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\centerline{MATH 110 -- Mathematics Through Time}
\centerline{Research Papers}
\centerline{October 27, 2013}
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\noindent
{\it General Information}
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As announced in the course syllabus, one of the assignments
for the course this semester will be research.  You
will be working on this individually and
the goals will be to prepare a paper of somewhere between
5 and 10 pages addressing a topic related to the historical
questions we are studying in the course.
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\noindent
{\it Schedule and Deadlines}
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\item{$\bullet$}  {\it On or before Friday, November 1} -- Inform me
{\it by email} which of the topics you want to work on.  Since each project 
area has several different directions that might be pursued (see the 
descriptions below), write up a short paragraph giving a description 
of the aspect(s) you would like to work on.  I hope to have a good mix
of different topics represented, so please be open to suggestions 
for changes if it looks as though that is not developing.
\item{$\bullet$}  {\it Papers will be due no later than 5:00pm on 
Monday, November 25}. 

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\noindent
{\it Some Guidelines}
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The goals of this assignment
are for you to collect information about your topic from the
various sources you find, for you to learn and assess that
information, and then for you to present your analysis of that
information.  The evaluation of your papers will be based on 
how well you have addressed the following guidelines and 
expectations:
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\item{$\bullet$}  You should distill the results of 
your investigations into a central argument.  A good 
research paper of this kind should be more than just a compilation of information 
from all the different sources you
consulted.  It should clearly show that you have thought independently
about the information you found, that you have weighed the evidence
for the various claims that were made in your sources, and that you have a
central theme or argument about your topic that you want to present.  It
is certainly permissible to say you disagree with points of view presented
in some sources, if you can explain why you think that and back up your
opinions with appropriate evidence.
\item{$\bullet$}  The paper should be {\it well organized} and the writing
should give the reader a clear indication where you are heading with your
central argument at all times.
\item{$\bullet$}  Pay special attention to the first few paragraphs that
will serve as an introduction.  Catch the reader's
attention, explain the significance of the topic or theme you will
discuss.  Say what you will do in general terms, without going into all the
details from the start.
\item{$\bullet$}  Also pay special attention to the final few
paragraphs, which will serve as a conclusion for your
paper.  Don't overstate the
importance of your findings, and be honest if there are limitations.
You might discuss how your investigations
could be continued in further research.
\item{$\bullet$}  Give proper credit to sources you consulted that contributed
to your ideas about the questions you studied.  Use footnotes or endnotes to identify
direct quotations from your sources, and also to indicate the sources
that contribute to specific points you are making even when you are not using 
direct quotations.  I don't care so much
about the exact form of citations that you use as I do that you are following some 
standard convention with care.  
\item{$\bullet$} In a {\it References} section at the end, include all books, articles, and
websites you used in the preparation of the work.  For books, give the author(s), title, publisher,
place and year of publication.  For articles, give the author(s), title, journal name, volume, 
year, and pages.  For any websites, give the full URL, the author (if that can be determined), 
and the date you consulted it.
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\item{$\bullet$}  Be clear, concise, and correct in your writing.
Aim for {\it no typos, misspellings, or grammatical problems}.  But even
more importantly, each paragraph should have a clearly evident purpose
in relation to your main argument.
\item{$\bullet$}  Use figures, graphs, etc. sparingly in the main
text.  (If you want to include more of these, that can be done in an additional
Appendix section at the end.)
\item{$\bullet$}  Proofread your work carefully and have an ``impartial''
reader or readers look at it and give you comments.  This can be other students
in the class or me.  Be prepared and willing to {\it revise} your work
based on the comments you get.  Of course, this means that the writing
must not be put off until the evening of November 24(!)  Be sure you
get started early enough so that the input can be put to productive use.
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\noindent
{\it Project Ideas}
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{\it Topic 1 -- Eleanor Robson's Reinterpretation of Plimpton 322}
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As we mentioned briefly in class, Eleanor Robson is a current scholar of old 
Babylonian mathematics whose work has gone a long way toward
putting that mathematics into a convincing historical and cultural
context.  She has provided, in particular, an
interesting and novel ``take'' on what Plimpton 322 might have meant
to its creators.  See the link from our course homepage to an 
article she published on this in 2002 in the {\it American Mathematical
Monthly}.  If you were intrigued by the Old Babylonian Mathematics
that we studied, this would be a very good topic to consider.
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{\it Topic Area 2 -- Further consideration of Euclid's Elements}
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There is, of course, much more to say about Euclid.  In particular, since we looked
only at Book I, there are many different topics that could be formed by 
considering the contents of the remaining 12 books, II - XIII.  For instance,
students who have taken MATH 243 (the Algebraic Structures course), or others
interested in number theory, will see that this side of mathematics is developed in 
Books VII, VIII, and IX, through properties of the prime numbers, the ``Euclidean
algorithm'' for the greatest common divisor, and many other related topics. 

Along a different line of inquiry, some historians have proposed that the 
overall goal of the {\it Elements} was to build up to constructions of the 
five {\it regular polyhedra} or {\it Platonic solids} -- the regular tetrahedron, 
the cube, the regular octahedron, and the regular dodecahedron and icosahedron.
Euclid devotes Book XIII primarily to these figures and finishes with an 
argument that these five are the {\it only} regular polyhedra.  The article by
Bill Casselman on the course web site (from the Notices of the American Mathematical
Society) is a good place to start if you are interested in that side of this.
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{\it Topic Area 3 -- Jesuit Mathematicians, Astronomers, and Missionaries}
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In our discussion of non-Euclidean geometry, we will mention
some of the contributions of Girolamo Saccheri, S.J., who tried to prove
Euclid's fifth Postulate from the other four Postulates.   A number of other Jesuit
mathematicians, astronomers, and missionaries have been involved in
various ways in other mathematical and historical topics we have touched on.

For instance, Christopher Clavius, S.J. was a geometer and astronomer
from the early years of the Society of Jesus who played a major role in
mathematics education in Europe and prepared an edition of Euclid suitable
for use as a school textbook.  Clavius was also a key figure in the
development of our current Gregorian calendar system.

Matteo Ricci, S.J. (a student of Clavius) was one of the early Jesuit missionaries
in China.  He facilitated a rich scientific, calendrical, mathematical, and religious 
exchange between Europe and the last of the Ming dynasty emperors of China.
(He was also well-known for his techniques of committing information to memory
and a biography called {\it The Memory Palace of Matteo Ricci} by Jonathan Spence
is a fascinating, though non-chronological, portrait of Ricci and his time in
China.)  Those contacts continued after Ricci's death through contacts between 
other Jesuits including  Johann Schreck, S.J. and Johann Adam Schall von Bell, S.J.
and emperors from the Qing dynasty.

Before entering China, Ricci also worked briefly in the Jesuit mission in Goa in
south India, where other Jesuit mathematicians had contacts with Indian mathematicians.
It was this connection that George G. Joseph posits as plausible evidence for
transmission of Indian precursors of ideas from calculus from Kerala to Europe in the
early 1600's in {\it The Crest of the Peacock} and other subsequent works.
David Mumford's somewhat more technical review of the book {\it Mathematics in India}
(authored by Kim Plofker) in a recent issue of the {\it Notices of the American Mathematical
Society} may also be of interest, although it does not deal with the suggested
``Jesuit connection.''   {\it The Mathematics
of Egypt, Mesopotamia, China, India, and Islam:  A Sourcebook} is
also an excellent resource for these topics.  Caution, though: because of the level of
the mathematics involved in this aspect of the topic (infinite series, Taylor
series of functions, and so on), I would recommend
a possible project about the suggested precursors of calculus topics
in Kerala mathematics only for students who have already completed
MATH 136 or 132 or the equivalent.

There is material for several distinct paper topics here -- three different ones at
least.  The first step would be to narrow down the focus to something suitable for
your paper.  If you are interested in pursuing a part of this, I would be happy to discuss the
options.
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{\it Topic Area 4 -- Mathematics, the arts, logic and other connections}
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We will barely scratched the surface of the art of M.C. Escher in
our discussion of his non-Euclidean tesselations of the plane.  His symmetry drawings
were just one facet of his work, although many other aspects were
also mathematically-inspired.  One nice project topic would be to investigate
some of the other mathematical aspects of Escher's work and the extent
to which he was aware of and in contact with the work of professional
mathematicians.  To what extent was he really ``doing mathematics''
or ``thinking mathematically'' in
 his art?  Several books and articles of Doris Schattschneider
would be the best places to start for this.  Her book {\it Visions of
Symmetry} is a great source for a discussion of Escher's
symmetry notebooks.  Her article {\it ``The Mathematical
Side of M.C. Escher''} from a recent issue of the {\it Notices of the
American Mathematical Society} is also very interesting.

I first learned about many of the ideas connecting Escher, Bach, mathematics,
and so on from another wonderful book called {\it G\"odel, Escher, Bach} by Douglas
Hofstadter.  We have not discussed who Kurt G\"odel was in this course.
Let me just say briefly that he was one of the great {\it logicians} of the 20th
century.  One of his most important contributions was a proof of a result called 
the Incompleteness Theorem.   This work has
had a tremendous influence on the way some people (mainly philosophers)
think about mathematics.  The Incompleteness Theorem says, in essence, that
in {\it every axiomatic mathematical system}  that is ``powerful'' enough
to describe the basic properties of addition and multiplication in
the integers, there are  {\it true statements that cannot be proved
to be true, and false statements that cannot be proved to be false}.  
(In a sense, you might think of this as a refutation of the whole idea of 
Euclidean mathematics -- any system like Euclid's axiomatic development 
of number theory in the later books of the {\it Elements} is necessarily 
incomplete in the sense that deductive proof
cannot uncover all the true statements about numbers.)   The
idea he used to derive this rather unexpected and unpleasant (at least
for mathematicians) result was a clever use of a sort of {\it self-reference}.
In intuitive terms, he showed there were ways to ``encode'' mathematical
statements and proofs as integers, and hence that it is possible to
work paradoxical statements like ``this statement is false''  (think about that one!)
right into the ``fabric'' of the theory of numbers.
Hofstadter discusses this in a very lively style, and draws connections between    
G\"odel's proof and some Escher prints and Bach music.  Hofstadter also proposes 
that perhaps this sort of self-reference might be the key to creating {\it artificial 
intelligence} -- a hot topic back in the 1980's when this book was written, and
an idea whose time may be coming again.  It's a very ``heady''
mixture, but understanding it is definitely worth the effort.
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{\it Topic Area 5 -- More on Islamic Mathematics}
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In class we will discuss some of George G. Joseph's ideas about the
ways Islamic mathematicians united and developed the legacies of Greek (and Indian)
mathematics during Europe's ``Dark Ages'' and then were instrumental in
the reintroduction of those ideas to Europe during the Renaissance period.
However, due to limitations of time, we will not be able to touch on
a number of important parts of this historical arc.
There are several different aspects of this story that would make for
very good final project topics.  First, the establishment of the library and
translation center known as the Bait al-Hikma (``House of Wisdom'')
in Baghdad during the rule of the Abassid caliphs was instrumental in
preserving the classic Greek texts for later scholars.  There is a recent
book called {\it The House of Wisdom} by Jonathan Lyons that deals with this.
A slightly older book called {\it Aladdin's Lamp: How Greek Science Came to 
Europe Through the Islamic World} by John Freely covers a lot
of the same ground.  Both of these books are quite complete, but somewhat dry,
discussions of who did what and when they did it.

In addition to the scholarly center at Baghdad, the Islamic civilization
of Al Andalus in present-day Spain became another great center of learning
under the Umayyad caliphs and their successors.
In particular, many of the translations of Greek classics
into Latin (which hastened their reintroduction into Europe) were done
in the city of Toledo in a rather amazing collaboration of Islamic and Christian
scholars in the 1100's and 1200's.  The two books mentioned above in
this area deal with some of this story.
Another book that deals with the context
for this contact and with the story of Al Andalus in general is
{\it The Ornament of the World} by Maria Rosa Menocal.  The BBC TV series
``When the Moors Ruled in Spain'' from 2005 is another (but inevitably slightly
superficial) treatment.  This aspect of this group of topics would be
great for anyone who wants to learn more about the culture that produced
the Alhambra in Granada and its mathematical accomplishments.

{\it The Mathematics
of Egypt, Mesopotamia, China, India, and Islam:  A Sourcebook} is
also an excellent resource for the actual mathematics connected with these topics.
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{\it Topic Area 6 -- More on Chinese Mathematics}
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In some cases, Chinese mathematicians
discovered results and techniques that were only developed much
later in Europe.  Moreover, there were more extensive trade contacts
between China and India, and between India and Europe than has sometimes
been realized.  So the whole development of our modern
understanding of mathematics almost certainly involves more exchanges
of ideas than than traditional histories often acknowledge.  There are 
a number of slightly more technical mathematical topics from Chinese history 
that would be interesting for people who might want to explore this area.  
Two ideas in
particular that might form the basis for good projects are
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\item{1.}  the result called the {\it Chinese Remainder Theorem} concerning
simultaneous congruences modulo several different integers, and
\item{2.}  Chinese techniques for solving simultaneous systems of linear
equations that anticipate the modern ideas of matrix form for systems
and Gaussian elimination 
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\noindent
 Either one of these (as well as several
other possible mathematical topics)  would be acceptable as
a paper by itself.  So there are again several different possible
topics here.  {\it The Crest of the Peacock} has good discussions
of both of these, so that is probably the best place to start.  {\it The Mathematics
of Egypt, Mesopotamia, China, India, and Islam:  A Sourcebook} is
also an excellent resource for these topics.  In
each case, the idea would be to explain the cultural and historical
context for the mathematics, then work through the details of what
the Chinese actually did, together with its relation to modern
methods.
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{\it Topic Area 7 --  A Topic of Your Choice}
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If there is another topic you would prefer to work on, I am open
to suggestions.  If you want to propose a topic
of your own, you {\it must get my approval} before starting to
work.  For the November 1 deadline above, write up a short
description of the topic or questions you want to look at and
how you want to try to address them.  I will let you know
as soon as possible whether you have my approval to proceed.
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