Monday 3 March 2014

Cantor counting

A month ago I had occasion to use Terence's all-embracing aphorism homo sum - humani nihil a me alienum puto, in the context of what should and should not be allowable in discourse.  Because something makes you feel uncomfortable is not sufficient reason to close up shop on its depiction or discussion.  I'd like to be able to lay claim to a narrower constituency which I won't affect to render in Latin, so Yoda-speak will have to do: "Scientist I am, beyond me nothing within its boundaries is".  Which makes it sufficiently obscure and oracular, but I've already admitted that wide chunks of physics send my brain teetering over the abyss or at least skittering down the inclined plane.

I'm teaching elementary math to our Yr1 biologists, or at least I am hosting a forum whereby the youngsters can use Khanacademy to practice their skills in geometry and algebra.  It reminds me that, when I was their age, I was quite nifty at mathematics and could knock off a bit a calculus, or trick about with set theory.  But I know that there are limits to what I can get my brain to accept.

Today is Georg Cantor's birthday. He was born on 19th February 1845 in St Petersburg Санкт-Петербург Petrograd Leningrad St Petersburg (they've changed the name a few times).  The 19th February was an example of the holdout that the Russians kept from embracing the Gregorian calendar until long after the rest of Europe.  19/02/45 was called 03/03/45 by everyone to the West of the cultural capital of Russia.  Cantor was smart and forced us to think on the nature of infinity in mind-melting ways so that since his time we recognise that there are some infinities that are bigger than others.

He showed first that the infinity of odd numbers is the same size as the infinity of all countable numbers although reason tells us that there must be twice as many of the latter.  He did this by explaining s l o w l y his idea of mapping.  He said that you could pair off the two sets of numbers two at a time
1=1 2=3 3=5 4=7 5=9 6=11 . . . ad infinitum
until they had all gone, so they must be equivalent in size.  Qualified Shazzam!?

If you can accept that you can push yourself to accept that rational numbers (fractions+integers to us) are also countable in the same way.  All you need to do is organise your data in a particular way so that you can tick the numbers off sequentially and know you're not missing any out. Cantor's insight was to set up all the fractions (which includes all the integers because 1 = 1/1; 2 = 2/1 etc.) in a grid and count them off along the blue diagonal against the counting numbers.  So far, so good?  Cantor then went on to apply similar reasoning to show that the points on a line were equivalent to the points in n-dimensional space.  This wrecked even Cantor's head and he famously wrote to his correspondent Dedekind "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!").

You might think that you (or at least mathematicians) can count any set of anything - but you can't.  And Cantor elegantly showed that the real numbers are not countable, that there must be some real numbers that fall between the cracks of the rationals . . . and so the infinity of reals must be bigger than the infinity of the countable.  Real numbers are the rationals (integers and fractions) PLUS the irrational numbers that cannot be represented by a fraction.  Although 22/7 is a damned good approximation for the ratio between the  circumference and diameter of a circle, any fule kno that this ratio is represented by π Pi 3.1415926...  Another beautiful real is the golden ratio ϕ Phi 1.6180339...  Last year, I laid out a bizarre connexion between the three most famous irrationals.

As with the diagonalised fractions, Cantor challenged us to give him any ("big as you like, big as you can imagine, ALL of them") set of real numbers and he would write them in a list one above the other (for convenience just dealing with the digits after the deci-point):
71828 18284 59045 23536 ..... (e)
14159 26535 89793 23846 ..... (π)
61803 39887 49894 84820 ..... (ϕ)
77777 77777 77777 77777 ..... (7/9)
Cantor then showed that we couldn't have included ALL the real numbers in the list because he could imagine/create/write a number that was different from the first number in its first digit AND different from the second number in its second digit AND different from the third number in its third digit etc.  That argument could be re-employed in the new N+1 list, with another number.  Sooooo, the infinity of real numbers is bigger than the infinity of rationals.

So far my head isn't piled up in a heap at the bottom of the cliffs of insanity, but beyond this much of mathematics quickly wings off towards the horizon leaving me behind in a pedestrian grounded world knowing my limits.

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