390 Quick
Answers 28 April
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have draft feedback now
Think
of exam topics. Reminder: look back at the questions
on the last exam for creative ideas for the final.
Reminder: you need 2x6 or 3x4 for _both_ post 1600 and
across all history. That is 4, 5, or 6
questions. Maybe diversity for all history?
"When free thought flourished, as did mathematics. When free
thought was... discouraged, mathematics struggled against the
crushing weight of the wrought iron cages which trapped all of
society."
For
the reactions due by Sunday, you are also writing 5 course
reactions (in the place of reading reactions, because you will
have *finished the book!* by then). These are reactions to
what you learned in the course. Big picture thoughts on
history of mathematics. Maybe large takeaways.
As
one of your lecture reactions for either day (better for the first
one) you may make a request for what you want me to talk about on
Monday. I鈥檓 not promising that I can pull together anything,
but I鈥檒l try what I can. It will be an interesting day,
surely.
And,
if anyone wants to email reactions after Monday 鈥� I promise to be
happy for the feedback, and I will reply and respond for as long
as you like.
Truth: I enjoy reading your writing, and I will miss
it.
Our
final is 3:30-6p here 15 days from now. It will be
impossible to submit past 6p. Know that now.
Reminder
to all: we鈥檙e in the 20th century. Nothing that is
being discussed now is obvious. One of the big themes this
time is trying understand the foundations deeply. Each time
the answer is 鈥渋t鈥檚 more complicated than you expect.鈥�
Why do we stop in the 1950s? Here's an answer from the book
I've been reading:
John Keay
_India: A History_
2000
Atlantic Monthly Press
"What followed has not lacked purpose;
it is just that, with the direction
less clear, the record easily
degenerates into a year-by-year recitation
of events and statistics.
"For somewhere in mid-twentieth
century ... history at least blends into
the clamorous world of current
affairs. Time's locomotive slows and the
broad horizons of he past are obscured
as the starker shapes of a
high-rise foreground press
close. Gathering up his cherished omniscience,
the historian must get down from the
air-conditioned express with its
tinted windows, cross the tracks, and
elbow his way aboard a slower,
noisier train whose windows have no
glass and whose doors are never
closed. Here, where single-seat
occupancy is unknown and the luggage is
all sacks and bundles, uninterrupted
views are rare. The comfortable
generalisations come less readily,
less confidently.
"With nose pressed hard against a
knobbly reality, the historian soon
makes a disconcerting discovery:
he has been downgraded to the role of a
participating observer, just another
of those contemporary chroniclers
whose testimony he has so often found
wanting. Nor are their failings
now a matter of surprise. It's
not easy, one finds, to make sense of what
is happening before one's eyes.
Until distance lends depth and time
clarity, there is much to be said for
hanging back. Impressions now look
a safer bet than narrative, and the
thumbnail sketch must serve as
illustration."
I may keep pushing off a bit in our last days, and that will get
squared away on Monday. I promise.
Lecture
Reactions
Recap
geometry: Islamic mathematicians
challenged the fifth postulate, tried to prove it by
contradiction, and failed. Lobachevsky and Bolyai proved
results in hyperbolic (which is really the same as above, just
with a different goal). Beltrami first proved that it is
consistent.
The 4-colour theorem was first proven in 1976 by Appel and
Haken. It has been reproven a few times since then by other
computers. There is no printed proof.
Perelman
declined all notoriety from his discoveries and has since been
mostly a recluse. "I'm not interested in money or
fame, I don't want to be on display like an animal in a zoo. I'm
not a hero of mathematics. I'm not even that successful; that is
why I don't want to have everybody looking at me.鈥�
Reading
Reactions
.
Why are people now noticing that tables exist? We've seen
trigonometric tables going back 1500 years, and logarithm tables
from the beginning. The mathematics tables project was
fascinating in that human computers were "programmed" in
parallel. But the existence of tables is definitely not
noteworthy. I am pushing Blanch and the MTP into
Friday. We'll get to them, I promise.
We will talk about philosophy. Materialism is the belief
that all knowledge comes from sensory experiences. In
particular this including embracing intuitionism and hence
avoiding the rule of the excluded middle and therefore not using
proof by contradiction.
The size of the natural numbers is denoted by Aleph_0. I
will write this. This is not the symbol for the cardinality
of an arbitrary set, but it is the particular cardinality of the
naturals. The question of the continuum hypothesis is if
there is a set of size larger than the naturals and smaller than
the reals. The answer is 鈥渃ould be鈥� - both ways are
consistent with mathematics.
We
do still have .