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I'm particularly interested in the condition in mathematics. In this post, I find that the average number of authors for a mathematics paper is less than three. This suggests to me that, most of the time, new mathematics does not originate from a large group working hard on the topic. Similar case for other theoretical subjects, with philosophy being 91% sole author.

At the same time, I notice there is a growing consensus of opinion that science is tending to the collaborative side nowadays; however, this data suggests the opposite. Maybe I am missing something, so I ask: are there any benefits from networking and getting contacts for researching in mathematics?

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    I would not expect there to be a lot of commonality between math, physics, and economics. (And since when is economics considered a "hard science"?)
    – Buzz
    Nov 6, 2022 at 0:26
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    FYI, many-authored mathematics papers used to be extremely rare -- 5 or more authors virtually non-existent more than about 3 decades ago, and only in the last couple of decades or so did more than one author become fairly common. My guess is that the main driving force behind this in math is NOT the same as in other sciences (e.g. groups working in labs requiring significant funding and people with various expertise), but instead due to the rise in the easy and rapid means for communicating (e.g. email, perhaps better for math communication than telephone) and sharing of digital manuscripts. Nov 6, 2022 at 9:01
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    If I understand you correctly, you are contrasting the value of a quantity (“the average number of authors for a mathematics paper is less than three”) with its temporal derivative (“there is a growing consensus of opinion that science is tending to the collaborative side nowadays“). I see no contradiction here.
    – Wrzlprmft
    Nov 6, 2022 at 11:27
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    It is also worth noting that authorship in pure mathematics is subject to stronger criteria than in other fields.
    – Wrzlprmft
    Nov 6, 2022 at 11:30
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    Bear in mind that you can get your name on a science/engineering paper and contribute relatively little to it (I published a paper with three co-authors even though two of them contributed almost nothing).
    – Tom
    Nov 6, 2022 at 15:02

2 Answers 2

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There is nothing inconsistent between collaboration being (very) important and it happening only or primarily in very small groups.

Mathematics isn't like particle physics. You don't need billion dollar accelerators, with a huge support staff. Paper and pencil will do, though a good library also helps a lot.

The size of collaboration groups in math is increasing a bit, I surmise, because it is easier now to communicate. I grew up in a time when postal mail and travel/visiting were the only viable options. Now we have the internet. And before my time international mail was very slow and unreliable. I've collaborated with people from four continents, actually.

A lot (most?) of math collaboration is face to face around a table. Often it happens in a coffee lounge in a university department. A whiteboard to sketch out ideas, perhaps. But a lot is just a couple of people sitting down and one mentions an idea they have and the other makes some suggestion about how it might play out. This is much harder in a large group. Even at a conference, if someone brings up an idea to a large audience, the most likely thing is that one or two people will approach the speaker afterwards to share "thought". It doesn't become a large group "huddle". The coffee table might now be a zoom screen, of course, but still only a few people for the most part.

This is, I think, driven by the intensely internal nature of pure mathematics. I wouldn't predict that math would ever be done in a group of hundreds (thousands) as happens at CERN, for example.

Another factor is that it is harder, I think, in math to "divide up the work" with different members of a large group contributing different things. One may have a large overall goal, but no path to carry it out other than very intense focus at a very small part until that gets resolved, giving some insight into what the next steps might be. That isn't quite as true in computer science, and collaborations tend to be somewhat larger there.

One reason for the importance of collaboration in math is that when you work strictly alone it can be difficult to know what the currently important questions are. What is worth pursuing. Those sorts of discussions can, and do, happen in larger groups. But the follow up is more likely to be only a few, especially interested, people who are willing and able to do the deep dive.

So, essential collaboration doesn't necessarily mean in large groups.


I also question, not that it is especially important here, the classification of math with the sciences. The methodology is completely different and the nature of proof (math) vs. evidence (science) is profound.


In thinking about this again overnight, I'm reminded that every mathematical (almost) collaborates widely, but with our, possibly, long dead mathematical ancestors. Nearly every paper is based on some things from earlier work that may have set a direction or provided some insight.

And, mathematicians with doctorates can know their "genealogy", which is a trace back through their advisor's advisor, etc. See: https://www.mathgenealogy.org/index.php

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    (+1) I didn't carefully read your answer before writing my comment above, but I see that you've essentially included the points I made -- not needing people with various expertise that labs tend to require (and also, a point I didn't think about, greater difficulty in dividing up the work), and snail mail (or travel, or telephone) vs. email. Nov 6, 2022 at 9:06
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At the same time, I notice there is a growing consensus of opinion that science is tending to the collaborative side nowadays; however, this data suggests the opposite.

No, it doesn't. All it shows is that mathematics tends to have fewer authors per paper than other subjects. However, within mathematics the proportion of collaborative papers, the number of authors per paper and the number of collaborators per person have all been increasing over time: see data for the 20th century here.

The number of authors on a mathematics paper tends to be limited because it is difficult for lots of different people to make a meaningful contribution to a single paper (and the criteria for authorship are perhaps stricter than in some other fields). However, there are fewer overheads to doing mathematics and research groups can be much more fluid.

Consequently, while in some subjects increased collaboration might take the form of each project having more authors, in mathematics it takes the form of each author being involved in more separate projects with different collaborators, often at the same time. For example I am currently actively working on three projects with a total of six other people involved, but no overlap.

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