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

"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

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_ge...

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.

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.

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.

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).

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.

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.

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?

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.

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.

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.

>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?

Mathematical physics and engineering holds up to empirical scrutiny.

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.

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_correspon...

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.

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

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

Mathematics is convex to errors

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

Platonism ftw!

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.

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.