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I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Göde

I wonder how many other results in maths require an entire book length treatment (or multiple book length treatments)? The one that jumps out to my mind is Gödel's Incompleteness Theorem(s)[1]. I know there are at least a couple of complete books dealing exclusively with this result. [1]: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...

kevinventullo 5y ago
I would say that most hallmark results in theory-heavy mathematics (number theory, algebraic geometry, topology, …) could stretch to at least a book length treatment if you wanted a mostly self-contained proof. It sounds like the main difference with this theorem is that the writeup of the original proof was so complicated, terse, and error-prone, that very few experts even understood it.
karatinversion 5y ago
I don’t think the incompleteness theorems are nearly as wordy - this translation of Gödel’s original paper is under 40 pages after subtracting the introduction (not part of the original).
lou1306 5y ago
Godel's theorem is actually not that complex. Sure, you may think of "Godel, Escher, Bach", but that book is stuffed with sooo much material that is only tangentially related to the theorem. Of course the consequences of that theorem are legion, and they do warrant many books worth of discussion. But the proof itself can fit in a long-ish blog post, I think. Fermat's last theorem, on the other hand...
- pdpi 5y ago
  GEB is a book about intelligence/consciousness. The incompleteness theorem is incidental — it’s only there to illustrate the notion of self-referentiality, and I’m not sure it does a particularly good job at it.
- qweqwweqwe-90i 5y ago
  If you could prove the ABC conjecture then Fermat's last theorem becomes much less than a page of proof :)
  - lou1306 5y ago
    Speaking of ABC, some years ago HN was all abuzz with inter-universal Teichmuller theory, did that fizzle out or what?
    - qweqwweqwe-90i 5y ago
      https://www.math.columbia.edu/~woit/wordpress/?p=12220 https://www.math.columbia.edu/~woit/wordpress/?p=12220
- moomin 5y ago
  For those interested: Assume a formal language that can model the natural numbers. Consider the set of all provable statements. This is countable because it’s a subset of the list of strings of a finite number of characters, which is countable. Consider the statement “Statement N is false”. This is both a valid statement and cannot appear in the list. Godel proved significantly more than that, but that’s the basic result. It’s basically the same as the diagonal argument of real numbers being uncountable and it’s also basically the halting problem.
  - harmoat 5y ago
    Since these 3 results are so similar are they a subset of a more general concept?
    - deleted 5y ago
      [deleted]
    - moomin 5y ago
      Arguably the halting problem is just a subset of Godel where every theorem is "This does or does not halt". Someone's written a paper going the other way: https://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf https://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf The crucial trick is that you can plug in an entry as data to another "program/theorem". This is extremely interesting, because there are limited logics and limited computation models where Godel and the Halting Problem do not apply. Not so sure about the diagonal argument.
  - alisonkisk 5y ago
    It's "This cannot be proved.", where "this" is defined via the mathematical coding system he created. The intuitive idea is very simple, but Godel's achievement in this theorem was in finding a way to rigorously define the word "this".
  - deleted 5y ago
    [deleted]
srcreigh 5y ago
I suspect someone could write a book (or an HN comment at least) about the popular confusion about Godel's 1st incompleteness theorem. :-) In fact we (my brother and I) agreed that it bears a striking resemblance to Turing's halting problem: They both seem to show what computers or math can't do, but in fact both theorems only reveal limitations of certain models of computers and/or math. We both have undergrad degrees in math fwiw. Both theorems use a clever logical trick. For Godel's theorem, it is an encoding of "This sentence is false". For the halting problem proof, Turing first fixes the algorithm, asks the algorithm what its result will be, then switches the correct answer, forcing the algorithm to produce the incorrect answer. Would the proof completely fall apart if the algorithm could update itself to handle the new information? The only thing stopping it is that in the formulation of the problem, the algorithm is assumed to not be able to change with new information (ie it's not an "online" algorithm). Similarly, what if the model of math had mechanisms for recognizing paradoxes, or circular logical requirements, etc. I don't think either of those theorems would work anymore. Anyway, I'm not saying the Halting problem would be easy to compute with the right model. If we had an efficient algorithm to solve the halting problem, we could solve tons of math problems easily. We could write some code to enumerate any countably infinite sequence, checking for the presence of some condition, and make inferences based on if it halts or not. For example, we could solve the Collatz conjecture by just writing code to halt if any number's sequence doesn't eventually go to one, and checking whether that loop ever halts. My punchline is, I think it's beautiful that even the undecidable, uncomputable problems are only that way because we are lacking a better model. Or at least that's my current understanding of the situation :-)
- thethirdone 5y ago
  > Would the proof completely fall apart if the algorithm could update itself to handle the new information? The only thing stopping it is that in the formulation of the problem, the algorithm is assumed to not be able to change with new information (ie it's not an "online" algorithm). Simple put the halting problem means that you cannot make a program (a static piece of code) that can determine in a finite amount of time if another program halts operating on some input. I.E. program P takes program H and input I and answers if H run with I halts. I honestly don't know what part of the computational model could be changed to make that not true (besides Oracles or Church-Turing thesis being wrong). Also, I don't think "online algorithm" means what you think it means. Online algorithms have no more computational power than offline ones; they just can get started with only partial information. > Similarly, what if the model of math had mechanisms for recognizing paradoxes, or circular logical requirements, etc. I don't think either of those theorems would work anymore. I don't know what a model of recognizing paradoxes would look like. The 1st incompleteness theorem is based on only a few simple axioms and reasonable rules of deduction. There isn't much wiggle room to make a mathematical model that isn't bound by it, but is useful.
CamperBob2 5y ago
The most recent case similar to this one is Mochizuki's work on the ABC conjecture, which has still not reached the point where it's accepted by the majority of the community due to its complexity. There are lots of parallels between Mochizuki and Freedman. They both had to break a lot of new ground, and they both had trouble communicating their results to enough people to become part of the official canon of mathematics that can actually be taught to others. Freedman at least convinced a few key people, but AFAIK Mochizuki's work has never been understood by anyone not named Mochizuki. It may be complete BS, it may be earthshakingly brilliant, but either way it will evidently take someone's entire career to prove it.
- iainmerrick 5y ago
  I believe the latest news on Mochizuki’s work is that Peter Scholze and Jakob Stix identified a fatal flaw, and Mochizuki has not convincingly rebutted it. See e.g. the first item here: https://www.math.columbia.edu/~woit/wordpress/?p=12429 https://www.math.columbia.edu/~woit/wordpress/?p=12429 Both the claimed proof and the disproof have now been published in journals, and neither has been retracted! That’s a rather unusual situation to say the least. But I think people in the field generally agree that the proof is flawed and unfixable.
  - deleted 5y ago
    [deleted]
  - ntlang 5y ago
    Experts have been skeptical for quite some time. You can see discussion on the exact point Scholze (PS) brings up in the following blog post (the post and many comments are by prominent number theorists) https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ https://www.galoisrepresentations.com/2017/12/17/the-abc-con... One quote from the post I find worth repeating: > This post is not about making epistemological claims about the truth or otherwise of Mochizuki’s arguments. To take an extreme example, if Mochizuki had carved his argument on slate in Linear A and then dropped it into the Mariana Trench, then there would be little doubt that asking about the veracity of the argument would be beside the point. The reality, however, is that this description is not so far from the truth.
jcranmer 5y ago
The classification of finite simple groups [1] is apparently ~11-12 books long (when it's finished). Yes, several books for a single theorem. [1] https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl...
- Denvercoder9 5y ago
  I don't wanna know how much books the classification of finite complex groups is gonna take.
  - zarzavat 5y ago
    Not very much actually. Finite simple groups are kind of like prime numbers: any finite group can be broken down into simple subgroups similar to how any natural number can be broken down into a multiset of prime number factors. That's why the classification of finite simple groups is so important, it's essentially a classification of all finite groups. https://en.wikipedia.org/wiki/Composition_series#Uniqueness:_Jordan–Hölder_theorem https://en.wikipedia.org/wiki/Composition_series#Uniqueness:...
    - alisonkisk 5y ago
      Someone should make a modern recording of "Finite Simple Group of Order 2" https://m.youtube.com/watch?v=UTby_e4-Rhg https://m.youtube.com/watch?v=UTby_e4-Rhg
Aicy 5y ago
I did a University module on Godel' Incompleteness theotem and the complete lecture notes which gave all the lead up, explanation and the proof itself along with lots of othet related material was only about 50 pages in total if I recall correctly. Was a pleasure to do that module though!
throwaway81523 5y ago
The incompleteness theorem can be proved in a few lines given the right definitions up front. It's basically the same as the undecidability of the halting problem, which has a proof that is written as a poem: https://introcs.cs.princeton.edu/java/54computability/halting-poem.html https://introcs.cs.princeton.edu/java/54computability/haltin... There is a wonderful book by Torkel Franzén about the incompleteness theorem that is 100+ pages, but it is not purely about the proof. It's a popularization that discusses bogus ways philosophers have interpreted the theorem, and stuff like that. It is called "Gödel's Theorem: An Incomplete Guide to its Use and Abuse". Meanwhile Aschbacher and Smith's proof of the classification of quasithin groups really is a book of 1200+ pages: https://books.google.com/books?id=SbO2irSQHMEC https://books.google.com/books?id=SbO2irSQHMEC And of course the proof of the classification of finite simple groups has afaik never been published all in one place, though there are some book-length expositions of it referring back to the original papers. It is a set of papers spanning around 20 years and adding up to around 10,000 pages. It is sometimes called "the enormous theorem": https://en.wikipedia.org/wiki/Enormous_theorem https://en.wikipedia.org/wiki/Enormous_theorem
kccqzy 5y ago
Others have already mentioned it, but Gödel's incompleteness theorem doesn't need a book-length treatment. My undergraduate mathematical logic class covered the proof in a little more than a week.
fogof 5y ago
"Incompleteness ex Machina" by Sebastian Oberhoff, is a nice paper-length exposition https://arxiv.org/abs/1909.04569 https://arxiv.org/abs/1909.04569.
joppy 5y ago
It is pretty common for some pure mathematics papers to be pretty much book length anyway these days, say 80 pages or more.
mindcrime 5y ago
It's a bit late to add this, but I realize that I wasn't entirely clear above. When I say "results in maths (that) require an entire book length treatment" I don't mean a proof that is inherently book length. I mean, a proof that justifies (one or more) book length "downstream" treatments that explain, clarify, expand, etc., the base proof. Going back to Gödel's Incompleteness Theorem(s) for example... yes, the original proof is not that long, but for whatever reason, people have felt compelled to write entire books on the topic, after Gödel, to say more about the proofs. That kind of thing is more what I was thinking of, FWIW.

kevinventullo 5y ago

I would say that most hallmark results in theory-heavy mathematics (number theory, algebraic geometry, topology, …) could stretch to at least a book length treatment if you wanted a mostly self-contained proof. It sounds like the main difference with this theorem is that the writeup of the original proof was so complicated, terse, and error-prone, that very few experts even understood it.

karatinversion 5y ago

I don’t think the incompleteness theorems are nearly as wordy - this translation of Gödel’s original paper is under 40 pages after subtracting the introduction (not part of the original).

lou1306 5y ago

Godel's theorem is actually not that complex. Sure, you may think of "Godel, Escher, Bach", but that book is stuffed with sooo much material that is only tangentially related to the theorem. Of course the consequences of that theorem are legion, and they do warrant many books worth of discussion. But the proof itself can fit in a long-ish blog post, I think. Fermat's last theorem, on the other hand...

pdpi 5y ago
GEB is a book about intelligence/consciousness. The incompleteness theorem is incidental — it’s only there to illustrate the notion of self-referentiality, and I’m not sure it does a particularly good job at it.
qweqwweqwe-90i 5y ago
If you could prove the ABC conjecture then Fermat's last theorem becomes much less than a page of proof :)
- lou1306 5y ago
  Speaking of ABC, some years ago HN was all abuzz with inter-universal Teichmuller theory, did that fizzle out or what?
  - qweqwweqwe-90i 5y ago
    https://www.math.columbia.edu/~woit/wordpress/?p=12220 https://www.math.columbia.edu/~woit/wordpress/?p=12220
moomin 5y ago
For those interested: Assume a formal language that can model the natural numbers. Consider the set of all provable statements. This is countable because it’s a subset of the list of strings of a finite number of characters, which is countable. Consider the statement “Statement N is false”. This is both a valid statement and cannot appear in the list. Godel proved significantly more than that, but that’s the basic result. It’s basically the same as the diagonal argument of real numbers being uncountable and it’s also basically the halting problem.
- harmoat 5y ago
  Since these 3 results are so similar are they a subset of a more general concept?
  - deleted 5y ago
    [deleted]
  - moomin 5y ago
    Arguably the halting problem is just a subset of Godel where every theorem is "This does or does not halt". Someone's written a paper going the other way: https://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf https://www.andrew.cmu.edu/user/avigad/Teaching/halting.pdf The crucial trick is that you can plug in an entry as data to another "program/theorem". This is extremely interesting, because there are limited logics and limited computation models where Godel and the Halting Problem do not apply. Not so sure about the diagonal argument.
- alisonkisk 5y ago
  It's "This cannot be proved.", where "this" is defined via the mathematical coding system he created. The intuitive idea is very simple, but Godel's achievement in this theorem was in finding a way to rigorously define the word "this".
- deleted 5y ago
  [deleted]

srcreigh 5y ago

I suspect someone could write a book (or an HN comment at least) about the popular confusion about Godel's 1st incompleteness theorem. :-) In fact we (my brother and I) agreed that it bears a striking resemblance to Turing's halting problem: They both seem to show what computers or math can't do, but in fact both theorems only reveal limitations of certain models of computers and/or math. We both have undergrad degrees in math fwiw. Both theorems use a clever logical trick. For Godel's theorem, it is an encoding of "This sentence is false". For the halting problem proof, Turing first fixes the algorithm, asks the algorithm what its result will be, then switches the correct answer, forcing the algorithm to produce the incorrect answer. Would the proof completely fall apart if the algorithm could update itself to handle the new information? The only thing stopping it is that in the formulation of the problem, the algorithm is assumed to not be able to change with new information (ie it's not an "online" algorithm). Similarly, what if the model of math had mechanisms for recognizing paradoxes, or circular logical requirements, etc. I don't think either of those theorems would work anymore. Anyway, I'm not saying the Halting problem would be easy to compute with the right model. If we had an efficient algorithm to solve the halting problem, we could solve tons of math problems easily. We could write some code to enumerate any countably infinite sequence, checking for the presence of some condition, and make inferences based on if it halts or not. For example, we could solve the Collatz conjecture by just writing code to halt if any number's sequence doesn't eventually go to one, and checking whether that loop ever halts. My punchline is, I think it's beautiful that even the undecidable, uncomputable problems are only that way because we are lacking a better model. Or at least that's my current understanding of the situation :-)

thethirdone 5y ago
> Would the proof completely fall apart if the algorithm could update itself to handle the new information? The only thing stopping it is that in the formulation of the problem, the algorithm is assumed to not be able to change with new information (ie it's not an "online" algorithm). Simple put the halting problem means that you cannot make a program (a static piece of code) that can determine in a finite amount of time if another program halts operating on some input. I.E. program P takes program H and input I and answers if H run with I halts. I honestly don't know what part of the computational model could be changed to make that not true (besides Oracles or Church-Turing thesis being wrong). Also, I don't think "online algorithm" means what you think it means. Online algorithms have no more computational power than offline ones; they just can get started with only partial information. > Similarly, what if the model of math had mechanisms for recognizing paradoxes, or circular logical requirements, etc. I don't think either of those theorems would work anymore. I don't know what a model of recognizing paradoxes would look like. The 1st incompleteness theorem is based on only a few simple axioms and reasonable rules of deduction. There isn't much wiggle room to make a mathematical model that isn't bound by it, but is useful.

CamperBob2 5y ago

The most recent case similar to this one is Mochizuki's work on the ABC conjecture, which has still not reached the point where it's accepted by the majority of the community due to its complexity. There are lots of parallels between Mochizuki and Freedman. They both had to break a lot of new ground, and they both had trouble communicating their results to enough people to become part of the official canon of mathematics that can actually be taught to others. Freedman at least convinced a few key people, but AFAIK Mochizuki's work has never been understood by anyone not named Mochizuki. It may be complete BS, it may be earthshakingly brilliant, but either way it will evidently take someone's entire career to prove it.

iainmerrick 5y ago
I believe the latest news on Mochizuki’s work is that Peter Scholze and Jakob Stix identified a fatal flaw, and Mochizuki has not convincingly rebutted it. See e.g. the first item here: https://www.math.columbia.edu/~woit/wordpress/?p=12429 https://www.math.columbia.edu/~woit/wordpress/?p=12429 Both the claimed proof and the disproof have now been published in journals, and neither has been retracted! That’s a rather unusual situation to say the least. But I think people in the field generally agree that the proof is flawed and unfixable.
- deleted 5y ago
  [deleted]
- ntlang 5y ago
  Experts have been skeptical for quite some time. You can see discussion on the exact point Scholze (PS) brings up in the following blog post (the post and many comments are by prominent number theorists) https://www.galoisrepresentations.com/2017/12/17/the-abc-conjecture-has-still-not-been-proved/ https://www.galoisrepresentations.com/2017/12/17/the-abc-con... One quote from the post I find worth repeating: > This post is not about making epistemological claims about the truth or otherwise of Mochizuki’s arguments. To take an extreme example, if Mochizuki had carved his argument on slate in Linear A and then dropped it into the Mariana Trench, then there would be little doubt that asking about the veracity of the argument would be beside the point. The reality, however, is that this description is not so far from the truth.

jcranmer 5y ago

The classification of finite simple groups [1] is apparently ~11-12 books long (when it's finished). Yes, several books for a single theorem. [1] https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Classification_of_finite_simpl...

Denvercoder9 5y ago
I don't wanna know how much books the classification of finite complex groups is gonna take.
- zarzavat 5y ago
  Not very much actually. Finite simple groups are kind of like prime numbers: any finite group can be broken down into simple subgroups similar to how any natural number can be broken down into a multiset of prime number factors. That's why the classification of finite simple groups is so important, it's essentially a classification of all finite groups. https://en.wikipedia.org/wiki/Composition_series#Uniqueness:_Jordan–Hölder_theorem https://en.wikipedia.org/wiki/Composition_series#Uniqueness:...
  - alisonkisk 5y ago
    Someone should make a modern recording of "Finite Simple Group of Order 2" https://m.youtube.com/watch?v=UTby_e4-Rhg https://m.youtube.com/watch?v=UTby_e4-Rhg

Aicy 5y ago

I did a University module on Godel' Incompleteness theotem and the complete lecture notes which gave all the lead up, explanation and the proof itself along with lots of othet related material was only about 50 pages in total if I recall correctly. Was a pleasure to do that module though!

throwaway81523 5y ago

The incompleteness theorem can be proved in a few lines given the right definitions up front. It's basically the same as the undecidability of the halting problem, which has a proof that is written as a poem: https://introcs.cs.princeton.edu/java/54computability/halting-poem.html https://introcs.cs.princeton.edu/java/54computability/haltin... There is a wonderful book by Torkel Franzén about the incompleteness theorem that is 100+ pages, but it is not purely about the proof. It's a popularization that discusses bogus ways philosophers have interpreted the theorem, and stuff like that. It is called "Gödel's Theorem: An Incomplete Guide to its Use and Abuse". Meanwhile Aschbacher and Smith's proof of the classification of quasithin groups really is a book of 1200+ pages: https://books.google.com/books?id=SbO2irSQHMEC https://books.google.com/books?id=SbO2irSQHMEC And of course the proof of the classification of finite simple groups has afaik never been published all in one place, though there are some book-length expositions of it referring back to the original papers. It is a set of papers spanning around 20 years and adding up to around 10,000 pages. It is sometimes called "the enormous theorem": https://en.wikipedia.org/wiki/Enormous_theorem https://en.wikipedia.org/wiki/Enormous_theorem

kccqzy 5y ago

Others have already mentioned it, but Gödel's incompleteness theorem doesn't need a book-length treatment. My undergraduate mathematical logic class covered the proof in a little more than a week.

fogof 5y ago

"Incompleteness ex Machina" by Sebastian Oberhoff, is a nice paper-length exposition https://arxiv.org/abs/1909.04569 https://arxiv.org/abs/1909.04569.

joppy 5y ago

It is pretty common for some pure mathematics papers to be pretty much book length anyway these days, say 80 pages or more.

mindcrime 5y ago

It's a bit late to add this, but I realize that I wasn't entirely clear above. When I say "results in maths (that) require an entire book length treatment" I don't mean a proof that is inherently book length. I mean, a proof that justifies (one or more) book length "downstream" treatments that explain, clarify, expand, etc., the base proof. Going back to Gödel's Incompleteness Theorem(s) for example... yes, the original proof is not that long, but for whatever reason, people have felt compelled to write entire books on the topic, after Gödel, to say more about the proofs. That kind of thing is more what I was thinking of, FWIW.