390 Quick Answers 22 April


I will not have office hours during GREAT Day.  Make a plan to participate and tell us about it on Friday.

Work on your papers and final topics.  

I received an anonymous inquiry “If we turn in the same paper for our final paper that we turned in for our rough draft will we get the same grade.”  Thank you for the question.  This is an excellent use of the system.  Your grade will be deducted by at least a full letter grade if you make no changes.

The first attempt is your best effort without my support.  The second is your best attempt to follow my guidance from the first.

After GREAT Day, we only have chapter 11.  We’re getting there!  Remember how busy we were with mathematics in the 19th century in Europe?  The 20th century will be like that, not like the slow US chapter.  There is _NOT_ less mathematics being done in the 19th century.  Definitely not.  The US is backwards and trying to catch up, and Jeff and I are struggling to talk about the mathematics because it's sophisticated and we don't know what to do.  This is important - there is more mathematics in the 20th century than all of history before.  Do NOT miss this big point. 

Today we will have a special supplement on the history of linear algebra.

“Why aren’t we required to learn ___?”  Because there is only so much space for requirements.  However, there is and was no limit on what you _choose_ to do.  For those who are finishing - it was your choice to not do more.  For those not yet finishing - perhaps you will learn from the laments of others.  Look for interesting things to study.  There are plenty of opportunities.

Someone was surprised at the correlation of dates.  As always - keeping a timeline is _so_ important to see how different stories fit together.  Don’t take this lightly.  I have good news for you regarding this - chapter 11 is worldwide and chronological.  So, get your timelines fixed up now and you’ll be all set.

When I comment personally to your reading reactions - do people read my comments?

Statisticians often don’t think of themselves as mathematicians.  Their history is rather separate.  Probability is a branch of mathematics, and statistics is on its own, which is why we almost do nothing with statistics.  That being said, there are a couple of good GREAT Day projects on statistics (it is always a nice area for 390 projects, since we don’t touch it in class).  


Lecture Reactions


We understand so much more when we see the real line as part of the whole complex plane.  Merely thinking about real numbers does not allow you to see the big picture. 

Banneker did not create the first clock.  But he made one (one - he wasn’t mass producing them!) by hand and was the only one in his region.  We do not records of details of the clock, beyond what you have read.  Think about making a mechanical clock.  It’s not easy to imagine.  He surveyed and decided the boundaries and layout of Washington.  Remember that was the beginning, 1791-2, not with later buildings.  In many ways, he deserves to be more well-known. 

It's interesting to me the way you (pl) interpret Jefferson's statement.  To me it sounds dismissive and as if he wasn't very interested in mathematics.  Banneker seems very likely to have known more than Jefferson, who seems to think it is a waste to learn calculus (fluxions).

Bowditch and Adrain did consider what would mean for a star to be so massive that it captured its own light with its gravity.  While this is related to black holes, it’s not exactly the same and not understood the way it would be 100 years later.  

Yes, a good part of 10.1 was not USian.  That is because the US is still very thin.  There isn't much mathematics in 10.2 either. 

Hamilton took a path of invention to devise quaternions.  This is the way math is invented.  You have properties you want, as he did, and you see what would lead to those properties.  He wanted to create a mathematical model of three dimensions, and along the way he surprised himself to realise that to do so he needed four dimensions.  

I think the bridge story is that Hamilton was working on this for a long time but had the crucial insight while walking to the bridge.  That insight would be … what if ij = -ji … which leads to a fourth dimension of k.  

The fact that quaternions were more extensively studied by humans before vectors is fascinating.  Why did we decide on vectors?  Extend more easily to higher dimensions, no pesky negative sign, and simpler derivation and introduction (as I'm sure you would all agree).  Quaternions were a big deal when they were discovered, but they were mostly subsumed by vectors, largely because of leadership like Gibbs and others.  This is how they led to many modern ideas (vectors including dot products and cross products, gradient, divergence, curl), but also sound unfamiliar.

Somehow more than one person struggled to understand that knots are made with string.  I can't really explain that.  They also happen to be my area of research, and featured somewhat prominently in the weekend's meeting. 




Reading Reactions

"What is the significance of a journal? What makes them so necessary?"  They are a way to disseminate mathematics.  Also a way to know that mathematics is reliable - has been checked.  We have seen repeatedly that journals are central to building mathematics culture and community.  Having a place where work can be shared and read with confidence is the way that a community learns mathematics together. 

Interesting question when was first ever PhD in the world?  One answer that is fun "The first PhD (Doctor of Philosophy) degree was awarded to Al-Kindi (also known as Alkindus) in the 9th century. Al-Kindi was an Arab philosopher, mathematician, and scientist who lived from 801 to 873."  That may be a stretch.  More reasonable by 1150 in Paris.  Definitely by 18th century.  More here.

Somehow this is difficult for me to track down, but Gibbs may be the first PhD in mathematics in the US in 1863.  And in 1886 Winifred Edgerton Merrill became the first woman to earn a PhD in mathematics in the US.  

Yes, University of Chicago was and is a big deal.  It really defined US mathematics.

George Hill used infinite determinants and periodic differential equations in his study of the three-body problem in astronomy.  His proof showed that the moon is not potentially leaving sometime.  

Complex numbers show up in electricity and magnetism because they are the best way to mathematically model two dimensional phenomena.  I won’t be heading in the physics direction during lecture today.  

Definitely it remains important to keep up-to-date on what’s happening in Europe.  Everyone is trying to do that as best as is possible.  It’s easier to the end of the 19th century than it was to the beginning.  Not as easy as it is today.

Someone asked about colleges curriculum at this time.  I can't say much, but I have some local information.