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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 mathematicians 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 independtly occur to two, 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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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. – neofoxmulder Mar 12 '14 at 21:10
Elisha Gray and Alexander Graham Bell independently filed patent applications for the first telephone on the same day. – MJD Mar 12 '14 at 21:14
@MJD telephone? holy crap. Never even heard of elisha gray! – Sabyasachi Mar 12 '14 at 21:19
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. – MJD Mar 12 '14 at 21:22
@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! :) – neofoxmulder Mar 12 '14 at 21:34
up vote 8 down vote accepted

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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"Dedekind and Leibniz gave different constructions". typo? – Sabyasachi Mar 12 '14 at 21:34
that should be dedekind and cantor, no? – Sabyasachi Mar 12 '14 at 21:38
Yes, thanks, I've fixed it. – Michael Weiss Mar 13 '14 at 2:23

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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