Feedback by Amir Alexander on Opinion 54

I have now read your comments and I have to say I am very flattered. 
Thanks! It made for some pleasant readings for me, and I hope you will
manage to convince some of your colleagues that this approach is, at 
least, worth considering.

I like your way of dividing the argument into different levels, and I
was very interested in your "second" level, of promoting non-rigorous
approaches in mathematics. That, I think,  is really an angle that only 
a practicing mathematician can take. As a historian I see my job as 
descriptive rather than prescriptive, and I don't intend to tell 
mathematicians how to do their
jobs (nor would they be likely to take my advice, in any case). On the 
other hand, the evidence that non-rigorous "hand waving" approaches 
played a crucial role in the development of mathematics at various 
points is incontrovertible. I think it is important that mathematicians 
know this and take it into account in considering their research methods 
or the nature of the field.

Most mathematicians, as you say, are Platonist, but not half so much as 
most (though not all!) historians of mathematics. I suspect many 
hostorians feel a need to be "holier than the Pope" in this regard in 
order to keep the respect of practicing mathematicians and not be seen 
as woolly humanists. While a lot of very interesting contextualizing 
work has been done in the history of science in the past 20 years, the 
history of mathematics has to a large degree remained an enclave of 
"Whig history," supposedly because mathematics is essentially 
a-historical. My
book was written largely as an effort to change this, and suggest 
fruitful ways in which mathematics can be historicized without doing 
violence to its character.

I liked much of Hersh's review, especially since he seemed to accept all 
the points of my argument that I considered the most critical. He 
argreed that narratives can be identified as structuring mathematical 
systems, and also that such narratives may have analogues in the wider
historical culture. So far so good. But at the end of his review he 
simply refuses to draw any meaningful conclusions from this, a position 
that I found very puzzling.

His argument that mathematicians simply use any methodology that might 
work and then dress it up with popular contemporary rhetoric simply 
doesn't apply here. Hariot & co. weren't just playing around with 
different approaches at random, but were systematicaly developing and 
legitimizing methodologies that their predecessors would never have 
considered. They did this while promoting mathematics as a voyage of 
exploration, and importing this into the very heart of their new 
mathematical practice. If we consider the deep involvement of many of 
the mathematicians in the voyages themselves, it becomes highly unlikely 
that this "analogy" is a mere coincidence.

I definitely agree with you that the mathematical practice of problem 
solving "by any means necessary" is very much in accord with the 
exploration narrative. It is not, furthermore, a self-evident or 
timeless attitude, though it may appear so at present. Most 
mathematicians in Early Modern Europe, in fact, would not accept this 
"anything goes" attitude, but would insist that only proper methods 
(usually geometrical) are legitimate. The practical "problem solving" 
attitude that Hersh views as characterizing mathematics at all times, 
may in fact be handed down to us from that non-rigorous age of 
exploration mathematics. It should not, in any case, be taken for 
granted.

Finally, thank you for standing up for my right to "speak." Legitimacy 
is a major barrier to overcome for a non-mathematician writing in this 
field, and a mathematician's support can make all the difference.

Doron Zeilberger's Homepage