A question for the mathematicians out there:Are major discoveries in mathematics often discovered independently by multiple people at around the same time?Do major discoveries in mathematics logically follow from prior discoveries?The reason I ask is because in physics (and engineering) many major discoveries are independently discovered by multiple people. Probably because one discovery follows logically from another. If Albert Einstein hadn't invented special relativity, someone else would have. Is it the same way in pure math? If Robert Langlands or grigori perelman hadn't been born, would someone else have done what they've done at around the same time (or within a century anyway)?I think this is one of the things which distinguishes art from science. If someone hadn't written [insert your favorite novel here], then it would've never come to exist. On the other hand if Isaac Newton had never been born, we'd still have [X's] laws of motion today.

Pythagoras and Archimedes discovered lots of stuff like calculus but the Romans killed them. Then we got 1500 years darkness with people doing PhD in long division CX/VI etc. until Newton rediscovered the calculus.It's been a long time since we had a mathematician like Euler.Mathematics started to die when Bourbaki arrived. Just kidding. Serously,A nice book on the discovery process in maths is I think maths is an art and not a science. It is more chaotic, as Hadamard relates.

in most fields, there's a finite number of interesting problems and a boatload of people working on them

QuoteOriginally posted by: CuchulainnPythagoras and Archimedes discovered lots of stuff like calculus but the Romans killed them. Then we got 1500 years darkness with people doing PhD in long division CX/VI etc. until Newton rediscovered the calculus.Best short history of math ever! (And no kommas!)If one visualizes the entire body of all math past, present, and future as a kind of digraph of true statements linked by logical implication that lead from simple axioms, postulates, or definitions toward lemmas and theorems of increasing complexity and elegance, then mathematicians can be viewed as builders of the chains of logic and clusters of related statements. The greatest mathematicians find longer chains of reasoning that lead to more remote statements or start new clusters of math by creating new collections of axioms, postulates, or definitions. Lesser mathematicians find short chains that lead to incremental knowledge about the properties of already "discovered" areas of math. It would seem likely that lesser mathematicians would eventually replicate anything created by a greater mathematician but it might take a long time for their plodding incrementalism to reach a remote and elegant result.To the extent that all mathematicians build on to a base of pre-existing math, major discoveries would seem to follow from prior discoveries. To the extent that different mathematicians share the same pre-existing base, parallel discoveries seem inevitable. Moreover, if one mathematician makes an exciting discovery, many other mathematicians are likely to study the result and make parallel extensions of that discovery.

What I find amazing is that graph theory started from Euler's ruminations on the 7 bridges of Konigsberg.

QuoteOriginally posted by: CuchulainnWhat I find amazing is that graph theory started from Euler's ruminations on the 7 bridges of Konigsberg....and complaints about Euler's route when he acted as a tourguide!

QuoteOriginally posted by: albertmillsA question for the mathematicians out there:Are major discoveries in mathematics often discovered independently by multiple people at around the same time?Do major discoveries in mathematics logically follow from prior discoveries?The reason I ask is because in physics (and engineering) many major discoveries are independently discovered by multiple people. Probably because one discovery follows logically from another. If Albert Einstein hadn't invented special relativity, someone else would have. You're wrong about what you said about physics or any science for that matter. Sometimes paradigms can be shifted in different (if not opposite) directions and a single person can be a game changer. E.g. Newton vs Hook (rather than Newton after Hook) or Bohr, Pauli, Heisenberg and the rest of the Copenhagen school dolts vs Dirac and Einstein -- both negative examples.

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnWhat I find amazing is that graph theory started from Euler's ruminations on the 7 bridges of Konigsberg....and complaints about Euler's route when he acted as a tourguide!Maybe it should have been a commodious Hamiltonian vicus of recirculation, what?

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnWhat I find amazing is that graph theory started from Euler's ruminations on the 7 bridges of Konigsberg....and complaints about Euler's route when he acted as a tourguide!Maybe it should have been a commodious Hamiltonian vicus of recirculation, what?Hamilton did some of his best thinking on bridges he did. (The quaternion equation is a good example of graphiti theory, what?)

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnWhat I find amazing is that graph theory started from Euler's ruminations on the 7 bridges of Konigsberg....and complaints about Euler's route when he acted as a tourguide!Maybe it should have been a commodious Hamiltonian vicus of recirculation, what?Hamilton did some of his best thinking on bridges he did. (The quaternion equation is a good example of graphiti theory, what?)Those Trinity guys were vandals in the 19th century.

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnWhat I find amazing is that graph theory started from Euler's ruminations on the 7 bridges of Konigsberg....and complaints about Euler's route when he acted as a tourguide!Maybe it should have been a commodious Hamiltonian vicus of recirculation, what?Hamilton did some of his best thinking on bridges he did. (The quaternion equation is a good example of graphiti theory, what?)Those Trinity guys were vandals in the 19th century. Mountjoy prison is on the Royal Canal. Good that WRH was not caught.Actually there is a Hamilton walk organized by Prof. Tony O'Farrell whom I worked for all those years ago in Maynooth. Fiacre O' Cairbre who runs it was a numerical analysis student of mine.

QuoteThe idea of using four dimensions instead, and the way to do it, came to him in a flash, as he walked with his wife by the Royal Canal. Since it is one of the few major mathematical discoveries which is precisely located in time and circumstances, the event is very well-known in the international mathematical community, and people from all over the world know about "Hamilton's Bridge" Omnibuses are also good (H. Poincare)Quote"Just at this time I left Caen, where I was then living, to go on a geologic excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience' sake, I verified the result at my leisure.