tldr; humans change their theorems/lemmas to make them fit their initial model, i.e. add caveats.
Why doesn't mathematics collapse, though humans often make mistakes in proofs?
41–50 of 85 posts
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#42I’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).
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#43I 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.
I have done a little with 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% complet…
> 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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#44An 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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#45Ted 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.
Could you share more or summarize?
Here's a brief synopsis: http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#46If 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
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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#47Mathematics is convex to errors
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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#48Earlier 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
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.
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.
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#49Earlier quoted context omitted.
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.
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 here.
[1] - https://www.researchgate.net/profile/Costas_Drossos/post/Is_...
Re: Why doesn't mathematics collapse, though humans often make mistakes in proofs?
#50Ted 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.
Mathematicians' take on the same hypothetical situation: https://mathoverflow.net/questions/40920/what-if-current-fou.... Not quite as earth-shattering.