# Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences?

Is the number of "breakthroughs" in mathematics decreasing, as it is claimed to be in other sciences?

Background for the question:

1. Park, M., Leahey, E. & Funk, R.J. Papers and patents are becoming less disruptive over time. Nature 613, 138–144 (2023). https://doi.org/10.1038/s41586-022-05543-x

2. What Happened to All of Science’s Big Breakthroughs?

A new study finds a steady drop since 1945 in disruptive feats as a share of the world’s booming enterprise in scientific and technological advancement.

Has a similar analysis been conducted for the field of mathematics (or, say, pure mathematics)? If yes, how do the findings compare?

• I wonder if the pressure to publish means that people now will publish things that in time past wouldn't have been considered worth publishing. I know one older mathematician (but not retirement age) who only wanted to work on big, meaningful projects, and not publish incremental work. He saw this as a massive cultural failure in mathematics. Sadly, it ended his career—he was more-or-less pushed to retire early—but he was principled to the end on this matter. Jan 22 at 10:40
• It is important to note that the decline in "disruptiveness" (their term for breakthrough-ness) in research found by Park et al refers to the average publication. When it comes to the absolute number of breakthrough papers they report consistency over time. So it isn't that there is less breakthrough research, but rather that there is more non-breakthrough research. Jan 22 at 11:37
• The number of breakthroughs is going up; it’s the ratio of breakthroughs to scientific work that’s going down. Will you edit the question to state the comparison case more accurately? This may require writing out the main question in the text of the post, and not just in the title. Jan 22 at 12:38
• One major issue to keep in mind is the phrasing of "the field of mathematics". A breakthrough in enumerative combinatorics, no matter how impactful, will likely not impact the research on anyone working on monoidal categories (say). This degree specialisation is more of a modern phenomenon. "Breakthroughs in mathematics", at least when counting their impact on the subject as a whole, is about as meaningful as "breakthroughs in studying living things". Jan 22 at 12:42
• Scott Alexander has pointed out that one needs to be careful not to draw incorrect conclusions from this type of data. In particular, depending on how you define "breakthrough," a decrease in the number of breakthroughs doesn't necessarily mean that "progress is slowing down" (again, depending on what you mean by that). Jan 22 at 13:16

Q: Is the number of "breakthroughs" in mathematics decreasing?

To get some quantitative feel for the question I considered the Timeline of mathematics on Wikipedia. Not all entries are "breakthroughs", but most could be considered as such. Here is a plot of the cumulative number of entries since 1900. I do notice a kink in the slope around 1965, so based on this evidence on might conclude that, indeed, the rate of discovery has decreased somewhat. Or perhaps the 1960's was just an unusually productive decade.

Relative to the total output the discovery rate is obviously much smaller now than in the past.

• +1 for an interesting choice of data. Additional observation: This is the view of the past achievements as of January 2023. Possibly, some 1965–2015 achievements that are currently not listed, might be viewed more favorably 20 or 50 years later. There could be a myopic bias about the recent achievements. (To be fair, that bias could be negative or positive.) Jan 22 at 13:30
• I’m not going to start saying names, but I think (following Carl-Fredrik Nyberg Brodda’s comment) that anyone looking at the list would find that it overlooks a significant number of breakthroughs from their field, which may undermine the conclusion about the rate slowing down. Jan 22 at 13:39
• this could be an invitation to add missing entries to the Wikipedia list... Jan 22 at 14:15
• The Wikipedia list is indeed a bit curious: Computing digits of $\pi$ is mentionned several times but (for example) the discovery of the Jones-polynomial is missing. I think the later had far deeper implications than computations of digits for $\pi$. Jan 22 at 16:16
• The list is bizarre (not just by the inclusions, but also omissions) and reflects the level of mathematical sophistication (or lack of thereof) of people who edited the list rather than the actual development of mathematics. Jan 22 at 16:31

Quanta Magazine has an article on 2022's Biggest Breakthroughs in Math:

"In 2022, mathematicians solved a centuries-old geometry question, proved the best way to minimize the surface area of clusters of up to five bubbles and proved a sweeping statement about how structure emerges in random sets and graphs."

The full article describes an impressive year of results. This does not address this year in comparison to previous years' results, but it is difficult to read this review and feel that the year was in any way disappointing mathematically.

(Added). Just to give a sampling, here are excerpts from the Quanta article under "Geometry":

• Emanuel Milman and Joe Neenan found out the shape of clusters of bubbles that can most efficiently enclose three or four volumes — in any number of dimensions.

• Isabel Vogt and Eric Larson solved the interpolation problem: how many random points in high-dimensional space certain types of curves can pass through.

• Andras Máthé, Oleg Pikhurko and Jonathan Noel ... figured out how a circle can be cut up into visualizable pieces that can be rearranged into a square.

• Martin and Erik Demaine ... published a paper that shows how to take any polyhedron and fold it into a flat shape — as long as you allow infinitely many creases.

• Dusa McDuff and several collaborators found intricate fractal structures emerging when they tried to embed shapes called ellipsoids into something called Hirzebruch surfaces.

• Other mathematicians made progress toward proving the Kakeya conjecture...

Similar lists are presented under the headings: the Fields Medalists' research, Number Theory, Machine Learning, Topology, Random Structures. Among the latter is a short paper (a 6-page proof that pinpointed when structure emerges in random graphs), while one breakthrough resulted in a 912-page paper showing that slowly rotating black holes will keep on rotating until the end of time.

I think that the picture is complex, and I'm sure that the two (currently) top comments by David Roberts and Brendan McKay, although providing arguments in apparently opposite directions, are both spot on.

I believe there's also another phenomenon which is worth mentioning: we generally feel more comfortable in assigning the "breakthrough" mark to works that have aged well. Without the kind of pedigree that only historic evolution can give, it's more difficult to get consensus about how large the impact of a result will be. Of course there are exceptions, like if a celebrated conjecture is proven, but for evaluating the breakthrough character of new ideas, frameworks, connections...and so on, time is usually needed.

So my two cents: in the following centuries, a non-negligible set of results, which currently are somewhat lost in the clouds, will be considered breakthroughs, as it often happened in the history of math.

If it all, I would say that the number of breakthroughs is increasing. Here is more anecdotal evidence that supports the graph posted by Carlo Beenakker. On 23rd November 2022 Terence Tao posted on https://mathstodon.xyz/@tao/109390971278692349:

Maths at internet speed. On 16 Nov, Justin Gilmer https://arxiv.org/abs/2211.09055 makes a breakthrough on the #unionClosedSets conjecture, achieving a lower bound of 0.01 instead of the conjectured 0.5. The next day, Gil Kalai blogs about it at https://gilkalai.wordpress.com/2022/11/17/amazing-justin-gilmer-gave-a-constant-lower-bound-for-the-union-closed-sets-conjecture/ . Four days after that, three independent groups optimize the argument to $$\frac{3-\sqrt{5}}{2}=0.38$$: https://arxiv.org/abs/2211.11504 https://arxiv.org/abs/2211.11689 https://arxiv.org/abs/2211.11731 . (Via Rachel Greenfeld)

It is quite easy to see why innovation is slowing down in many academic fields, as compared, say to the sixties. The sixties were a time where the population of students increased a lot in many countries, and so the number of teachers in university increased accordingly. In higher education, teaching positions come with research duties and so the number of people doing academic research increased pretty fast. China is a country where this phenomena is taking place at the moment. For many other countries however, this is over.

• It is a very bizarre way of describe reality. Research positions usually come with teaching duties. I highly doubt that teaching positions with research duties often lead to breakthroughs. Jan 22 at 14:37
• With the caveat that I'm skeptical of that what is being measured is an accurate reflection of "breakthroughs" but it seems perfectly plausible that having a greater number of academic positions would allow for more people to continue to contribute to research even if their "productivity" was not high or they were working in unfashionable areas. Yitang Zhang is a contemporary example. Jan 22 at 15:30
• I do not find this bizarre. But then I got a position at that time. With much more freedom than is customary nowadays. Yes, we also had to teach. Times do change. Jan 23 at 8:32
• This answer would explain why a field of research would slow down, but not whether maths is doing that right now...
– AnoE
Jan 23 at 8:51
• @VladimirDotsenko I think the point was that the new academics were only hired because there was a bump in student numbers, and hence more academics were needed. As a by-product, the number of researchers therefore increased, so the amount of research increases. This explains why the sudden acceleration in the 60s tapered off, but does not explain why (if it is the case as in the graph in another answer), the totla output regressed to the mean. Jan 24 at 11:41