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While browsing the Net for some articles related to the history of the Whittaker-Shannon sampling theorem, so important to our digital world today, I came across this passage by H. D. Luke in The Origins of the Sampling Theorem:

However, this history also reveals a process which is often apparent in theoretical problems in technology or physics: first the practicians put forward a rule of thumb, then the theoreticians develop the general solution, and finally someone discovers that the mathematicians have long since solved the mathematical problem which it contains, but in "splendid isolation."

Other interesting examples?

(Matrices and Bohr's Quantum Mechanics of course. Someone could elaborate on the sampling theorem if they wish. )

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@Tom Copeland As I feared, I misread it. Hence, the deletion. –  Sniper Clown May 21 '12 at 5:49
The General Theory of Relativity almost fits this, except for the fact that somebody pushed Einstein into the direction of Riemannian Geometry before he could reinvent it. I remember reading that Einstein was pleasantly surprised that mathematicians already had developed a general theory in which his ideas fitted, exactly the "splendid isolation". Of course, this was used to develop his general solution to the problem, so it's not strictly an example. –  Jan Jitse Venselaar May 21 '12 at 11:54
@Tom: I think that the tale of Minkowski and Einstein was on Special Relativity (Minkowski died in 1909, 6 years before Einstein published his General Theory of Relativity). By Einsteins college buddy you probably mean Marcel Grossmann, who indeed pointed him to Riemannian Geometry (along with Levi-Civita), and probably was more attentive in math classes, becoming a full professor of Mathematics. The way I understood it, Einstein was never taught differential geometry during college. As I said, it isn't a good example, but Einstein commented that there was a good theory waiting for him. –  Jan Jitse Venselaar May 21 '12 at 12:52
Why does the question have a link on the term "splendid isolation" that has nothing to do with mathematics? –  KConrad May 29 '12 at 4:30
(cont.) My words "that was where the author got it" was a quick way of saying "from the historical usage as presented in the Wiki article." The Wiki article has a reference to Splendid Isolation? Britain and the Balance of Power 1874-1914 published in 1999. The question mark suggests there are nuances to the meaning and context (that may be evading you). For me it adds meaning to his choice of words. Until you have a more substantive argument .... –  Tom Copeland May 30 '12 at 3:57
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Cormack and Hounsfield received the 1979 Nobel prize in medicine for their work on CT scans. Cormack, a physicist, published his mathematical work on this in 1963, to essentially no response. Hounsfield, an engineer, built the first CT scanner in 1971 unaware of Cormack's work. Cormark included the following in his Nobel prize speech: "If a fine beam of gamma-rays of intensity $I_0$ is incident on the body and the emerging intensity is $I$, then the measurable quantity is $g = \ln(I_0/I) = \int_L f ds$, where $f$ is the variable absorption coefficient along the line $L$. Hence if $f$ is a function in two dimensions, and $g$ is known for all lines [...], the question is: Can $f$ be determined if $g$ is known? This seemed like a problem which would have been solved before, probably in the 19th century, but a literature search and enquiries of mathematicians provided no information about it. Fourteen years would elapse before I learned that Radon had solved this problem in 1917."

Fourteen years after Cormack's work means 1977, so Radon's work was rediscovered by the people involved with creating CT scan technology only after CT scan's had been around for several years. (Search on "Radon transform" for more information.)

Radon's work was rediscovered multiple times:

1) Cramer and Wold (1936) in probability theory,

2) Ambartsumian (1936) in astronomy,

3) Bracewell (1956) in astronomy,

4) De Rosier and Klug (1968) in chemistry.

In fact, Radon's basic idea was worked out before Radon, by Funk (1916) and Lorentz (1905). This work of Lorentz was unpublished, but a formula he found is mentioned in a paper by Bockwinel in 1906. More on this history is in Cormack's survey paper Computed tomography: some history and recent developments, pp. 35--42 in "Computed tomography: Proceedings of Symposia in Applied Mathematics" 27, AMS, 1983.

Shortly before the work of Cormack, Olendorf (a medical doctor in LA) published a paper in 1961 describing a crude CT scanner he had built out of household parts, such as model railroad tracks (!) but it went unnoticed. Hounsfield acknowledged it, but Olendorf was not included in the Nobel prize list with Cormack and Hounsfield. He once said in an interview "I think Professor Cormack was selected [for the Nobel prize] because he worked out all the line integrals mathematically. [...] I didn't provide any mathematical treatment of it, and that apparently carried a lot of weight with the Nobel committee. See http://en.wikipedia.org/wiki/William_H._Oldendorf for more on his story.

The mathematical and engineering concepts in CT scan technology, with applications to medical imaging, were worked out in an obscure journal in Kiev by S. T. Tetelbaum in 1957-58, before Olendorf!

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Is it known whether or not Funk or Lorentz were inspired by the Cauchy-Crofton formula? –  Ryan Budney Jul 4 '12 at 5:01
@Ryan: I have no idea. –  KConrad Jul 5 '12 at 1:14
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One example that springs to mind are the Dirac equation and Clifford algebras. Dirac wanted to take the square root of the Klein-Gordon equation, and calculations showed that he needed 4 "numbers" $\gamma_i$ such that $\gamma_i \gamma_j + \gamma_j \gamma_i = 2\eta_{ij}\text{Id}_4$ with $\eta$ the $4\times 4$ diagonal matrix of the Minkowski metric. He found 4 complex $4\times 4$ matrices which satisfied these equation. Later physicists found that a general theory of such matrices was given in the 19th century, the theory of Clifford algebras.

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Another nice example. What motivated Clifford? –  Tom Copeland May 21 '12 at 12:40
@Tom: Wikipedia says that Clifford used it to study motions in non-euclidean spaces and on the Clifford-Klein space. Maybe it also arose as a generalization of the quaternions, which were quite trendy at the time. –  Jan Jitse Venselaar May 21 '12 at 14:21
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In 1954 Chen-Ning Yang and Robert Mills discovered nonabelian gauge fields in a physical context (in order to understand the strong force), only to realize later that the same notion has been discovered in 1950 by Charles Ehresmann in a purely mathematical context. Related notions, e.g., Cartan connections, has been known to mathematicians for many years before 1950.

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When Kepler was trying to work out the orbits of the planets, he wrote something to the effect of, "If only they were ellipses!" as he knew the Greeks had worked that theory out 1500 years earlier. Of course, eventually he convinced himself that they actually were ellipses. Is this the kind of thing you have in mind?

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Maybe close enough (?). The ancient astronomers had observed the orbits of the planets and had come up with rules of thumb to predict them long before the theoreticians (Kepler and his predecessors) came along and tried to give some conceptually accurate mathematical rules. Greek/Egyptian mathematicians worked on the conics without applying the ellipses to the planets. Kepler struggled with the numbers and math until he realized the relation to ellipses. Newton connected the physics with the ellipse. Shall we say the Greek mathematicians worked in "splendid isolation?" –  Tom Copeland May 21 '12 at 10:06
On the other hand, Kepler's laws of motion were really "rules of thumb." It took a Newton to prove them mathematically with his newly created calculus and inverse square law of gravitation. –  Tom Copeland May 21 '12 at 14:53
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Quantum mechanics of Born, Heisenberg, and Jordan.

From Physics in my Generation (Springer, 1969) by Max Born:

"In Gottingen we also took part in the attempts to distill the unknown mechanics of the atom out of the experimental results ... The art of guessing correct formulas ... was brought to considerable perfection ...

This period was brought to a sudden end by Heisenberg ... He cut the Gordian knot ... he demanded that the theory should be built up by means of quadratic arrays ... one must find a rule ... for the multiplication of such arrays ...

By consideration of known examples discovered by guesswork, Heisenberg found this rule ...

Heisenberg's rule left me no peace, and after a week of intensive thought and trial, I suddenly remembered an algebraic theory that I had learned from my teacher, Rosanes, in Breslau. Such quadratic arrays are quite familiar to mathematicians, and are called matrices ...

[Born writes down the now iconic qp-pq=iħ.]

My excitement over this result was like that of the mariner who, after long voyaging, sees the land from afar..."

Edit (Mar 2014): In addition, according to Harold Davis in The Theory of Linear Operators (Principia Press, 1936, pg. 199), the commutator [q,p]=1 "was apparently first studied by Charles Graves as early as 1857." Davis goes on to use the commutator to get some "normal ordering" results obtained by Graves and to expand on them.

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Heaviside's operational calculus, used by electrical engineers to work with differential equations, predates its mathematically accepted justification by decades. The same can be said about Dirac's delta function, which is used together with it. Of course, to some extent the operational calculus is a repackaging of the Laplace transform, but that is not all there is to it.

One might argue that in this case mathematicians' splendid isolation worked the in the opposite direction.

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Does his work on induction fit the sampling theorem scenario? –  Tom Copeland May 21 '12 at 23:46
You mean the equations of transmission lines? I do not know. –  Pait May 23 '12 at 18:34
Actually, Heaviside's successes influenced Bromwich, who corresponded with Heaviside, to investigate the Laplace transform and its inverse as a means of interpreting Heaviside's methods. –  Tom Copeland Mar 9 at 17:34
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