Dirichlet’s principle

This morning, 13 April 2015, the Financial Times has a reference to mathematical method in one of its articles on economic affairs.

Briefmarke 2013-05 Gustav Lejeune Dirichlet.png.1295844

The author, Wolfgang Munchau, makes in my view a valid point about economics, but gets his mathematical reference wrong. He quotes Richard Dedekind as writing that mathematics should shift “to put thoughts in the place of calculations”, and argues (truly, in my view) that economists today must learn to do the same.

The phrase Munchau quotes came not from Dedekind, and not from the end of the 19th century, as Munchau also has it, but from Johann Dirichlet in 1851. Dirichlet was the successor to the great Carl Gauss as professor of mathematics at Göttingen, in Germany, then and until 1933 the world centre of mathematics. (Apparently Dirichlet’s name should properly be pronounced like this, though I’ve always heard it pronounced DIHreeshlay).

Henrik Kragh Sørensen explains the development:

“From the mid-18th century to the mid-19th century, the style of mathematical research in analysis underwent changes from a formula-centered approach epitomized by L. Euler (1707–1783) to a concept-centered style presented in works of G. P. L. Dirichlet (1805–1859) and G. F. B. Riemann (1826–1866).

“The transition manifested itself in multiple aspects of the mathematical enterprise including notations, questions, results, methods, and techniques. It was felt by the active and creative mathematicians of the nineteenth century who spotted a difference between the computational machinery associated with the formula-centered approach and the decidedly mental analysis belonging to the concept-centered approach.

“We find this distinction seized upon for instance in Dirichlet’s obituary of C. G. J. Jacobi (1804–1851) where Dirichlet noticed:

the constantly increasing tendency of the new analysis to put thoughts in the place of calculations…”

Sørensen notes that Dirichlet’s thought was developed by Hilbert and Minkowski.

Hilbert: “I have tried to avoid the large computational apparatus of Kummer such that also here Riemann’s principle should be observed, according to which one should conquer proofs not by computations but solely through thoughts”.

Minkowski formulated what he called the “second Dirichlet principle” as “problems should be conquered with a minimum of blind calculations and a maximum of enlightening thoughts”.

You’ll find Hilbert’s and Minkowski’s statements of that principle, in different translations, stuck up outside 2B6. The idea was further and far-reachingly developed by Emmy Noether. The credit for first stating it should however go to Dirichlet.

Sørensen: http://www.sciencedirect.com/science/article/pii/S0315086004000916

For Dirichlet’s “first” principle, click here. Or possibly Minkowski had this one in mind.