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\centerline{Montserrat 108N -- Mathematics Through Time}
\centerline{Paper Assignment 3}
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\noindent
{\it General Information}
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Your third larger-scale formal writing assignment for the course will be
an essay on one of the topics described below.  
Your paper should be prepared using MS Word or equivalent
software, {\it approximately 5 double-spaced pages
in length} (but don't worry if you go a bit over or are 
a bit short -- the length is not the main point).   
You will be saving the paper as a .doc file
and submitting it by email to me by email to:
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\centerline{\tt little@mathcs.holycross.edu}
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\noindent
The due date/time is {\it 5:00 p.m. on  Tuesday, December 14}.
Note: this is different from what the course schedule on the 
class web site originally said.  This paper will be the
final assignment for the semester and will (in effect)
replace the final examination originally planned.
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\noindent
{\it Description}
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Unlike the first two essays this semester, this one
{\it will be} primarily a ``research paper.'' The
idea will be to read the suggested sources carefully
to determine answers for the questions posed in the 
paper topic you choose, then write up your findings
in your own words showing where in the sources you 
found the facts and ideas you are presenting  

\vskip 10pt
If you use quotations or other information, you should:
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\begin{enumerate}
\item  Identify direct quotations with quotes and a 
note in parentheses giving information where the quote
can be found within your source,
\item  List all your sources in a References (Bibliography)
section at the end of your paper:
\begin{itemize}
\item For books, give the 
author, the title, the publisher and year of publication.
\item For magazine or newspaper articles, give the author, 
the title of the magazine or newspaper, the date of publication, and 
the starting and final page of the specific article.  
\item  For web sites, give an author (if you can 
determine that), the full URL (web address) and the date you 
accessed it (since information on the web often changes!)
\end{itemize}
\end{enumerate}
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\noindent
{\it A Suggestion}
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I will be happy to discuss (or read a first draft of) your paper and give 
you some preliminary comments by email.  Or, you can come by my office hours
if you want to ``run your paper by me.''  Alternatively,  I think 
you may find it very helpful to have a first draft of your paper read
by another student in the class.  I can set up ``reading pairs'' 
if you are interested.
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\noindent
{\it Evaluation}
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I will provide written comments on your work, and
assign two grades for each paper -- one for how well your conclusions
are presented and supported (in other words, for how convincing your
arguments are), the other for how well your writing follows the standards
for formal written English. (For instance, how well is the paper
subdivided into paragraphs each addressing a particular item in your
overall argument?  Are the paragraphs arranged in a logical, recognizable
sequence?  Are the sentences within each paragraph ordered well?
Are they grammatically correct?  Are there awkward sentences?  
Are there overly flowery, overly colloquial, or incorrectly-used
words or phrases?  Is punctuation used correctly?  Are there spelling and/or 
typographical errors?)   
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\noindent
{\it Topic 1 -- The Final Word on Plimpton 322?}
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Recall that earlier in the semester we discussed several different
interpretations of the Old Babylonian mathematical tablet 
called ``Plimpton 322.''  In recent years, primarily through the work of 
a historian of ancient Mesopotamia named Eleanor Robson, another 
interpretation besides the ones we discussed has become arguably the 
most convincing one.
For this paper topic, you would read a summary of this work that Robson
wrote for a mathematical audience in the {\it American Mathematical
Monthly} -- follow the link from the course homepage to download a copy.
This will be your primary new source (in addition to the class notes and
Rudman's book {\it The Babylonian Theorem}). You might also look at the 
article ``Sherlock Holmes in Babylon"
by R. C. Buck listed in the biblography of Robson's article to understand
some of Robson's criticisms of that style of mathematical history.
In your essay, you should be sure to address the following questions:
\begin{enumerate}
\item What is Robson's main point about the proper way that mathematical
history should be done?  Why does she criticize other work on 
Plimpton 322 on this score?
\item Why specificially does she reject the idea that Plimpton
322 is a sort of trigonometric table?  
\item Why does she reject the idea that Plimpton 322 was 
primarily a number-theoretic investigation of Pythagorean triples
along the algebraic lines suggested by Otto Neugebauer?
\item What does she suggest about the {\it correct} historic interpretation
of Plimpton 322?  (And how is that connected with other texts we
considered earlier like the YBC 6967 tablet?)
\item Are you convinced by her arguments?  (Not everyone is.)  
\end{enumerate}
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\noindent
{\it Topic 2 -- Number Theory in Euclid's {\bf Elements} and Beyond}
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In class we read Book I of the {\it Elements}, which
leads through properties of triangles and parallelograms to 
an ingenious proof of the Pythagorean theorem.  The {\it Elements}
contains much more than this sort of geometry, though.  For 
this topic, you would discuss some of the properties of the 
whole numbers (or integers) developed in Books VII through IX
of the {\it Elements}.  Your primary source here will be
the text of those books and the commentaries from David Joyce's 
web site that we used before (use the link from the course homepage).  
You may also consult other modern mathematics texts if you 
like, but that is not necessary.  For the purposes of this
topic, I would suggest you concentrate on just the following
specific topics (among all the topics covered in the 
propositions in those books):
\begin{enumerate}
\item How are Euclid's definitions at the start of Book VII
a sort of ``marriage'' of geometry with properties of numbers.
How are they different from the way we usually think of numbers
now?
\item Propositions 1 and 2 in Book VII give Euclid's version
of integer division and what we now call the {\it Euclidean
algorithm} for finding the greatest common divisor (or as
Euclid says, ``greatest common measure'') of two numbers.
How does this work?  How would we write this in modern
notation?  Work out an example (use Euclid's
method to find the greatest common divisor of 244 and 128, say) 
and include it in your paper.
\item Proposition 20 in Book IX is one of the most famous
results in the {\it Elements}, and it is often considered
to be one of the most insightful and beautiful proofs in
all of the mathematics we know.  What does it say, and
how does the proof work?  
\item Understanding how the prime numbers are
distributed among all the positive integers is still
a major area of mathematics.
What are some of the questions about primes that
have not yet been answered?
\end{enumerate}
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\noindent
{\it Topic 3 -- The Platonic Solids}
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In class we read Book I of the {\it Elements}, which
leads through properties of triangles and parallelograms to 
an ingenious proof of the Pythagorean theorem.  The {\it Elements}
contains much more than this, though.  For 
this topic, you would discuss the famous and beautiful 
solid geometric figures known as the {\it Platonic solids} studied in Book XIII
of the {\it Elements} -- known today as the {\it tetrahedron}, {\it cube}, {\it octahedron}, {\it dodecahedron}, and
{\it icosahedron}.  A primary source here will be
the text of Book XIII and the commentaries from David Joyce's 
web site that we used before (use the link from the course homepage).  
For some of the other questions below, you will need to identify other
sources.  I, or our science librarian, Ms. Barbara Merolli, will be happy to 
help you with this if you need it.  
\begin{enumerate}
\item The name ``Platonic solid'' refers to the ancient
Greek philosopher Plato, of course.  Why was Plato interested
in these figures, and what role did he think they played in 
the mathematical structure of the universe?  Was he correct?
\item In the Remark after Proposition 18 of Book XIII, Euclid sketches
a proof that these five solids are the only solid figures, all of whose
faces are regular (i.e. equilateral and equiangular) plane polygons,
{\it all of the same type}.  How does his argument go?
\item There is another argument for the fact that there are 
only five of these solids that just uses
a relation between the numbers of vertices, edges, and faces, discovered
by Leonhard Euler (1707-1783).  How does that argument go?
\item Where do the Platonic solids appear in nature?
\item What role did the mathematician and astronomer Johannes Kepler
think the Platonic solids played in the mathematical structure of 
the universe?  Was he correct?
\item Given the slightly ``wacky'' ways some very famous philosophers
and scientists have thought about the Platonic solids, is it possible
that the idea that the universe is designed along mathematical
lines is overrated?
\end{enumerate}
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\noindent
{\it Topic 4 -- Hypatia of Alexandria}
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One of the final famous figures associated with the ``University''
at Alexandria in Egypt was a (female) mathematician and astronomer
named Hypatia.  During her lifetime, she was a well-known teacher
who also apparently helped her father, Theon, prepare some of 
his works, including possibly a famous edition of the {\it Elements} 
of Euclid.  However, the main reasons Hypatia is remembered today are 
probably
\begin{itemize} 
\item the fact that she is one of the very few female scholars
known from antiquity, and 
\item the nature of her death -- she was brutally murdered by 
a mob in 415 C.E. 
\end{itemize} 
For this topic, you would research what is known about her life
and death.  
\begin{enumerate}
\item Who were some of the students she taught and why did
they revere her as a teacher?
\item What role did the fact that Hypatia was a follower of
the old pagan religion play in her death?  Why was the emerging
Christian church opposed to her teachings?
\item What was the role of Bishop Cyril of Alexandria in her death?
\item (Optional)  There is a recent film called {\it Agora} that
presents the story of Hypatia from a rather opinionated perspective.  If
you would like to turn your essay into a film review, that would
be OK.  I can loan you a DVD of this film if you want to pursue
this option.  If you do, you will need to make your essay more than just
a ``thumbs up/thumbs down'' judgment on the film as a film, though.  
You will still need to explore what is known about the actual history and 
think about how well and how accurately the film follows what is known.  
\end{enumerate}  
Some sources for this topic are on reserve in the Science Library;
consult with Ms. Merolli for these.  
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\noindent
{\it Topic 5 -- Geometry and Philosophy}
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Starting from the time of the ancient Greeks, Euclidean geometry, and later non-Euclidean geometry as well, have had a tremendous influence in the development of philosophy.  In the book {\it The Mathematical Experience} by Davis and 
Hersh (on reserve in the Science Library), there is a section called 
``The Euclid Myth'' that summarizes a lot of this in relatively readable
form.  The idea of this project would be to grapple with some of the issues
raised there and to try to understand 
\begin{itemize}
\item why Euclidean geometry came to 
be seen as an exact description of the physical universe, and 
\item how (or whether) the discovery of non-Euclidean geometry has
changed that.  
\end{itemize}
For the essay, try to address the following points:
\begin{enumerate}
\item What is the point of the story of Socrates questioning the slave
boy from the dialogue known as the {\it Meno}?  How did Plato use
this in his philosophy?
\item What are the competing schools of {\it rationalism} and 
{\it empiricism} that Davis and Hersh discuss?  
\item What were Kant's ideas about geometric knowledge?  What is
meant by his categories of ``synthetic'' and ``analytic'' 
{\it a priori} knowledge? 
\item How are Kant's ideas viewed today?  Does the fact that
there are other mathematically consistent geometries besides
Euclidean geometry invalidate what he was trying to say?  
\item Are there implications here for what mathematics means and
what it is good for?
\end{enumerate}    
You may also want to read some of the other
parts of that chapter in Davis and Hersh to understand some 
of the issues in the ``crisis in foundations'' in early 
20th century mathematics.  
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