Â鶹´«Ă˝ÍŶÓ

390 Quick Answers 4 April

Paper drafts due by Monday classtime.  I will be returning the first couple with comments next week.  Please have _someone_ else read you paper before handing in.  Focus focus - write about mathematics in detail about mathematics.  More details, not more topics. 

We get ... busier, mostly starting on Friday.  There's just more and more mathematics.  We are also deeply in the "middle third" of the course.  The beginning mathematics is presented so differently that it's unfamiliar; the end mathematics is so sophisticated that it's unfamiliar.  But, the middle the mathematics should be content that you know well - around the time of calculus - mostly presented in the ways you know it.  (If you're now saying "there are so many words" then you don't read your textbooks very much.  Words are necessary to explain steps, but we do have most of the notation at this point in history.)  Enjoy it while it lasts.  We're getting close to the end of mathematics that is familiar to you. High School mathematics tends to run through the mid-17th century.  Undergraduate mathematics tends to run through mid-19th century, but soon familiarity will start depending on what classes you've taken.  20th century and beyond is mostly learned in graduate school. 


Lecture Reactions

Good question - Berkeley did have a background in mathematics and had written elementary works on the subject. I don't know of other cricitisms of Newton in this direction.  There were of course criticisms between the Newton and Leibniz camps. 

We will put deMoivre's theorem into another form today via Euler's work. 

This is a recurring issue that students have … combinatorics (any "how many ways" questions, e.g. combinations and permutations) is not probability.  It is often _used_ in probability, sometimes it's the only place that _you_ see it used, but it is not probability.  There are several diverse mathematical contexts where one wants to know "how many ways".  They are as much probability as multiplication is probability.  Then, the next level … probability is not statistics.  Probability is solely finding the chances or odd of events.  Statistics is making inferences from actual data.  We can see that probability was developed later than other mathematics.  Statistics later than that, and we don't much follow that story, largely related to the fact that statisticians don't think of their work as mathematics.  



Reading Reactions

This is my reaction … ohh, here Suzuki is willing to admit that Leonardo of Pisa worked with the Fibonacci sequence … hmm.  I hadn't noticed that before.

Hm, maybe Suzuki doesn’t tell the full l’HĂ´pital story.  Guillaume Antione was the Marquis de l’HĂ´pital (Marquis is the rank of a noble).  You know him as l’HĂ´pital.  He paid Johann Bernoulli to teach him calculus.  Bernoulli agreed to this with the condition that l’HĂ´pital _not_ publish it.  l’HĂ´pital then went to write the first calculus book. 

Almost never does anyone name something after themselves. 

Euler reproved Fermat’s little theorem.  We reprove to understand better, and to seek better understandings.  “Euler published an average of 800 pages per year. 800 pages. That explains why this guy keeps popping up in college level math: he basically contributed to all of it.”  - quote from student last year.  Yeah, that’s Euler.  And that’s the right way to think of him.  His work is staggering.  It sounds exaggerated like fiction.  It’s not. 

The story about Euler and a + b/n =x and god is very likely fiction.  Euler was religious and also obviously seriously mathematical.  He wouldn’t make such frivolous statements about either.  

Yes, we finally have a woman.  Agnesi is not the first chronologically.  We will see that on Monday.  The reason we haven’t seen them is systemic sexism - societies built to make in impossible for them to contribute.  They had to fight to participate.  We will see how each of them fights to break down the barriers.  That fight is an important part of the story. 

Lambert proved that Ď€ is irrational.  That’s great, and we’ll talk about it.  I will not talk about al-Haytham and al-Khayyami’s quadrilaterals again under European names (and Saccheri).  They don’t deserve to be mentioned here.   Yes, it is a shame that your Geometry professors call theme these names.  There's really no news there, people are still trying to prove the fifth postulate, like al-Haytham did, and as a consequence are considering alternatives to Euclidean geometry. 

We will talk about Lagrange next time.  He fits better with France for me.  (Suzuki is also not done with him.)