Fear
29 - 31 May - and some stuff about counting
A couple of nights back after we'd retired to bed
we were kept awake by the sound of singing or drumming from somewhere very close
to the house; "very close" here meaning either right outside or on the next roof
terrace or both. Paula went downstairs and I lay in bed knowing that she was
capable of going outside and telling them all to shut up, and quietly hoping
that she wouldn't. Well, she shut the window and banged on it meaningfully a
couple of times, and immediately
the drums were joined by louder drums
augmented with a front line of the type of trumpet that they have round here.
Since it's unlikely that they had spare armed musicians on stand-by in case
anyone cut up rough this was almost certainly a coincidence. But the music was
very loud and there was a brief instant when I had to remind myself that it
wasn't threatening - a little like when lions were padding right by our car and
I had to resist a momentary impulse to close the window. In this instance I
ascribe the moment of fear to the fact that pretty much every person I've seen
photographed in the UK press and identified as Arabic over the past few years
had some connection with brutal acts of terrorism. In contrast, the people
we've actually met here have all been really friendly. In reality, the noise was
in all likelihood a festivity associated with a birth or a circumcision or some
other event that's happily celebrated here. At about 1 am I peeked out of the
window and saw girls as young as three or four hanging out with older women;
within about half an hour it was all
over.Another topic for which people
can cultivate a fear is maths. Yesterday I had a very interesting email from my
friend Steve Mildenhall in response to my entry about Andrew Wiles etc.
(Memories of the Blair
Administration). In that blog I claimed that
only a few people in the world were capable of understanding the proof of
Fermat's Last Theorem. It turns out that Steve is one of them. This doesn't
disprove my assertion, by the way: apart from being extremely smart and
hard-working, Steve covered just the type of maths needed in his PhD - and was
prepared to spend 3 months doing little other than reading the Wiles proof. In
his email Steve noted that his all time favourite maths result is that you can't
count the real numbers. One of the beauties of this result is that I can
explain it to you. "Count" means just
what you'd expect: mathematicians reassuringly say that a set is
countable
if you can number the elements, starting with
1 then 2 and so forth. Key point is that you don't have to reach a number where
you
stop
counting for the set to be countable - this describes infinite
sets.Let's put aside for a moment what
"real" means here, and why you can't count the real numbers. There are plenty
of sets of numbers that you
can
count. For example, the set of even numbers is countable: 2 is the first, 4 is
the second, 6 is the third etc. The set of all positive and negative whole
numbers (..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...) is called the
integers. While you may think that there are twice as many of these as there
are counting numbers (1, 2, 3 etc) it turns out that you can still count them.
For example, you can go 0, 1, -1, 2, -2, 3, -3, 4, -4 etc. and every one of the
integers will show up on the list that you make.
More surprisingly, the set of all
numbers that you can write as a fraction - like 3/4 or 22/7 - is also countable,
though this requires a bit more guile to show. First, you have to show a way to
lay them all out. A good way way is to make a grid with the numerator (the
number on the top of the fraction) along the y-axis and the denominator (the
number at the bottom of the fraction) along the x-axis. Then every rational is
given by one of the grid points.Having
laid the numbers out on a grid you can count them by starting in the corner and
working outwards like
this: The
squiggly line gives you the order in which you count the numbers onto your list
(if it troubles you that some of these are equal - like 2/2 and 3/3 - you can
skip over them after their first appearance).
Well you may wonder if there's
anything that you
can't
count. There is. The numbers like those above that can be expressed as
something over something else are called the
rationals.
In contrast, the entire set of numbers that can be expressed using decimal
notation is called (for reasons that you'll be questioning by the end of this
blog) the
reals.
Many people know that pi is an example of a number that can't be expressed
precisely as a fraction - 22/7 being an approximation. It's easy to prove that
the same is also true of the square root of two. (I wont do it because I'd need
to use something that would start to look like an equation.) Now we get to the
interesting argument that a mathematician called Cantor found to prove that the
reals aren't countable, which runs as
follows....Let's suppose that someone
claims that they
did
find a way to count the real numbers and got them into a list like we did for
the evens and the integers. Just to make it concrete, lets assume that the
first four numbers in the list
are:something.2451890...something.7615208...something.3212122...something.8734261...something.4545452...Cantor
pointed out that for any such list you can make another real number that isn't
on. Start it with 0. and then set the first digit after the decimal place to be
anything that isn't 2 (thus making it different from the first number on the
list), the second digit after the decimal to be anything except 6 (thus making
it different from the 2nd number), the 3rd digit after the decimal to anything
except 1 (thus making it different from the 3rd number) etc etc. Carrying on
with this ad infinitum ensures that the number you make isn't the same as any of
the numbers on the list, and so the list
can't
cover all of the reals as claimed.This
is Steve's favourite result - I
told
you I could explain it! Many of Cantor's contemporaries thought that this was
very fishy.It shows that there's more
than one kind of infinity and that in a sense the infinity of the reals (called
aleph-1 by mathematicians) is "bigger" than the inifinity of the rationals
(called aleph-0). The way that we say this while avoiding saying that one type
of infinity is bigger than another (which could sound like nonsense) is to say
that aleph-1 has greater
cardinality
than aleph-0. There is a continuity hypothesis that there aren't any infinities
with cardinality between aleph-0 and aleph-1, but there is no known logical
proof for this.When I explained this
to Zoe this afternoon (over mint tea and Sprite at the Nejjarine museum) I
likened it to when you learn about monkeys: when you're young you know that
monkeys are brown hairy animals that swing in the trees. Then when you get
older you learn that monkeys come in vervet, squirrel, spider and other types.
They all generally have the monkey characteristics that you first learned but
there is more nuance than you first appreciated. Same with
infinities.The infinities higher than
aleph-0 are really
weird.
There is a famous result with a geeky name (the Banach-Tarski paradox - Mike
said I should mention it) that proves that you can take a "real" sphere, split
it into 4 parts and a dot in the middle and then re-assemble it into two spheres
each
identical
to the original. This provides a good hint that the
reals
aren't real.Another sign of the
weirdness of the reals is to try to find some? Probability tells us that if you
pick a number at random on the number line it will, certainly, be a real that's
not rational. However, you can easily show that the set of decimal-style
numbers for which you can provide an algorithm to say how to determine which
digit is at each place of the decimal expansion - or even the wider set of
numbers for which you can give any
sort of description - is countable. So the
uncountability of aleph-1 is provided by numbers for which in principle there
can be no description.There's a lot
more that could be said about this but for now I'll hint that the point where it
goes funny is where you're allowed to keep making a free choice of the next
digit ad infinitum...Do you find all
of this scary, boring or
interesting?This morning I went to get
a paper and some vegetables, which is pretty much part of my daily routine, and
wore my shorts in the medina for the first time. All the time I was out I
didn't see another adult's knees and it felt really uncomfortable; shows how
quickly you can fall into the small habits of cultural acclimation.
I read in the paper that the French
have voted
Non
to the "constitution". I never would have expected that. The confusion into
which Europe is now propelled will be entertaining and Corsica may be an
interesting place from which to watch what happens next - we'll be there at the
weekend.
Posted: Wed - June 1, 2005 at 01:04 AM
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Published On: Feb 08, 2006 06:20 PM
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