19 ms·

Why doesn't mathematics collapse, though humans often make mistakes in proofs?

256 points by mathgenius 7y ago

danharaj 7y ago
"Once in a while, I like to indulge in an informative anecdote concerning the genesis of the proof. The criterion was found by the end of 1985; then I remained more than six months making circles around the « splitting tensor ». One nice day of August 1986, I woke up in a camp in Siena and I had got the proof: I therefore sat down and wrote a manuscript of 10 pages. One month later, while recopying this with my typewriter, I discovered that one of my lemmas about imperialism was wrong: no importance, I made another one! This illustrates the fact, neglected by the formalist ideology, that a proof is not the mere putting side by side of logical rules, it is a global perception: since I had found the concept of empire, I had my theorem and the faulty lemma was hardly more than a misprint." - Jean-Yves Girard, The Blind Spot
- triska 7y ago
  Complementing this situation, there are also cases where a proof hinges on a critical corollary or theorem where "no importance, I made another one!" did not or does not come as easily. One recent example is Corollary 3.12 in Mochizuki's series of papers on Inter-universal Teichmüller Theory. This single corollary is the main topic of "Why abc is is still a conjecture" by Peter Scholze and Jakob Stix: http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf Quoting from the paper: "We are going to explain where, in our opinion, the suggested proof has a problem, a problem so severe that in our opinion small modifications will not rescue the proof strategy." An argument can be made that hence, this is no proof at all, and indeed that is the conclusion of the paper. However, mistakes in other suggested proofs were also found in the past, and they could sometimes be salvaged with significant new insights and additional work. An example for this is Andrew Wiles's proof of Fermat's Last Theorem. Quoting from https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_...: Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error. He states that he was having a final look to try and understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that ...
  - hyperpape 7y ago
    Mochizuki’s paper seems like a debatable example. Many mathematicians have been baffled by the proof or ended up feeling that it was not that promising, even prior to attempting to do a detailed analysis. If it’s unrecoverable, then the community will look good for its hesitation. https://news.ycombinator.com/item?id=15971802 https://news.ycombinator.com/item?id=15971802
    - triska 7y ago
      Take for instance Wu-Yi Hsiang's suggested proof of the Kepler conjecture as a less controversial example. Quoting from https://en.wikipedia.org/wiki/Kepler_conjecture https://en.wikipedia.org/wiki/Kepler_conjecture: "The proof was praised by Encyclopædia Britannica and Science and Hsiang was also honored at joint meetings of AMS-MAA." Which is followed, after a few sentences, by: "The current consensus is that Hsiang's proof is incomplete."
      - new2628 7y ago
        Or Kempe's (mistaken) proof of the 4 color theorem.
Someone 7y ago
If builders of a huge castle place some stones incorrectly, does it collapse completely? No. Such errors may be discovered soon and taken down, or stand for decades, sometimes with lots of stuff built on top of it, and sometime parts will collapse, but the overall structure of the castle is sound, and, sooner or later, errors will be corrected. And of course, sometimes, somebody decides to start building an entire new wing to the castle using new rules, as, for example, happened when non-Euclidean geometry was discovered/invented.
- alephnan 7y ago
  On the analogy of castles, I asked a professor whether Godel’s Incompleteness theorems affects their work. She told me Mathematics is this great castle, where in one quadrant you have the number theorist, in another the topologists. Every decade or century, the logicians would come running up from the basement with results confronting the foundations of math, but everyone else continued with their routine
  - pasabagi 7y ago
    Perhaps the point is that the basement of a castle is also called a dungeon.
  - bitL 7y ago
    Yeah, swapping out currently dominant logic with another one would redo the whole castle, like when a castle built in Gothic architecture is rebuilt into Baroque with some parts left intact. However, mainstream is so dominant and "better logics" either don't have sufficient manpower, are more complicated, are weaker or impractical, so that they never develop unless they solve something extraordinary quickly.
  - pmoriarty 7y ago
    My memory from the logic classes I've taken was that after Godel, mathematicians largely abandoned Hilbert's, Russell's and Whitehead's ambition of founding mathematics on logic, and largely grew disinterested in foundational issues as a whole. As such, mathematics doesn't have a solid foundation, and most contemporary mathematicians seem to be ok with that, as long as it works and they can get interesting results.
juskrey 7y ago
Mathematics is convex to errors
- webdva 7y ago
  Supposing that the definition of convexity that you're referring to is that of, for a set C with x_1 and x_2 being elements of C and for any scalar b between zero and one inclusively, [b * x_1 + (1 - b) * x_2] always being a member of C, I presume that you mean that errors, namely those kind of errors that are the subject of this thread's discussion, in mathematics will always symbolically be elements of some solution set of a finite number of linear equalities and inequalities (or the intersection of a finite number of halfspaces and hyperplanes).
  - lonelappde 7y ago
    It is informal and means that mistakes hurt less than non-mistakes help. A hypertechnical article that gives the gist: https://www.edge.org/conversation/nassim_nicholas_taleb-understanding-is-a-poor-substitute-for-convexity-antifragility https://www.edge.org/conversation/nassim_nicholas_taleb-unde...
- gwern 7y ago
  It seems very curious to analogize mathematics results to random variables. At the least, there's a lot going into that analogy which might be worth unpacking and justifying to explain why logical inference, with its elaborate and precise chains of truth-preserving deductions, potentially collapsing at a single counterexample, can be considered remotely equivalent to something like a dice roll.
1e-9 7y ago
I think it wise to be skeptical of any new result coming from a complicated proof until it has been proven multiple ways by multiple people. If multiple approaches are not easily attained, then passing the test of multiple counterexample attempts can at least provide some degree of confidence. If something new is significant, it tends to attract that kind of scrutiny pretty quickly. I think that is the saving grace of the mathematics community.
alephnan 7y ago
The simpler and more elegant the proof, the less likely we are to commit an error
mitko 7y ago
Or does it...? Case in point, the fifth axiom of geometry, attributed to Euclid, that through a point there is exactly one line parallel to a given line has been shown to be unprovable, and Lobachevski and Rieman have built alternative geometric theories assuming that axiom wrong. I find it hard to believe that Math is that different from physics or other sciences in its robustness. Some advantages that math has had over other sciences is low resource consumption (pen and paper), easier reproducibility(thinking), and it has gotten a long head start of 2-3 thousand years.
- doubleunplussed 7y ago
  Axioms shouldn't be provable (other than from themselves) - if the fifth axiom of geometry had turned out to be provable from something else, then it would be have just been renamed a theorem instead of an axiom.
  - fao_ 7y ago
    That's the wrong way to think about it, I think. It's more correct to think of axioms as being preconditions. We say, "Here is what happens if these things are true", and then specify what happens, but that doesn't deprive us from independently checking whether those conditions are true. Just because something has been proven does not stop it from being an axiom. It's contextural.
nitwit005 7y ago
The more boring answer correctness is testable. If I present a proof that two equations are equal, you don't have to trust me. You can try some values and see if they work, look for a contradiction, try to produce your own proof, or any other test you can think up.
diego 7y ago
Why would it even mean for mathematics to collapse? It's obvious that if a branch is based on rules that don't make sense, at some point that branch will break. But it only breaks to the root node, which is fine. But why would the whole edifice collapse?
booleandilemma 7y ago
Isn’t this like asking why society doesn’t collapse, when the software that drives it is filled with bugs? My impression was: any result in modern mathematics critically depends on another result, and that result depends on some other result... This is like how most large software systems depend on various 3rd party libraries. I guess the answer for both questions is that the things we build, whether they be mathematics or software, work well enough most of the time that you don’t notice the problems unless you’re paying attention.
contravariant 7y ago
It probably helps that quite a lot interesting mathematics is the result of figuring out exactly why something doesn't work.
paulpauper 7y ago
Mathematical physics and engineering holds up to empirical scrutiny.
- codesushi42 7y ago
  Exactly this. Mathematics provides an imperfect model that needs to be adjusted for engineering problems in the messy, real world.
  - krastanov 7y ago
    Maybe true about physics (as physics does strive to provide approximate models of reality), but mathematics is abstract and for most mathematicians any application to reality is coincidental. You might enjoy reading "The unreasonable effectiveness of mathematics" (it is many decades old, but recently a lot of CS essays copy the style and title).
    - perl4ever 7y ago
      "The unreasonable effectiveness of mathematics" just always struck me as similar to commenting on how noses are made to fit glasses.
      - deleted 7y ago
        [deleted]
  - waterhouse 7y ago
    While we're on the subject: The engineer thinks his equations approximate reality. The physicist thinks reality approximates his equations. The mathematician doesn't care.
yters 7y ago
Platonism ftw!
aaron695 7y ago
The same reason 'the' Stack Overflow works amazingly and no other stacks do at all. [Small exceptions SF, SU and subs of SO, and work means goes towards a truth, not, makes ad $]
gone35 7y ago
Collapses are rare indeed, but they do happen. Cf. the collapse of the Italian School of Algebraic Geometry [1], the cleanup of which took all the efforts of Grothendieck et al. [1] https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry#Collapse_of_the_school https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...
- jquery 7y ago
  That's pretty incredible, thanks for sharing. Hard to believe decades of mathematical research collapsed on itself.
- liuyao 7y ago
  Great example. Weil is said to be the one who was deeply immersed in both the geometric tradition of the Italians and the functional perspective of Riemann, so he was in a unique position to see many analogies and deep connections. (He proved small pieces of it, and laid the roadmap for Grothendieck.) These are the kind of “ideas and understanding” that Schulman is talking about. What collapsed were the proofs, and some if not all of the ideas survived and got painted over with the new language. Sadly algebraic geometry has this bad reputation of being obtuse and opaque, that it has become devoid of geometry. Mathematics need to be more open about ideas that are not rigorous, not just left unspoken between the experts themselves.
jokoon 7y ago
I have never understood why proofs are always required. Usually when you have an unproven theorem and it works often enough, to me it's good enough for most of what you're doing, for example in applied math. Of course if you find places where an unproven theorem doesn't work, it becomes interesting to why it doesn't work, and it's usually a big discovery, but it doesn't really disprove a theorem, it just helps to refine it. I remember that in school teachers were often insisting about giving a proof to something, but to me it often made enough sense and it was often right, and it was often frustrating to have teachers tell you "no". It felt like a burden of proof.
- deleted 7y ago
  [deleted]
- im3w1l 7y ago
  If some claim is not proved but believed to be true it's called a conjecture. If a claim is known to be false but "works often enough" it's called a heuristic.
- alanbernstein 7y ago
  Well, broadly speaking, pure mathematics is picking some axioms, and then proving rigorous results based on them. If that doesn't sound like a fun game to you, you don't need to play it or watch it.
  - jokoon 7y ago
    Not saying it's not fun, but to me proofs don't seem to help me understanding mathematics. Not saying proofs are not important, but it seems that proving a theorem is important at a higher levels of mathematics, not in high school or university.
vbezhenar 7y ago
I think that mathematicians should strive to use formal language proofs verified by computer. It'll allow to avoid situations when it's unclear whether this proof is correct or not and it'll allow people not to waste time on checking whether that proof is correct.
- drchewbacca 7y ago
  I have done a little with http://us.metamath.org/ http://us.metamath.org/ and I am sure within 10 years every professional mathematician will use formal proofs. The two prerequisites are 1) making a fluid and intuitive interface for inputting proofs which can also display human readable proofs for any theorem known and 2) creating a database of all known mathematical proofs into which new results can be inserted. Both tasks are ~50% complete. Metamath, for example, has all of mathematics formalised up to a basic undergraduate level. It has some reasonable editing software, but not close to good enough for mass adoption. There are also plenty of other systems being developed. The time will come though relatively soon.
  - matheist 7y ago
    > I am sure within 10 years every professional mathematician will use formal proofs. I'll take that bet. 1:1 odds, up to $50? Shall we say, every mathematician employed at an ivy league university math department (postdoc level or above) has published at least one paper which employs proof checking for at least some claim? (provided that they have published at all.) So I win if I can find at least one professor or postdoc employed at an ivy league university math department on August 18, 2029, who has published at least one mathematics paper, and who has not published any mathematics paper which contains any formal proof. That's a much weaker claim than yours, so it ought to be pretty generous to you.
    - parthpatel1001 7y ago
      I just wanted to note how precise of a bet this is. I will be stealing the exact definition of conditions and contexts to which the bet applies, to future bets of my own.
malicioususer11 7y ago
because "maths" will always be the consensus. you cant break it, only make it not work. >:3
jimmcslim 7y ago
Ted Chiang’s short story “Division by Zero”, about a mathematician who discovers the foundations of mathematics are inconsistent, would appear to be relevant to this discussion.
- ShamelessC 7y ago
  Could you share more or summarize?
  - teraflop 7y ago
    Here's a brief synopsis: http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf194 http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...
- dwohnitmok 7y ago
  Mathematicians' take on the same hypothetical situation: https://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent https://mathoverflow.net/questions/40920/what-if-current-fou.... Not quite as earth-shattering.
galaxyLogic 7y ago
Think about Curry Howard isomorphism which says programs are proofs and data-types are theorems. A lemma is then like sub-routine. If there is something wrong in the sub-routine it can often be fixed without requiring any changes to the callers of the sub-routine. It might be the case that the subroutine can never produce a value needed by its callers without changing the types of arguments fed to it, and in such a case it is not enough to fix the subroutine. But in many cases just the lemma-subroutine needs to be corrected, and the rest of the program, rest of the proof can be used as is. https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
- guerrilla 7y ago
  The vast majority of mathematics is classical, not intuitionistic, just so you know.
sabujp 7y ago
tldr; humans change their theorems/lemmas to make them fit their initial model, i.e. add caveats.
qubex 7y ago
I’d venture that the reason that mathematics doesn’t risk imploding is because ultimately it can be treated as a fairly empirical body of knowledge, and maps accurately to the world as we observe it (or rather, the implications of pure mathematics, when applied by the sciences, give results that are not ad odds with reality as we observe it to be).
westoncb 7y ago
An idea related to this helped me get a grasp on writing proofs when I was first struggling to teach myself mathematics. The idea was roughly: A particular formalized proof (or lemma) is one of many ways of expressing/representing a more general concept that grounds some mathematical idea. It seems to mirror the way you could have the general concept of "having gone to the store and purchased bananas" in your head but express it many different ways through natural language. If humans were doing mathematics by thinking purely in terms of the theorem space of some particular formal system, then 'collapses' should be expected after minor errors: if you take one wrong turn, your error should be compounded in any further progress. But that does not appear to be how we do mathematics. Instead we're operating in a more general space of concepts, and we do not do math by proceeding linearly in our thoughts from the beginning of a proof to the final conclusion--we assemble the conception piecemeal until we get some feel for the sufficient overall integrity/coherence of a general conception, and then secure it by building up a kind of formal carapace around it. But if part of the proof/carapace is wrong, it's like a malformed section of armor that needs to be replaced: the whole structure isn't going to collapse because of it (not that that couldn't happen). Thinking about things that way gave me a better view (I think) on how to use and think about formalizations when doing mathematics.
ggggtez 7y ago
>senior undergraduate student in mathematics Yikes. The student is only now getting around to first year ideas that everything they know might be a sham. I mean... come on. I mean, what does the student think will happen? We'll all realize that pi=4? Planes will fall out of the sky because physics finally caught up with a mistake in a proof?
colechristensen 7y ago
Many answers which I think miss the big question. Mathematics doesn't collapse like a house of cards because the more "important" a proof is to the foundations of math, the more frequently it is tested. A counterexample to a proof is an easy way to find errors up the chain. In the other direction is that many things that might be considered "errors" in the foundations could be looked at more like choices. Others mentioned Euclidian vs non-Euclidian geometry. There was a sort of error of the idea that Euclidian geometry was the only geometry which was fixed not by tearing it down but creating new geometries.
harry8 7y ago
There is no proof in science and we observe it doesn't suffer from wholesale collapse. Things being proved in mathematics are unlikely to be wildly wrong even without proof. Fermat's theorem being an example of one that took a very long time to be proved and basically nothing changed when it was. We were already convinced it was true by brute force failure to find a counter example. This lack of proof but search for counter-example is basically the scientific method. Is A equivalent to B? Prove it? Or write a for-loop and test it for values you care about. If you think you've proved it you probably write the for loop anyway (where such a thing is feasible). So do we actually /need/ proof in math at all or is it just good fun and properly satisfying for mathematicians? Perhaps these are two extremes and case-by-case we could plot them somehwere between the two. Some right up hard against one side or the other. So is P=NP? There is no proof. You probably aren't going out on a massive limb to have a view on the matter without it.
robpal 7y ago
Pretty funny to see Simpson's name showing up here. I have kept two souvenirs from my academic life: the original manuscript of my thesis and Simpson's very eloquent and overwhelmingly positive review of it. Regarding the topic, I'd say that mathematics, as a whole, is a set of intuitions and ideas, rather than strict proofs. I've spent a fair amount of time working my way through some of Kontsevich's ideas and attended many of his talks -- he rarely (if ever) gives any proofs, my educated guess is that he proposes some extremely profound ideas and his collaborators grind through the details (not always). And it still works just fine. I remember Kenji Fukaya saying once, talking about a PDE-heavy theorem: "We are trying to keep the details of the proof under 500 pages and it's not sa easy". The main issue here is that people who don't do professional, academic research in mathematics are unaware of the complexity of modern maths. It has evolved immensely during last 50 years, the theories are just layers and layers of foundational work one has to assimilate before getting any work done. Rigorous verification takes years and no one in the academia is being offered a job for proof-reading of existing papers. One should also remember that referees of the papers are not being paid, it's considered "work for the community" and it's hard to blame them for not reading the papers in detail. Mathematics are anti-fragile.

256 points by mathgenius 7y ago

"Once in a while, I like to indulge in an informative anecdote concerning the genesis of the proof. The criterion was found by the end of 1985; then I remained more than six months making circles around the « splitting tensor ». One nice day of August 1986, I woke up in a camp in Siena and I had got the proof: I therefore sat down and wrote a manuscript of 10 pages. One month later, while recopying this with my typewriter, I discovered that one of my lemmas about imperialism was wrong: no importance, I made another one! This illustrates the fact, neglected by the formalist ideology, that a proof is not the mere putting side by side of logical rules, it is a global perception: since I had found the concept of empire, I had my theorem and the faulty lemma was hardly more than a misprint." - Jean-Yves Girard, The Blind Spot

triska 7y ago
Complementing this situation, there are also cases where a proof hinges on a critical corollary or theorem where "no importance, I made another one!" did not or does not come as easily. One recent example is Corollary 3.12 in Mochizuki's series of papers on Inter-universal Teichmüller Theory. This single corollary is the main topic of "Why abc is is still a conjecture" by Peter Scholze and Jakob Stix: http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf Quoting from the paper: "We are going to explain where, in our opinion, the suggested proof has a problem, a problem so severe that in our opinion small modifications will not rescue the proof strategy." An argument can be made that hence, this is no proof at all, and indeed that is the conclusion of the paper. However, mistakes in other suggested proofs were also found in the past, and they could sometimes be salvaged with significant new insights and additional work. An example for this is Andrew Wiles's proof of Fermat's Last Theorem. Quoting from https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_...: Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error. He states that he was having a final look to try and understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that ...
- hyperpape 7y ago
  Mochizuki’s paper seems like a debatable example. Many mathematicians have been baffled by the proof or ended up feeling that it was not that promising, even prior to attempting to do a detailed analysis. If it’s unrecoverable, then the community will look good for its hesitation. https://news.ycombinator.com/item?id=15971802 https://news.ycombinator.com/item?id=15971802
  - triska 7y ago
    Take for instance Wu-Yi Hsiang's suggested proof of the Kepler conjecture as a less controversial example. Quoting from https://en.wikipedia.org/wiki/Kepler_conjecture https://en.wikipedia.org/wiki/Kepler_conjecture: "The proof was praised by Encyclopædia Britannica and Science and Hsiang was also honored at joint meetings of AMS-MAA." Which is followed, after a few sentences, by: "The current consensus is that Hsiang's proof is incomplete."
    - new2628 7y ago
      Or Kempe's (mistaken) proof of the 4 color theorem.

Someone 7y ago

If builders of a huge castle place some stones incorrectly, does it collapse completely? No. Such errors may be discovered soon and taken down, or stand for decades, sometimes with lots of stuff built on top of it, and sometime parts will collapse, but the overall structure of the castle is sound, and, sooner or later, errors will be corrected. And of course, sometimes, somebody decides to start building an entire new wing to the castle using new rules, as, for example, happened when non-Euclidean geometry was discovered/invented.

alephnan 7y ago
On the analogy of castles, I asked a professor whether Godel’s Incompleteness theorems affects their work. She told me Mathematics is this great castle, where in one quadrant you have the number theorist, in another the topologists. Every decade or century, the logicians would come running up from the basement with results confronting the foundations of math, but everyone else continued with their routine
- pasabagi 7y ago
  Perhaps the point is that the basement of a castle is also called a dungeon.
- bitL 7y ago
  Yeah, swapping out currently dominant logic with another one would redo the whole castle, like when a castle built in Gothic architecture is rebuilt into Baroque with some parts left intact. However, mainstream is so dominant and "better logics" either don't have sufficient manpower, are more complicated, are weaker or impractical, so that they never develop unless they solve something extraordinary quickly.
- pmoriarty 7y ago
  My memory from the logic classes I've taken was that after Godel, mathematicians largely abandoned Hilbert's, Russell's and Whitehead's ambition of founding mathematics on logic, and largely grew disinterested in foundational issues as a whole. As such, mathematics doesn't have a solid foundation, and most contemporary mathematicians seem to be ok with that, as long as it works and they can get interesting results.

juskrey 7y ago

Mathematics is convex to errors

webdva 7y ago
Supposing that the definition of convexity that you're referring to is that of, for a set C with x_1 and x_2 being elements of C and for any scalar b between zero and one inclusively, [b * x_1 + (1 - b) * x_2] always being a member of C, I presume that you mean that errors, namely those kind of errors that are the subject of this thread's discussion, in mathematics will always symbolically be elements of some solution set of a finite number of linear equalities and inequalities (or the intersection of a finite number of halfspaces and hyperplanes).
- lonelappde 7y ago
  It is informal and means that mistakes hurt less than non-mistakes help. A hypertechnical article that gives the gist: https://www.edge.org/conversation/nassim_nicholas_taleb-understanding-is-a-poor-substitute-for-convexity-antifragility https://www.edge.org/conversation/nassim_nicholas_taleb-unde...
gwern 7y ago
It seems very curious to analogize mathematics results to random variables. At the least, there's a lot going into that analogy which might be worth unpacking and justifying to explain why logical inference, with its elaborate and precise chains of truth-preserving deductions, potentially collapsing at a single counterexample, can be considered remotely equivalent to something like a dice roll.

1e-9 7y ago

I think it wise to be skeptical of any new result coming from a complicated proof until it has been proven multiple ways by multiple people. If multiple approaches are not easily attained, then passing the test of multiple counterexample attempts can at least provide some degree of confidence. If something new is significant, it tends to attract that kind of scrutiny pretty quickly. I think that is the saving grace of the mathematics community.

alephnan 7y ago

The simpler and more elegant the proof, the less likely we are to commit an error

mitko 7y ago

Or does it...? Case in point, the fifth axiom of geometry, attributed to Euclid, that through a point there is exactly one line parallel to a given line has been shown to be unprovable, and Lobachevski and Rieman have built alternative geometric theories assuming that axiom wrong. I find it hard to believe that Math is that different from physics or other sciences in its robustness. Some advantages that math has had over other sciences is low resource consumption (pen and paper), easier reproducibility(thinking), and it has gotten a long head start of 2-3 thousand years.

doubleunplussed 7y ago
Axioms shouldn't be provable (other than from themselves) - if the fifth axiom of geometry had turned out to be provable from something else, then it would be have just been renamed a theorem instead of an axiom.
- fao_ 7y ago
  That's the wrong way to think about it, I think. It's more correct to think of axioms as being preconditions. We say, "Here is what happens if these things are true", and then specify what happens, but that doesn't deprive us from independently checking whether those conditions are true. Just because something has been proven does not stop it from being an axiom. It's contextural.

nitwit005 7y ago

The more boring answer correctness is testable. If I present a proof that two equations are equal, you don't have to trust me. You can try some values and see if they work, look for a contradiction, try to produce your own proof, or any other test you can think up.

diego 7y ago

Why would it even mean for mathematics to collapse? It's obvious that if a branch is based on rules that don't make sense, at some point that branch will break. But it only breaks to the root node, which is fine. But why would the whole edifice collapse?

booleandilemma 7y ago

Isn’t this like asking why society doesn’t collapse, when the software that drives it is filled with bugs? My impression was: any result in modern mathematics critically depends on another result, and that result depends on some other result... This is like how most large software systems depend on various 3rd party libraries. I guess the answer for both questions is that the things we build, whether they be mathematics or software, work well enough most of the time that you don’t notice the problems unless you’re paying attention.

contravariant 7y ago

It probably helps that quite a lot interesting mathematics is the result of figuring out exactly why something doesn't work.

paulpauper 7y ago

Mathematical physics and engineering holds up to empirical scrutiny.

codesushi42 7y ago
Exactly this. Mathematics provides an imperfect model that needs to be adjusted for engineering problems in the messy, real world.
- krastanov 7y ago
  Maybe true about physics (as physics does strive to provide approximate models of reality), but mathematics is abstract and for most mathematicians any application to reality is coincidental. You might enjoy reading "The unreasonable effectiveness of mathematics" (it is many decades old, but recently a lot of CS essays copy the style and title).
  - perl4ever 7y ago
    "The unreasonable effectiveness of mathematics" just always struck me as similar to commenting on how noses are made to fit glasses.
    - deleted 7y ago
      [deleted]
- waterhouse 7y ago
  While we're on the subject: The engineer thinks his equations approximate reality. The physicist thinks reality approximates his equations. The mathematician doesn't care.

yters 7y ago

Platonism ftw!

aaron695 7y ago

The same reason 'the' Stack Overflow works amazingly and no other stacks do at all. [Small exceptions SF, SU and subs of SO, and work means goes towards a truth, not, makes ad $]

gone35 7y ago

Collapses are rare indeed, but they do happen. Cf. the collapse of the Italian School of Algebraic Geometry [1], the cleanup of which took all the efforts of Grothendieck et al. [1] https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry#Collapse_of_the_school https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...

jquery 7y ago
That's pretty incredible, thanks for sharing. Hard to believe decades of mathematical research collapsed on itself.
liuyao 7y ago
Great example. Weil is said to be the one who was deeply immersed in both the geometric tradition of the Italians and the functional perspective of Riemann, so he was in a unique position to see many analogies and deep connections. (He proved small pieces of it, and laid the roadmap for Grothendieck.) These are the kind of “ideas and understanding” that Schulman is talking about. What collapsed were the proofs, and some if not all of the ideas survived and got painted over with the new language. Sadly algebraic geometry has this bad reputation of being obtuse and opaque, that it has become devoid of geometry. Mathematics need to be more open about ideas that are not rigorous, not just left unspoken between the experts themselves.

jokoon 7y ago

I have never understood why proofs are always required. Usually when you have an unproven theorem and it works often enough, to me it's good enough for most of what you're doing, for example in applied math. Of course if you find places where an unproven theorem doesn't work, it becomes interesting to why it doesn't work, and it's usually a big discovery, but it doesn't really disprove a theorem, it just helps to refine it. I remember that in school teachers were often insisting about giving a proof to something, but to me it often made enough sense and it was often right, and it was often frustrating to have teachers tell you "no". It felt like a burden of proof.

deleted 7y ago
[deleted]
im3w1l 7y ago
If some claim is not proved but believed to be true it's called a conjecture. If a claim is known to be false but "works often enough" it's called a heuristic.
alanbernstein 7y ago
Well, broadly speaking, pure mathematics is picking some axioms, and then proving rigorous results based on them. If that doesn't sound like a fun game to you, you don't need to play it or watch it.
- jokoon 7y ago
  Not saying it's not fun, but to me proofs don't seem to help me understanding mathematics. Not saying proofs are not important, but it seems that proving a theorem is important at a higher levels of mathematics, not in high school or university.

vbezhenar 7y ago

I think that mathematicians should strive to use formal language proofs verified by computer. It'll allow to avoid situations when it's unclear whether this proof is correct or not and it'll allow people not to waste time on checking whether that proof is correct.

drchewbacca 7y ago
I have done a little with http://us.metamath.org/ http://us.metamath.org/ and I am sure within 10 years every professional mathematician will use formal proofs. The two prerequisites are 1) making a fluid and intuitive interface for inputting proofs which can also display human readable proofs for any theorem known and 2) creating a database of all known mathematical proofs into which new results can be inserted. Both tasks are ~50% complete. Metamath, for example, has all of mathematics formalised up to a basic undergraduate level. It has some reasonable editing software, but not close to good enough for mass adoption. There are also plenty of other systems being developed. The time will come though relatively soon.
- matheist 7y ago
  > I am sure within 10 years every professional mathematician will use formal proofs. I'll take that bet. 1:1 odds, up to $50? Shall we say, every mathematician employed at an ivy league university math department (postdoc level or above) has published at least one paper which employs proof checking for at least some claim? (provided that they have published at all.) So I win if I can find at least one professor or postdoc employed at an ivy league university math department on August 18, 2029, who has published at least one mathematics paper, and who has not published any mathematics paper which contains any formal proof. That's a much weaker claim than yours, so it ought to be pretty generous to you.
  - parthpatel1001 7y ago
    I just wanted to note how precise of a bet this is. I will be stealing the exact definition of conditions and contexts to which the bet applies, to future bets of my own.

malicioususer11 7y ago

because "maths" will always be the consensus. you cant break it, only make it not work. >:3

jimmcslim 7y ago

Ted Chiang’s short story “Division by Zero”, about a mathematician who discovers the foundations of mathematics are inconsistent, would appear to be relevant to this discussion.

ShamelessC 7y ago
Could you share more or summarize?
- teraflop 7y ago
  Here's a brief synopsis: http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf194 http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...
dwohnitmok 7y ago
Mathematicians' take on the same hypothetical situation: https://mathoverflow.net/questions/40920/what-if-current-foundations-of-mathematics-are-inconsistent https://mathoverflow.net/questions/40920/what-if-current-fou.... Not quite as earth-shattering.

galaxyLogic 7y ago

Think about Curry Howard isomorphism which says programs are proofs and data-types are theorems. A lemma is then like sub-routine. If there is something wrong in the sub-routine it can often be fixed without requiring any changes to the callers of the sub-routine. It might be the case that the subroutine can never produce a value needed by its callers without changing the types of arguments fed to it, and in such a case it is not enough to fix the subroutine. But in many cases just the lemma-subroutine needs to be corrected, and the rest of the program, rest of the proof can be used as is. https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...

guerrilla 7y ago
The vast majority of mathematics is classical, not intuitionistic, just so you know.

sabujp 7y ago

tldr; humans change their theorems/lemmas to make them fit their initial model, i.e. add caveats.

qubex 7y ago

I’d venture that the reason that mathematics doesn’t risk imploding is because ultimately it can be treated as a fairly empirical body of knowledge, and maps accurately to the world as we observe it (or rather, the implications of pure mathematics, when applied by the sciences, give results that are not ad odds with reality as we observe it to be).

westoncb 7y ago

An idea related to this helped me get a grasp on writing proofs when I was first struggling to teach myself mathematics. The idea was roughly: A particular formalized proof (or lemma) is one of many ways of expressing/representing a more general concept that grounds some mathematical idea. It seems to mirror the way you could have the general concept of "having gone to the store and purchased bananas" in your head but express it many different ways through natural language. If humans were doing mathematics by thinking purely in terms of the theorem space of some particular formal system, then 'collapses' should be expected after minor errors: if you take one wrong turn, your error should be compounded in any further progress. But that does not appear to be how we do mathematics. Instead we're operating in a more general space of concepts, and we do not do math by proceeding linearly in our thoughts from the beginning of a proof to the final conclusion--we assemble the conception piecemeal until we get some feel for the sufficient overall integrity/coherence of a general conception, and then secure it by building up a kind of formal carapace around it. But if part of the proof/carapace is wrong, it's like a malformed section of armor that needs to be replaced: the whole structure isn't going to collapse because of it (not that that couldn't happen). Thinking about things that way gave me a better view (I think) on how to use and think about formalizations when doing mathematics.

ggggtez 7y ago

>senior undergraduate student in mathematics Yikes. The student is only now getting around to first year ideas that everything they know might be a sham. I mean... come on. I mean, what does the student think will happen? We'll all realize that pi=4? Planes will fall out of the sky because physics finally caught up with a mistake in a proof?

colechristensen 7y ago

Many answers which I think miss the big question. Mathematics doesn't collapse like a house of cards because the more "important" a proof is to the foundations of math, the more frequently it is tested. A counterexample to a proof is an easy way to find errors up the chain. In the other direction is that many things that might be considered "errors" in the foundations could be looked at more like choices. Others mentioned Euclidian vs non-Euclidian geometry. There was a sort of error of the idea that Euclidian geometry was the only geometry which was fixed not by tearing it down but creating new geometries.

harry8 7y ago

There is no proof in science and we observe it doesn't suffer from wholesale collapse. Things being proved in mathematics are unlikely to be wildly wrong even without proof. Fermat's theorem being an example of one that took a very long time to be proved and basically nothing changed when it was. We were already convinced it was true by brute force failure to find a counter example. This lack of proof but search for counter-example is basically the scientific method. Is A equivalent to B? Prove it? Or write a for-loop and test it for values you care about. If you think you've proved it you probably write the for loop anyway (where such a thing is feasible). So do we actually /need/ proof in math at all or is it just good fun and properly satisfying for mathematicians? Perhaps these are two extremes and case-by-case we could plot them somehwere between the two. Some right up hard against one side or the other. So is P=NP? There is no proof. You probably aren't going out on a massive limb to have a view on the matter without it.

robpal 7y ago

Pretty funny to see Simpson's name showing up here. I have kept two souvenirs from my academic life: the original manuscript of my thesis and Simpson's very eloquent and overwhelmingly positive review of it. Regarding the topic, I'd say that mathematics, as a whole, is a set of intuitions and ideas, rather than strict proofs. I've spent a fair amount of time working my way through some of Kontsevich's ideas and attended many of his talks -- he rarely (if ever) gives any proofs, my educated guess is that he proposes some extremely profound ideas and his collaborators grind through the details (not always). And it still works just fine. I remember Kenji Fukaya saying once, talking about a PDE-heavy theorem: "We are trying to keep the details of the proof under 500 pages and it's not sa easy". The main issue here is that people who don't do professional, academic research in mathematics are unaware of the complexity of modern maths. It has evolved immensely during last 50 years, the theories are just layers and layers of foundational work one has to assimilate before getting any work done. Rigorous verification takes years and no one in the academia is being offered a job for proof-reading of existing papers. One should also remember that referees of the papers are not being paid, it's considered "work for the community" and it's hard to blame them for not reading the papers in detail. Mathematics are anti-fragile.