15 ms·

New math book rescues landmark topology proof

372 points by theafh 5y ago

phaedrus 5y ago
The part I find most interesting is that (as I understand the article), Freedman's proof relies on a meta-proof that details not worked out in the base proof could be ignored: "But there were places where he couldn’t quite complete the picture — as if he were creating a portrait and there were some aspects of his subject’s face he couldn’t see. His last move, then, was to prove that those gaps in his picture — the places he couldn’t see — didn’t matter from the standpoint of the type of equivalence he was after. That is, the gaps in the picture could not possibly prevent the Casson handle from being homeomorphic to the disc, no matter what they contained."
- dodobirdlord 5y ago
  A common approach to proving a broad claim when there’s no straightforward proof is to figure out all of the entities that have the potential to produce a counterexample, break them up into a finite list of categories, and go through category by category proving, often with very different approaches depending on the category, that each category cannot contain a counterexample. A major theorem proved in this style was the classification of finite simple groups. Not only was it done in this style, it frequently serves as a major component in other proofs of this style, since theorems about other kinds of groups can sometimes be re-expressed as claims about finite simple groups, and then checked category by category. https://en.m.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.m.wikipedia.org/wiki/Classification_of_finite_sim... EDIT: This is also part of why sometimes the first proof of a theorem will be very long, but someone will later use the insight of the first proof to produce a much shorter proof. The first proof will basically be a proof that the general claim breaks down into a (possibly very long) list of separately probable lemmas, followed by a proof of each of the lemmas. These may not overlap very much, leading to proofs that are very long and take a lot of background to understand. Once the result is known to be true, it may be worth the effort for the original author or someone else to refactor the “proved it can’t be disproved” proof into a much shorter direct proof.
leephillips 5y ago
If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.
- taeric 5y ago
  I realize I have little chance of actually being able to understand all of it. Still, I'd gladly have bought it at a lower cost. :(
- mindcrime 5y ago
  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...
hxksleo 5y ago
I'd be interested in whether proofs like these will be formalized in proof assistants that can be checked with computer code, so that it removes doubt of error. It's something in math I'd like to see become more prevalent.
- zozbot234 5y ago
  It can be done, this is exactly what homotopy type theory is for.
- thaumasiotes 5y ago
  > I'd be interested in whether proofs like these will be formalized in proof assistants that can be checked with computer code, so that it removes doubt of error. This does not remove doubt of error.
- openasocket 5y ago
  It's possible, but difficult. Proofs written by humans tend to not include a bunch of relevant details and assumptions. They consist of lines like "We have some property/object/logical statement X, then by theorem Y we have Z", but often don't state how exactly they are using theorem Y to get result Z. There's often some additional algebraic or logical manipulation, or some simplification of terms, or implicitly using some other theorem or lemma that seems obvious to the writer. Getting a computer to automatically find proofs of statements is difficult (impossible in general), and I wouldn't be surprised if converting a standard human-written proof into a formal proof system is just as hard.
- mahogany 5y ago
  Something at this level is still far away, but it should be possible in theory. Last time I checked, there was still a lot of work to be done with formalizing geometric objects in Lean (the only proof assistant I have experience with). We are more in the stage of formalizing proofs from undergraduate books, whereas the proof of the 4-dimensional Poincare conjecture is multiple orders of magnitude more difficult. From my understanding and limited experience, formalizing complex objects is quite difficult. A long proof where the objects are integers or algebraic relations might be easier than even defining a manifold. For example, with Lean, it was a lot of work to even define a perfectoid space -- which, to be fair, is a complicated object -- much less, say anything interesting about it. If someone has more experience with other proof assistants, I'd be interested to hear about how far away a proof like Freedman's would be. My understanding is that each one (coq, agda, lean, etc) has certain drawbacks or benefits and can describe some concepts more easily than others, but "high-level" proofs are few and far between for all. For example, Coq has a verification of Feit-Thompson Theorem which is really cool, but I don't think there are many complex objects in there. As far as I know, it's groups, linear algebra, generator and relation computations -- all of which are pretty well handled by computers. On the other hand, to even begin a verification of Fermat's Last Theorem, you would have to first build up the entire scaffolding of modern (well, second half of the 20th century at least) algebraic number theory, quite a feat in itself.
spekcular 5y ago
This has been a long time coming. There is an infamous MathOverflow thread about the proof (also linked in the article), with the following comment summarizing the state of affairs ~10 years ago: > There is no other evidence. In fact there is absolutely no evidence what so ever. I have never met a mathematician who could convince me that he or she understood Freedman`s proof. I attempted to read that monstrosity of a paper a number of times by myself and quite a few times with a group of other mathematicians. We never were able to finish checking all of the details. Such seminars always ended before we could make it through even half of his paper. No other expositions on the subject seem to be any better. It is truely an odd state of affairs that after all of these years no one has managed to write a clear exposition of this so called proof, and that no one seems to question the claim that there ever was a proof. I remember thinking as a young mathematician either this "proof" is sheer nonsense or someone will eventually write out a clear and detailed explanation. As of April of 2011 I have understood that the so called proof is full of errors and they can not be fixed. I mentioned this to several mathematicians during the summer of 2011 and I believe these conversations are directly linked to the dialogue seen here on math overflow. Also on that thread, Brendan Guilfoyle raised the issue of writing a complete book-length treatment of the proof. I found the following response an interesting commentary on incentives in academia. > Promotion committee: "What have you done in the last 5 years?" > B.G.: "I've been writing a book about a paper published in 1982." > Promotion committee: "How is that groundbreaking overachieving hyped research?" > B.G.: "It is not, but I am preserving the knowledge of the elders." > Promotion committee: "Who told you that was a good idea?" > B.G.: "A guy on the internet, and 24 people liked his suggestion." > Promotion committee: "We will call you, don't call us. Good bye."
- auntienomen 5y ago
  That comment on incentives is funny, but I'm not sure it captures the situation here: The mathematicians who wrote The Disc Embedding Theorem have been quite successful. All are on the tenure ladder (or tenured already) and publishing novel research in a field they revitalized. I can see how sinking time into this is risky, but that's generally true of choosing a field of research. I think the authors deserve credit for doing it well. Likewise, Peter Teichner should get a little credit for helping to create the space for them to do so.
martincmartin 5y ago
So who gets credit for proving it? Freedman, the original author from 1981; or the authors of this book? If Freedman's proof has so many gaps, should it be considered more of a proof outline, or a motivation for believing a conjecture, and the current book be considered the actual proof? Or does Freedman get the glory? In that case, there's not much incentive for people to do what these book authors did. Which I suppose is why it took 30 years, and is only happening now because they "were captivated by the beauty of Freedman’s proof and wanted to give it new life. "
- auntienomen 5y ago
  It's not a strict exclusive-or choice. Credit will accrue to both Freedman and the authors of this book, because both advanced the state of the subject. And eventually perhaps further credit to someone who publishes a machine-verifiable proof, or finds a simpler way to understand the theorem.
- pfortuny 5y ago
  Well, thank God lots of scientists (mathematicians in this case) understand that transmission of knowledge is as important (or more so) as novelty. And that is their incentive.
  - ChrisKnott 5y ago
    This distinction has been called an "open exposition problem", versus an "open research problem". Once Freedman's proof was accepted, it was no longer an open research problem; on the publication of this book, it will (hopefully) no longer be an open exposition problem. The phrase was coined by Timothy Chow in his paper A Beginner's Guide To Forcing [0]; > "All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves." [0] https://arxiv.org/abs/0712.1320 https://arxiv.org/abs/0712.1320
- 5y ago
danabo 5y ago
The book in question is called "The Disc Embedding Theorem" (https://global.oup.com/academic/product/the-disc-embedding-theorem-9780198841319 https://global.oup.com/academic/product/the-disc-embedding-t...)
lostmsu 5y ago
Math is the most formal science there is. Why don't serious math journals require programmatically checked proofs for all publications, and instead rely on what essentially is a code review for a very large change request with extremely convoluted code to be vetted bug free by only a handful of experts? P.S. Reviews are still needed to check novelty etc.
- HenryKissinger 5y ago
  > Math is the most formal science there is. Can you prove that?
  - robbedpeter 5y ago
    Formality is technically the degree of mathiness applied to a domain, since formal logic and any other framework of formalization decomposes to mathematics. So... it's a tautology. Math is the most mathy math. Biology or psychology are less mathy maths. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... Having provablility holes in the formal system doesn't imply the existence of other systems that could be decomposed from currently known or unknown symbols. In this universe, information at the level of string theory seems to describe the limits, so math, as she is wrote, is as formal as it gets. The map is not the territory, so math, as a map, will always be less "formal" than reality itself. It's pretty firmly in second place, though.
    - ttctciyf 5y ago
      But if you only have maps and no territory, can you still be said to be doing cartography? :)
- nerdponx 5y ago
  Is it possible to even check such a proof programmatically? Do we have that kind of technology? Do you also have to review the proof checking code to make sure you didn't make any errors in transcription from the original paper?
maxFlow 5y ago
Makes you wonder what's the most impactful thing you could build given the possibility of seven years of isolation and single-minded focus. Would be interesting to know, at which point did Freedman decide to commit full time to his research---based on what, an intuition? And how did he decide to stop at 7 years and not 4, 9 or 10? This is advanced stuff for me but I'll still be getting a copy of _The Disc Embedding Theorem_ to support this kind of niche research. Findings like these could be stepping stones towards bigger discoveries in the future.
doublerabbit 5y ago
I really wish I could get math to stick. I just finished my National 5 maths (rough equivalent of a US High School Diploma) night-class today and all I ever seem to understand is how, but not why. I'm the one asking "why is that that." And today was recapping on trinomial, simplifying fractions. Simple I expect to anyone with a mathematical mind, but to me, it's just an insane implosion which leaves me exhausted. I have notes, but even still. What is the secret behind it? How do people get so good with Maths? Is it truly study,study & study? .. I hadn't even herd of the terminology of co-efficient until today. Let alone something like this.
- curtisf 5y ago
  High school math does not look much like research math. High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ... Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving_ that that characterization is actually correct) For example, the definition of "continuous" you learn in pre-calc is probably something like: if you compute a function's limit, you get the same thing as computing the function. The topologist's definition is that the function's inverse map preserves openness. These are basically unrecognizable as being the same thing, but in fact they are. And the second doesn't actually reference calculation or even anything identifiable as numbers! I believe this is a significant cause of not understanding "why" despite understanding "how". As an example, I remember in pre-calculus we spent at least an entire unit on partial-fraction-decomposition. Partial fraction decomposition is a strange algebraic manipulation that is basically only important to integrating rational functions -- something you don't do until at least a year later, and which doing by hand is basically pointless because computers are better at it than humans. They also require very different kinds of intuition. Building intuition is hard, but important for people studying math. Working through many different examples and problems and applications can help you build intuition. But I don't really have any advice for how to make math stick better, but just want to share that the mathematics you learn in high school is not representative of the entire field. If you ever want to get a taste of "higher math" without having lots of prereqs, point-set topology and group theory are two (very different!) fields that are also fairly accessible.
korbonits 5y ago
What about this guy's book? https://math.uchicago.edu/~dannyc/books/4dpoincare/4dpoincare.html https://math.uchicago.edu/~dannyc/books/4dpoincare/4dpoincar...
- codethief 5y ago
  Relevant: https://mathoverflow.net/questions/87674/independent-evidence-for-the-classification-of-topological-4-manifolds#comment881086_87674 https://mathoverflow.net/questions/87674/independent-evidenc...
cschmidt 5y ago
I do find it amazing that mathematicians seem to have almost infinite time to work on problems, and so much freedom to choose what they're working on. How great that a big group of people can collectively decide to spend years creating a book like this.
- Koshkin 5y ago
  What is amazing to me is that most people think they do not have such freedom.
  - vlovich123 5y ago
    Most people don’t as doing so requires a level of talent, intelligence, and work ethic/determination that’s in the upper tail of human distribution. Paradoxically, many people who could probably do underestimate their talents while others who can’t/should t overestimate theirs. The universe is funny in that way.
- paulpauper 5y ago
  The US ,global economy is huge. colleges have billion of dollars of endowments and tuition. More than enough $ to go around to fund it
dls2016 5y ago
My buddy had “beat a Fields medalist at a foot race” on his CV for a while after completing the Pier to Peak half marathon faster than (a ~60 year old) Prof Freedman. I’d run to Goleta beach playground with my kids and would see Freedman wander down from Station Q and bust out a few chin ups on the monkey bars. I hope I’m in that kind of shape!
kurthr 5y ago
"I probably didn’t treat the exposition of the written material as carefully as I should have,” said Freedman. That's a pretty sad (if self aware) statement for someone who gets paid $1M/yr to find arrangements which might make quantum decoherence effects lower for computation ... or not. The editors certainly won't make that much, but at least there's an afterward by Freedman.
throwaway81523 5y ago
I had not realized the 4-dimensional case was so hard that the much more famous 3-dimensional case was "another" contender for the hardest. I had thought it was sort of like the 4 color map theorem on stereoids, where first it was proved in 5+ dimensions (Smale, 1961), then 4 dimensions (Freedman, 1982), and finally 3 (Perelman, 2006), getting harder and harder at each step.
- civilized 5y ago
  And the proofs of this one fact are all completely different depending on the dimension. I wonder if there are any prospects of a unified proof for all dimensions.
  - throwaway81523 5y ago
    Smale's proof for d>=5 was unified. The d=4 and d=3 cases were special and those are special dimensions for a lot of reasons. "Low-dimensional topology", i.e. d=3 or d=4, is a big subject in its own right, because of those issues. At least for now, the prospect of unifying them with higher dimensions is somewhat remote. Basically in higher dimensions, there is more space to maneuver and things don't get in each other's way as much, so many proofs become easier. As well, many things are true in high dimensions and false in low dimensions, in which case unification is obviously impossible.. See also "curse of dimensionality" about why things are different for high dimensions.
yejin83810 5y ago
https://web.archive.org/web/20210910012837/https://tmpfiles.org/dl/110636/thediscembeddingtheorembystefanbehrensboldizsarkalmarminhoonkimmarkpowellarunimaray.pdf https://web.archive.org/web/20210910012837/https://tmpfiles....
willhinsa 5y ago
I read the title as "New math book rescues landmark topology prof", and I was wondering how on earth a math book pulled that off!
EVa5I7bHFq9mnYK 5y ago
Funny, how the same conjecture required completely different and unrelated proofs for 3, 4 and >5 cases.

372 points by theafh 5y ago

The part I find most interesting is that (as I understand the article), Freedman's proof relies on a meta-proof that details not worked out in the base proof could be ignored: "But there were places where he couldn’t quite complete the picture — as if he were creating a portrait and there were some aspects of his subject’s face he couldn’t see. His last move, then, was to prove that those gaps in his picture — the places he couldn’t see — didn’t matter from the standpoint of the type of equivalence he was after. That is, the gaps in the picture could not possibly prevent the Casson handle from being homeomorphic to the disc, no matter what they contained."

dodobirdlord 5y ago
A common approach to proving a broad claim when there’s no straightforward proof is to figure out all of the entities that have the potential to produce a counterexample, break them up into a finite list of categories, and go through category by category proving, often with very different approaches depending on the category, that each category cannot contain a counterexample. A major theorem proved in this style was the classification of finite simple groups. Not only was it done in this style, it frequently serves as a major component in other proofs of this style, since theorems about other kinds of groups can sometimes be re-expressed as claims about finite simple groups, and then checked category by category. https://en.m.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.m.wikipedia.org/wiki/Classification_of_finite_sim... EDIT: This is also part of why sometimes the first proof of a theorem will be very long, but someone will later use the insight of the first proof to produce a much shorter proof. The first proof will basically be a proof that the general claim breaks down into a (possibly very long) list of separately probable lemmas, followed by a proof of each of the lemmas. These may not overlap very much, leading to proofs that are very long and take a lot of background to understand. Once the result is known to be true, it may be worth the effort for the original author or someone else to refactor the “proved it can’t be disproved” proof into a much shorter direct proof.

leephillips 5y ago

If I understand this article, it’s about a 500-page book that is devoted to one proof of one theorem in topology. I find that amazing. And cheers to Quantum Magazine for regularly publishing popularizations of research mathematics. I know many, at times, take issue with their simplifications and framing, but they’re trying, where almost no one else with their reach covers these areas at all.

taeric 5y ago
I realize I have little chance of actually being able to understand all of it. Still, I'd gladly have bought it at a lower cost. :(
mindcrime 5y ago
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...

hxksleo 5y ago

I'd be interested in whether proofs like these will be formalized in proof assistants that can be checked with computer code, so that it removes doubt of error. It's something in math I'd like to see become more prevalent.

zozbot234 5y ago
It can be done, this is exactly what homotopy type theory is for.
thaumasiotes 5y ago
> I'd be interested in whether proofs like these will be formalized in proof assistants that can be checked with computer code, so that it removes doubt of error. This does not remove doubt of error.
openasocket 5y ago
It's possible, but difficult. Proofs written by humans tend to not include a bunch of relevant details and assumptions. They consist of lines like "We have some property/object/logical statement X, then by theorem Y we have Z", but often don't state how exactly they are using theorem Y to get result Z. There's often some additional algebraic or logical manipulation, or some simplification of terms, or implicitly using some other theorem or lemma that seems obvious to the writer. Getting a computer to automatically find proofs of statements is difficult (impossible in general), and I wouldn't be surprised if converting a standard human-written proof into a formal proof system is just as hard.
mahogany 5y ago
Something at this level is still far away, but it should be possible in theory. Last time I checked, there was still a lot of work to be done with formalizing geometric objects in Lean (the only proof assistant I have experience with). We are more in the stage of formalizing proofs from undergraduate books, whereas the proof of the 4-dimensional Poincare conjecture is multiple orders of magnitude more difficult. From my understanding and limited experience, formalizing complex objects is quite difficult. A long proof where the objects are integers or algebraic relations might be easier than even defining a manifold. For example, with Lean, it was a lot of work to even define a perfectoid space -- which, to be fair, is a complicated object -- much less, say anything interesting about it. If someone has more experience with other proof assistants, I'd be interested to hear about how far away a proof like Freedman's would be. My understanding is that each one (coq, agda, lean, etc) has certain drawbacks or benefits and can describe some concepts more easily than others, but "high-level" proofs are few and far between for all. For example, Coq has a verification of Feit-Thompson Theorem which is really cool, but I don't think there are many complex objects in there. As far as I know, it's groups, linear algebra, generator and relation computations -- all of which are pretty well handled by computers. On the other hand, to even begin a verification of Fermat's Last Theorem, you would have to first build up the entire scaffolding of modern (well, second half of the 20th century at least) algebraic number theory, quite a feat in itself.

spekcular 5y ago

This has been a long time coming. There is an infamous MathOverflow thread about the proof (also linked in the article), with the following comment summarizing the state of affairs ~10 years ago: > There is no other evidence. In fact there is absolutely no evidence what so ever. I have never met a mathematician who could convince me that he or she understood Freedman`s proof. I attempted to read that monstrosity of a paper a number of times by myself and quite a few times with a group of other mathematicians. We never were able to finish checking all of the details. Such seminars always ended before we could make it through even half of his paper. No other expositions on the subject seem to be any better. It is truely an odd state of affairs that after all of these years no one has managed to write a clear exposition of this so called proof, and that no one seems to question the claim that there ever was a proof. I remember thinking as a young mathematician either this "proof" is sheer nonsense or someone will eventually write out a clear and detailed explanation. As of April of 2011 I have understood that the so called proof is full of errors and they can not be fixed. I mentioned this to several mathematicians during the summer of 2011 and I believe these conversations are directly linked to the dialogue seen here on math overflow. Also on that thread, Brendan Guilfoyle raised the issue of writing a complete book-length treatment of the proof. I found the following response an interesting commentary on incentives in academia. > Promotion committee: "What have you done in the last 5 years?" > B.G.: "I've been writing a book about a paper published in 1982." > Promotion committee: "How is that groundbreaking overachieving hyped research?" > B.G.: "It is not, but I am preserving the knowledge of the elders." > Promotion committee: "Who told you that was a good idea?" > B.G.: "A guy on the internet, and 24 people liked his suggestion." > Promotion committee: "We will call you, don't call us. Good bye."

auntienomen 5y ago
That comment on incentives is funny, but I'm not sure it captures the situation here: The mathematicians who wrote The Disc Embedding Theorem have been quite successful. All are on the tenure ladder (or tenured already) and publishing novel research in a field they revitalized. I can see how sinking time into this is risky, but that's generally true of choosing a field of research. I think the authors deserve credit for doing it well. Likewise, Peter Teichner should get a little credit for helping to create the space for them to do so.

martincmartin 5y ago

So who gets credit for proving it? Freedman, the original author from 1981; or the authors of this book? If Freedman's proof has so many gaps, should it be considered more of a proof outline, or a motivation for believing a conjecture, and the current book be considered the actual proof? Or does Freedman get the glory? In that case, there's not much incentive for people to do what these book authors did. Which I suppose is why it took 30 years, and is only happening now because they "were captivated by the beauty of Freedman’s proof and wanted to give it new life. "

auntienomen 5y ago
It's not a strict exclusive-or choice. Credit will accrue to both Freedman and the authors of this book, because both advanced the state of the subject. And eventually perhaps further credit to someone who publishes a machine-verifiable proof, or finds a simpler way to understand the theorem.
pfortuny 5y ago
Well, thank God lots of scientists (mathematicians in this case) understand that transmission of knowledge is as important (or more so) as novelty. And that is their incentive.
- ChrisKnott 5y ago
  This distinction has been called an "open exposition problem", versus an "open research problem". Once Freedman's proof was accepted, it was no longer an open research problem; on the publication of this book, it will (hopefully) no longer be an open exposition problem. The phrase was coined by Timothy Chow in his paper A Beginner's Guide To Forcing [0]; > "All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves." [0] https://arxiv.org/abs/0712.1320 https://arxiv.org/abs/0712.1320
5y ago

danabo 5y ago

The book in question is called "The Disc Embedding Theorem" (https://global.oup.com/academic/product/the-disc-embedding-theorem-9780198841319 https://global.oup.com/academic/product/the-disc-embedding-t...)

lostmsu 5y ago

Math is the most formal science there is. Why don't serious math journals require programmatically checked proofs for all publications, and instead rely on what essentially is a code review for a very large change request with extremely convoluted code to be vetted bug free by only a handful of experts? P.S. Reviews are still needed to check novelty etc.

HenryKissinger 5y ago
> Math is the most formal science there is. Can you prove that?
- robbedpeter 5y ago
  Formality is technically the degree of mathiness applied to a domain, since formal logic and any other framework of formalization decomposes to mathematics. So... it's a tautology. Math is the most mathy math. Biology or psychology are less mathy maths. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... Having provablility holes in the formal system doesn't imply the existence of other systems that could be decomposed from currently known or unknown symbols. In this universe, information at the level of string theory seems to describe the limits, so math, as she is wrote, is as formal as it gets. The map is not the territory, so math, as a map, will always be less "formal" than reality itself. It's pretty firmly in second place, though.
  - ttctciyf 5y ago
    But if you only have maps and no territory, can you still be said to be doing cartography? :)
nerdponx 5y ago
Is it possible to even check such a proof programmatically? Do we have that kind of technology? Do you also have to review the proof checking code to make sure you didn't make any errors in transcription from the original paper?

maxFlow 5y ago

Makes you wonder what's the most impactful thing you could build given the possibility of seven years of isolation and single-minded focus. Would be interesting to know, at which point did Freedman decide to commit full time to his research---based on what, an intuition? And how did he decide to stop at 7 years and not 4, 9 or 10? This is advanced stuff for me but I'll still be getting a copy of _The Disc Embedding Theorem_ to support this kind of niche research. Findings like these could be stepping stones towards bigger discoveries in the future.

doublerabbit 5y ago

I really wish I could get math to stick. I just finished my National 5 maths (rough equivalent of a US High School Diploma) night-class today and all I ever seem to understand is how, but not why. I'm the one asking "why is that that." And today was recapping on trinomial, simplifying fractions. Simple I expect to anyone with a mathematical mind, but to me, it's just an insane implosion which leaves me exhausted. I have notes, but even still. What is the secret behind it? How do people get so good with Maths? Is it truly study,study & study? .. I hadn't even herd of the terminology of co-efficient until today. Let alone something like this.

curtisf 5y ago
High school math does not look much like research math. High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ... Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving_ that that characterization is actually correct) For example, the definition of "continuous" you learn in pre-calc is probably something like: if you compute a function's limit, you get the same thing as computing the function. The topologist's definition is that the function's inverse map preserves openness. These are basically unrecognizable as being the same thing, but in fact they are. And the second doesn't actually reference calculation or even anything identifiable as numbers! I believe this is a significant cause of not understanding "why" despite understanding "how". As an example, I remember in pre-calculus we spent at least an entire unit on partial-fraction-decomposition. Partial fraction decomposition is a strange algebraic manipulation that is basically only important to integrating rational functions -- something you don't do until at least a year later, and which doing by hand is basically pointless because computers are better at it than humans. They also require very different kinds of intuition. Building intuition is hard, but important for people studying math. Working through many different examples and problems and applications can help you build intuition. But I don't really have any advice for how to make math stick better, but just want to share that the mathematics you learn in high school is not representative of the entire field. If you ever want to get a taste of "higher math" without having lots of prereqs, point-set topology and group theory are two (very different!) fields that are also fairly accessible.

korbonits 5y ago

What about this guy's book? https://math.uchicago.edu/~dannyc/books/4dpoincare/4dpoincare.html https://math.uchicago.edu/~dannyc/books/4dpoincare/4dpoincar...

codethief 5y ago
Relevant: https://mathoverflow.net/questions/87674/independent-evidence-for-the-classification-of-topological-4-manifolds#comment881086_87674 https://mathoverflow.net/questions/87674/independent-evidenc...

cschmidt 5y ago

I do find it amazing that mathematicians seem to have almost infinite time to work on problems, and so much freedom to choose what they're working on. How great that a big group of people can collectively decide to spend years creating a book like this.

Koshkin 5y ago
What is amazing to me is that most people think they do not have such freedom.
- vlovich123 5y ago
  Most people don’t as doing so requires a level of talent, intelligence, and work ethic/determination that’s in the upper tail of human distribution. Paradoxically, many people who could probably do underestimate their talents while others who can’t/should t overestimate theirs. The universe is funny in that way.
paulpauper 5y ago
The US ,global economy is huge. colleges have billion of dollars of endowments and tuition. More than enough $ to go around to fund it

dls2016 5y ago

My buddy had “beat a Fields medalist at a foot race” on his CV for a while after completing the Pier to Peak half marathon faster than (a ~60 year old) Prof Freedman. I’d run to Goleta beach playground with my kids and would see Freedman wander down from Station Q and bust out a few chin ups on the monkey bars. I hope I’m in that kind of shape!

kurthr 5y ago

"I probably didn’t treat the exposition of the written material as carefully as I should have,” said Freedman. That's a pretty sad (if self aware) statement for someone who gets paid $1M/yr to find arrangements which might make quantum decoherence effects lower for computation ... or not. The editors certainly won't make that much, but at least there's an afterward by Freedman.

throwaway81523 5y ago

I had not realized the 4-dimensional case was so hard that the much more famous 3-dimensional case was "another" contender for the hardest. I had thought it was sort of like the 4 color map theorem on stereoids, where first it was proved in 5+ dimensions (Smale, 1961), then 4 dimensions (Freedman, 1982), and finally 3 (Perelman, 2006), getting harder and harder at each step.

civilized 5y ago
And the proofs of this one fact are all completely different depending on the dimension. I wonder if there are any prospects of a unified proof for all dimensions.
- throwaway81523 5y ago
  Smale's proof for d>=5 was unified. The d=4 and d=3 cases were special and those are special dimensions for a lot of reasons. "Low-dimensional topology", i.e. d=3 or d=4, is a big subject in its own right, because of those issues. At least for now, the prospect of unifying them with higher dimensions is somewhat remote. Basically in higher dimensions, there is more space to maneuver and things don't get in each other's way as much, so many proofs become easier. As well, many things are true in high dimensions and false in low dimensions, in which case unification is obviously impossible.. See also "curse of dimensionality" about why things are different for high dimensions.

yejin83810 5y ago

https://web.archive.org/web/20210910012837/https://tmpfiles.org/dl/110636/thediscembeddingtheorembystefanbehrensboldizsarkalmarminhoonkimmarkpowellarunimaray.pdf https://web.archive.org/web/20210910012837/https://tmpfiles....

willhinsa 5y ago

I read the title as "New math book rescues landmark topology prof", and I was wondering how on earth a math book pulled that off!

EVa5I7bHFq9mnYK 5y ago

Funny, how the same conjecture required completely different and unrelated proofs for 3, 4 and >5 cases.