3 May Quick Answers

You finished the book, you read the _entire_ book!  (well, someone did at least).  Speaking of which, our department has a survey about reading in mathematics.   I would think you have a thing or two to tell them.  Here 'tis.

Well, it’s comes to this.  I will take time for exam discussions both today and Monday.  And I’m looking for other things to talk about.  I have several ideas.  We’ll see how much I manage to fit into our remaining time.  Any is all good. 

Remember for your last reactions you need 5 to today’s lecture (video) and 5 to the entire course.  If you want to include one suggestion for Monday that can count.
I think you currently only have 56.5% of your course determined.  The last two components are significant.  You have a serious need to not slack now, because with that weight remaining it is easy to be very detrimental.  On the other hand, this also means that you have a significant chance remaining to make a difference.  Quick arithmetic:  if you have a final worth 20% remaining (in another class) a 5 point improvement on the final counts for 1 point improvement in the course.  Since we have almost 50% remaining, a 2 point improvement on both would count for a 1 point improvement in the course.  Related to this:  Your “actual current average” is.  I have dropped one zero for those who have one.  I have not dropped any other reactions - I will do that for Monday.  It won't have much of an effect (probably under 0.5). 

I have regular office hours as long as we have regular classes.  I remain happy to talk to people from this class. In particular, I happy to offer opinions on either of the two remaining aspects - final paper or plans for final exam.  Do you want anything for office hours next Thursday?  Friday is uncertain for me now, and one extra day probably doesn't help for this class. 

I will not be giving feedback on the paper, since it will be finished.  I will judge against my prior comments.  Remember if you do nothing the grade will go *down*.  




Lecture Reactions

Re: Borel, what does compactness mean aside from closed and bounded?  One answer is that every sequence has a convergent subsequence.  That's about the least technical other answer. 

You should have done this in proofs class, but |N| = |Z| = |Q| < |R|.  It has been proven that "there could be" and "there could not be" a set of size in between.  Both are consistent with mathematics as we know it.  We'll need some somehow external reason to decide as a community between these two options.  Chris Leary gave a talk in which people seem to want to say there is exactly one size between them. 

What is the current opinion on computer made proofs like the 4-colour theorem?  I would say … probably accepting, moreso than not.  Definitely it's reassuring that it is independently verified by different programs. 

I am glad that so many appreciated the models.  If any want to know how they work, I will be happy to talk more about them.  They are central to my geometry course, which is quite different from that of my colleagues.   There are lovely ways to crochet them.  Here’s a way to make a (different) paper one:  https://www.youtube.com/watch?v=HaIV2Cj9Ewk

Each of the fractals has an infinite number of iterations.  Do not stop until done, or none of the results are true.  All of our examples in that segment were fractals.  Fractals have non-integral dimensions.  We'll look at Sierpinski's triangle again to make sure you see how there are 3 copies. 

The Koch snowflake is reminiscent of a snowflake, but is not a real snowflake, just like the Menger sponge is not a real sponge.  That being said, there is a sense in which snowflake formation is a self-similar process which produces fractal-like objects. 

Yes, in algebraic topology there are cases where we have negative or infinite dimensions.  

Gödel also proved (happily) that everything that can be proven is true.  True is based off a truth-table analysis, and provable is based on whether you can string together a sequence of statements according to proof rules.  The true unprovable statements are true according a truth-table like analysis, but there is no string of statements to prove them.

The key to Gödel’s argument is coding - a way that the mathematical symbols can be coded as numbers.  And then relations among symbols can be coded as relations of numbers.  And one relation is that the symbols could be a proof of a statement.  So, suddenly statements about proofs can be numerical statements.  

I don’t think Gödel can identify all the unprovable statements.  That feels unknowable.  



Reading Reactions

Hilbert’s third problem was the first of his problems to be solved.  It relates to an earlier known result that some see in geometry class:  every polygon in the plane can be cut into finitely many pieces and rearranged into a square of equal area (hence any two polygons of the same area can be cut and rearranged into each other).  Max Dehn proved that this is not possible for polyhedra.  Dehn used algebra to prove that there is no way to cut a regular tetrahedron (a pyramid with a triangle base) into pieces and reassemble this into a cube.  

Please be aware that Suzuki is rightly mocking “aryan mathematics”.  

E. T. Bell wrote a book called _Men of Mathematics_ in 1937.  Unfortunately it is one of the best known books about history of mathematics.  I say “unfortunately” because it includes many made-up stories (e.g. Gauß summing 1 to 100) that have no evidence whatsoever.  It also includes the story that Galois’ duel was over duMotel.  Suffice it to say it is not a very reliable source.

The ergodic theorem is far from what I know about.  A quick exploration gives me something about a dynamical system (something that's changing over time) having the same average over time as over space. 

Somewhere I lost track of Gertrude Blanch.  I want to say more, this is a reminder. 

Oh, what is differential geometry?  I will talk about this a little, that's a nice question.  This I actually know something about. 

A Turing machine is more of a thought experiment than an actual machine.  I will try to put something about one here.  Films about famous mathematicians:  Imitation Game (Turing), The Man Who Knew Infinity (Ramanujan), A Beautiful Mind (Nash).  They each have some disconnect with reality (I think the middle is the most reliable as I know some of the mathematicians consulted), but they do bring public attention. 

I contacted Jeff once and mentioned that the class wanted him to write a sequel.  I think he dismissively laughed at the idea.