2 May
Quick Answers
You finished the book, you read the _entire_ book! (well,
someone did at least).
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).
Your final paper is due by on Wednesday at 3:30p (class
time). Our final exam is Tuesday 13 May at 3:30p (class
time). You will have until 6p to write your essays.
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. I
will have regular office hours Thursday evening and an office hour
session 1-3p on South 336 on that day also. I will not be
available after Thursday before the exam.
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 assignment is not completed.
A topic for Monday - how to incorporate history in teaching.
I will say some about this.
Lecture
Reactions
Everything is not suddenly great after Noether. There's plenty
of reasons left that fighting needs to happen. I guess the
most important point there is with both her and Maryam Mirzakhani
anyone who says "women can't" needs to be faced with both of these
examples.
The Banach-Tarski paradox leaves us in a very odd position.
Maybe the conclusion is reconciled by saying "there's a theoretical
decomposition, but it's not practical, so we can have both
mathematics and the world around us".
You
should have done this in proofs class (or you will for those who
haven't yet), 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.
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. This _is_ actually
important and valuable - to prove that our way of proving things
actually works. The proof is to check that the proof-steps
agree with truth tables. The true unprovable statements are
true according a truth-table like analysis, but there is no string
of statements to prove them. The existence of endless
unprovable statements is _proven_ and a true part of
mathematics.
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. One way to view
Gödel's incompleteness work is that self-referential statements
are problematic.
Constructivists deny proof by contradiction. They do not make
decisions for others, but for themselves. If you prove
something that way, they will say to you "you haven't proven it for
me."
Reading Reactions
Please
be aware that Suzuki is rightly mocking “aryan
mathematics”. I think most of you got this.
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.
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. GauĂź 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.
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. It is interesting that the book is now over 15 years
old, and "the past 50 years" is growing closer to the past 75
years.