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. )

  • $\begingroup$ @Tom Copeland As I feared, I misread it. Hence, the deletion. $\endgroup$ – Sniper Clown May 21 '12 at 5:49
  • $\begingroup$ 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. $\endgroup$ – Jan Jitse Venselaar May 21 '12 at 11:54
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    $\begingroup$ @Jan, also read the last two paragraphs of en.wikipedia.org/wiki/William_Kingdon_Clifford. $\endgroup$ – Tom Copeland May 22 '12 at 5:45
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    $\begingroup$ Why does the question have a link on the term "splendid isolation" that has nothing to do with mathematics? $\endgroup$ – KConrad May 29 '12 at 4:30
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    $\begingroup$ (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 .... $\endgroup$ – Tom Copeland May 30 '12 at 3:57

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 Bockwinkel 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, Oldendorf (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 Oldendorf 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 Oldendorf!

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    $\begingroup$ Is it known whether or not Funk or Lorentz were inspired by the Cauchy-Crofton formula? $\endgroup$ – Ryan Budney Jul 4 '12 at 5:01
  • $\begingroup$ @Ryan: I have no idea. $\endgroup$ – KConrad Jul 5 '12 at 1:14

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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    $\begingroup$ Another nice example. What motivated Clifford? $\endgroup$ – Tom Copeland May 21 '12 at 12:40
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    $\begingroup$ @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. $\endgroup$ – Jan Jitse Venselaar May 21 '12 at 14:21
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    $\begingroup$ There is a related, earlier example, which is the Pauli spin matrices, which are isomorphic to quaternions. $\endgroup$ – Ben Crowell Dec 14 '16 at 19:39

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.


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.

Edit (Jan 2015) Charles' brother John Graves discovered the octonians (octaves, see Wikipedia) in 1843 and is credited by Hamilton in encouraging his search for the quaternions.

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    $\begingroup$ Very surprising and interesting story! It means that great physicists like Bohr and Heisenberg were not completely familiar with multiplication of matrices. Or do I understand wrongly this passage ? In general relativity for example, multiplication of matrices (and tensors) is everywhere... $\endgroup$ – Joël Jan 5 '15 at 14:54
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    $\begingroup$ And in classical mechanics with the non-commuting Euler-angle matrices for rotations in 3-D, with which they must have been familiar, so, looking at the notes in the Wikipedia article on matrix mechanics, maybe the difficulty was in making the connection between what was initially regarded as an infinite "Fourier" series expansion for transition spectra and a pair of infinite matrices representing non-commuting ops. It seems Born was prepared by earlier work to make the explicit connections to algebraic manipulations of infinite matrices. $\endgroup$ – Tom Copeland Jan 17 '15 at 17:12
  • $\begingroup$ See this reference arxiv.org/abs/quant-ph/0404009 from the Wiki article. $\endgroup$ – Tom Copeland Jan 17 '15 at 18:10
  • $\begingroup$ The arxiv paper is "Understanding Heisenberg's 'Magical' Paper of July 1925: a New Look at the Calculational Details" by Aitchison, MacManus, and Snyder (pg. 4-5). $\endgroup$ – Tom Copeland Apr 27 '15 at 21:52
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    $\begingroup$ @Joël: It means that great physicists like Bohr and Heisenberg were not completely familiar with multiplication of matrices. In 1925, even vector notation was quite new. In Einstein's 1905 paper on special relativity fourmilab.ch/etexts/einstein/specrel/www , he writes out every vector equation as three equations involving components. In the 1922 edition of Millikan's Practical Physics, the word "vector" does not appear in the index, and a force is represented as a directed line segment, with a notation like AB. (Segments differing by a displacement are considered inequivalent.) $\endgroup$ – Ben Crowell Dec 14 '16 at 19:35

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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    $\begingroup$ 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?" $\endgroup$ – Tom Copeland May 21 '12 at 10:06
  • $\begingroup$ 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. $\endgroup$ – Tom Copeland May 21 '12 at 14:53
  • $\begingroup$ @TomCopeland -- Richard Feynman quoted Newton's proof about ellipses in one of his books. Newton didn't use calculus but only the pure ancient Greek method. $\endgroup$ – Włodzimierz Holsztyński Jan 5 '15 at 1:23
  • $\begingroup$ Newton had to conform to the tradition of mathematical proof of his times (and perhaps was hoarding his new method). Anyway, read more deeply about Newton and the calculus. I believe, he, like many innovators, had already tasted the backlash of conservatism and was not so naive to believe he could use a new mathematical method to introduce his modern science, not both at the same time. That's my recollection from readings years ago. $\endgroup$ – Tom Copeland Jan 5 '15 at 1:46
  • $\begingroup$ @Wlodz: See also the comments in the preface of Needham's "Visual Complex Analysis" on geometry and Newton's calculus. $\endgroup$ – Tom Copeland Jul 6 '15 at 0:04

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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    $\begingroup$ Does his work on induction fit the sampling theorem scenario? $\endgroup$ – Tom Copeland May 21 '12 at 23:46
  • $\begingroup$ You mean the equations of transmission lines? I do not know. $\endgroup$ – Pait May 23 '12 at 18:34
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    $\begingroup$ 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. $\endgroup$ – Tom Copeland Mar 9 '14 at 17:34
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    $\begingroup$ More on Bromwich's thoughts on the Heaviside op calc: archive.org/stream/theoryoflinearop033341mbp#page/n29/mode/2up $\endgroup$ – Tom Copeland Feb 27 '16 at 20:59
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    $\begingroup$ Some more on the history of op calc and the Laplace transform in "Some highlights in the development of algebraic analysis" by Synowiec eudml.org/doc/209068 $\endgroup$ – Tom Copeland Aug 19 '16 at 12:03

Rooted trees and numerical methods for differential equations.

Excerpt from "What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations" by McLachlan, Modin, Munthe-Kaas, and Verdier:

"Robert Henry ‘Robin’ Merson (1921–1992) was a scientist at the Royal Aircraft Establishment, Farnborough, UK, who was invited along with more senior numerical analysts to a conference on Data Processing and Automatic Computing Machines at Australia’s Weapons Research Establishment in Salisbury, South Australia. It seems like a long way to go for a conference in 1957. However, the UK was still performing above-ground atomic bomb tests in South Australia at that time and the Australian government was very keen to be a part of the emerging era. Merson’s work is bound up with one of the most significant events of 1957, the launch of Sputnik 1 on 4 October 1957, and the tale of Farnborough’s involvement is told in detail by one of the key participants, Desmond King-Hele, in his book A Tapestry of Orbits. The short version is that with the aid of a large radio antenna hastily erected in a nearby field, and some calculations of Robin Merson, within two weeks they had an accurate orbit for Sputnik 1. This allowed them to estimate the density of the upper atmosphere and (after Sputnik 2) the shape of the earth. Robin Merson became an expert in practical numerical analysis and orbit determination.

Merson’s paper explains clearly the structure of the elementary differentials ... and, crucially, shows how they are in one-to-one correspondence with rooted trees. He also introduces various basic operations on rooted trees. This development, perhaps regarded initially as a bookkeeping device for finding and keeping track of the different terms, has over time become central to the combinatorial and algebraic study of B-series.

As it happens, the required mathematics and structures had already been discovered a century earlier by Arthur Cayley in 1857.

... Cayley needed trees for exactly the purpose we are using them here—to keep track of how vector fields interact when applied repeatedly to one another—and this purpose was then forgotten for a hundred years. The need for better numerical integration methods arose quite soon, towards the end of the 19th century, and the required tools for a complete theory were already present, but they had been forgotten."

The paper goes on to explain the connections to pre-Lie algebras and work by Vinberg, Gerstenhaber, and several other contemporary researchers. However, it doesn't mention the work of Charles Graves in 1857 on iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis).

  • $\begingroup$ Actually, Grave's work, published in "A generalization of the symbolic statement of Taylor's theorem" in the Proceedings of the Royal Irish Academy, Vol. 5, (1850-1853), p. 285-287, preceded Cayley's. $\endgroup$ – Tom Copeland Dec 16 '16 at 1:04
  • $\begingroup$ As testimony of the continuing interest in rooted trees in mathematical physics, see the table on p. 39 of "Wilsonian renormalization, differential equations and Hopf algebras" by Krajewski and Martinetti (arxiv.org/abs/0806.4309) and a companion presentation "Wilsonian renormalization and Connes-Kreimer algebras." $\endgroup$ – Tom Copeland Dec 16 '16 at 2:48
  • $\begingroup$ See also mathoverflow.net/questions/168888/… $\endgroup$ – Tom Copeland Jan 20 '17 at 21:40
  • $\begingroup$ See also Blasiak, "Combinatorial Route to Algebra: The Art of Composition & Decomposition" arxiv.org/abs/1008.4685 $\endgroup$ – Tom Copeland Apr 6 '18 at 14:56
  • $\begingroup$ Also note "Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries" by Connes and Kreimer arxiv.org/abs/hep-th/9904044 $\endgroup$ – Tom Copeland Jan 4 at 19:40

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