Attempts to comment on these questions will clearly depend on the
samples which a respondent can draw upon - samples taken in the
one or the other cultural climate: the world is not homogenous.
And as climates change with generations, answers will also depend
on age.
Moreover, behind the questions new ones arise: what are the
criteria, the measures by which mathematicians appreciate a
colleague's field of work ? Inherent difficulty ? The stories of the
tower of underlying supplementary theories which have to mastered ?
Hearsay or fashion of the day ? The influence of a 'school' in
spreading the tastes of its founder ?
* *
*
As for (a), I am not certain what to understand by a philomath, but
I shall assume that it means a person doing mathematics as an
amateur and without formal university training. In that case, we
have to limit the period under consideration to the last two
centuries at most, since before that such formal training is
difficult to evaluate: would Tschirnhaus or de l'Hospital to be
counted as amateurs ?
The more one proceeds towards the present, the larger the
amount of technical preparation grows that is required to work on
a topic investigated by university mathematicians. Yet the
Zentralblatt lists 27 published articles on group theory between
the 1930ies and the 1950ies, written by Otto Gr"un - a man who
had not studied at a university at all and held no university
position.
Also, I well remember that until the 1960ies every mathematics
department at a German university had one of its members assigned
to handle the correspondence with amateurs: angle trisectors,
circle squarers, Fermatists ... Of course, their approaches to
such aims had to be faulty, but often there were rather ingenuous details
in the one or the other geometric construction. It seems to be
an indication of the change of times that such correspondence
appears to have dwindled away over the last thirty years, possibly
due to the detractions of interest provided by television and
the liberalization of gendered pursuits ... But maybe the fun to
play with a PC , e.g. the popularity of the 3x+1 - sequence, may
bring about new interests.
* *
*
As for (b), does HM appear before the later 18th century, the
time of A.A.Kaestner and Montucla ? The mathematics of the Ancients,
at least, for more than a thousand years appears to have been
considered less as historical, but as still sufficiently alive to
be exposited through commentaries. During the 19th century,
historical themes were studied by professors and teachers of
mathematics, both at universities and at the Gymnasium [please,
do not confuse with an American "highschool"], and by interested
laymen - Boncompagni ! They all appear as amateurs: with
exception of the laymen, their first occupation was mathematics,
for which they drew their salaray, and HM was the pleasure of
their spare time. [ Clearly, there are amateurs and amateurs, and
it may well be doubted that the lexicographical pursuit of the
first printed appearance of a mathematical term has historical
value - what is of interest is the history of an idea, a concept,
not that of a name, and in a particular language at that. The
participation of teachers from the Gymnasium seems to have come
to an end during the last decades; Joseph Ehrenfried Hofmann
here is still remembered.]
Mathematicians generally will appreciate other mathematicians,
occupying themselves with HM , as mathematicians first and as
historians second. Dehn, Hellinger, Reidemeister, van der Waerden
were appreciated as mathematicians; that they also wrote about
history was noticed only by those who themselves had an interest
in HM , and this certainly did not diminish the appreciation
awarded to them.
Quite similarly, mathematicians are likely to notice professionals
in HM (who do not work in mathematics themselves) only if they
have already developed an interest in history. That then may
profit from the professionals' erudition: their bibliographical
work and the resulting analysis of conceptual connections. [ One
of the first places to institutionalize HM seems to have been
Moritz Cantor's chair at Heidelberg one hundred years ago. But
institutes expressedly devoted to HM and HS, the history of
science, seem to have appeared here only after the last war. The
one established at Tuebingen about 1970 has been closed recently
by a not-so-insightful university president.]
And the majority of mathematicians will continue to consider HM
as a vast quarry, filled with the petrifications of former life
forms. A quarry of references and quotations, useful to dig up for
use in the preface or the afterword of a larger publication. This
is what makes the popularity of books such as those by Mrs.
Robinson's sister: that they provide anecdotes, e.g. about the
evil little dwarf at Berlin who persecuted poor Cantor, about his
"God made the natural numbers ... " or about somebody else's
"this proof is not mathematics but theology" - without ever
attempting to analyze what this 'theology' consisted of. The
persons to counteract this deplorable superficiality would be the
professionals of HM .
Yet the professionalization of HM leads to a dilemma of its own.
The old saying "what is not examined, that is not attended" would
suggest that HM be established as a field of study by itself,
with credited courses and examinations. But to read the mathematics
of the 19th century, a student has to know French and German; to
read that of the 17th and 18th century he has to know French and
Latin, and to read that of the Ancients he needs to read Greek.
With school systems decaying progressively, the average pupil
does not learn these languages (unless they happen to be his own);
so he would have come to learn them as a student. On the other
hand, while school mathematics (were it to include geometry ... )
might suffice to understand the mathematics of the centuries
before the 17th, sound mathematical training is required to
understand the mathematics of Huygens, Gregory, ... , not to
mention that of Gauss, Dirichlet, ... : a full contemporary study
of mathematics up to a master's degree. It seems impossible to
fit these requirements into the time limits given in advance by
university courses of study.
It appears that in North America another way around this
dilemma has come to be cultivated recently: credited
courses in HM for undergraduates, without knowledge of foreign
languages on the student's as well as on the instructor's side,
and without knowledge of the mathematical background either.
Fifty years ago, Carl Boyer published a thought-inspiring book on
the history of the calculus and its conceptual development - a
splendid survey, one would have hoped to lead to further,
detailed and systematic studies of the medieval, renaissance and
early baroque sources. Instead, about thirty years later
'textbooks' on HM began to appear, among them one by Boyer which
occasionally has been mentioned on this list. Nicely printed books
on quality paper, no doubt, brief surveys which, read during the
idle hours of a mathematician, will give him a first glance of
what happened in his field. They will provide the background for
intelligent conversation at the mathematics department's Christmas
party, are "invitations" to look at HM : but the actual act of
looking oneself they cannot replace. Scholarly presentations of
(sections of) HM they are not. Some of them even seem to have so
widely appealing a mathematical content that they can be succesfully
marketed to the audience addressed by Penguins.
Courses based on such books will usually not be followed by more
thorough ones; their students will have touched the surface, but
will neither have the occasion nor the linguistic means to proceed
from third-hand accounts to the real matter. Yet from the
mathematicians' point of view, the most pernicious effect of
these courses is caused by the credits earned for attending them:
they will draw the mentally disadvantaged, draw them to stories about
Galois and away from the 'tedious computation' of Galois groups,
tell them about Kronecker's influence in university politics, but
leave them to believe that his only remembered mathematical
achievement was the invention of the 'Kronecker delta".
W.F.