[HN Gopher] Mathematicians Solve Decades-Old Classification Problem
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Mathematicians Solve Decades-Old Classification Problem
Author : theafh
Score : 111 points
Date : 2021-08-05 15:22 UTC (7 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| timestampgalore wrote:
| can't wait till this problem has to be solved within 20 mins in a
| tech interview
| m3kw9 wrote:
| Can this be used to classify molecules, which also have near
| infinite chemical spaces?
| ducttapecrown wrote:
| No.
| FredPret wrote:
| ...turns out that dress is white and gold
| spywaregorilla wrote:
| What is the day to day of someone trying to solve potentially
| unsolveable problems? How much of it is just starting blankly at
| a chalkboard?
| tzs wrote:
| Julia Robinson [1], who played a crucial role in resolving
| Hilbert's Tenth Problem, was once asked by the personnel
| department at her university to submit a description of what
| she did. She gave them a description of her typical work week:
| Monday: Try to prove theorem Tuesday: Try to prove
| theorem Wednesday: Try to prove theorem Thursday:
| Try to prove theorem Friday: Theorem false
|
| [1] https://en.wikipedia.org/wiki/Julia_Robinson
| ellis-bell wrote:
| a huge part of it is trying to solve / formulate _easier_
| versions of the problem, or problems that are similar or
| related to the original problem. Or making some stronger
| assumptions to get rid of the clutter / all of the moving
| variables and distill it down to the smallest form a human
| brain can handle! For many of the "huge" unsolved problems,
| there tends to be a program of "dominos" or "ledges" you hope
| to work on in some order that will make the original problem
| fall.
|
| that way you don't just meander idolly from day to day, but
| instead gain some intuition for the central problem (and of
| course have publishable work to appease the grant gods / the
| university).
|
| trying to code up some of the work to experiment is also
| useful, but that can be a research problem of its own :-)
| eigenket wrote:
| A good start is to try easier problems that will hopefully
| provide some insight into the hard problem. I.e. if you have
| some theorem you want to prove about all matrices of a certain
| form then do it for 2x2 and 3x3 matrices first or something.
|
| Straight up numerical examples help me a lot as well, so a
| first approach is to write some code that generates specific
| cases I can play around with.
|
| Obviously the usefulness of this varies a lot depending on the
| field and the problem.
|
| Personally I really like coming up with counterexamples to
| stuff, i.e. if someone has an idea for something they think is
| true I'm really good at coming up with random examples that
| break it, so if I'm trying to prove something occasionally I
| take a break and try to work out what a counterexample would
| look like and this often provides insight into why
| counterexamples can't exist.
| wizzwizz4 wrote:
| > _A good start is to try easier problems that will hopefully
| provide some insight into the hard problem. I.e. if you have
| some theorem you want to prove about all matrices of a
| certain form then do it for 2x2 and 3x3 matrices first or
| something._
|
| Other times, it helps to go the other way. If you have a load
| of numbers to work with and can't see a pattern, replace them
| with variables; there will probably be more patterns in your
| derivation if you do that, and you'll be able to simplify
| more easily.
| spywaregorilla wrote:
| I don't think I've worked on any problems that took more than
| a couple days to solve. And yeah that feels intuitive. But I
| suppose I would have thought these long-unsolved general case
| math problems were mired in emergent properties at large
| numbers that made these kinds of approaches impractical.
| Smaug123 wrote:
| Well, "these kinds of approaches" isn't really a binary
| thing. When you're looking at a problem, you try a big mix
| of lots of different approaches - maybe you try making the
| problem smaller, and you get something that you could
| possibly crack if you made _this_ extra assumption, and
| then your friend talks about their own unrelated work and
| you think "hold on, there's a bit of an analogy there",
| and you go back to your books and discover that a handy
| missing piece is actually Lemma 3.6 of some famous text,
| and you take a lot of showers, and eventually maybe the
| walnut shell has softened enough that you can peel a bit of
| it off.
| locao wrote:
| So THAT'S why Brazil is quickly destroying the Amazon. No forest,
| no need to classify species.
| paulpauper wrote:
| seems like unsolved problems are being solved at a breakneck
| speed
|
| https://www.google.com/search?q=site%3Ahttps%3A%2F%2Fwww.qua...
| [deleted]
| magneticnorth wrote:
| I _love_ this series of Quanta 's. They have truly excellent
| coverage of real, modern mathematics results of the type that
| are very hard to explain outside of their own field.
|
| Speaking as a former mathematician, I don't think problems are
| being solved any faster lately, but we hear about it more
| outside of academia because this is something that Quanta has
| started covering.
|
| As an example, you could look only at Saharon Shelah, one of
| the co-authors of this result and a giant in the fields of
| model theory and set theory. He has spent a career settling
| long-standing open problems, including Whitehead's problem and
| Morley's problem. Until this Quanta series, Shelah's work
| didn't get much coverage outside mathematics as far as I know,
| but results of this caliber are not atypical for him.
| https://en.wikipedia.org/wiki/Saharon_Shelah#Academic_career
| jjgreen wrote:
| Shelah's productivity is astonishing, but he uses the _worst_
| notation possible, his preprints are practically unreadable
| as a result (not that I 'm clever enough to understand them).
| tux3 wrote:
| That begs the question of whether the unorthodox notation
| is a catalyzer for the results he gets, just a coincidence,
| or if just thinking differently in general is the reason
| for both. =)
| jjgreen wrote:
| I suspect that he has some alternate level of perception,
| sees the beauty of the underlying mathematical structures
| directly without seeing the ugliness of the notation that
| distracts dull-witted mortals like me.
| Epa095 wrote:
| It raises the question, it doesn't beg it (sorry for the
| boring off-topic comment).
| https://www.writersdigest.com/write-better-
| fiction/begging-t...
| DemocracyFTW wrote:
| This literally begs the question whether the prescriptive
| approach is to be preferred over the descriptive one.
| tux3 wrote:
| Thank you! I think I was actually corrected on that once
| before, but maybe this time it'll stick =)
| mhb wrote:
| A nice substitute might be begets. It is so similar, I
| wonder if that is where the confusion arose to begin
| with.
| mhh__ wrote:
| I think notation is completely misunderstood by some
| fields, and that physics in particular really shows the
| benefits of it (i.e. With tensors and feynman diagrams we
| can nearly stop ourselves from writing down the wrong
| answer through notation), but academics are just as
| idiosyncratic as everyone else so I'd guess something
| more human.
| dmix wrote:
| I bought ~5 different kindle books on logic as a
| prerequisite for understanding another book I originally
| bought on Probability Theory (E. T. Jaynes). Every logic
| book I bought used a different notation and none of them
| matched the notation used in the book I wanted to read
| either, which is how I ended up with 5 of them.
|
| You'd think something as straight forward as (boolean)
| logic would converge on a standard notation. But I'm not
| familiar with the mathematics field to really criticize
| this. Regardless I found it to be a large barrier-to-
| entry which over complicated something when I was hoping
| to find direct analogies not needing constant
| 'translation'.
| thechao wrote:
| In programming language theory there is a unified
| notation for operational semantics, but most papers use
| nonstandard and/or incorrect interpretations of the
| notation. It's a glorious minefield when you go to try to
| implement the papers -- you inevitably end up talking to
| the authors & helping them debug issues, even years
| later.
| bakuninsbart wrote:
| I agree. What I saw with the mathematics students during
| our Bachelors (I did CS, so adjacent, but still far away)
| was that they would have a _really_ hard time in the
| beginning coming from school, but after a few semesters
| they would transcend notations and it would then become
| mostly a thing of preference. The thing is that all of
| these guys were already really talented and engaged in
| mathematics, and it was still quite hard combined with
| the fundamental difference of how we do mathematics in
| university compared to school.
| Bayart wrote:
| I've got terrible mathematical education, but Feynman
| diagrams are something even a clown like myself can
| easily wrap his head around. They're really an
| exceedingly clever device, end "clever" doesn't do it
| justice.
| mhh__ wrote:
| They don't even have to be Feynman diagrams - theoretical
| (rather than mathematical) physicists are usually pretty
| lazy. We make up notation for us rather than the reader.
| Mathematicians, I find, are a lot worse but still
| bearable. Engineers though, oh boy, seem to actively
| enjoy using notation to obscure the material in the books
| I have read.
| OscarCunningham wrote:
| Can you give an example paper? I want to see what you mean.
| jjgreen wrote:
| Picked pretty-much at random, but typical of what I mean:
| https://arxiv.org/pdf/1802.01137.pdf
|
| The use of the filled club makes the whole thing look
| like it's been sneezed on by someone who's just come off
| a shift in a coal-mine.
|
| There are 1087 published research papers here if you'd
| like to browse ... https://shelah.logic.at/paper-list/
| alserio wrote:
| Ah, how to feel totally inadequate...
| Smaug123 wrote:
| Shelah is a _powerhouse_ , in fairness. A 10x
| mathematician.
| sorokod wrote:
| It is said about Saharon Shelah that colleagues are
| reluctant to share research problems with him since he
| solves them on the spot.
| random_upvoter wrote:
| I'm reminded how Newton invented modern physics after spending
| a year(?) in seclusion because of the pest raging in London and
| I wonder if something similar has been going on with the Covid
| lockdowns.
| Koshkin wrote:
| I wonder if computers have something to do with this.
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