In "splendid isolation" - MathOverflow

In "splendid isolation"

2012-05-20T23:59:29Z

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. )

Answer by Gerry Myerson for In "splendid isolation"

2012-05-21T06:15:03Z

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?

Answer by Jan Jitse Venselaar for In "splendid isolation"

2012-05-21T12:05:52Z

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.

Answer by Dmitri Pavlov for In "splendid isolation"

2012-05-21T19:08:56Z

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.

Answer by Pait for In "splendid isolation"

2012-05-21T22:19:51Z

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.

Answer by Tom Copeland for In "splendid isolation"

2012-05-28T19:47:44Z

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..."

Answer by KConrad for In "splendid isolation"

2012-05-29T00:25:11Z

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!