Dear Mr Tragesser,
Many thanks for your clarification -- your title question (*) seems
to be (at least to me) sufficiently clear so as not to load the list
with posts that in the end are not about historical matters.
As Walter Felscher remarked in one of his latest sharp contributions
to our forum, "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."
No matter the weather, when we *attempt* to spell out the so-called
RULES OF THUMB by which historians of mathematics recognize some BIT
OF CULTURE as being mathematical (or of mathematical flavour), we are
confronted by the necessity of 'framing' our terms. In any case, in
the following, do not expect that your thought-provoking question meets
any definite answer.
Obviously enough, the scope of the discussion -- the so-often-forgotten
framework -- will call upon the following enquiries:
- How do we take the word "history"?
- What do we mean by "mathematics"?
- What does the term "history of mathematics" signify?
- How do we handle the concept of "culture"?
These are hard questions, and reactions to them will clearly depend on
WHO drops the answer (naturally, I am assuming all responses as coming
from scholars). As you are well aware of, it is plainly not true that
there is just one royal philosophical path to history - the same goes
to mathematics.
In his well-known "History of Mathematics" (1923), David Eugene Smith
remarks:
"If we consider history as a narration of recorded events,
we shall have one course laid out for us; but if we look
upon it as a relation of incidents which probably happened
even before the advent of the human race, then our course
is a different one."
"If" clauses... "If we consider history as ... if we look upon it..."
But not all in Smith is in conditional form. From the beginning, Smith
definitely shows his 'philosophical stripes'... This is quite clear,
for instance, when he states: "With the advent of the human race there
developed an opportunity for mathematics to show itself more consciously."
Now ... what are these RULES OF THUMB by which Smith recognizes
mathematical concepts? ... Although he does not spell out these rules,
he provides sufficient information to the point. And in doing so,
Smith could not help escaping from the _philosophical web_ of the
"meaning" of mathematics:
"When we attempt to define 'mathematics' we find ourselves
encircled by unexpected limitations and these limitations are
still more in evidence when we change the term to 'elementary
mathematics'. If mathematics means that 'abstract science
which investigates deductively the conclusions implicit in
the elementary conceptions and numerical relations', as the
Oxford Dictionary defines it, then the history of mathematics
cannot, strictly speaking, go back much earlier than the time
of Thales (c. 600 B.C.), a relatively modern writer if we
consider the antiquity of the race. Such a limitation, however,
would not be a satisfactory one, for it would withdraw from our
consideration those early steps in the development of the
science..."
In accordance with his philosophical standpoint, Smith discards the
"niceties of definition", and takes a broader scope "seeking to tell
the story of the genesis of mathematics even before the period in
which science, as define above, began to exist". In doing so, he leads
the reader back "not only to the days when the human race was young,
but to ages immediately antedating its appearance upon the earth, and
even farther."
It goes without saying that Smith's philosophical attitude might bother
many modern scholars, but surely not all of them. I happen to believe
-- but please, do not press me to provide solid evidence -- that there
are many philosophers that subscribe Spencer's ideas of the properties
of space:
"It is impossible to imagine how the marvellous space-relations
discovered by the Geometry of Position came into existence. The
consciousness that without origin or cause, Infinite space has
ever existed and must ever exist, produces in me a feeling from
which I shrink."
I leave the time-relations and the 'thin' (atomic?) structure of time
for another occasion.
Clearly enough, Smith was an outstanding historian of mathematics that
saw mathematics everywhere. His paragraph "Cosmic Figures" is a case
in point:
"When we consider the birth of a solar system like ours,
and point a telescope at a nebula in the actual throes
of such a stupendous effort, we are impressed by the fact
that one of the great cosmic forms is the spiral, a curve
not scientifically studied until late in the evolution of
human intelligence."
He shows similar enthusiasm in his "Advent of Life":
"Passing from the preorganic era of the earth's existence,
when mathematics was manifest in the spirals of the nebulas,
in the courses of the planets and the comets, and in the
crystallizing habits of the minerals, all of which give
meaning to the statement, often attributed to Plato, that
'God eternally geometrizes' we find new interest in the
history of mathematics with the advent of life upon our
planet."
Smith believes that the feasibility of the recognition of mathematical
forms did not have to await the advent of the human race, since it can
be acknowledged also within plant life (and goes on to quote several
examples, such as phillotaxy or leaf arrangement, structure of the
pineapple, Golden Section of ferns, etc.).
Besides, Smith does not hesitate about recalling the possibility of
the recognition of such mathematical concepts as form, number, and
measure with the advent of animal life (animals that know 'cardinality',
spiders and regular polygons and webs, bees and maxima and minima, ...).
"No beast is so stupid as not to know that a straight line is the
shortest path between two points, and few birds fail to observe the
principle of symmetry in the structure of a nest."
In discussing the origins of mathematics, other authors have considered
*animal psychology* as a source which might suggest something about the
way in which mathematics arouse as a *human activity*. In his fine book
"The History of Mathematics: A Brief Course" (1997), Roger Cooke says
that:
"Certain ways of coping with the problems of life that may
be called 'mathematical' are shared by human beings and other
mammals and birds, namely, distinguishing numbers and shape,
the fundamental elements of arithmetic and geometry."
And after referring to experiments carried out by Koehler [birds that
associate numbers ] and Pavlov [dogs that distinguish ellipses of very
small eccentricity from circles], Cooke remarks that his aim in telling
these stories is that:
"... certain aspects of reality that we may call arithmetical
or geometrical must be dealt with by all living organisms.
Organisms possessing at least rudimentary cognitive abilities
are therefore capable of learning to use these properties
of the world about them. In particular, the perceptual ability
needed to create mathematical concepts is not uniquely human."
Cooke's textbook, which (curiously) has not been mentioned much on
this list, begins with an entire chapter on the origins of mathematics
[pp. 5-23] which I recommend.
---------
Ivor Grattan-Guinness, in his latest book "The Norton History of the
Mathematical Sciences: The Rainbow of Mathematics" (US edition 1998),
notes that his intellectual enterprise differs significantly from
many others histories of mathematics in several respects. One of these
is the following:
"I take _the word 'history'_ to relate to the question 'What
happened in the past?'; by contrast, mathematicians (and
scientists in general, and even a distressing number of
historians) take history to mean 'How did we get here?' The
difference between these two questions is worth pondering.
Answers to the second one draw _only_ on those parts of the
past that have led to our present situation; while a perfectly
respectable form of research, they can give quite mistaken
impressions about the aims and purposes of historical figures,
and the priorities they saw in their own work."
I agree - the difference between these two questions is worth pondering,
but whereas Grattan-Guinness criticizes the second question, he does
not seem to object anything to "What happened in the past?". Obviously
enough, though, I would merely remark that in *most* cases "what
happened in the past, really?" simply is unanswerable (in any field)!
And in *all* cases our understanding of what we think in fact did
occur in the past is subjective.
---------
The important part of Robert Tragesser's question
"What are the "rules of thumb" by which the historian of
mathematics recognizes some bit of culture as being mathematical?"
is the word "culture". Robert (if I may be informal) has wisely tried
to direct us in a "cultural" direction. Listmember Marcia Ascher would
say, I think, "Let us take a step toward a global, multicultural view
of mathematics."
And the important word here is "multicultural", in an attempt to
"introduce the mathematical ideas of people who have generally been
excluded from discussions of mathematics." The concept of "culture" is
not easy to handle, and has no universally agreed-upon definition. A
widely accepted "rule of thumb" about the cultural view of mathematics
is that we should strive to understand the mathematical notions of
others in their time-space context. This might be difficult to grasp,
and we must keep in mind that in order to fully appreciate their
*concepts* we are limited by our own mathematical and cultural
*pre-concepts*.
Greetings from sleepy Montevideo,
Julio Gonzalez Cabillon