390 Quick
Answers 5 April
Now
we wait for drafts. It’s my turn to work. It will take
time. For those who think you are presenting at GREAT Day.
If you are impatient for written feedback, I am happy to talk with
you if you come to visit during office hours. Make
sure that your rehearsals are 10-12 minutes long.
While
your project is in my hands - this is a good time to work on
starting final exam preparations. Remember, it is like the
midterm only twice so. 2-3 topics post 1600, and 2-3 topics
that do span across the divide.
I
think today’s reading may be the longest of the semester.
Congratulations - you did it.
I
saw this on Nova last
night … Halley (who is mentioned about the moon in today's
reading) was the first person to accurately predict the location
of an eclipse in 1715, using Newton's work to analyse the position
of the moon. Some of the history in Nova was a bit
overstated. Here's
a better source for history. And all the reading we've
been doing about the position of the moon is essential to
predicting eclipses … now if only the clouds will cooperate.
Please use appropriate glasses when watching the sun.
Lecture
Reactions
Please remember - Bernoullis _need_ a first name. You can't
avoid it.
e
is not aptly called Euler’s number. (wow, but I find it
often so online, ugh!) Euler’s constant is the limit as n
goes to infinity of the sum of the harmonic series to n terms -
the natural log of n. e was named by Euler, either for
"exponent" or just because it was the next letter.
There
are lots and lots of unknown conjectures in mathematics.
Goldbach’s conjecture is one of the oldest and simplest to
state. There are endlessly many. It is what research
mathematicians work on - asking and answering new questions.
Primes are the fundamental building blocks of natural numbers by
multiplication - this is the content of the fundamental theorem of
arithmetic: unique prime factorisation of positive natural
numbers. Why is it hard? Basic
idea - prime numbers aren’t really about adding.
Are there older unsolved problems - the oldest I know - “there are
no odd perfect numbers” is roughly 2500 years old.
Agnesi’s
challenges: had to be tutored at home, and debate scholars
at home. She was paraded as a spectacle by her
father. She needed to use her father’s money to
publish her book. She asked to become a nun, and her father
denied her, but she convinced him to agree to her living a simple
life and avoiding social engagements. Appointed to an
academic chair, but could not teach.
Do
I think there are other “secret” women in mathematics around this
time? Not in Europe is my guess. Not who were
contributing to new mathematics. That were thinking about
mathematics - sure. Our course has mostly been about new
mathematics. And the reason no one was is only because the
system prevented them.
Reminder:
post the proof that π is irrational. Here are
several. Here is a proof for e and π.
The one for π is a summary of Niven's proof.
Reading
Reactions
I will not talk about it given the vast amount of material today,
but here is Clairaut's
paper submitted when he was 12. I saw that he learned to
read by reading Euclid.
d’Alembert’s
probabilty error is common and tempting. For those of you
who have made the mistake of thinking there are three outcomes
when flipping two coins. Yes, it is incorrect, but it is
understandable. Before d’Alembert’s limit definition, we
have seen some ideas in Newton, also struggling.
What
did Chatelet do? Best known for translating Netwon’s
Principia (published after her death - for many years the only
translation into French). Wrote a general audience
explanation with Voltaire about Newton’s works. She wrote on
the propogation of fire. “She attempted to integrate
Cartesian, Newtonian, and Leibnizian ideas. On the philosophic
side the themes she discusses are free will, God's power and role,
and the nature of space, matter, and force.” I also believe
Chatelet predates Agnesi slightly. So, if I were to
pick a “first”, I would pick Chatelet. Chatelet surely used
her connections with high-placed academic men. I do believe
discussing Chatelet’s personal life _is_ relevant to show how she
used them to fight through the barriers. There are different
paths. Her path was different than Agnesi’s, very, but some
path was necessary. Could she have accomplished what she did
without? It’s not clear, and it’s _not_ her fault, but the
system. When I say “could she”, I mean would the system
allow her?
We
will see a decimal clock. Why not decimal time? People
are too hesitant to make logical choices for sentimental
reasons. The same goes for the world calendar which would
unquestionably be better and more stable (same calendar each year
[with leap year accommodation as needed]). There is
absolutely no technical reason to not use decimal time. The
French calendar was about breaking from tradition and cultural
associations. A deep theme of the revolution.
Jeff
clearly shows us that Gauß is his favourite. This time I
won’t follow him, it’s too far afield.
We
talked about this with Alcuin. Please do not continue to
tell the Gauß summing 1 to 100 story (which Alcuin _did_ talk
about). It is almost certainly false. Feel free to
tell my colleagues the same. This is not known as Gauß’s
formula by anyone who knows the history. The story likely
comes from ET Bell, who we will discuss later who is infamous for
writing fictional stories about mathematicians. (And other
more honest fiction.)