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{\it Mathematical Journeys:  First day, 9/2/15}
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{\it CHQ Theme:}
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How then shall we live when the journey may be as important as the origin or the destination?
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While many other species (think migratory birds or monarch butterflies, for instance) regularly
undertake almost incredible travels, human beings are quite distinctive too for the extensive 
journeys we have made. From our origins in Africa, over the course of tens of thousands of years, 
we have spread over the whole earth and adapted to extremely different habitats with the help of 
technology that allows us to journey ever faster and to new places.  
Our species has in fact been molded and transformed by those journeys and the history of 
those movements is preserved in our genetic diversity and the 
variability of our features.  We have even set foot on the Earth's Moon and are perhaps 
poised to move farther out into our solar system if we can muster the will and the 
resources to do so.   

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On another level, too, every human life is a journey.  With luck, we start out from a safe 
home with caring parents and a loving family.  We gradually make our way into wider and 
wider communities; we have all sorts of different experiences; we grow in some ways, or 
perhaps we do not.  And after many years (if luck is still with us), we reach that one 
destination that we must all experience eventually.  Looked at this way, of course, you might 
say that our personal journeys are {\it everything} to us -- they are our whole lives.  
%And 
%some would even say they are all that we really have knowledge of because they are the only 
%things we experience directly.  (Others would disagree about this, of course!)  But in any case 
But this means that the first part of our cluster theme, the ``How then shall we live?'' is 
the most basic and important question we all must ultimately address if our lives are to 
have meaning and purpose.  

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Many of the experiences we have along the way are
also journeys of one sort or another.  Those experiences can be some of the most
influential ones we have because they can lead to conscious or unconscious choices of how 
we live. So in the CHQ cluster of Montserrat this year, through our common readings (this semester: 
Homer's {\it Odyssey}, Mark Twain's {\it Adventures of Huckleberry Finn}, and Cheryl Strayed's {\it Wild}) 
and our co-curricular activities, all of our seminars will be addressing this theme of the journey.  
Our first common reading (Homer's {\it Odyssey} next week) is really the 
prototype of the journey story in our literature, so it's a perfect
place to start.  The second cluster common reading, {\it Huckleberry Finn} by Mark Twain,  
has been seen as the source of most of distinctively American 
literature.  We will see how Huck and Jim's trip down the Mississippi
River in pre-Civil War America is another sort of epic journey.
{\it Wild}, our final common reading, is a contemporary story of a woman's
attempt to hike the Pacific Crest Trail solo, to find herself and put
her life back together.  All of these focus on the experience of journeys 
(often more than origins or destinations) and the way they
can lead us to address that core human question: ``How then shall we live?''

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Right about now, I know some of you might be thinking -- ``What's up with all this?? I thought this 
was supposed to be a mathematics course!''   (Be honest!)   Well, the short answer is that this
{\it will be}, partly, a mathematics course, but it will not be {\it only} a mathematics course, 
and it will almost certainly be different from any mathematics course you have taken before.
After all, it's a part of the Core Human Questions Cluster of Montserrat!  We'll be
looking at one particular sort of mathematical work that ties into our theme in what I 
hope will be a challenging and thought-provoking way.  

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Let me try give you some initial ideas about what I have in mind.  Mathematics {\it can be 
taught} as a collection of rote procedures for calculating things without much (or even any)
indication why the procedures work or what they mean.  And you probably have experienced this, 
since on some level, this is actually the way the subject is often taught, especially in 
grade school and high school!  At the same time, calculators and mathematical software have become
important tools because they can do almost any low-level operation millions of times faster 
and more accurately than any human being.  So it's perhaps understandable that we have introduced 
them early into mathematical education, sometimes without the underlying knowledge needed to 
make sense of the masses of numbers they produce.  But the experience of being turned into 
an unthinking and uncomprehending button-pusher (and then being severely penalized and made 
to feel stupid for making the almost inevitable resulting small errors in carrying out processes 
that are not really understood) is not a pleasant one for most students.  So if 
you asked most people in the US today
which subject in school they liked the least, mathematics would win hands down.  This
is especially discouraging for those of us who love the subject because most of us
would agree that those rote calculations are {\it not what the subject is really about
anyway!}

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So, if that's not what mathematics is about, what is it about?  Well, I would say math is really about
finding patterns in quantity (numbers) and arrangements (geometry), about creatively 
{\it figuring things out}, ultimately about learning to use the power of our brains and
not just accepting things without justification.  In fact, the word ``mathematics'' itself 
comes from words related to the Greek verb $\mu\alpha\nu\theta\acute{\alpha}\nu\omega$ (``to learn by inquiring,''
or sometimes ``to understand.'')
So here's one possible (rather extreme) answer to our 
question ``How then shall we live?'' -- Don't accept anything on faith.  Always ask why? how? who benefits?
Warning: This is much harder than it might sound and it can make you very unpopular. 

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If you think about it, if this is what mathematics is really about, then we also
need to have ways to convince others that the things we have figured out, or that 
we think we understand, are actually true.  And this is where the idea of {\it proof} 
comes into play.  At the foundation level, a proof is an argument that
a statement is true.  It should {\it convince} the reader or hearer -- so in a sense
someone who discovers something new in mathematics, and then
the reader or hearer who learns about it, has to experience a journey from 
one state of knowledge to another.  The origin is what we knew before, the destination 
is something new (previously unknown), and the proof can be seen and experienced as 
a sort of journey.  At its heart, this isn't so different
from persuasive speaking or writing in many other areas of human thought (including
some of the writing we will be doing for this seminar!).  So 
it's perhaps not so surprising that the people who first experimented with the 
ideas of democratic self-government (where citizens had to convince one another about
the right actions to take) and trial by jury for deciding law cases 
-- the ancient Greeks -- were also heavily involved in developing the idea of 
mathematical proof.  

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After reading the {\it Odyssey}, we'll start off by looking at what a well-known
20th-century English mathematician, G.H. Hardy, said about his life and
the subject of mathematics in his book {\it A Mathematician's Apology}.  Hardy's life
story raises a number of interesting questions about ``How then shall we live?''
In his book, Hardy also highlights the way ingenious proofs can be 
seen as the lifeblood of mathematics.  He will lead us to the major focus of 
the first section of our course.  Much as 
the {\it Odyssey} has served as a prototype for stories about journeys, the 
{\it Elements} of Euclid has served as a prototype for textbooks of mathematics, right
down to the present day.  As we will see, the section we will study in detail -- Book I -- is 
a very tightly-organized sequence of theorems and proofs leading to a wonderful 
proof of a familiar result from plane geometry that all of you have seen 
(although you may not know a proof for it).  


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The second mathematical portion of the course will be devoted mainly to 
the question: How do we {\it find proofs}?  We will have seen by this point
in the course that this step in the process is not addressed at all in Euclid's 
work. Many, many generations of students have found this frustrating, to say the 
least.  And indeed, having only the ultimate destination or {\it finished 
product} in the form of a polished proof, {\it \`a la Euclid}, is arguably not the 
most productive way to learn mathematical ideas, although teachers
of mathematics have different opinions about what might replace it(!)  In this part 
of the course, we will look at the ``messier'' cycle of conjectures (informed 
guessing), attempted proofs, counterexamples, and refutations by which actual
proofs come into being through the interaction of groups of real people.     

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If any of you go on to major in mathematics, you will see that upper-level
books and courses start to look an awful lot like what we're going to be
doing in this course, though with more complicated and intricate patterns and proofs 
to find and to learn.  I hope some of you will do that, but even if this is the 
last (even partly) mathematics course you ever
take, I hope you find it to be a mind-changing experience and that you 
gain a better appreciation for what the subject is all about.  


\bye
