A revolution in mathematics? What really happened a century ago and why it matters today (Q2883508)

The article under review is divided into four sections. In Section~1, ``Revolution'', the author briefly describes changes that occurred in mathematics between 1890 and 1930. The two salient features implemented during this time were, first, ``precise definitions'', i.e., definitions that completely support mathematical reasoning without any reference whatsoever to pre-mathematical notions or intuitions, and, second, ``logically complete proofs'', i.e., proofs that do not fill gaps by reference to pre-mathematical intuitions or facts. Section~2, ``Obscurity'', is about an observation the author made, namely, that math educators still base their work on pre-modern, nineteenth-century mathematics. His explanation is that mathematicians failed to produce what he calls ``proxies'', i.e., certain outcomes that, while resulting from a revolutionary change in methodology (which in itself may be too complex to be communicated easily), draw and focus the attention. An example of what the author has in mind would be man's descendence from apes as a proxy for evolution theory. In Section~3, ``Core at risk'', the author expresses his concern that funding will be funneled away from university mathematics to math education. This, he believes, happens basically for two reasons, public ``pressure for more highly visible research'' and educators being ``better organized, more coherent, and far more powerful politically''. Section~4, ``Solutions for education'', repeats a suggestion the author has made earlier [New ICMI Stud.\ Ser.\ 15, 231--257 (2012; Zbl 1247.97017)], namely, to conceive of the methods of post-revolutionary mathematics ``as profoundly rich resources rather than alien threats''. NEWLINENEWLINENEWLINE Remarks. While the article provides for an interesting read and touches on some important topics not much discussed elsewhere (e.g., the role of axioms as being opaque to non-users but highly effective for users), it's still a popular piece: you want to speak about the ``osteoporosis of the skeleton that supports the muscles of science'' when donors with deep pockets are sitting in the first row of your audience. It's also an opinion piece and as such exhibits the shortcomings you would expect: the author doesn't summarize what is known and what new insights he is going to add, nor does he go beyond stating opinions based on anecdotal evidence. For a fuller account of the author's reasoning the reader has to turn to his unpublished manuscript [``Contributions to a science of contemporary mathematics'', \url{http://www.math.vt.edu/people/quinn/history\_nature/nature0.pdf}]. Even though this can be a brief review only, I feel two claims shouldn't go unchecked. First, the author declares constructive mathematics to be dead (p.~35), a common but, I think, nonetheless blatantly wrong statement to make. Second, the author's key notion of reliability doesn't mesh well with empirical findings (see, e.g., [\textit{C. Geist} et al., Texts Philos. 11, 155--178 (2010; Zbl 1209.01112)]), which is somewhat ironic since the author himself helped to spawn such empirical studies with a paper he co-authored with \textit{A.~Jaffe} [Bull. Am. Math. Soc., New Ser. 29, No.~1, 1--13 (1993; Zbl 0780.00001)].