Karl Menger

“This result should make geometers realize that (contrary to the traditional view) the fundamental notion of curvature does not depend on coordinates, equations, parametrizations, or differentiability assumptions. The essence of curvature lies in the general notion of convex metric space and a quadruple of points in such a space”

Taken from AMS.

Menger was a philosopher and mathematician. He pioneered work with less concern for variables, and names; and more concern for forms, and deep structures. Nowadays that computers are ubiquitous, his prescient notions are more and more relevant. I am reading his "Calculus: A Modern Approach," and sicm (Structure and Interpretation of Classical Mechanics). I see similarities; Menger was aware of the need, and the ways to automatize mathematics. He wrote:

"Automation is impossible ``in the classical notation.'' In the automatized calculus one obtains specific results from general theorems by replacing function variables by the designation of specific functions, and numerical variables by the designations of specific numbers."

Reading the preface by Bert Schweizer, and Abe Sklar, to the 2007 Dover Edition, I can see that Menger was above the heads of the American mathematical establishment. The establishment ended up following antiquated notions of variable; to the detriment of countless engineers, and technicians. Menger was able and willing to train the working class at the Illinois Institute of Technology, in Chicago, but maybe he thought too much, for the taste of the anti-intellectual bosses of industry, and their stooges in the House and Senate of the US government, that set policy in the fifties.

Better late than never, nowadays more and more American students are learning Calculus from books like that of Spivak.

I am teaching Mechanics this term, with the equally enlightened book by Sussman, and Wisdom quoted above.

From the ASM paper we can further read:

"That his book was ignored saddened Menger’s later years. When Menger addressed the foundations of dimension theory, topology, projective n-space, or differential geometry, attention was paid by the best mathematical minds of his generation: Hahn, Brouwer, von Neumann, Gödel. The failure of the calculus endeavor strained his relations with the mathematical community.

Menger has been described as a fiery personality. As a junior faculty member at IIT in the 1960s, I found him gracious, charming, and vivacious. Menger was solicitous of students. From his early days in Vienna onward he invited students and faculty to his home. In Chicago it included a tour of his decorative tile collection, which lined the walls of his living room. And he sometimes invited doctoral students for early morning mathematical walks along Lake Michigan.

His office was a showplace of chaos, the desktop covered with a turbulent sea of papers. He knew the exact position of each scrap. On the telephone he could instruct a secretary exactly how to locate what he needed. Once, in his absence, a new secretary undertook to “make order”, making little stacks on his desk. Upon his return, discovering the disaster, he nearly wept, because “Now I don’t know where anything is.

”Menger liked America. He even liked the Marx Brothers. I once met him emerging from the Clark Theater in Chicago, where “A Night at the Opera” was playing. Still suffused with laughter, all he could say was, “Funny! Funny!”

Though born into a family with ties to the Austrian crown, Menger did not like establishments. His work shows that he could shed traditional ways as called for. He was a peerless mathematician and an independent and original spirit.

Seymour Kass"