Here’s an obvious thought: more scientists, more discoveries; fewer scientists, fewer discoveries. But is that true? Let’s take a look at a few papers studying two (traumatic) upheavals in the scientific labor force. #ThursdayThreads #ProgressStudies.
First; Nazi Germany's 1933/1940 dismissal of Jewish (and other) academics. Because the dismissals varied by university, this can be used as a (grim) experiment on the impact of rapidly reducing your scientific labor force.
Waldinger (2016) tracks the fallout on German chemistry, physics, and mathematics departments over the ensuing decades. He finds, no surprise, departments that dismissed more faculty had fewer publications and fewer citations in the ensuing decades. 
Meanwhile, @PMoserEcon, @AlesVoena, and Waldinger (2014) uses the same event to study the impact of an unexpected influx of scientific talent. A small but significant number of talented German chemists moved to the USA in areas where US science was weaker.
While German chemistry departments that dismissed scientists experienced a decline in scientific publications, in the USA an influx of these scientists was associated with a patenting boom in the fields where they were most active.
These shifts had long-lasting impacts. In Germany, Waldinger shows departments that lost star scientists were unable to attract replacements with the same research productivity (presumably because good scientists want to work in departments where there are other good scientists).
As late as 1980 (the last year studied) the negative effect of dismissals was just as strong as in 1940! For comparison, he shows allied bombing also reduced research output, but the effect disappeared within 10 years.
In the USA, the patenting boom was not due to the German chemists inventing on their own. And it can't be explained by more productive US scientists either: MVW find US chemists already working in these fields were actually less likely to patent after the talent influx.
Instead, it's like the new chemists were seeds of a new industry. The patent boom came from pulling new people into the field, not from augmenting the research productivity of existing scientists.
Second event: the collapse of the USSR led to a large exodus of talented Soviet mathematicians. Over 300 came to the USA, and Borjas and Doran (2012) studies the impact on US mathematicians. academic.oup.com/qje/article/12…
This was a big change: during the Cold War, there had been minimal collaboration between US and Soviet mathematicians. US mathematicians didn’t cite much Soviet work, since much of it was presented in books that were not translated.
So what happens when a country experiences a surge in mathematical talent? Is there a corresponding surge in mathematical proofs and theorems? In fact, no. BD cannot reject the hypothesis that the total number of publications and citations was unchanged.
What happened? BD basically argue Soviet mathematicians replaced US ones and there was no net increase in the number of research mathematicians. They paint a picture of tough career prospects for young US academics working in fields experiencing a surge of talent:
publications⬇️, citations⬇️, moving to lower ranked universities ⬆️, staying in research ⬇️– they even find worse outcomes for the graduate students exposed Americans advised!
(Aside: apparently mathematics keeps amazing records including records on every doctorate in mathematics issued going back to the 14th century)
Whereas with the German chemists, the inflow of talented scientists led more American inventors to enter those fields, Borjas and Doran (2015) find Americans moved out of fields where Soviets were strong. sites.hks.harvard.edu/fs/gborjas/pub…
What explains the difference? One obvious explanation is the availability of research positions. US universities and the private chemical industry were expanding through the 1940s, so for chemistry, maybe there was a lot of capacity to employ new chemists for research.
In contrast, the total number of math positions in academia didn't move much during years new talent came to America. My intuition is that there isn't the same scope to do research in the private sector for theoretical math (but I have no evidence of that).
If the number of research positions is fixed, then of course an increase in qualified candidates will result in fewer people finding positions.
That said, it remains puzzling that the increased competition for research positions didn't obviously result in more heavily cited research. And why did new chemists move into fields of immigrant strength, but mathematicians move out?
I can think of various explanations, but I don't really know. One possibility could be that sometimes research is self-catalyzing (each discovery creates more avenues for inquiry than it closes) and other times it's not.
Chemistry certainly seems to have been poised for major growth in the 1930s. In hindsight, we can see it was a golden age. Maybe that just wasn't the case with math?
So is it true that more scientists => more discovery? The answer seems to be "it depends."
Appendix: research links here - fabianwaldinger.com/research
