The same is true of most (all?) scientific discoveries. A good evidence for it the fact that most important breakthroughs were made independently, within the space of months or years, by two or more people/teams that didn't collaborate with each other.

My takeaway is that science is very incremental, and discoveries depend on what the state of scientific knowledge is at any given time, and not on particular smart individuals.

I also believe the same applies to social progress - e.g. if Martin Luther died as a child, someone else would start the Reformation; if Marx and Lenin decided to pursue art instead of economics/politics, someone else would invent communism anyway, etc.

Its very tempting to believe that because most of math/science (e.g.) is incremental, that all math is incremental. Once in a long while I stumble onto something really new, a theorem or a proof which simply has no precedent. These things make you go "wow how did someone think of that?" - and it is the ability to do that which distinguishes a genius from a regular and competent mathematician.

It's very easy to dismiss all ideas as being incremental, but I think there exist real, rare sparks of genius which leapfrog the field forward, and these sparks of genius only get deeper and more mysterious the more you think of it. My only reaction to these ideas is a certain sense of awe.

Someone mentioned this earlier - "The Innovators" by Walter Isaacson is pretty much this thesis exactly. Progress comes from building on previous work, usually collaboratively.

Now, I think it's fair to say that there are "geniuses" and it's perhaps a little uncharitable to claim that we shouldn't look up to them at all - but there is certainly more than one dimension to progress.

This sort of determinism seems wrong. It implies that if you had a Monte Carlo simulation engine of the universe and ran it 1,000 times beginning at the 20th century through today, you would expect those iterations to look a lot like our world. Only with some of the names and people being different, but fundamentally, everything else would still have happened.

That seems unlikely, no?

Yeah, sort of. I'd say it's quite likely.

The thing about how engineering and the markets work - they are attracted towards local optimas. So I'd expect each iteration to e.g. have cars that are more-less like we know. I'd expect it to have soft drinks; maybe not Coca Cola, but something very similar and recognizable to us, etc.

Incremental, or low cost/risk? A more general concept. Because sometimes what seems non-incremental, is just a change in economics.

At a history of biology colloquium, a sociology of science person, was trying to fit a narrative of new perspectives unlocking previously unconsidered questions. And the pushback was, no, the dominant effect was the cost of various tools dropping by multiple orders of magnitude. Which changed previously intractably (and thus uninteresting) questions, into potentially fruitful ones (worth pursuing).

One of the many problems with patents, is non-obviousness being incorrectly assumed time-invariant. Granting multi-decade monopolies for "you did it this year, that's cute - everyone else was waiting until next year, when it would be easy", which doesn't promote the progress of science and useful arts.

Although it's true that there are many examples of multiple independent discoveries (supporting the zeitgeist theory of invention), this does not fully explain the nature of discovery and invention.

Consider that many of the examples, when examined carefully, fail to maintain the independence constraint. If the inventors were not in communication with each other but were consulting the same sources, or studying the same past failures (which was often) the case for independence becomes weaker. Nonetheless, one can still make the argument that since all the pieces were in place, that the inventor was less important. Three things stick a wrench in this: people who were truly ahead of their time, when there are many qualified yet with only a few succeeding and when there are outstandingly prolific individuals.

1) A notable example of someone ahead of their time is Hermann Grassmann, who toiled for many years in relative obscurity, quietly nursing a zoo of groundbreaking ideas. Gregor Mendel is another example. His discoveries took 30 more years to be independently rediscovered by 3 different individuals when the time was more ripe.

2) At each point in time, one can argue that many individuals met the criteria, from an intellectual and knowledge standpoint, to produce an invention yet only a handful do. There were plenty of people just as or even more intelligent than Einstein. Some even got very close to making the breakthroughs he did but none ever quite did. It seems genius individuals are able to grasp enough of a space to take steps that are locally suboptimal but end up escaping local optima for far better optima. There's a great deal of chance to this and it's not an issue of just out of the box thinking and high IQ (but sadly for the rest of us, it's likely still an innate quality).

3) It's difficult to look at the works of Gauss and Euler and not think there was something special about them. Genius individuals tend not to be one hit wonders and usually have something of an outsized effect on their field. While it's true that there is a rich get richer effect, the ability of geniuses to generate novel ideas and fields of investigation slightly outpaces their peers. And when advantages multiply, which is likely the case for productivity, these unique individuals end up dominating.

In the language of machine learning, geniuses seem able to sample more widely and are better at learning to learn. Even though it's likely that they're only a bit better at those things and they don't think qualitatively differently than the average person, it still ends up making a large amount of difference.

All in all, 2) impacts 3) which makes it so multiple independent discoveries can more readily occur. This doesn't mean the importance of environment can be discounted. Genius only appears under conditions of stability and sharing, the availability of gifted mentors and a rich set of freely shared and interacting ideas.

The thing about Einstein, at least, is that he formulated a precise, compact framework that expressed a variety of results that had arrived already in his time.

Without Einstein (and maybe Shannon and other geniuses), all of the results might still be here but might be summarized in a much more difficult and confusing fashion. Perhaps someone later would discover the compact formulation but quite possibly would not be able get it accepted due to the more verbose and confusing formulations already being established.

Maybe not exactly equivalent but some claim that using a constant tau, equal to 2 times pi, would simplify a lot of math - of course good luck changing that now.

See: https://www.scientificamerican.com/article/let-s-use-tau-it-...

> Einstein ... formulated a precise, compact framework that expressed a variety of results that had arrived already in his time

Einstein had an especially productive 1905, publishing three groundbreaking papers among others [1].

His paper on Special Relativity certainly consolidated the work of Poincaré, Lorentz, Minkowski and others, much as you say.

His paper on the photoelectric effect, while building on Planck's work, outright contradicted the earlier work of Maxwell and was a crucial contribution to quantum mechanics.

His paper on Brownian motion, on the other hand, built on very little earlier work and provided the first work offering a method to count molecules; moreover, the paper was the foundation for Perrin's demonstration of the physicality of Dalton's atomic theory.

Each of these three papers had enormous immediate impact, and we still use much of the mathematical forms he introduced in each.

Einstein did not stop with these 1905 papers.

> be summarized in a much more difficult and confusing fashion

Einstein certainly proposed a number of write-downs that were neither easy nor unconfusing. General Relativity (GR) is perhaps a good example of one of those in a theory that has proven to be highly successful. Solving the field equations of GR requires solving a system of ten nonlinear partial differential equations (which in itself would exceptionally difficult even before confronting the explosion into the thousands of elliptical, hyperbolic and undefined terms in the PDEs' couplings in general coordinates). Indeed, we're still confused about how one can decide whether a given solution to the field equations is unphysical and derivative questions like whether a physically plausible solution to the field equations can always be studied using the initial value formulation or something similar.

The mathematical structure of GR is complete and self-consistent, and the notation Einstein developed was extremely concise, but it would be wrong to think that the tersest notation (omitting indices and constant factors) G = T makes GR only as complex as tersest notation (omitting constant factors and in coordinates in which momentum vanishes[2]) E = m of Special Relativity.

---

[1] In 1905 he published more than twenty other papers too, several of which are only "lesser" in that they do not figure in the minds of non-specialists.

[2] People tend to underestimate the complexity of Special Relativity; with suitable choices of coordinates one can use it extremely liberally in flat spacetime, including where acceleration is non-negligible (i.e., where there is obviously non-uniform motion in the systems under study). It's only the presence of real gravity that wrecks the globality of Special Relativity; but even where there is real gravity -- which is never exactly uniform or linear -- Special Relativity is still valid locally).

Could be, at the same time I would argue that those who had disproportional impact almost by definition thought very differently from others (defying convention is tough: imagine what is must feel like to question the multi century reign of Newton as a lowly patent clerk).

Very few minds have done the work to come within reach of the conclusions that these guys have been able to draw. From there I suppose it ought to be possible to approximate the likelihood of a Eureka moment probabilistically.

I'd be interested whether future scientific progress is so complex that it cannot be attributed to a single mind (some indicators are rising numbers of contributors for influential papers).

I think this is precisely what 'sillysaurus3 is arguing against. Studying relativity a bit closer in its historical context shows that most relevant groundwork for the theory was already laid down by other scientists. Einstein might have been the first to connect those dots and publish, but I seriously doubt that he was magically smarter than all the other interested people. That is, if he didn't do it, someone else would, probably at most few years later.

He wrote three groundbreaking papers in 1905; Special Relativity was just one of them. He continued writing groundbreaking papers for decades after that. Even his unsuccessful investigations produced interesting insights.

Sure, "someone else" may have eventually written down much of the content of any one of these three papers, but probably not all three and not all in the same year.

Notably, he could have won the Nobel prize for any of those important 1905 papers (and did so for the photoelectric effect, rather than Special Relativity). In the same year he also had more than twenty other papers published, several of which were intersting, and all of which are eclipsed by the famous one on relativity.

In objectively quantifiable terms, the count, rate, and accuracy of his published papers over the years is simply exceptional.

A bit more detail here: https://news.ycombinator.com/item?id=14823796

I was hoping someone would mention Newton. I think if we measure Genius by (time of discovery before that discovery would have happened without the individual), Newton would rank as the greatest Genius in human history.

Most apparent with Darwin's theory of natural selection: He had to rush to publish because someone else had stumbled upon the same thing while he was still analyzing his data and compiling it together.

Is that true? That's very interesting. The common story is that he had to rush to publish it because he was worried about his health.

Yes, GP is correct: https://en.wikipedia.org/wiki/Alfred_Russel_Wallace#Natural_.... Where is that story common, I've never heard it.

From Darwin himself: https://archive.org/stream/originofspecies00darwuoft/origino...

My work is now (1859) nearly finished; but as it will take me many more years to complete it, and as my health is far from strong, I have been urged to publish this Abstract.

I remembered this quote, yet failed to recall the very next sentence:

I have more especially been induced to do this, as Mr. Wallace, who is now studying the natural history of the Malay archipelago, has arrived at almost exactly the same general conclusions that I have on the origin of species.

Whoops. That's some impressive integrity!

Yes it is true. Alfred Russel Wallace was on the other side of the world doing his own independent work and looked to Darwin for advice about his own theories. ARW is a bit of an unsung hero with regard to what he contributed to this part of Biology (at least to those that are not deeply involved in Biology).

https://en.wikipedia.org/wiki/Charles_Darwin#Publication_of_...

It steam engines when it's steam engine time, and I guess it theory-of-relativities when it's theory-of-relativity time.

It's still pretty remarkable that Einstein/Shannon were the ones to discover their respective theories though. Can you imagine what it must have been like for them? I've always envied genius. Genius always know exactly what it's supposed to do and just does. Recently I'm wondering whether I'm truly cut out for my chosen field (programming).

If Einstein or Shannon hadn't discovered their respective theories, they would have discovered something else.

Not sure what that does for you, but it also seems true.

if Einstein or Shannon hadn't discovered their respective theories, someone else would have Maybe

It probably wouldn't have taken very long, either. It actually could take decades, centuries, millenia. Think of all the discoveries lost to antiquity that had to be rediscovered. Think of Eulers findings that would take decades for some othe mathematician to figure out afresh

Well, Fermat's last theorem is one such example. How long did it take to rediscover a proof?

I believe the consensus among mathematicians is that Fermat either didn't have a proof or had an incorrect proof.

Any statement of the modern proof would require hundreds of pages and results not available to Fermat. So it's likely Wiles didn't rediscover the proof but simply discovered it.

What makes you think so?

Throughout history, many ideas have been discovered independently at the same time. Newton/Leibniz is the most famous example.

There are so many people looking over so much of the idea landscape that this seems like the norm rather than the exception. And if it's the norm, that means for any given groundbreaking idea there are likely a few people within a few years of discovering it.

It's impossible to know for sure, but it has interesting implications if it's true. For example, publishing first is more important than being correct in every detail.

I meant specifically for Einstein and Shannon. I’m not so sure anyone was right on the heels of Einstein, and at least physics seems to be full of discoveries that nobody even anticipated.

Lorentz and Minkowski were developing special relativity at the same time as Einstein (albeit in a mathematical context). And Hilbert arguably produced general relativity prior to Einstein as well.

> but it seems true.

It's definitely true. Here's a proof (?):

1) The universe has a static (if high) and objective number of rational facts that can be stated about it.

2) Anyone can discover these facts via accident, intuition, trial-and-error, experimentation or direct observation.

3) We have a smallish but more or less constant percentage of people who are obsessed with discovering these various facts as their life's work and who are gradually teasing out these facts from the mountain of mystery that lies before us.

4) Therefore, if Shannon hadn't discovered some of these facts, eventually someone else would have.

I would contest step 1.

If we assume that rational means "true", then it is clearly false:

If statements A and B are true, then the statement "A and B" is true. In fact, if the statement A is true, then "A and A" is true, so if there is at least one true statement, there are infinitely many.

If by "rational" you mean "true and meaningful", well, then we have to define meaningful.

The strongest definition of meaningful that I can muster is "contains information that cannot be logically deduced from previously known rational statements". But then by Godel incompleteness (assuming that we accept arithmetic as rational), there will always be a (true) statement we can make that is independent of our previous statements.

So while I think the discovery was more or less inevitable, there's certainly a flaw in your proof!

"A and A" is not a meaningful statement, though; it's a tautology, no? if you exclude those, what do you have?

It's not a tautology, because its truth relies on the truth of A. A true tautology would be something like "A or not A".

However, I address your concerns in the second part of my comment, as "A and A" is derivable from "A", and so is excluded.

5) Where 'eventually' means months to years at most. Not sure about Shannon, but to the extent I'm familiar with discoveries in physics, they're usually pretty incremental - at some point the state of the field becomes ripe for a breakthrough, and the first person to connect the dots is the one that gets credited as genius, even though their direct contribution is pretty small.

Everyone stands on the shoulders of giants, where that giant is composed of everyone who stood on the shoulders of the previous giants, ad recursionum

Someone discovering something or inventing something is a rare event. I believe its the confluence of hard-work, timing, luck, good health, inspiration from peers' work, etc. Its even more unlikely that TWO people would be in that exact same situation. Maybe they're 'destined' to be the person whose name gets mentioned in the textbooks.