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 $]
Why doesn't mathematics collapse, though humans often make mistakes in proofs?
21–30 of 85 posts
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#22If 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 ne…
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
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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#23"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 typew…
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
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_...:
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 ...
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#24Collapses 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...
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#25Or 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…
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.
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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#26Mathematical physics and engineering holds up to empirical scrutiny.
Exactly this. Mathematics provides an imperfect model that needs to be adjusted for engineering problems in the messy, real world.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#27"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 typew…
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.kuri…
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
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#28Earlier quoted context omitted.
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.kuri…
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
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:
"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."
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#29I 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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#30I 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 ref…
[deleted]