A diagram of a Klein bottle, one of the many mathematical objects named after German mathematician Felix Klein. Image: Vierkantswortel2, via Wikimedia Commons.

I just finished reading a set of lectures the great mathematician Felix Klein delivered at the 1893 World's Columbian Exposition in Chicago. The lectures are now in the public domain, and you can download them for free here. (Unfortunately, not all the mathematical notation survived digitization, so a good amount of creative interpretation—also known as guessing—is necessary.) The appendix to the set of lectures, "The development of mathematics at the German universities," is a translation of a section Klein wrote for a publication about German universities that was prepared for the exposition. This is an excerpt from Klein's essay.

"In conclusion a few words should here be said concerning the modern development of university instruction. The principal effort has been to reduce the difficulty of mathematical study by improving the seminary arrangements and equipments. Not only have special seminary libraries been formed, but study rooms have been set aside in which these libraries are immediately accessible to the students. Collections of mathematical models and courses in drawing are calculated to disarm, in part at least, the hostility directed against the excessive abstractness of the university instruction. And while the students find everywhere inducements to specialized study, as is indeed necessary if our science is to flourish, yet the tendency has at the same time gained ground to emphasize more and more the mutual interdependence of the different special branches.

"Here the individual can accomplish but little; it seems necessary that many co-operate for the same purpose. Such considerations have led in recent years to the formation of a German mathematical association [Deutsche Mathematiker-Vereinigung, a much more lovable DMV]. The first annual report just issued (which contains a detailed report on the development of the theory of invariants) and a comprehensive catalogue of mathematical models and apparatus published at the same time indicate the direction that is here to be followed.

"With the present means of publication and the continually increasing number of new memoirs, it has become almost impossible to survey comprehensively the different branches of mathematics. Hence it is the object of the association to collect, systematize, maintain communication, in order that the work and progress of the science may not be hampered by material difficulties. Progress itself, however, remains—in mathematics even more than in other sciences—always the right and the achievement of the individual."

Reading this reminded me that the more things change, the more they stay the same. People were trying to improve mathematics education, and mathematicians felt like the pace of new research was dizzying. Many mathematicians today might disagree with the last sentence, though. Large collaborative efforts such as the Polymath projects have made a great deal of progress in several different fields, and smaller collaborations make breakthroughs much more often than lone geniuses laboring in isolation, as ubiquitous as that image is. On the other hand, if a mathematician's goal is to increase her understanding of mathematics, then perhaps progress is still an individual endeavor.