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This is a meta question. No this isn't a meta question about site, this is a meta question about maths itself.

It has been observed quite a lot of times, that around some point in history,maybe with a gap of five or six years, the same result is independently discovered by two different mathematicians, and a dispute arises as to whom the discovery should be attributed to. It happened with Newton and Leibniz. It happened with Gauss and Bolyai. Why does this happen?

Given the large breadth of mathematics (or any science for that matter) what are the odds that two different mathematicians derive the same thing within such short times of each other. Clearly a mathematician's progress and work is heavily influenced by mathematical research going on at that time, but I am not talking about small papers here. Huge, groundbreaking discoveries like calculus and non-euclidean geometry independently occur to two even sometimes three mathematicians at the same time.

Why?

I would assume that there was some other discovery, in maths or otherwise, that promted multiple mathematicians to think in a specific way, and a few of these mathematicians came upon a new result. What were these discoveries in the cases of calculus and non-euclidean geometry then?

And as a more general question, this seems to remind one of the truism, "great men think alike", how true is it in this case then? And why?

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    $\begingroup$ IMHO there is no guarantee that someone will not use portions of your work without properly giving you credit. I don't blame Andrew Wiles for working in secret , to the annoyance of his peers , for that reason. Especially if the result is 'obvious' many people will use it without a proper reference IMHO. $\endgroup$ – neofoxmulder Mar 12 '14 at 21:10
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    $\begingroup$ Elisha Gray and Alexander Graham Bell independently filed patent applications for the first telephone on the same day. $\endgroup$ – MJD Mar 12 '14 at 21:14
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    $\begingroup$ @MJD telephone? holy crap. Never even heard of elisha gray! $\endgroup$ – Guy Mar 12 '14 at 21:19
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    $\begingroup$ Anyway, I brought that up because it is a common phenomenon in science and engineering generally, not just mathematics, and it might be useful to consider mathematics as only a special case of a more general phenomenon. As another example, you might consider Joseph Swan's invention of the incandescent light bulb. But there are many, many examples. $\endgroup$ – MJD Mar 12 '14 at 21:22
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    $\begingroup$ @MJD, good one! There are other examples that come to mind like Thomas Alva Eddison and Nikola Tesla (Again the guy with 3 names wins :) L'Hospital and Bernoulli , though Bernoulli never complained because he was paid well. The Gauss Bolyai incident sounds 'fishy' to me. IMHO , I find it hard to believe that Bolyai and his son would 'steal' Gauss's work and then send it back to Gauss for review! :) $\endgroup$ – neofoxmulder Mar 12 '14 at 21:34
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The same thing happens in science generally. The science historian Thomas Kuhn wrote a famous essay about this phenomenon, "Energy conservation as an example of simultaneous discovery", in The Essential Tension; you may want to take a look at it.

As long as we believe that mathematics exists in some sense independently of people, I think it's not so surprising. Take the discovery of calculus. The basic problems of calculus (finding a tangent, finding the speed of a moving object, finding areas) had been around for a long time. In some form, the ancient Greeks worked on these problems. In the generation before Leibniz and Newton, algebra reached pretty much its modern form, at the hands of Fermat, Descartes, and some others. To a very large extent, calculus is what you get when you mix together the classic problems with the symbolic techniques of algebra, and stir vigorously.

As another example, look at the constructions of the real numbers: Cantor and Dedekind. Mathematicians like Euler, the Bernoullis, Lagrange, and Laplace took the calculus and developed it extensively. Inevitably, the logical problems and fuzzy spots came to the surface. Already with Gauss, Cauchy, Abel, and others you can find complaints about the lack of rigor. So there was a perceived need for a more precise definition of what the real numbers "really were". On the one hand, it's not surprising that the previous generations hadn't worried too much about this: they were having too much fun exploiting the legacy of Newton and Leibniz, and the problems hadn't become acute. A perceived need, and a couple of geniuses: voila, a solution.

Note however that Dedekind and Cantor gave different constructions. For that matter, Newton's calculus differed in many ways from Leibniz's. This is generally true of simultanous discovery, when it's examined more closely. Kuhn discusses this in detail.

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  • $\begingroup$ "Dedekind and Leibniz gave different constructions". typo? $\endgroup$ – Guy Mar 12 '14 at 21:34
  • $\begingroup$ that should be dedekind and cantor, no? $\endgroup$ – Guy Mar 12 '14 at 21:38
  • $\begingroup$ Yes, thanks, I've fixed it. $\endgroup$ – Michael Weiss Mar 13 '14 at 2:23
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Why are mathematical results discovered by multiple people independently ?

Why is the sun discovered by multiple people independently ? Why do two people who look in the same direction see the same thing ? What happens when you take the Taylor series formula for the exponential function, and switch the base and the exponent in the numerator ? You rediscover the Bell numbers, whose roots date back to medieval Japan. What happens when you try to introduce a symbolic notation for nested radicals, similar to $\displaystyle\sum$ and $\displaystyle\prod$ , for instance, and then you write a negative quantity for its order ? You rediscover the fact that nested fractions are nested radicals of order $-1$, about a century after Herschfeld. What happens when you take the binomial theorem, and place a non-natural quantity for its exponent ? You rediscover the binomial series, centuries after Newton. What happens when you play around with definite integrals whose integrand does not possess an elementary anti-derivative, and you start focusing your attention on $\displaystyle\int_0^\infty e^{-x^n}dx$ ? You rediscover the expression for the $\Gamma$ function centuries after Euler and Gauss, by zooming in on its behaviour for $n=\dfrac1N\in(0,1)$. Etc. And the list could go on $($and on, and on$)$. It's all just one giant inter-connected web of lies, uhm, I mean, truth. ;-)

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Because there is some probability of it happening. There is a trend of research and logical implications that are required to be fulfilled before more progress can be made in some specific direction.

For example, suppose two men each have a garden. For the last year a severe drought has plagued the lands and little food exists. One day though it must rain because nature dictates it to be so(if it didn't rain life on this planet would cease to be life and we could not even discuss these things, and so there is always a change of a downpour in a drought). Both these men's gardens will be replenished with water and they will both bear the fruits of the rain. The other men who gave up will not have anything to show.

When some discoveries are created by some men, it was done on the backs of others... and then new men jump on their backs. It's how things are done. Because literally millions upon millions of humans work on the same things constantly there will be simultaneous events. Two men might read the same paper independently and come up with the same idea that thousands of others didn't... because of a whole host of reasons... because the two men happen to think alike or had similar experiences.

It's like asking why Pi is 3.141592... It is because it is. It had to be something and so something was chosen. The universe works in the way it does because that is the way it has worked.

I have rediscovered countless mathematical theorems in my journey through mathematics by simply generalization some result... only to then find out later that it was something already discovered by someone else. This is how it is for many. They are a dollar short and a day late, but this happens to everyone. In fact, many people discover things and it was never published and someone else took the "credit".

As science becomes more specialized and all the low hanging fruit have been picked, the "gems" require more devoted pickers who are willing to climb the tree of knowledge. The good news is that the tree is always bearing fruit and for those interested in it they learn to look, they can get it... of course, there is always someone who is trying to get their first.

E.g., it's basic mathematics and probability. You will have some people who beat others and by probability there will have to be some likelihood of both beating each other(meaning it occurs together).

E.g., suppose the probability of someone discovering some important thing in any given moment is p.

This means you have a p chance and so do I. If we run this experiment a bunch of times most of the time One of us will discover it first... but there will be some occurrences where we will discover the same thing on the same day. One can make this more explicit by studying it mathematically(e.g., a Poisson process). It is the same as when two light bulbs would go out at the same time, or two fail safes would go out... it is unlikely but it happens.

Given that most of the people who spend their entire lives trying to discover something and they want to be famous, they will go for the largest most ripe fruit on the tree... hence you have millions of people trying to prove the RH... So every day people are working on it and it will just so happen that there is a chance two people will figure it out on the same day.

But lets get real: These things actually don't happen instantly. A discovery is not a light bulb that goes off but many years of research. It's more like a race where you have the top 10 finishers... but the top 3 are really trying to win. If they see one guy pushing it then they will try harder and try to beat him.

What this means is that if one guy is about to publish a paper(which takes time) and someone else hears about it they will work hard to publish theirs first. These things actually don't happen simultaneously both are given credit because they both essentially came in first. There is no photo finish in these things because it's not so much about personal glorification as it is about scientific progress.

So it's more of an illusion. Two people don't actually stub their toe on the exact same couch model at the exact same time on the exact same planet wearing the exact same type of clothes at the exact same.... (each exact makes the likelihood far less)...

It just seems like it happens "exactly" because all the details are missing. There is really no such thing as "perfection". Even a baseball player who hits a home run doesn't hit the ball perfectly. He just hit's it perfectly enough for everyone to cheer.

So, you can essentially blame it on your ignorance, and as you learn what really is going on there will be other ignorant people who will ask the same question and have to learn, we all ask the questions and that is how we learn. That is, we all are constantly discovering things. You can bet that there were probably 10 people on this planet that was thinking of the exact same type of thing(but not exactly of course) that you were when you posted it... At least probabilistically.

Why it really happens? Well, you will have to ask God that...

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