
Why doesn't mathematics collapse, though humans often make mistakes in proofs? - mathgenius
https://mathoverflow.net/questions/338607/why-doesnt-mathematics-collapse-down-even-though-humans-quite-often-make-mista
======
danharaj
"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
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_...](https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem):

 _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
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
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
Or Kempe's (mistaken) proof of the 4 color theorem.

------
gone35
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...](https://en.wikipedia.org/wiki/Italian_school_of_algebraic_geometry#Collapse_of_the_school)

~~~
jquery
That's pretty incredible, thanks for sharing. Hard to believe decades of
mathematical research collapsed on itself.

------
Someone
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
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

~~~
dwohnitmok
AFAICT mathematical logic has always felt like a curious side show in the
greater math community. If you were to ask professional mathematicians to list
off all the axioms of ZFC they'd probably shrug if they didn't get all of
them. They probably wouldn't know about the more exotic parts of set theory
such as large cardinals nor would they would probably know too much about
alternate foundations of mathematics. They might not be familiar at all with
the specifics of model theory.

The logical foundations of mathematics have always had a much smaller impact
on the day to day lives of mathematicians than their foundational nature might
suggest.

~~~
heyitsguay
That's because formal logic is a (relatively) recent attempt to formalize the
intuitions behind the math that people were already doing. Doing mathematics
doesn't _really_ depend on foundational logic in the same way that, e.g., a
web app depends on transistor physics.

As somebody once said, if we ever found a contradiction in the ZFC axioms, we
wouldn't throw out math, we'd just throw out ZFC.

~~~
fnrslvr
> Doing mathematics doesn't really depend on foundational logic in the same
> way that, e.g., a web app depends on transistor physics.

Web apps don't depend on transistor physics at all, though. An important
consequence of Turing universality is that computer science is not a subfield
of electrical engineering.

Also, mathematics underwent very dramatic transformations during the late
1800s and early 1900s alongside the early development of formal logic and set
theory. To say that logicians were merely "formalizing the intuitions behind
the math that people were already doing" strikes me as misguided at best. The
mathematics that rose to prominence in that era was very different from what
preceded it, often controversially so.

------
westoncb
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.

------
colechristensen
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.

------
qubex
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).

------
1e-9
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.

------
vbezhenar
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
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
> 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
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.

------
diego
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?

------
jimmcslim
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
Could you share more or summarize?

~~~
teraflop
Here's a brief synopsis:
[http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...](http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumber=mf194)

------
robpal
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.

------
booleandilemma
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.

------
ggggtez
>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?

------
paulpauper
Mathematical physics and engineering holds up to empirical scrutiny.

~~~
codesushi42
Exactly this. Mathematics provides an imperfect model that needs to be
adjusted for engineering problems in the messy, real world.

~~~
krastanov
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
"The unreasonable effectiveness of mathematics" just always struck me as
similar to commenting on how noses are made to fit glasses.

------
nitwit005
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.

------
galaxyLogic
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...](https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence)

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

------
harry8
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.

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

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

------
juskrey
Mathematics is convex to errors

~~~
webdva
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
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-
unde...](https://www.edge.org/conversation/nassim_nicholas_taleb-
understanding-is-a-poor-substitute-for-convexity-antifragility)

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

------
yters
Platonism ftw!

------
mitko
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
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_
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.

------
jokoon
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.

~~~
alanbernstein
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
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.

