390 Last Day Quick Answers 6 May

Most important thing today - any questions about your exam.  But, I think did most of that last time.  Are there questions different from Friday?  Any new topics? Please remember - the next week is important.  Take it seriously.   And please oh please let's not have issues about academic integrity.  You may access _NO_ materials during the exam.  If you violate this, you will fail the _course_ and be reported for an academic integrity violation.  We'll be back here in 168 hours.  Bring computers and plan your time carefully, the exam _will_ end promptly at 6p.

I have now dropped lowest reactions for everyone (including a zero for the one before the first class if you didn't do it).  My goal is to get papers processed by Friday and again have a more updated current average then.  

If you miss sending me reactions, then write an email.  I will reply to all, and we can continue it as long as you like.  Of course - the only reason I keep saying it is that I will miss them. 

I will be available regular office hours and Thursday 2-4p in Welles 123 (across the hall). 


Lecture Reactions

Arrow's theorem says that there _cannot_ be an ideal voting system for 3 or more candidates.  We must pick the failings we choose to accept. 

"how easily would today’s computing capabilities solve the Enigma code?"  Quite very, I imagine.  Also remember this - the word coding means some very different things.  Enigma is about cryptography, secret codes.  Computer programming is sometimes called coding, that is very different.  Gödel coding is kinda like secret coding, but there's nothing secret about it - it's a way of representing mathematics by numbers. 

It seems many people were surprised that there is more equality and less discrimination under communism.  That _should_ be obvious, if you know more about communism than propaganda that says it is bad.  

What does it mean that fractals have infinite iterations?  I'm back to one of the examples …

I was asked to say more about Japanese mathematics.  I will do that. 

I asked Jeff about why he didn't include Japanese mathematics.  He replied in a way that helps to understand his choices throughout:

One of the problems I struggled with this (in this and other contexts) is the question of influence: How much influence did a person or culture have on the overall development of mathematics? So we can trace a pretty clear line from modern mathematics back to Euclid, etc., and we can see the influence of Indian mathematics on Islamic mathematics, which then influences Renaissance Europe. The challenge is getting beyond India. It seems there was some influence of Chinese mathematics on Indian mathematics, but the evidence gets a little conjectural at that point. (For example, al Kashi's method of roots is similar to Liu Hui's method, but there are some idiosyncrasies that suggest it was a independent invention) With Japanese mathematics, you're looking at an even more tenuous connection to the mainstream. So while there are some very neat problems and solution, their influence on the development of mathematics is a lot harder to identify.



Course Reactions

For any not graduating who want to know more about topology, there is a course next semester.  I last taught it 3 semesters ago. 

I am very glad that so many take away the important point that there is more mathematics happening now than ever in history.  This is very important for you all to know.  I am also very pleased that many take away a large appreciation of all of mathematics.  That is definitely one of my goals.  I am also very proud that this course inspires more interest in history.  It is an interest deep in my family, and I like that others can see it here. 

I am grateful that we can look at the history of women in mathematics and see what they each contributed to making the study of mathematics open to women today. 

"I am surprised that we did not go through more proofs in this class. I feel like a large amount of the mathematics I learned in college was mainly centered around learning how to solve a proof. So, I am surprised that we did not talk much about how proofs themselves emerged and the different methods that are used to solve them."  I thought I did.  Let me talk about this more ...
    1.    In the first half of the class, it was nice to see mathematics from a variety of cultures and places and see how culture impacted mathematics. However, in the second half of the class, I felt like we reverted back to the typical eurocentric view of history. Was there really no math going on anywhere but Europe and the US after 1600?

No, of course not.  Here are the learning outcomes:

Trace the development and interrelation of topics in mathematics up to the undergraduate level,
Discuss mathematics in historical context with contemporary non-mathematical events,
Analyze historical mathematical documents - interpret both the concepts of the text and the methods of mathematics, and
Identify significant contributions in mathematics from women and from outside of Europe.

The first one means that I need to focus on the mathematics that you have learned.  I need to give you the history of mathematics you know.  Yes, most of that comes from Western civilisation.  Yes, there is other valuable mathematics happening elsewhere, but it doesn’t always feed into what you learn.  

There is more to absolutely everything that we talked about.  I chose to just do the surface of absolutely everything, breadth over depth.  Remember you can learn more about everything.  If you ever want help doing so, please ask.  

Learning more into modern times is not learning history, it’s learning modern mathematics.  That is what graduate school is all about. 

Talk some about course format and my design choices.  

What do I see big picture:  humans take time to figure things out, and it happens slowly.  Greek mathematics is not amazing and is built on slaves.  Islamic mathematics deserves more attention.  There is more mathematics than you think.  Mathematics doesn’t appear from magic or from supernatural experiences - people struggle to figure this out.  

4. How many times have you taught this class? In what ways have you taught this class differently throughout those years, or what do you think you will change after teaching it this time.

Someone asks of examples of disagreements in mathematics.  Usually they are when something is disproven, and then it is settled.   Beyond that, you have philosophical disagreements which do not get settled.  This ranges from quaternions vs. vectors (which was mostly settled, but not because one was right or wrong), to formalism vs. intuitionism.

If you were talking to someone who was not a math major about Pythagoras, how would you describe him and the Pythagorean theorem?

Is there anything within mathematical history that you have learned that has was surprising or changed your perspective?