Jan Arnoldus Schouten

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Jan Arnoldus Schouten (August 28, 1883, Nieuwer-Amstel –January 20, 1971, Epe) was a Dutch mathematician. He was an important contributor to the development of tensor calculus and was one of the founders of the Mathematisch Centrum in Amsterdam.

Schouten was a descendant of eminent shipping magnates. He started in electrical engineering, but became fascinated by the power and subtleties of vector analysis.

Schouten received his Ph.D. degree from Delft University of Technology in 1914. His dissertation, written under the direction of J. Cardinaal, applied his "direct analysis," modelled on the vector analysis of Josiah Willard Gibbs and Oliver Heaviside to higher order tensor-like entities he called "affinors." The symmetrical subset of affinors were tensors in the physicists sense of Woldemar Voigt. Entities such as axiators, perversors, and deviators appear in this analysis. Just as vector analysis has dot products and cross products, so affinor analysis has different kinds of products for tensors of various levels. However, instead of two kinds of multiplication symbols, Schouten had at least twenty. This made the work a chore to read, although the conclusions were valid. Schouten later said in conversation with Hermann Weyl that he would "like to throttle the man who wrote this book." (Karen Reich, in her history of tensor analysis, misattributes this quote to Weyl.) Weyl did, however, say that Schouten's early book has "orgies of formalism that threaten the peace of even the technical scientist." (Space, Time Matter, p. 54). Roland Weitzenböck wrote of "the terrible book he has committed."

Schouten independently discovered in 1915 what is now known as the Levi-Civita connection. Schouten's derivation is generalized to many dimensions rather than just two, and Schouten's proofs are intrinsic rather than extrinsic, unlike Tullio Levi-Civita's. Despite this, since Schouten's article appeared almost a year after Levi-Civita's, the latter got the credit. Schouten was unaware of Levi-Civita's work because of poor journal distribution and communication during World War I. Schouten engaged in a losing priority dispute with Levi-Civita. Schouten's colleague L. E. J. Brouwer took sides against Schouten. Once Schouten became aware of Ricci's and Levi-Civita's work, he embraced their simpler and more widely accepted notation. Schouten also developed what is now known as a Kähler manifold two years before Erich Kähler.^{[citation needed]} Again he did not receive full recognition for this discovery.

Schouten's name appears in various mathematical entities and theorems, such as the Schouten tensor, the Schouten bracket and the Weyl-Schouten theorem.

He wrote Der Ricci Kalcul in 1922 surveying the field of tensor analysis.

In 1931 he wrote a treatise on tensors and differential geometry. The second volume, on applications to differential geometry, was authored by his student Dirk Jan Struik.

Schouten collaborated with Élie Cartan on two articles as well as with many other eminent mathematicians such as Kentaro Yano (with whom he co-authored three papers). Through his student and co-author Dirk Struik his work influenced many mathematicians in the United States.

In the 1950s Schouten completely rewrote and updated the German version of Ricci-Kalkül and this was translated into English as Ricci Calculus. This covers everything that Schouten considered of value in tensor analysis. This included work on Lie groups and other topics and that had been much developed since the first edition.

Later Schouten wrote Tensor Analysis for Physicists attempting to present the subtleties of various aspects of tensor calculus for mathematically inclined physicists. It included Paul Dirac's matrix calculus. He still used part of his earlier affinor terminology.

Schouten, like Weyl and Cartan, was stimulated by Albert Einstein's theory of general relativity. He co-authored a paper with Alexander Aleksandrovich Friedmann of Petersburg and another with Václav Hlavatý. He interacted with Oswald Veblen of Princeton University, and corresponded with Wolfgang Pauli on spin space. (See H. Goenner, Living Review link below.)

Schouten was an effective university administrator and leader of mathematical societies. During his tenure as professor and as institute head he was involved in various controversies with the topologist and intuitionist mathematician L. E. J. Brouwer. He was a shrewd investor as well as mathematician and successfully managed the budget of the institute and Dutch mathematical society. He hosted the International Congress of Mathematicians in Amsterdam in early 1954, and gave the opening address.

[edit] Works by Schouten

Grundlagen der Vektor- und Affinoranalysis, Leipzig: Teubner, 1914.
On the Determination of the Principle Laws of Statistical Astronomy, Amsterdam: Kirchner, 1918.
Der Ricci-Kalkül, Berlin: Julius Springer, 1924.
Einführung in die Neuen Methoden der Differentialgeometrie, 2 vols., Gröningen: Noordhoff,1935-8.
Ricci Calculus 2d edition thoroughly revised and enlarged, New York: Springer-Verlag, 1954.
With W. Van der Kulk, Pfaff's Problem and Its Generalizations, New York: Chelsea Publishing Co., 1969.
Tensor Analysis for Physicists 2d edn., New York: Dover Publications, 1989.

[edit] Works about Schouten

Albert Nijenhuis, "J A Schouten : A Master at Tensors", Nieuw archief voor wiskunde 20 (1972), 1-19.
Karin Reich, History of Tensor Analysis, [1979] transl. Boston: Birkhauser, 1994.
Dirk J. Struik, "Schouten, Levi-Civita and the Emergence of Tensor Calculus," in David Rowe and John McCleary, eds., History of Modern Mathematics, vol. 2, Boston: Academic Press, 1989. 99-105.
Dirk J. Struik, "J A Schouten and the tensor calculus," Nieuw Arch. Wisk. (3) 26 (1) (1978), 96-107.
Dirk J. Struik, [review] Die Entwicklung des Tensorkalküls. Vom absoluten Differentialkalküt zur Relativitätstheorie , Karin Reich, Historia Mathematica, vol 22, 1995, 323-326.
Albert Nijenhuis, article on Schouten in Dictionary of Scientific Biography, Charles Coulston Gillispie, ed.-in-chief, New York: Scribner, 1970-1980, 214.
Dirk van Dalen, Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer 2 vols., New York: Oxford U. Press, 2001, 2005. Discusses disputes with Brouwer, such as over publication of early paper and priority to Levi-Civita and conflict over editorial board of Compositio Mathematica.
Hubert F. M. Goenner, Living Reviews Relativity, vol 7 (2004) Ch. 9, "Mutual Influences Among Mathematicians and Physicists?"