July 2004 - Posts
I was reading a book on Database foundations, recently. There were three foundational notions presented: a Relational Algebra, a Tuple Calculus, and a Domain Calculus. Each of these could be viewed as a formal language for expressing items in the database and for doing calculations on them like queries, joins, intersections, and what not. A sketch was given of a proof that all three are “equal” in power. I'd like to do some riffs on that theme in the near future, but today's topic, really, is “why call one of them an algebra and the others calculi?”
I don't know whether there is a consensus “diagnostic” that can unambiguously tell whether a thing is a calculus or an algebra. It could just be up to “inventor's discretion:“ if you invent something, you can call it anything you like. But I do know some examples of each thing. Things like lambda calculus, pi calculus, predicate calculus, and even college calculus are called “calculi.” Things like groups, rings, fields, vector spaces, and Clifford algebras are called “algebras.” Both of them consist of axioms and rules for transforming expressions. Both of them insist on “closure,” so that if you do some calculations (transformations) with items “in” the calculus or algebra, you end up with items “in” the calculus or algebra.
I can sense, possibly, a little difference in focus. Seems like the focus of a calculus is on the expressions. You have rules for making them longer (composition, for instance) and rules for making them shorter (reduction, for instance). The things that expressions represent (like procedures) are of secondary interest. In an algebra, the focus might be on “theorems” rather than on “expressions.” Now, really, there is little difference between a “theorem” and an “expression,” at least formally. A theorem is something you get when you apply the transformation rules to an expression correctly. But in an algebra, you're really looking for theorems, often using intuition, which is more like magic or spiritual inspiration than it is like calculation.
Well, I guess all I've done here is either confuse myself more or ramble on something that doesn't amount to a hill of beans. I do think that if there IS anything interesting to say about the difference, it will be found in Category Theory, an branch of metamathematics that reasons about things like sets and algebras and calculi from a higher perch in heaven. I have a nice book by Benjamin Pierce on Category Theory. I'll be reading some of it if time allows while I'm at the race track this weekend testing out my old restored Corvette race car.
Cheers till Tuesday.
Ok, rather than trying to blog Math in place, let me just direct you to this little snippet: http://home.comcast.net/~brianbec/Multiplying32bitNumbersWithoutOverflow.htm
Let me know how you like it :) I hate having to redirect you in this way, but it's the best I've been able to do so far. Onward and upward.
When I write, I use mathematical notation all the time. This need has
been a source of continual frustration my entire life. In the 1980's,
I chose Macintoshes whilst all about me were choosing PCs, because
with Macs I could print math. But that choice isolated me,
computer-wise, from my colleagues. In recent years, I've written some
papers using Mathematica (http://www.wolfram.com) and MathType
(http://www.mathtype.com). These tools permit me to write magnificent
paper documents, but sharing the documents electronically has been all
but impossible. The mathematical fonts tend to be copyrighted, so
software will not copy them. Asking my readers to find, buy, and
install copyrighted fonts manually is just too much. While some people
smarter than I, like Peter Ogden, have managed to convert and post
some of my papers (http://phors.locost7.info), I've spent more time
fiddlie-futzing with fonts and markup than I have spent actually
writing those papers, only to meet with failure and frustration. The
best I could do, in the end, was scan the printed form into
bitmaps. Maybe someday I'll post them in that form.
The state of mathematical presentation today is so poor that it kept
me away from blogging these last five months. It's frustrating,
tiring, distracting, and terribly annoying.
I'm not the only one, by the way. One of the finest minds of our time,
no less than Donald E. Knuth, spent about 10 of his prime research
years working on nothing BUT mathematical typesetting. His frustration
with mathematical presentation while creating the Art of Computer
Programming books was so profound that he decided to apply that
selfsame art to slay the dragon forevermore. The result was TeX
(http://www.tug.org) and history.
But, today, I'm going to give it another go. I installed the free
MathPlayer download (http://www.mathtype.com/en/products/mathplayer/)
into my browser, Internet Explorer 6. This lets me look at Web pages
created with various tools supporting the MathML standard. This is a
step forward. It's not integrated with ASP .NET blogging, yet, so I
don't think I can use it to put math in my blog, yet. The reason is
that I can't access the HTML header of the blog to insert the MathML
namespace reference (FRUSTRATION)!
But I'll be working on getting math in here. There has to be a way.
Maybe I have enough energy for just one more push on this problem.
I'd really just rather work on the math, but what's the point if I
can't share it?
If you've got MathPlayer installed, you MIGHT be able to see the
following rendered:
I have a hair-raising story about gummy bears. Gummy bears made eight
of my associates crash their cars on a race track. No one was hurt,
but everyone learned to RESPECT and FEAR the gummy bears.
One of my hobbies is throwing myself around racing circuits inside
automobiles. A racing circuit is a dedicated bit of road with corners
and straight bits. A driver has to put on the brakes and steer and
unsteer here and there. Racing circuits are a little different from
"ovals," where it's often not necessary to use the brakes at all and
you're pretty much steering all the time. A really famous racing
circuit is http://www.sebringraceway.com/track.html . You can see a
professional driving a lap of Sebring here
http://www.davidfarmerracing.com/lap-sebring.wmv .
The activity of driving around a racing circuit requires "traction," a
phenomenon whereby rubber pneumatic tires adhere to asphalt. Other
hobbyists have the same requirement, but in different forms. One of
the other kinds of hobbyist is called a "drag racer." Drag racers
don't use the whole circuit, just a straight bit at the front, which
is about 1/2 a mile long. They also use a "taxiway" to go slowly back
to the beginning of the straight bit. The circuit drivers use a corner
at the beginning of the straight bit and another corner at the end of
the straight bit.
So picture in your mind the drag racers using the straight bit, going
up it, two at a time, really really fast, and coming back via a
separate taxiway really really slow (or in pieces), and the circuit
drivers also using the straight bit, but turning off it at the corner
at the end, going around some back "hot-road" sections with corners
that the drag racers never see, and getting back on the straight bit
at the corner at the beginning over and over again, really fast, but
not really really fast.
The circuit drivers (like me) do laps as quickly as they can. The drag
racers go from beginning to end of the straight bit as fast as they
can, We all share part of the track (the straight bit), but we're
never on the track during the same session, because the circuit
drivers go round and round in a periodic fashion, whereas the drag
racers sit at the starting line fiddling with their engines and what
not, then a "christmas tree" of lights goes red, yellow, yellow,
yellow, GREEN and let them go. Since the "go" times of the drag racers
are not periodic, and since the drag racers go much, much faster than
the circuit drivers (because the drag racers don't turn corners fast),
it would not be feasible to have circuit drivers and drag racers on
the track during the same session.
Ok, gummy bears.
Drag racers use the phenomenon of traction to help them go really
really fast in the shortest amount of time possible. This is called
"acceleration." To get the most acceleration possible, drag racers use
some tricks to make the rubber of their tires more sticky. One of
these tricks is liquified gummy bears (I don't know if it's ACTUAL
gummy-bear candy from the candy store, but it smells EXACTLY like the
candy gummy bears). Before staging their automobiles in front of the
christmas tree for launching, the drag racers "paint" their tires with
liquified gummy bears. They then hit the gas and spin the tires really
really fast, liquifying the rubber and mixing the liquified rubber
with the liquified gummy bears and making a tremendous amount of noise
and smoke that makes everyone laugh unconrollably and spill popcorn
(if you've never been to a professional drag race, you should go. It
does cause uncontrollable laughter when these burnouts happen). When
this liquified mix of rubber and gummy bears cools down just a little
bit, it is really really sticky, much stickier than just rubber on
ordinary tires like the ones the circuit drivers use (we're going to
get back to the circuit drivers in a minute). Christmas tree says GO
GO GO and a pair of drag racers zip down the straight bit as fast as
they can leaving huge streaks of semi-liquid rubber plus gummy bears
on the asphalt. During a session, dozens, if not hundreds, of drag
racers will do this. At the end of a session, a very considerable
amount of gummy bears and rubber will be on the asphalt of the
straight bit of the track. When gummy bears cools off completely, it
is still really really sticky and you pretty much can forget about
ever getting it off the asphalt.
Ok, circuit drivers.
Remember, I said the circuit drivers enter the straight bit via a
corner and leave the straight bit via a corner? This means they are
trying to drive across parts of the streaks of gummy bears at an
angle, rather than straight down the streaks the way the drag racers
do. You see, as the gummy bears accumulate on the race track, the drag
racers are helping each other out. The more gummy bears put down by
the early racers, the happier are the later racers. But gummy bears
create a challenge for the circuit racers. When you are trying to
cross the streaks of gummy bears at an angle and turning in a
curve-like way, you are encountering alternating bits of asphalt alone
and asphalt with a thick layer of gummy bears. The problem is that the
asphalt with gummy bears is much much stickier -- most of the time --
than the asphalt, so the circuit drivers have to plan for this.
You're going to feel "stick" "slide" "stick" "slide" as you drive
across the drag-racers' launching area while driving the circuit. If
you're not prepared for the "slide" parts, you might lose control and
crash into the wall. This would not be good. But there is more.
Ok, we're in Seattle or Portland or Spokane or Vancouver, etc.
It rains here much of the time, even during the summer. And when it
doesn't rain, it drizzles or is just foggy. Guess what? Remember I
said gummy bears are much much stickier than asphalt, most of the
time? The one time they're not stickier than asphalt is when they're
wet, even just a little bit wet. When they're wet, gummy bears are
more slippery than soap on black ice. In fact, they're probably the
most slippery thing you could possibly put on asphalt. And they don't
go away, because, remember, the drag racers put down A LOT of gummy
bears on the asphalt. If the circuit drivers get a foggy or drizzly
morning, their whole day might be spoiled because the gummy bears make
the track very very unsafe to drive. In fact, one day I remember eight
drivers crashing on the circuit and totaling their cars. Silly me, I
tried to drive, too. I didn't crash, but on a section of the track
where, in the sun, I would very safely go 140 miles per hour over and
over again, I could barely go 20 miles per hour without spinning
out. It's that bad. Everyone who tried to drive that day pulled the
slippery mix of wet gummy-bear "soap" all the way around the circuit,
so that before long, the entire circuit was as slippery as the front
bit.
Moral of the story:
Don't drive on road circuits on foggy mornings when the drag racers
have been there the day before, because of the fearsome horror of
gummy bears.
Hello, all. I'm back after being away from blogging for some time. There is quite a backlog of topics to go over and pick up. I'm putting Luna and Scheme on the back burner for a while. I'm working on Clifford algebras, pseudorandom number generation, Monte-Carlo simulation, and Rubik's cube at the moment. In the meantime, take a look at one of my favorite pseudoblogs,
http://math.ucr.edu/home/baez/TWF.html . Really, really good.
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