Mathematical knowledge is a dynamic entity that often goes down improper or uncertain paths before the self-correcting nature of “many eyes and many minds” asserts itself. In fact, the “new” development tactic of “crowdsourcing” is not new to the mathematician or scientist. As Krantz points out, centuries ago when there were few mathematicians, results were tightly held and publication brought the risk of losing your livelihood. However, starting several centuries ago, publication became the norm. From then on it was a rare paper that did not issue some form of challenge for further work. That trend has been expanded to include preprints, where raw work is “immediately” published before it is refined. This was crowdsourcing before it had that name.
As Krantz moves through the history of the concept of a proof you come to the hard realization that the only weakness in the absolute certainty of mathematics is in the quality of the proof. In fact, I came to the conclusion that it is one place where philosophy and mathematics align, for philosophy often deals with the “what can we know and how can we know that we know it?” issues.
In many instances mathematical proofs have become so long and complicated that one can legitimately ask questions as to whether the proof itself is understandable. Furthermore, there are proofs in which some or all of the results were obtained or checked by computer and there is the inherent uncertainty of that tactic as well. It is a fascinating area of mathematics, and one that must be resolved, for in mathematics, the proof is the entire game. Weaken the certainty of proof and progress is at risk.
Krantz’s book is entertaining, can be read by the early undergraduate and puts forward some serious issues. There are few math books that are useful and valuable reading for all mathematicians, but this is one of them.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.