Thoughts about the Monthly in late December of 2002


The referee's objection about my "ungracious" writing (presumably in connection with several historically misleading articles in the Monthly) is nonsense.  My viewpoint is that Kalman and White as well as the editor and referees involved with that paper were merely extremely superficial.  Anyone involved could have easily found that the bibliography of the Kalman-and-White paper (click here) was incomplete by making an electronic search based on "circulant matrices."   However, electronic searches under key words are themselves superficial.  For example, circulant matrices were frequently used before they were given a name or given other names.   

The 1883 paper by A. Legoux (click here) and the 1883 paper by A. Lodge (click here) contain the main ideas of the Kalman and White paper; they appeared in print well before the Monthly was initiated or the Mathematical Association of America was founded.  Moreover, other matrices than circulant ones work just as well for the subject of the Kalman and White paper.  Namely, see the 1883 second-edition textbook by G. Dostor (click here) for cubic equations or the 1880 paper by Lauro Clariana Ricart (click here) for biquadratic equations.  Moreover, there was no mention of the 1882 paper by Arthur Cayley (click here) in which he used a 5 x 5 circulant matrix in connection with a quintic equation.  

The referees of the Kalman and White paper should have contributed to making it more instructive by pointing out how the early study of circulant matrices in the 1880's led naturally to the study of other group matrices and then to group representations.   

Even when expository articles are written by knowledgeable researchers, historical motivation  may be absent.   (Editors may  encourage omissions because of "space constraints.")  As a typical example, in the article "What is a Group Ring?" by D. S. Passman in the American Mathematical Monthly 83 (1976), pages 173-185,  the author begins with the usual historically-unmotivated definition of a group ring that can be found in algebra textbooks (e.g., see the one by Serge Lang).  There is no mention that group rings arose naturally from the older concept of group matrices.  

In my view, the Passman article could have been made more interesting for most readers of the Monthly if it had been one page longer and Passman had been permitted to start with concepts already familiar to Monthly readers.  For instance, he could have begun by pointing out that: the n x n circulant matrices over a field are closed under addition, multiplication, and scalar multiplication of matrices; circulant matrices are merely one of the many types group matrices; the definitions of addition, multiplication, and scalar multiplication for matrices give a natural structure for corresponding rings of group matrices; and a slight alteration of notation then gives the group rings.  That motivates naturally the definition of multiplication in a group ring.