390 Quick
Answers 11 April
I
have read over 1/3 of the papers, including all submitted before 7
April (more, in fact!). It would be an outrageously bad idea
to not read my feedback very promptly after I post it. I am
slowing down now, as I have other commitments. I am also
trying to update actual current average when or shortly after I
read drafts. Watch for that.
I鈥檓
going to start asking for final topics now. Oh, and also
it鈥檚 a good time to see what we have left. You will finish
the book! And, I will add in some extra things as we get
closer to the end.
For anyone who is interested to know more history of mathematics
at 麻豆传媒团队, here's a (somewhat low quality) .
Not only is there more mathematics as we go forward, but the rate of
increase is increasing. Someone used the word "exponential"
and people use that carelessly, but it is actually relevant
here.
I like that many are saying "this was my project and I know more
than Suzuki says." Yes, you're experts.
Lecture
Reactions
d'Alembert
is making progress with limits. He's not the end of the
story. We'll see more today
Gau脽's 17-gon was the first constructible regular polygon that
hadn't yet been constructed. 3 is Euclid's first
theorem. 4 is easy. 5 is more challenging, but in
Euclid. The rest are by bisecting known ones: 6, (7 is
impossible), 8, (9 is impossible), 10, (11 is impossible), 12, (13
is impossible, as are 14-15), 16 is bisecting 8, and that leaves
17.
Projective
geometry is based originally on what we see. We _see_
parallel lines intersect at a vanishing point. At
some point one realises that "never intersecting" or "equidistant"
are not practical definitions of parallel as they are impossible
to check. A more practical definition and relating more
nicely to projective geometry is "has a common
perpendicular." Again my condolences if this was not in your
geometry course.
Regarding
metric time: After the day is divided into 10 鈥渉ours鈥�, then
each 鈥渉our鈥� is divided into 100 鈥渕inutes鈥� and each of those into
100 鈥渟econds鈥�. Metric time was introduced at the same point
in history as the rest of the metric system and makes as much good
sense as the rest of the metric system. It is much easier to
add 8.07667 to than to add 8:04:36. Having all places
be powers of ten for time is just as helpful as it is for any
other measurement. Someone observed that basically all the
world was happy to adopt decimalised (metric) money - probably
because everyone must do computations with it. Why not
everything else?
Reading
Reactions
The end of our polynomial story comes today (and next time).
Yes, Augustus de Morgan is the one associated with de Morgan's
"Laws" of set theory. That work was related to mathematical
logic, and moving algebra in a purely symbolic direction. He
also did work with trigonometry and complex numbers.
Boole did not introduce the fundamental concepts of mathematics
(e.g. commutative and distributive), they have been used through all
of our history. He used them in a different context. We
will see his work.
It
might slip by in the scope of our main stories for today, but
Liouville proving the existence of transcendental numbers is
significant. Algebraic numbers are numbers that are a
solution to polynomial equations with integer coefficients.
They include rationals, but also include things like 鈭�2 which is a
solution to x^2 - 2 = 0. Transcendental numbers are numbers
that are not solutions to polynomial equations. It turns out
(this is years down in the story) that like you know (I hope) that
most real numbers are irrational, in fact most real numbers are
transcendental. Liouville first proved that the 1/10^{n!}
series (I will write it) is transcendental. I think I know
three non-trivial transcendental numbers (although apparently the
natural log of any positive rational number other than one is
transcendental according to Suzuki according to Liouville).
One more detail on this topic, because I think we won鈥檛 come back
to it, Lindemann proved that 蟺 is transcendental in
1882. This is plenty sufficient to show that the circle
can鈥檛 be squared.
Maybe more than anyone else, Cauchy is responsible for the
limit-style in which you were taught calculus.
Why were logic and mathematics separate? Mostly because they
come from separate historical origins. Mathematics was the
quadrivium (all of it). Logic was part of the trivium which
belongs more to humanities (along with grammar and rhetoric).
Journals are becoming more a topic. Yes, papers are
peer reviewed for journals.
Are
we seeing specialisation in mathematics? Yes,
definitely. We鈥檙e getting closer to the last
universalists. That鈥檒l come at the dawn of the 20th
century. I鈥檒l make a big deal about it.
For
those of you who want Weierstra脽, he鈥檚 German, so he鈥檒l show up
next time. I promise.