunity cost to the time. By the time a continuous math student hits measure theory and understands why they need a(nother) different definition of an integral, most calculus students have been happily chugging along calculating things they need, blissfully ignorant of the "problems" with the integrals they barely remember being defined.

hannob 5y ago

Given that you seem to know this space, I have been wondering: Was there ever any major flaw in an important mathematical proof that was taken as a given, and subsequently plenty of papers building on that turned out to be wrong as well? Like I don't mean "this proof has a gap in its justification, but the thing claimed likely is still true". I mean "we thought someone had a proof for X, but actually later someone else showed X isn't true".

Penyngton 5y ago
I'm not sure if this quite matches your request (I suppose much hinges on the word "important"), but here's a story of a fairly famous and long-standing error: https://www.jamesrmeyer.com/ffgit/godel-error-2.html https://www.jamesrmeyer.com/ffgit/godel-error-2.html Personally, I'd say the fact that someone bothered to discover it was false fifty years after it was published shows its importance. The short version is that Gödel claimed in a paper that he had an algorithm to determine satisfiability for a certain logic, but the algorithm actually worked for a slightly different logic instead. People basically took Gödel's word for it for fifty years until a logician called Goldfarb found the mistake. EDIT: As a disclaimer, I don't have any association with or endorsement of the website I've linked to. I found it just now because I needed a source for the story that I've known for many years (I think I first heard it in person, but it's definitely contained in the book `The Classical Decision Problem' by Börger, Grädel and Gurevich). I skimmed the post and it seemed to cover the relevant details, but that's the extent of my knowledge of the site and its contents.
dynrzk 5y ago
Somewhat famously, symplectic geometry had its fair bit of foundational issues.
Sniffnoy 5y ago
This[0] may be a good place to start, but I'm not sure if it contains proper examples of what you say (of serious collapse). E.g., I expect a lot of people probably relied on the false result mentioned by Matheus, but most of them could probably be repaired with the corrected statement. So it seems like this probably hasn't happened? OTOH, there was the Italian school of algebraic geometry, but that was more than one single flaw... [0] https://mathoverflow.net/questions/35468/widely-accepted-mathematical-results-that-were-later-shown-to-be-wrong https://mathoverflow.net/questions/35468/widely-accepted-mat...

oolonthegreat 5y ago

wanted to drop a link to buzzard's very interesting talk https://youtu.be/Dp-mQ3HxgDE https://youtu.be/Dp-mQ3HxgDE starting at 13:00 he talks about that frailness. also loved the bit about "elders" at 40:37

deleted 5y ago

[deleted]