For just about any academic field F and number X greater than ~100, there are a lot more F researchers than there were X years ago, and this remains true if you restrict to researchers above a given level of quality/competence Q.

This observation seems pretty straightforward/uncontroversial to me, and yet it has quite a few consequences that seem quite nontrivial. Before getting to those, allow me to explain the claim in more detail.

The Claim

For concreteness, let's let F be "maths" and let X be "519".

Well, the world population is about 15 times what it was in the year 1500, therefore there's 15 times as many researchers (at any given level of quality). Right?

Of course not. Let's go through what's wrong with that.

Most notably, pretty much all the mathematicians who were doing cutting-edge research in 1500 lived in Europe. Today, they almost all live in Europe, the US, Canada, Australia, Japan, and some other small developed countries. Every now and then a prodigy like Ramanujan comes along, but by and large, the developing world does not have the economic/pedagogical/cultural means to raise many of these, or even "discover" them when they do come along, so I will round these off.

The point is, we should be comparing the population of Europe in 1500 to the combined current population of all those regions that have the infrastructure to reliably pump out mathematicians today.

Population of Europe in 1500: 88,000,000

Population of Europe, US, CA, AU, JP in 1998: 1,280,000,000

...which gives us a quotient of 1280/91 ~= 14.54. I will round this up to account for the "other small developed countries" and also the fact that talented 3rd world prodigies occasionally make their way in academia in a way that wasn't possible in 1500. So by our reasoning so far, the talent pool of researchers has grown by a factor of 15 in that time. But a few more corrections are still in order.

One consideration is whether, due to better nutrition, schooling, economic conditions, etc., the average person in the developed world today is actually smarter than the average European in 1500. The Flynn effect is the observation that average IQ scores have risen substantially (over 10 points) in the past century, but it is controversial whether this represents a real gain in intelligence or test-taking familiarity (or, of course, a combination of these). If the former, our factor of 14 goes into the triple digits (especially if one sets the quality bar Q high). But I don't feel like reading up enough to take a stance on this, so I'll be conservative and assume the null hypothesis. The point is, there's no reason to believe people today are innately "dumber".

The other point is, we've so far focused just on the talent pool, rather than who's actually doing math research full-time, now and then. It seems likely to me that on a per capita basis, there are many more math researchers now than then, considering the academic infrastructure we have today. This raises the question of how the average researcher quality used to be. There are essentially two models one can have of how someone becomes a math researcher in 1500:

a) Because they had all 3 of intellect, interest in math research, and the means to support themselves (via personal wealth or benefactors).

b) Because they truly were the crème de la crème of their time

Personally, I find (a) more plausible: if you were an aristocrat with a passion for mathematics, then probably that's what you'll do with your time as long as you had the requisite amount of intellect, on par with any math undergrad/grad today. If you were middle class, then probably you'd have to find a benefactor and this process would be correlated at least somewhat with talent. But if you were socially awkward, or a peasant, then your prospects would seem bleak, so ultimately I suspect this would be far less meritocratic than our current system. Unfortunately, this would take some tedious research to sort out, and I have no idea how I'd quantify the results. Also, I'm not positive there's proportionally more researchers now than then, so I'll be conservative again and assume the null.

I've just given a couple ways our factor could be significantly greater than 14, but I see no good reason why the number would be less than 14.

If F is a field besides maths, the same conclusions seem to apply as far as I can tell, except perhaps more strongly (e.g. more developing countries around the world today produce top artists/poets/musicians, even though the best were largely restricted to Europe in 1500).

If we let X vary, this does change the picture. For instance, if we are comparing to Europe up to 1700, the factor remains above 10. By 1800, other countries such as the US enter the picture, but even 100 years ago we still have a factor of 2.

Some Consequences: How Good Were "The Greats"?

If we are inclined to say that Newton was the greatest physicist of his century, then the world should have produced about 10 physicists of his caliber in the past century (well, more like 5-10, since the OECD population a century ago was half what it is now). So:

a) Pick your 10 best physicists who did most of their work 1920-2019. If they'd been born in Newton's place, they'd have done just as well in advancing physics (pinning down the exact counterfactual is tricky, but "you know what I mean").

b) Pick the #1 physicist of this decade. Let's say for concreteness, Kip Thorne. He'd also do just as well.

c) Pick the #1 physicist of the past century, say, its Niels Bohr. He'd do even better.

d) If Newton grew up today, he'd probably win a Nobel, but "Newton" would be about as much of a household name as "Thorne" (and less than "Bohr").

This game is even more amusing to play in other fields:

If we are inclined to say that Shakespeare was the greatest poet of his century, then in expectation, the world should have produced someone equal to his ability in this decade. So let's take the best poet of this decade. I dunno how to actually do this, but of the last 10 Nobel Laureates in literature, apparently 3 do poetry. Now take max{Herta Müller, Tomas Tranströmer, Bob Dylan}, and that person should be as good a poet as Shakespeare...

I notice I'm confused. Or more accurately, I notice I've derived a weird conclusion from seemingly valid reasoning and anodyne premises. It's probably worth examining in more detail.

I assumed that a population of ~100 million people should produce a Shakespeare every century on average. One could argue that Shakespeare was even more special than that, and if one "reran the simulation" of Renaissance Europe a few times with different "random seeds", then we wouldn't get another Shakespeare equivalent. This seems believable up to a point, depending on how much one is willing to argue for Shakespeare being significantly better than anyone before or since. But then we can make the same argument on Goethe, Poe, or Whitman, and then one can no longer credibly argue for Goethe-, Poe-, or Whitman-exceptionalism. At some point, the outside view of "how strange that most of the best people just happened to be among the first to have lived chronologically" kicks in.

It could also be objected that maybe we are "dumber" now in literature. The only plausible ways I see this happening:

a) Literature used to be a more central part of culture/education than today

b) Living in that time gave Shakespeare and other authors more colorful experiences to inform their writing. Or at least, they seem more interesting to us than fiction based on modern society.

Lastly, one could object to my assertion that max{Herta Müller, Tomas Tranströmer, Bob Dylan} is the best poet of the decade (or some recent decade, since Nobels are usually awarded pretty late in one's career). For one thing, one could argue that the best poet of a given decade won't be that likely to win a Nobel. This could be the case (I have no idea), but then the question becomes why the field is unable to recognize its top performers. Maybe the Nobel committee is idiosyncratically bad but I'm not sure what else there is to go on (book sales seems worse). I'm sure there's also an amount of "intellectuals in any field are hard to appraise while they're alive, so their contemporaries (rationally) err on the side of not lionizing them too highly".

(I am curious if people find the Thorne ~= Newton conclusion less or more controversial as Tranströmer ~= Shakespeare?)

Further Notes on Data Sources, and Another Calculation

For the population of Europe at different centuries, I used this table (which ends at 1998), manually adding up the 3 subtotals. Note this includes European Russia, even though I don't think Russia was producing many intellectuals until ~1700s, but that's just a guess. So besides the Flynn effect and increased meritocracy, this is a third way that my estimate of the increase of researcher quality are potentially conservative. Maybe the region we should've started with in 1500 was non-Russian Europe, or even just "Western Europe": the UK, France, the Low Countries, Italy, Switzerland, Austria, Germany, Denmark, and Poland (because Copernicus). Let's see what happens when we do this:

Population of "Western Europe" in 1500: 51,000,000

Population of "Western Europe" in 1998: 342,000,000

And then we only get a factor of 342/51 ~= 6.7. But if we are inclined to say that the US, Canada, and Australia are now doing about as good as "Western Europe" in per capita researcher production, then we get a population of roughly 700 million, for a factor of 14. And then there are other places that certainly add to the quantity, but I'm not convinced they're on par with "Western Europe" in per capita terms: Japan, Korea, Russia, eastern Europe, etc. It seems roughly correct to me to say that ~1/3 of the world's researchers were born/raised in "Western Europe", ~1/3 in the US, and ~1/3 in the rest of the world. This would push our factor up to ~20.

Of course this factor declines as the centuries pass after 1500, in a way that would have to be quantified on a case-by-case basis. It's worth noting, though, that even though Western Europe's population has less than doubled since 1900, the US population has more than quadrupled.

For those interested in such calculations, here's another which is tedious to do but whose result which be interesting to look at:

Most lists of the best physicists, or best authors, or anything in between, almost entirely contain people from over 100 years ago. If, for example, you have a list of the n best physicists, the chance that all of them just happened to live among the first p proportion of people to live since physics became its own field is p^n. For instance, suppose you have a purported list of the top 10 physicists, and all of them were born before 1900. Let's say "physics" became a thing around 1500, and that a physicist would have to be born before 1970 to be recognized today. Then one would have to calculate the "area under the curve" to find the number of people born 1500-1900, and the number of people born 1500-1970, and then p is the quotient of these. If p is, say, 0.7, then .7^10 ~= .028, which is sufficient for us to reject the null hypothesis that the list is not biased against modern physicists. Actually calculating p would take some work though.

Also, there is the issue of finding the actual number of researchers in centuries past, rather than making inferences based on the talent pool or the top-end as I've done in this post. Currently, I welcome suggestions for how to estimate this number and/or how it has increased over time.