Why doesn't mathematics collapse, though humans often make mistakes in proofs? • Hacker News Board

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

#51

Post by Jweb_Guru » Mon, Aug 19, 2019, 1:28 AM UTC

pmoriarty wrote: post #49

dwohnitmok wrote: post #48

Earlier quoted context omitted.

I agree with the rest of what you wrote, but doesn't have a solid foundation seems a bit far. Mathematicians hand-wave that their proofs can be reduced to ZFC set theory and despite reasonable wariness about that hand-waving, it has so far been borne out by all efforts at deeper inspection all the way up to full machine-checked formalization.

Although I am starting to get out of my depth, my impression was that ZFC itself is (per Godel's results) inconsistent or incomplete. Here is some research to this effect: If this is indeed the case (as it seems to my unprofessional first glance to be) then mathematics does not in fact have a solid foundation, despite its appeal to ZFC. I'd love to hear a professional logician weigh in on this, if there are any on he…

The short answer is "incompleteness is not a big deal" and the slightly longer answer is "if it's inconsistent, the proof of inconsistency is devilishly hard to find." If it does exist, the inconsistent part seems likely to be sufficiently esoteric that it could be "built around."

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

#52

Post by dwohnitmok » Mon, Aug 19, 2019, 1:30 AM UTC

pmoriarty wrote: post #49

dwohnitmok wrote: post #48

Earlier quoted context omitted.

I agree with the rest of what you wrote, but doesn't have a solid foundation seems a bit far. Mathematicians hand-wave that their proofs can be reduced to ZFC set theory and despite reasonable wariness about that hand-waving, it has so far been borne out by all efforts at deeper inspection all the way up to full machine-checked formalization.

Although I am starting to get out of my depth, my impression was that ZFC itself is (per Godel's results) inconsistent or incomplete. Here is some research to this effect: If this is indeed the case (as it seems to my unprofessional first glance to be) then mathematics does not in fact have a solid foundation, despite its appeal to ZFC. I'd love to hear a professional logician weigh in on this, if there are any on he…

ZFC is incomplete. There are many celebrated results that are independent of ZFC (continuum hypothesis, large cardinals (a lot aren't even known to be consistent with ZFC), Suslin's Hypothesis, etc. However, by Godel we know this is true for any reasonable foundation of mathematics (i.e. anything capable of expressing the usual version of arithmetic). So it is not something that particularly troubles either logicians or other mathematicians.

Godel's results also show that we cannot use ZFC to prove that ZFC is consistent. However, it is strongly suspected that ZFC is consistent.

In practice, very few of the results independent of ZFC affect day to day mathematical work unless you're a logician or set theorist (which honestly start to blend into each other).

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

#53

Post by whatshisface » Mon, Aug 19, 2019, 1:56 AM UTC

dwohnitmok wrote: post #46

alephnan wrote: post #7

Earlier quoted context omitted.

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

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…

I wonder what fraction of web programmers understand transistor physics?

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

#54

Post by perl4ever » Mon, Aug 19, 2019, 2:20 AM UTC

Someone wrote: post #3

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 ne…

Well, this is "hacker news", and I imagine there are quite a few people around who have plenty of first hand experience with programming but not so much mathematical proofs. And there are a lot of examples of "a few stones placed incorrectly" taking down a huge edifice when it comes to software.[1]

So it may be true that math doesn't work like that, but I don't think it's obvious why. Many people have suggested that software would be better if habitually proved correct, but then, why is it not 100% necessary that proofs be perfect? Why should proofs be more resilient than software?

[1]https://gcn.com/articles/1998/07/13/software-glitches-leave-...

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

#55

Post by perl4ever » Mon, Aug 19, 2019, 2:22 AM UTC

alephnan wrote: post #7

Someone wrote: post #3

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 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 impression is that today we know that there can be no single foundation, which means that there is no way to undermine all areas at once.

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

#56

Post by heyitsguay » Mon, Aug 19, 2019, 2:22 AM UTC

dwohnitmok wrote: post #46

alephnan wrote: post #7

Earlier quoted context omitted.

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

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…

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.

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

#57

Post by perl4ever » Mon, Aug 19, 2019, 2:26 AM UTC

krastanov wrote: post #40

codesushi42 wrote: post #26

Earlier quoted context omitted.

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

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).

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

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

#58

Post by dwohnitmok » Mon, Aug 19, 2019, 2:30 AM UTC

whatshisface wrote: post #53

dwohnitmok wrote: post #46

Earlier quoted context omitted.

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…

I wonder what fraction of web programmers understand transistor physics?

Well maybe not quite that extreme. It's more analogous to the fact that most professional programmers do not fully understand the programming language they use (nor honestly do they need to for the most part).

Nonetheless, it's perhaps surprising for a beginning student of pure mathematics when all they can see is the sky-high demand for rigor that "professional" pure mathematics is seemingly content to hand-wave its foundations. (For the most part this hand-waving is the postrigorous stage described by Terence Tao https://terrytao.wordpress.com/career-advice/theres-more-to-... rather than the pre-rigorous hand-waving of the student).

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

#59

Post by ggggtez » Mon, Aug 19, 2019, 2:33 AM UTC

>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?

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

#60

Post by colechristensen » Mon, Aug 19, 2019, 3:21 AM UTC

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.