(in reference to material from Solstice, Volume XV, Number 2, Winter 2004)

Hello Sandy,

I was reviewing your very awesome studies of the Ann Arbor Downtown in your
Virtual Downtown Study document - I was wondering when that study was done
(month, year) as I am interested in citing your work in a casestudy document?

I appreciate your help,
Melissa Marks


> Ms. Sandra Lach Arlinghaus,
> I am master's student at the University of Alabama.  I am trying to find a
> definition of the the feild of mathematical geography.  So far I have
> assertained that it might include geodesy, the earth's magnetic fields, the
> creation of projections, and the location of exact points of the earth, but I
> need a definition from an official source.  Could you point me to a web site,
> journal, book, or lexicon that properly defines the field.  Unfortuanlty the
> term has become obscure in today's world, such that even my professor's in
> geography have been unable to define the discipline, and cannot point me to
> any printed source to find the definition.  Any help you can render would be
> greatly appreciated.
> Thank you for your time,
> Heath Robinson


Dear Heath Robinson,

You ask an interesting question.  One reason I find it interesting is to
consider if one can ever "properly define" any field.  Let's consider "history"
for example, as a kind of neutral example.  Suppose we take a simple answer and
say "history is the study of the past."  Then someone can counter and say, what
is the "past"...the instant I say anything, it too belongs to the past...where
is there a starting point to the soon as one defines a starting point
to the past, it too belongs to the past and there is space, belonging to the
past, between that starting point and the present, and that space is now not
included in the definition. That is a kind of "Achilles and the Tortoise"
argument.  Some might find that view to be a bit "small" and to beg the
question.  Now back to the "simple" definition of history.  Others might say
that the given definition is too broad and does not make clear what belongs to
the field of history; after all, mathematics has a history of its own, but does
that mean that mathematicians who study and write about the history of
mathematics are therefore historians (of course not, they are writing about the
history ... the study of the past...of their own discipline from their own
vantage point or frame of reference).  Others might carry this idea further and
claim that the simple definition of history is wrong and that the reason it is
wrong is that since every field and every person and every thing has a history
of its own that therefore this definition claims all for history...that is an
error in logic.  Thus, we arrive at a situation in which we can talk for a long
time about this matter and probably never achieve universal agreement.  To me,
there are at least two facts that emerge from such discussions, however.
First, simple definitions that are broad are best:  they stimulate further
discusssion and thought and they err on the side of being overly inclusive
rather than on the side of leaving someone or something out.  Second, overlap
among fields, in terms of definition, is to be expected.  The world of academic
discourse cannot be partitioned into a set mutually exclusive, yet exhaustive,
categories, even though the curricular interests of university administrators
might wish it were so.  It is in these areas of overlap that so-called
"interdisciplinary" studies take place and are often overlooked.  A related,
but not derivative, observation is that any field of endeavor has both an
analytic and a synthetic approach.  Different fields have a different balance
on which dominates (at a particular time or at any time).  When the analytic
approach dominates, it appears easier to "define" what is done, because one can
look at the small pieces and focus on some of them in order to give specificity
to broad definitions such as the one above.  When one starts with the big
picture, in the synthetic approach, then that specificity is not in focus.

With these thoughts in mind, let's return to "mathematical geography."  An
obvious definition of it is that "mathematical geography studies the
mathematical components of geographical problems and issues: it lies in the
interface between geography and mathematics."  It is of course distinct from
geographical mathematics:  the adjective modifies the noun, by our linguistic
agreements in grammar.  So, with that sort of "simple definition" one is led to
ask "what is geography" and "what is mathematics."  As long as I know, and a lot
longer than that, people have been asking "but what really is geography?"  One
can spend a lifetime debating this and never have a good answer, as was the
case with "history".  Personally, I like the simple one:  "geography is the
study of the surface of the earth."  One can make all the same arguments about
it as one can make about the simple definition of history (or of any other
field).  Naturally, geographers do things like study patterns they observe on
the earth's surface (of course defining "surface" is rather like defining
"past" (above)).  If they use geometry to help them describe or understand
these patterns then they are using a mathematical approach to geography, and
hence that work could, if they chose, be properly classified as "mathematical
geography" (but not as geographical mathematics).  It could also be classified
as "geography" and depending on what the pattern is, as the "geography as x."
For example, Skinner's work on periodic marketing in China, and its relation to
classical central place theory, might fall under any of:  "geography,"
"marketing geography," "mathematical geography," "geography of China,"
"cultural geography," "spatial analysis," and so forth.  The category one
chooses for anything is not unique.

Another simple definition of geography is the Kantian one, that makes use of
relativity, in a way:  "history is the study of time; geography is the study of
space."  AGain, one is led to question but is time the domain only of history
(and the corresponding about geography).  Again, there is an error in logic
here...the definition does not claim it to be the domain of only those
disciplines.  Of course, what is in the overlap area, in space/time, is of
critical importance not only in history and in geography, but in other fields
as well (such as physics).  Once again, the same discussion as above applies

Yet another simple definition, that I might class as being a bit "hostile" in
that it clearly begs the question is: "geography is what geographers do"
(naturally a bad "definition").  This sort of definition can be elucidated
mentioning a few key concepts that many geographers focus upon (but they are
not the only ones who do so):  scale, hierarchy, and so forth--the Education
Community has identified a set for the training of social studies
teachers...again, the same kinds of discussions can take place around
identifying any set of one is definitive and no one is unique.

In any event, mathematical geography, from the viewpoint of grammar, is the
study of the mathematical attributes of geography.  ("Mathematical" is the
adjective modifying the noun "geography.")  ONe is then left to consider "what
is geography" and "what constitutes a mathematical approach."  There are
probably more attempts to define "geography" than there are geographers and
there is more written on the topic than is probably useful.  What constitutes a
mathematical approach might be learned from a study of literature:  but, one
issue here is that the mathematics used must be correct (within the logic
system agreed upon).

Thus, I'm sure your library has much information on the "what is geography"
issue.  As to references about mathematical geography, I'll offer you a few, so
you might see some example.  To me the clearest example of interaction between
mathematics and geography, in using a mathematical tool to solve a geographical
problem, is in Eratosthenes of Alexandria's use of Euclidean geometry to measure
the circumference of the Earth.  So, I'd start there, move forward through the
work of Varenius, to consider the Konigsberg Bridge problem, the four color
theorem and the Jordan curve theorem (and their implications for geography) to
wherever your mind leads you.  You note that the term mathematical geography
has become "obscure"--out of vogue, perhaps, but certainly NOT
represents a great tradition that has endured at least from the time of the
Library at is, however, a difficult field---one must have
extensive training in both mathematics and in geography.  In that regard, it is
most like "theoretical physics"...there are not that many practitioners because
of the need to acquire so much mathematical background with accompany small
tangible reward for doing so (mathematical geography might, from some
standpoints, also be called "theoretical geography"--it is not, today, by most
of us because of some unfortunate incidents in the past (that I do not
personally remember but others still alive do)...that is, if one views "theory"
as being composed of a set of theorems deduced using some mutually agreed-upon
system of logic).

Anyway, those are a few thoughts...I don't know what your level of interest is,
but I'm happy to discuss the matter further with you.  There is also a good
deal of work posted on the website of the Institute of Mathematical Geography:
some is heavily geographical, some heavily mathematical, and some lies in the
interface that is mathematical geography. ( and other

Please let me know how I might be helpful.  Thank you for your interest!

Best wishes,

Sandra Arlinghaus.