The article suggests this test to establish that math is running out of interesting research: "Take a fairly generalist journal, like the Journal of Algebra (take a topic in which you have expertise — my doctoral thesis was in algebra). Look at some of the papers. How many of them are truly interesting to you?".
But it seems to me fairly likely that applying the same test to math journals from 100 or 200 years ago would produce similar results. Most published papers will not be of great interest to any particular one person.
No way, if you take a generalist algebra journal from 50 years ago, chances are many more graduate students interested in abstract algebra will be able to understand and appreciate the nature of the research. I mean, the best and most interesting papers from algebra were published between 1950-1990, IMO.
Is the test about ability to understand or about finding it very interesting? Of course it's easier to understand what has already become old hat and popularized in the field.
I bet Salami slicing[1] explains part of it. Before publish or perish culture, It was more common to see more complete papers, that gave a holistic view of the subject, easier to understand and appreciate. Now those papers are sliced into N least publishable units, less interesting and harder to get the big picture in isolation, but hey, why to have one paper when you can have five?
Again, it's not clear to me that if you asked 1950s mathematicians to look at math journals in the 1950s, they would react very differently. You haven't presented any evidence of this.
It's certainly true that math is a more fragmented specialized field today than it was in the eras of Euler or Gauss, but without some more concrete evidence or objective claim, I don't know what is so bad about math journals today compared to 1950.
For what it's worth, I also have a PhD in mathematics and I also ultimately left academia with some disappointment at the gap between what it is and my sense or fantasy of what it once was or could be.
Do tell more... does it have to do with hyperspecialization, not being able to get fluent in a large enough proportion of the field as was the case in Euler's time, say?
No, while it may have been fun to be a generalist in Euler's era, that wasn't bothersome to me. To be clear, the issues I found in academia had little to do with math specifically, and affected academia broadly. The usual issues you've likely heard about dwindling ability to make a comfortable career of it without a great deal of luck.
The 20th century was just a completely abnormal period, where people decided to change their understanding of math at the same times that physics, statistics, and engineering were demand more and more different ideas fro it. On top of that, CS was created and branched from math.
Is that based on the average level of maths at the time? I would argue that there are far more mathematicians now who understand the results from that period now than there were then because our mathematical literacy, especially in higher education and in the developing world has increased significantly.
The fact that those results are easier to understand is because of our increased literacy. Trigonometry was the cutting edge of maths at one point and math literacy was even less then. Now it’s material for tweens.
My take on this is that each narrow field goes through an S curve with a very exciting exponential beginning followed by a long tail of slow and boring progress. Then there is a new field with a new S curve.
The current hotness is experimental and theoretical investigations of large language models. This is just maths, and the papers coming out recently have been amazing.
I’ve read papers showing things like that neurons pack information into nearly-orthogonal spaces with beautiful geometric symmetries.
Just yesterday there was a paper showing that the middle layers of a deep language model can be interchanged and still work!
I think this is a great point. The author seems to have a selection bias. All the “great” problems in maths are the ones that have remained hard to solve.
All the stuff in the middle has been solved and then taught and is no longer “interesting”. Or, are we build on those results the new problems are a bit further from the fundamentals so you have to look in specific more specialized domains to find new areas.
What the author seems to forget is that most of the stuff we take for granted now were at one point the cutting edge of maths and obscure to all but the leading mathematicians of the time.
But it seems to me fairly likely that applying the same test to math journals from 100 or 200 years ago would produce similar results. Most published papers will not be of great interest to any particular one person.