> “Mathematics used to be a much more individual activity,” Minsky said. “Maybe in the next year or so, we’ll be back to some solo projects.”
Seems that most research used to be more individual. I'm not sure why research has shifted towards being collaboration-heavy and I don't think that this has necessarily improved the quality of science. In my experience, papers with many authors tend to be worse. (Could be field specific.)
A few months ago I got reviews back for a paper I wrote which had only me as the author and one of the reviewers was suspicious of this and brought it up in the review. They didn't say what they were suspicious of but I'd guess that they assumed that the work couldn't have been done by an individual. I replied that the work was done one my own and anyone with a small contribution not worthy of authorship was mentioned in the acknowledgements. I'm still waiting to hear if that would satisfy the reviewer.
In many experimental sciences, a lot of research projects now require an effort that a few authors cannot handle. There are some discussions if this means that instead of being creative, people are merely scaling up, which is probably true in some cases, but often strong empirical results that enable "real-world" progress require such efforts. But that's not Math. I work (partly) in a field of theoretical CS which is Math-heavy and attracts mathematicians and there I also feel the indirect effects of Covid-19. As a junior in the field, I need to rely on the feedback of more senior people, and these people move fastest on the whiteboard/with pen and paper, in physical meetings. The good thing about these collaborations is that we juniors can provide fresh ideas and the seniors can shoot down the bad ones quickly (before a long, formal review process) and make sure the good ones are executed with the necessary rigor.
(Note: I find the comment of "your" reviewer strange as well.)
Many current problems are simply too complex to be solved by a single person, often because it requires expertise from very different fields. High energy physics, biology and neuroscience come to mind. There is not a single person who understands all aspects of CERN. Likewise there isn’t a single person who understands how to 1. Build a large scale Electron microscope to acquire high res images of an entire brain 2. Analyze hundreds of terabytes of data to make these images usable and 3. Analyze the results which would require highly specific domain knowledge of neural circuits or other biological features of the system you are looking at. My experience is opposite to yours. You are lost as a lone researcher in many fields and the best papers these days are written by enormous collaborations.
The parent explicitly talks about science in general. There is a tendency of some (often of those outside of science) that feel like science is getting worse over time and nothing of value is found anymore. The parent seems to echo this sentiment and gives very shallow anecdotes to make his point. Still it is the top comment as of writing. It’s just plain wrong in my opinion.
I didn't argue that science is getting worse in general and that nothing valuable is being done now, and that's not what I believe. I could see how someone could infer the former from the number of authors per paper increasing and me thinking that papers with many authors tend to be worse. My experience, at I said, could be field specific. (I work in fluid dynamics.) My view is more that science has never been particularly well done as a whole, though some recent things like the reproducibility crisis in social science and verification and validation in engineering have been promising.
That some projects require a large amount of expertise that no single person is likely to have is true. But there are disadvantages to having more authors as well, e.g., "design by committee" immediately comes to mind. And I don't think a majority of current projects require expertise that no individual could have to an extent that would explain why collaboration is expected in science. Ultimately I think a mix of the two would be ideal for science in general, with a larger contribution from individuals than is seen at present.
Also: If my example is a shallow anecdote than your CERN example is as well. I don't have hard data on any of this and you didn't mention any either. I'd be happy to look at any hard data that exists, but I don't know if there is any.
Others have mentioned the size of projects, which might be true.
Another factor is the incentive structure. There's immense pressure to obtain grants (due to indirect fund income) and focus on h factors increases this as well. The best case strategy given both of those incentives is to have large groups of people who each submit as many grants and papers as possible, and put each other on each submission. That way, even with a high grant failure rate each will bring some money in, and self-citations to a paper will be multiplied by the number of authors.
> No one I talked to had gotten much new math done. They were preoccupied with the news, distracted by kids at home, and pulled between online Zoom meetings.
So, it's a time-management issue, nothing to do with math here.
I have a genuine question, as someone with an interest in mathematics but alas, no natural ability in the area.
Is there some kind of 'map' or list of mathematics areas, problems, etc. that professional mathematicians consult for ideas when conducting research? Likewise physics?
It strikes me that we've collectively had at least 5,000 years of math-related research stretching back to ancient civilisations. Surely most discoveries have been made? How much more math ore is there to mine?
When I read Richard Feynman claim (paraphrasing) that the joy of physics, math is the theoretical discovery of new knowledge without applications, I wonder where on earth today's mathematicians are looking for new ideas and discoveries?
And how do they get research funding? As a CS researcher, funding is hard to come by if I don't propose a tangible product. When I say, e.g., let's re-examine the relational model and extend it, or do something interesting in AI, the question is always the same: what is the product? Mathematicians on the other hand never create a 'product', so I'm confused who, why and where they get their money?
Lots of questions, I know, but I'm missing something here.
Background in pure math here. Generally researchers learn about what problems are fashionable as they talk to others in the field. As a PhD student, your advisor should give you ideas for problems to work on. Solutions to old problems tend to open up new lines of inquiry. It's a potentially infinite process, with the only limit being the capacity of the human mind.
The odd perfect number problem has been unsolved for thousands of years:
Why would anyone want to know the answer to this problem? The truth is that the motivation for most pure math research is purely aesthetic. You could also ask arts or english departments "How can you apply your paintings or novels?!?". Some mathematicians (mostly geometers and topologists) are inspired by problems in physics, but most aren't. In truth you never know what structure or theorem might have some future application; number theory was totally "useless" until modern cryptography made (some of) it useful.
For just a tiny taste of one small area of modern math, I dare you to click on any of the links here:
We are flush with structures to investigate.
The idea that "most discoveries have been made" is nonsensical in a domain where the discoveries to be made are literally infinite. To give a hint as to the infinite nature of mathematical inquiry, you probably are familiar with the idea of a function mapping a number to another number. A good deal of modern math is involved with much higher order functions; we can have functions that map functions to numbers, functions to functions, functions to spaces, and so on and so on. And then we can consider functions between those functions (and so on). Category theory is an attempt to give a framework to some of these "meta" relations. It should be obvious there is no limit to these structures and no limit to the number of problems one could pose about them.
I only really know about the company-sci touching parts, but —
Mathematics didn’t have a strong notion of proof until the late 1800s/early 1900s. Contradictory “proofs” in calculus created a need for a more rigorous system of reasoning. (It took until the 1950s to formalize what an ‘infinitesimal’ is — but those were used in early calculus prior to limits.)
That gave us electronic computers, as the effort to reduce proofs to things that could be mechanized bore fruit. Every program is a proof!
Since then, the idea of proof and structuring mathematics has been radically redefined — introducing things like category theory and type theory as alternatives to set theory, while studying the impact of certain axioms and developing tools around “reverse mathematics”, which is fitting a set of axioms to the theorems you want to be true.
Modern mathematics is working on the study of logic topology, extending our reasoning tools to deal with complexities around proving equality in the hopes we can automatically verify mathematics. These tools overlap heavily with AI research.
The applications here are what you might guess: new AI methods, data analysis methods, verification of software, etc. The DOD has paid a fair bit to support that research into software verification, for instance. The NSF and others fund “big data” analysis.
I’m less familiar with other parts of mathematics, but —
The study of knot theory has impacts for physical sciences, from particle physics (anyons) to fluid dynamics. In particular, there’s some work to be done in higher order knot theory and computational knot theory (ie, how to be efficient). MSFT is building a quantum computer based on this — and has suggested that higher order knots might not require as expensive of hardware.
In number theory, we still don’t know much about almost all real numbers. Things like Chaitins constant exist — and such uncomputable, normal numbers form the bulk of the reals — but we don’t really know how to get our hands on them. In less exotic research, elliptic curves are used in cryptography. There’s some work I don’t quite understand in building out a homomorphic encryption — where we can operate on encrypted data.
In many areas, we’re still working through the algebra-geometry correspondence, which we got hints of 400 years ago but only formalized once topology and category theory were invented — and still are building tooling around.
And there’s lots of areas I have no idea about — but I assume are being similarly productive.
Ones that I know of, but can’t comment on: fractals, differential equations, bifurcation theory, and chaos theory.
If you're interested, there's a paper on a more "extreme" situation that mainly related to Soviet mathematics developing solely within USSR apart from the rest of the world (see the "Luzin affair").
Yes, it's one of the more reputable economics journals. Modern economics seeks to answer research questions (e.g. impact of research collaboration and global knowledge diffusion) using quasi-experimental settings (e.g. the unexpected fallout of the Luzin affair to mathematics research). Would be surprised to find this in a math journal if anything!
My reaction to the various pictures, for what it's worth, was "what a beautiful place - perhaps it needs more blackboards, but I would love to do maths there".
You could also fit a large assortment of gourmet meats into a stack of shoeboxes, but it's preferable to display them in a way that whets the observer's appetite.
I don't know, I find a bunch of yellow-spined Springer texts to be appetizing just shelved together. I didn't realize the shelves weren't empty until I saw the top comment and went back to look.
I'm not sure how big Oberwolfach's library is, but the first pic looks more like a (somewhat fancy) journals/new acquisitions display than like primary storage.
Seems that most research used to be more individual. I'm not sure why research has shifted towards being collaboration-heavy and I don't think that this has necessarily improved the quality of science. In my experience, papers with many authors tend to be worse. (Could be field specific.)
A few months ago I got reviews back for a paper I wrote which had only me as the author and one of the reviewers was suspicious of this and brought it up in the review. They didn't say what they were suspicious of but I'd guess that they assumed that the work couldn't have been done by an individual. I replied that the work was done one my own and anyone with a small contribution not worthy of authorship was mentioned in the acknowledgements. I'm still waiting to hear if that would satisfy the reviewer.