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
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).
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. =)
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.
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.
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.
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'.
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.
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.
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.
See https://languagelog.ldc.upenn.edu/nll/?p=2290 for some details. I would summarize by saying, there were words involved (i.e. the Latin "petitio" and Greek "αἰτεῖσθαι") that have multiple meanings (a modern analogue being "plead", which can mean "ask"/"beg", but can also mean "plead to the court [some statement of fact]"—i.e. to make a claim), and the word "beg" comes from a translator selecting one of the wrong meanings. Also, "the question" is really a poor choice to mean "the original point to be proven"/"the conclusion"/"the point in question".
I am strongly against prescriptivism telling people that a bad translation that makes no sense is correct. I recommend "assuming the conclusion" as a correct name.
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.
... and you passed it, but then they tell you that you missed the job criteria, because you don't have 10 years kubernetes experience, which is a hard requirement
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 :-)
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
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.
> 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.
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.
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.
My record for one I've actually solved is like 6 months working on a problem (on and off). I'm actually more of a physicist than a mathematician (although my work is all about proving theorems and lemmas), so the objects I work with tend to be at least possible to play around numerically with for special cases.
i read polya's book as an undergrad to my detriment. 10 years later and having "solved" several problems, i can authoritatively say that it is not how difficult problems are solved. the real "how to solve it" for problems that aren't exercises is closer to what feynman said in one of his books: a good physicist [or mathematician] has 5-10 problems on their mind at all times and when they learn of a new technique they apply it to their problem. such learning happens by reading monographs, papers, and going to conferences. that is to say that math (or theoretical physics) is socially produced/developed.
Many 'external' actors are actually destroying the Amazon: Brazil's incompetence whilst voting (putting a crazy in charge); USA's inability to embargo Brazil if it keeps devastating the forest; Amazon is not located 'just at' Brazil, there are at least 5 other countries in the region; lack of international preoccupation with the forest;
Sure, I should have said "the crazy in charge", not Brazil itself. And, although you're right about Amazon existing beyond Brazil, its Brazilian territory that is being devastated right now.
USA is not the world regulator, but I agree that an worldwide embargo is fully justified in this case.
Finally, it was just a joke about the example the author used in the article.
This seems related to computational complexity theory, but in a more ‘structural’ notion as it deals not with the algorithms that are doing the comparisons but whether or not objects even have attributes you can differentiate by.
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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
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'.
I am strongly against prescriptivism telling people that a bad translation that makes no sense is correct. I recommend "assuming the conclusion" as a correct name.
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/
Holy crap. I've got about a dozen articles in preparation. Not a one of them is in Shelah's league
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 :-)
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.
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.
I think “How to solve it” https://en.wikipedia.org/wiki/How_to_Solve_It) is worth reading to get an idea of what it might involve.
It has lots of hints on approaches that may teach you something about the problem at hand and eventually may help you solve it.
I could go on and on and on...
Edit: 'on' to 'in' charge.