Reuben Hersh has written often of the history, culture, and deeper nature of mathematics. And he is, famously, a NON-Platonist... one who believes mathematics is more a by-product of the human mind than a real extant part of the physical Universe. One of his classic works (with Philip Davis), that most of you have likely read, was "The Mathematical Experience," which Martin Gardner reviewed quite critically back in 1981. I enjoy reading Hersh, and I've been re-reading this particular volume on its 30th anniversary, but having said that, his writing sometimes seems to skim the surface of the material he is tackling. The content bounces back-and-forth between regurgitation of standard pedagogical material and more interesting, but not always convincing, arguments of deeper philosophy. I'm sometimes reminded of the old Wendy's commercial, "Where's the beef?" in reacting to certain subjects broached in this book that don't seem fleshed out as fully as they deserve (the second half of book though is richer than the first half, and Hersh provides plenty of good references for "further study"). Perhaps I just miss the nuance of Hersh's stance on some matters, but I usually find Gardner's arguments more persuasive and articulate. Having said that, more and more respected mathematicians these days seem to be moving toward the minority non-Platonist stance that Hersh has long expounded, so the debate is hardly settled (...indeed, it is probably more UNsettled than ever!).
I mention all of this only because of recently stumbling across this informal response, I'd not seen before, from Hersh to Gardner's original review, and it makes for interesting reading:
http://cs.nyu.edu/pipermail/fom/1997-November/000128.html
The below page links to an Edge interview where Hersh further spells out his notions:
http://edge.org/3rd_culture/hersh/hersh_p1.html
"The Mathematical Experience" remains a classic mathematical opus, with a great breadth of math subject matter (and there is a newer, updated version which I don't own, so not sure how much it differs from the original), and I'm definitely finding it worth a re-read decades later.
1 comment:
Phylosophical views of Gardner and Hersh were opposite, as you wrote. But I am not sure that non-Platonist mathematicians are growing in percentage: they are just noisier than before :-)
(disclaimer: I am Platonist, and I think that mathematics is an empirical science: you do not invent theorems :-) )
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