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\centerline{MONT 107Q -- Thinking About Mathematics}
\centerline{Final Projects}
\centerline{March 6, 2017}
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\noindent
{\it General Information}
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As announced in the course syllabus, one of the assignments
for the seminar this semester will be a final project.  You
will be working on this project in {\it teams of 2} and
the goals will be to prepare a roughly 15-page research paper and
an oral presentation of approximately 25 minutes to the class
on your project.  
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{\it Schedule and Deadlines}
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\item{$\bullet$}  {\bf On or before Friday, March 17} -- Inform me
{\it by email} which general topic you want to work on and with whom
you will be working.  Since each project area has several different
directions that might be pursued (see the descriptions below), 
write up a paragraph giving a description of the aspect(s) 
you would like to work on.  If you need assistance in forming ``teams,'' I will
be happy to help with that.  Ideally, each group will work
on a different topic, although as noted below the areas
are large enough that if more than one group wants to try
that, there will be ways to ``split up'' the topic into
several parts.   We can discuss the possibilities before this is due if you would
like.
\item{$\bullet$}  {\bf Friday, April 7} -- Each group will submit, 
{\it by email}, a bibliography of the sources to be used for your
project.  You should identify {\it at least six books, articles, or 
web sites, including no more than two web sites}, that will be relevant.
For each of your sources, write up a short paragraph giving a rough description
of how that source relates to your main topic, what kind of information you 
will take from it, and how you will be using it (including a preliminary
estimate of how reliable you think the information there is).  
The project descriptions below contain some first places to look, but 
you should plan to spend some time searching for additional sources.  This
annotated bibliography will count for 10\% of the final project grade.  
\item{$\bullet$}  {\bf Between April 10 and April 21} -- Each team
will meet with me during office hours (or at another time
if that is not convenient) for a progress report and a chance
to discuss any questions that have come up as you have started to
work on the project.   I will be happy to meet to talk over any
aspect of the project at other times too, of course.
\item{$\bullet$}   The presentations will be given in the final meetings
of the seminar -- May 1, 3, 5, and 8.  We will have two
presentations each of those days; the exact scheduling
will be determined later.  
\item{$\bullet$}  The oral presentation will receive a separate grade,
and will count for 30\% of the total on the final project.
\item{$\bullet$} {\bf Monday, May 8} -- All final project papers will be due 
(via email) by 5:00pm.  The grade for the paper component will account for 
60\% of the total on the final project.   This assignment will take the place 
of a final examination for this course.

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{\it Other Information}
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\item{$\bullet$}  See the day-by-day course schedule for the other assignments
for the course so that you can plan your work for the project.  
\item{$\bullet$}   Ms. Merolli, our
Science Librarian, will be visiting our class during the week of 
March 20 to introduce herself and give some introductory information
about using the library resources to identify sources for your
project.  She will be more than
happy to assist you in the important
process of assembling the resources you will use. 
\item{$\bullet$}  The presentations can be done with PowerPoint, or
without visuals, as you prefer.
\item{$\bullet$}  I will ask each group to do a ``dry run'' of their
presentation with me at least one day before you go in front of the class.
The purpose of this is to give you some feedback about what
is working and what is not, and to give you some practice to minimize
the effect of ``nerves'' when the time comes for the real thing.

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\noindent
{\it The Final Project Paper}
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This writing assignment will be different from the other
writing assignments on {\it All the Pretty Horses} and the {\it Metamorphoses}.
The goals of this assignment
are for you to collect information about your topic from the
various sources you find, and then present your analysis of that
information.
The evaluation of your project
reports will be based on how well you have addressed the following
guidelines and expectations:
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\item{$\bullet$}  Distill 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.  (In a longer thesis,
it would be expected that another section reviewing
the most relevant contributions of previous work on related subjects
would be included -- that kind of full literature review is {\it not expected
for this assignment}.)  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.
\item{$\bullet$} In a {\it References} section at the end, include all books, articles,
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.
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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 one of the
other teams 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 afternoon of May 8(!)  Be sure you
get started early enough so that the input can be put to productive use.
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\noindent

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{\it Project Ideas}
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{\it Area 1 -- Jesuit Mathematicians, Astronomers, and Missionaries}
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A number of Jesuit
mathematicians, astronomers, and missionaries have been involved in
various ways in mathematical and historical topics we have touched on this year.

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. He was instrumental in the inclusion of mathematics
in the curriculum of the schools founded by the first generations of Jesuit priests
and he authored a version of Euclid's {\it Elements}
that was used in Jesuit schools all over Europe. 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 worked with Chinese scholars and produced a Chinese translation of part of Euclid.
(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 an 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.  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 completed or are currently enrolled in
MATH 134, or 136 or the equivalent (Calculus 2).   

There is material for several distinct project topics here -- three different ones at
least.  The first step would be to narrow down the focus to something suitable for
your project.  If you are interested in pursuing a part of this, discuss the
options with me before starting work on your bibliography.
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{\it Area 2 -- Mathematics, the arts, logic and other connections}
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Another sort of ``mathematical thinking'' can be 
seen in the art of M.C. Escher.  We looked at the symmetry groups of 
symmetric strip patterns last semester.  Escher used many related ideas
in a series of drawings illustrating ``tilings'' of the plane by 
repeating motifs.  These drawings 
were just one facet of his work; many other aspects were
also mathematically-inspired.  One nice project topic would be to investigate
some of the 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 Escher's
symmetry notebooks.  Her article {\it ``The Mathematical
Side of M.C. Escher''} from an 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.   Kurt G\"odel 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 Area 3 -- 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 (available in several different 
versions on YouTube) 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 Area 4 -- 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.  
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 algebra
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 project by itself.  So there are at least two 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 Area 5 -- What do Mathematicians Think about Mathematics?}
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The beginning chapters of the book {\it Loving and Hating Mathematics}
by Hersh and John-Steiner are a rich source if ideas for more open-ended
projects about different topics related to the culture of research 
mathematicians, how different mathematicians think about the subject and
why they find it attractive, how they collaborate with colleagues (or not), etc.
Another interesting book they mention several times is the memoir
{\it A Mathematician's Apology} by G.H. Hardy, a well-known British
mathematician from the late 19th and early 20th century.  Hardy has
some very provocative ideas about why humans do mathematics and 
what its ultimate purpose is.  This topic (group) is probably most 
appropriate for the more philosophically-minded.   Some thought will be necessary 
here to narrow the focus down to a manageable project topic.  I
would be happy to discuss options with you if you would like that.

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{\it Area 6 -- 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 March 17 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.
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