(in reference to material from

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

Quoting gavynn@comcast.net:

> 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

ONE ANSWER ALREADY E-MAILED TO THE AUTHOR OF THE NOTE ABOVE; OTHERS...??

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 past...as 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

here.

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 concepts...no 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 obscure...it

represents a great tradition that has endured at least from the time
of the

Library at Alexandria...it 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. (http://www.imagenet.org
and other

URLs).

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

Best wishes,

Sandra Arlinghaus.