# Undiscovered for a long time before it is realised it is the same concept developed under different names.

Mathematics has been described as the giving of the same name to different things, but sometimes different names are given to the same thing.

Can you give examples of concepts where researchers in different areas have used the same concept under different names for a long time before it is discovered they are talking about the same thing ?

Wavelets might be an example.

EDIT: In response to Willie Wong's comment. I was thinking of the book "The World According to Wavelets. The Story of a Mathematical Technique in the Making" by Barbara Burke Hubbard.

Here's some quotes from page 26: "I have found at least 15 distinct roots of the theory, some going back to the 1930s", Meyer said. "David Marr, who worked on artificial vision and robotics at MIT, had similar ideas. The physics community was intuitively aware of wavelets dating back to a paper on renormalization by Kenneth Wilson, in 1971". Littlewood and Paley developed wavelet-like techniques. Alberto Calderon developed a continuous version of wavelets. Yet other researchers developed wavelets-which they called "self-similar Gabor functions"-to model the visual system. Jean Morlet developed wavelets as a tool for oil processing.

From page 40: "Multiresolution approximation and wavelets", Mallat. The paper made it clear that work that existed in many different guises and under many different names -- were at heart all the same.

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Can you clarify on why you think wavelets are an example? –  Willie Wong Jun 28 '10 at 12:10
This highly depends on the language! As I mentioned in one of similar questions before, "continued fractions" are known in Russian as both "непрерывные дроби" ("continuous fractions") and "цепные дроби" ("chain fractions"). Do you specify the language? –  Wadim Zudilin Jun 28 '10 at 12:41
Another "cultural" example: Cauchy-Bunyakovsky (in Russia and around) and Cauchy-Schwarz (overseas). ;-) –  Wadim Zudilin Jun 28 '10 at 12:44
It could be any language. Different languages would be a source of groups of researchers working independently of each other. –  Roy Maclean Jun 28 '10 at 12:53
Related question - mathoverflow.net/questions/15731/cryptomorphisms –  François G. Dorais Jun 28 '10 at 14:13

From your example with wavelets, I interpret your question as asking about convergent evolution of sorts: the idea doesn't have to be developed into a full blown mathematical theory in all of its roots, but that some similar shapes and structures can be found?

Then, a few items from geometry

• Yang-Mills theory / Gauge theory / Cartan connection
• The "DeTurck Trick" in proving well-posedness of Ricci flow is essentially the same as the Harmonic coordinates used in Riemannian geometry and the de-Donder-gauge or wave(map) coordinates in general relativity. (I've also heard claim that the DeTurck trick was used before DeTurck in renormalization group flow in physics, but that may be historically inaccurate.)
• On a slightly shorter (and more competitive) time frame, the whole notion of Lorentz-Fitzgerald contraction and Minkowski space was separately realized by different groups in physics and maths. Recall that Einstein famously didn't believe in this geometrization of space-time until he finally found it useful for the general theory.
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The use of harmonic co-ordinates to kill gauge invariance (by the diffeomorphism group) was introduced in a paper of DeTurck and Kazdan for prescribed Ricci curvature. I don't recall whether it was used in general relativity before or after this, but I believe whoever did it second was aware of the other. –  Deane Yang Jun 28 '10 at 16:54
And I"m not sure what you mean by "Yang-Mills theory / Gauge theory / Cartan connection", but I would say that Yang and Mills managed to develop in their own unique way the ideas of a principal G-bundle, connection, and curvature without knowing about the earlier work of Cartan and others on this. –  Deane Yang Jun 28 '10 at 16:55
And, to combine your first two items, the idea of using a gauge condition ("harmonic gauge", "Coulomb gauge", "harmonic co-ordinates") was something that originated in physics (Maxwell's equations), was extended by physicists to more general gauge theories such as Yang-Mills, and then exploited by mathematicians only more recently. However, my recollection is that everybody was aware of the precedents when they did this. For example, DeTurck and Kazdan understood what they were doing for the diffeomorphism group was the same as what others were doing for the gauge group in Yang-Mills. –  Deane Yang Jun 28 '10 at 17:00
@Deane: your second comment is precisely why I put that as an answer: Yang-Mills described their theory independently of earlier work of Cartan, which is the convergent evolution asked for by the OP. About your first comment: I believe De Donder was either 30s or 40s? I don't have my bibtex file handy at the moment, but he definitely predates DeTurck. –  Willie Wong Jun 28 '10 at 21:59
@Deane: Also, about the third comment, it was more of the terminology. In the relativity community it is ALWAYS known as the harmonic gauge condition, whereas in the Ricci flow community I see the phrase "DeTurck trick" more often. This I believe is the phenomena of two different groups of scholars calling the same thing by different names as asked for by the original poster. –  Willie Wong Jun 28 '10 at 22:00

The concept of an $r$-cover-free family of sets was studied independently in at least three different communities: combinatorics, group testing, and information theory, and of course was called by different names (superimposed codes, $ZFD_r$ codes, etc.). I myself independently rediscovered the concept and was inclined to call it a "$k$-Sperner family" before I discovered Ruszinkó's paper (J. Combin. Theory Ser. A 66 (1994), 302–310). Ruszinkó did a fine job of tracking down the literature in all the different fields, and proved one of the basic theorems in the subject.

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Denis-Charles Cisinski mentioned in an answer that Heller's homotopy theory of homotopy theories is the same theory as Grothendieck's theory of derivators.

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