390 Quick
Answers 12 April
I
hope that I have read 1/4 of the papers (more, in fact!). It
would be an outrageously bad idea to not read my feedback very
promptly after I post it.
I’m
going to start asking for final topics now. Oh, and also
it’s 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.
I would be pleased for more compelling reactions that would
inspire more inclusion in quick answers.
Lecture
Reactions
I
spell Gauß as he spelled it. We will see this letter again
on Monday with Weierstraß.
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 “hours”, then
each “hour” is divided into 100 “minutes” and each of those into
100 “seconds”. 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
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).
Invariants
are well named, as objects that don’t change. (Sylvester
was famously good at naming things.) They are useful to
compute when attempting to recognise mathematical objects.
For equations, the discriminant is something that doesn’t change
when translating the equation from left to right.
Journals are becoming more a topic. Yes, papers are
peer reviewed for journals.
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 √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’t come back
to it, Lindemann proved that π is transcendental in
1882. This is plenty sufficient to show that the circle
can’t be squared.
Are
we seeing specialisation in mathematics? Yes,
definitely. We’re getting closer to the last
universalists. That’ll come at the dawn of the 20th
century. I’ll make a big deal about it.
For
those of you who want Weierstraß, he’s German, so he’ll show up
next time. I promise.