The reality is murkier. Huge names like Lorentz and Poincare had worked on relativity before Einstein. Gregory and Barrow had proved the fundamental theorem of calculus before Newton, not to mention Leibniz's work.
If you mythologize the past it's easier to look around and wonder why we don't have immortals like Zeus or incredible warriors like Achilles anymore. But the truth is science always proceeds by steps that look small at the time and it's often only in retrospect that things seem amazing and unprecedented.
Semi-relatedly, I used to attend a seminar with a well-known Russian mathematician who would often chime in with Russian/Soviet priority over historical results mentioned by visiting speakers. The cold war created two mathematical cultures that had limited contact. So famous European and American mathematical results from, say the 40s to the 90s often had Soviet versions worked out independently in journals nobody here had ever heard of and written in a language they can't read.
So this is all just a way of saying that empirically, the big kahuna theory of mathematical development seems more fiction than reality. And it should be treated with skepticism when you hear things framed in those terms.
[0] Not that their achievements aren't great. They are all incredible. I just mean they seem more incredible than they are if we forget all the other incredible achievements. Shoulders of giants etc etc.
But math keeps advancing at a good clip, there are tons of brilliant young people. Physics IIUC is largely gated on waiting for new experimental results. Einstein, Poincare, Newton and many others arrived at a time when the experimental physics data was perplexing and theoretical advances were needed to clear up the picture.
The reverse is true now if I understand. People are just waiting for more funding to build larger accelerators or whatever big expensive astronomy things they need to test the many predictions we have from the various mathematical attempts to unify quantum mechanics and relativity.
Biology seems to be moving pretty briskly. I can't speak to chemistry or any of the other sciences.
EDIT: I should add that when I left academia about 8 years ago, the mood was that academia was increasingly hostile to smart people who just wanted to do research. The move toward running universities like businesses has pushed out a lot of people who would be candidates for the next Einstein or Poincare. Poincare happened to be independently wealthy, so wealth may again be more of a factor than it was during the last century.
Exactly these people are more likely to be spending their time speed running Mario.
Which doesn't seem at odds with the grandparent post, given that we shouldn't expect throwing more people at the problem to accelerate discoveries.
I think it's really an empirical question how it works. My own take is that it takes both. You need the ideas floating around and you need an obsessive synthesizer.
A music example might be Bob Dylan. Folk music could have been a passing local fad, but Dylan studied all of it, distilled it down, and produced a version that seems different in kind than what came before. Without Dylan, we could look back on that era as being like the swing revival of the 90s. Just a period where a kind of music was popular again but with no real lasting change.
Similarly, there could be areas of math that are ripe for a Newton figure to come in and clarify everything. But that field could wait 100 years in the backwaters of math journals before anyone took notice. So you can't directly jump from the ideas floating around to the synthesis, you do need some sort of figure who will read and understand everything and put all the pieces together into a coherent piece of writing that people can read.
Labs and researchers are given funds based on "impact".
As we all know, when a metric becomes the goal, the metric gets gamified.
This is very hard to fix.
Gonna give you an example. As soon as you move in any direction in science you're entering a niche. Pretty much everything is its own niche where there's a limited number of people really able to review your paper.
No, there's not thousands of experts dedicating their life to helicoidal peptides interactions with metal layers. It's an extremely small number of people. They all know each other and are gonna review each other's papers regularly.
Solar-powered water splitting to produce hydrogen and oxygen? Again, extremely small club.
Perovskite? Graetzel solar cells? Bigger club, but still, at the end of the day the relevant luminaries are a handful again.
You think it's much different for quaternions or advanced complex numbers analysis?
And those small clubs set the rules and standards, and there is really not much way to have oversight on those small clubs.
Plenty of terrible science is published every day, I can confidently say 90 to 95% of experiments in Physics or Chemistry are impossible to reproduce (numbers people get by taking outlier results or even more often by tweaking the data).
When it comes to softer sciences like psychology it's even worse. It's crap.
It's sad, but I think the world desperately needs a *free* alternative to the biggest publishers out there like Nature or ACM or all these things. That free alternative has to be funded by universities and governments globally. This entity should only allow papers that present clear experiments that are reproduced elsewhere or under supervision.
This would greatly enhance the quality and reliability of papers.
It feels like it should be. Maybe the following is just the naivety of a novice looking at experts they don't understand.
Learning numbers, integers, rationals, and real numbers, introduced so much power with each step, to the point that every child is expected to know the basics about real numbers. Maybe not what they are theoretically, but the basic ability to do something like .5 * pi or root(2)^4. They likely never dig any deeper, most don't really get the idea of never ending never repeating decimals, but they are able to work with them on a simple level. For most fields, an introduction only requires this level of math.
When one does move to the complex numbers, it opens up far more possibilities. Fourier transform is everywhere, to the extent many use it without having the math to understand how it does what it does. More complex problems, and I mean outside of the field of math, only have general solutions if you allow for complex numbers to be used. These are difficult, mapping from C to C requires 4D to represent and thus are much harder to visualize, but people still struggle through it.
When I realized that complex numbers weren't the end, but the second step in a tower of infinite height, I wondered what otherwise unsolvable problems needed higher levels from that, just like how many problems needed complex numbers. The difficulty working with them grows, though somewhat given C to C is already 4D, we have already reached the limit of 3D viewing power.
Yet they are rarely used, and higher levels used even less. Maybe there is something fundamental that makes them less useful, far weaker than complex numbers. But from a viewpoint of a novice, I find that surprising.
The amount of additional power you get from real numbers is zero, and in fact it isn't possible to work with them because specifying an irrational number takes an infinite amount of time and space.
They give you additional power in the sense that they let you give names to the exact answers to certain equations, but not in the sense that you can manipulate those equations any better than otherwise.
> When I realized that complex numbers weren't the end
But... they are the end. That's the point. We call that the Fundamental Theorem of Algebra: every solution to every equation takes the form of a complex number.
Quaternions do not generalize from complex numbers in the sense of the progression of the numeric tower from integers to rationals to reals. That is known to stop at the complex numbers. They generalize from complex numbers in the sense of functions that scale and rotate two-dimensional images. Quaternions are a way of expressing functions that scale and rotate three-dimensional images. They are familiar to most people who have taken calculus under the name "vector cross product".
All I said is that as soon as you release a paper you fall in a niche of some topic (unless the paper is super generic) where the number of people with the expertise allowing them to review it is super small.
Quaternions or complex analysis where just examples where, even if not strictly experimental and more theoretical, if you're pushing the boundaries of knowledge the number of people able to review what you wrote and concluded isn't that big.
There are research problems to solve in industry, with more prestige, more pay, and so on.
I'm certain his contemporaries said the same about Pythagoras.
You're not wrong about how much more complex things have gotten, certainly.
But you don't know what you don't know. Another Ramanujan or von Neumann could be right around the corner.
Just look at all the wasted potential in our education system - it's impossible to quantify just how much effort is profoundly wasted.
This article points directly to that wastage. We've tolerated an academic system that's hated by just about everyone except publishers, in an era where publishing is as close to free as it could be.
Now, start extending that limit by integrating computers? The options are literally unimaginable.
But that all kinda misses the point, which is, there's a lot of low lying fruit that we're leaving unpicked.
> I'm certain his contemporaries said the same about Pythagoras.
Pythagoras is best known as the center of a mystery cult. What did he bring a century forward?
https://en.m.wikipedia.org/wiki/Pythagoras#In_mathematics:
“Although Pythagoras is most famous today for his alleged mathematical discoveries, classical historians dispute whether he himself ever actually made any significant contributions to the field.
[…]
The Pythagorean theorem was known and used by the Babylonians and Indians centuries before Pythagoras, but he may have been the first to introduce it to the Greeks. Some historians of mathematics have even suggested that he—or his students—may have constructed the first proof. Burkert rejects this suggestion as implausible, noting that Pythagoras was never credited with having proved any theorem in antiquity.”