Sunday, 19 June 2011

The first theoretical probability analysis was for one die. The second was for two. Both are distinct monuments in the history of ideas.

Cardano is given credit with being the first to correctly calculate a theoretical probability.  I disagree.  FN David, whose book seems to be a key reference in many subsequent books on the history of probability, gives Cardano this credit while being aware that for many centuries before, empirical calculation of the likelihoods of dice outcomes (and outcomes from other kinds of randomisation machines) were known.

Specifically, the people who took the given astragalus and chipped and rendered it into a die must have been working towards a theoretical goal of equi-probability.  They must get credit.  They kept chipping away until they reached their goal - a tolerable or measurable closeness to a theoretical model of equi-probability.  This may seem to trivial to be a calculation, but it surely was.  And it was even perhaps literally a calculation too, since the word has its origin in the latin work for a small counting pebble made of the easily manipulable limestone.

What Cardano did was correctly showed how you really needed to enumerate all possible combinations of complex outcomes, to see through equivalence classes to complete enumerations.  Someone else, lost forever to history, introduced the idea that the fraction of elementary outcomes over the universe of elementary outcomes was a significant fraction.  We had up until then been lulled into making the wrong choices between permutations and combinations due to the similarity of two (or more) dice.  If humanity had started rolling identically crafted randomisation machines with clear and distinct markings on them, perhaps we would have got it earlier.  If the pair of dice were marked with 1 to 6 on the first and 7 to 12 on the second then it might have clicked earlier.  But it didn't.  And Cardano (or his side-kick Ferrari) get the credit for disambiguating combinations and permutations.  I'll do another posting on combinations and permutations, which are always badly explained in all probability books I've read.  It is still, I think, quite a non-intuitive idea for us to understand.

What Cardano's work has enabled, however, is the possibility of the correct analysis of the repeat of the same experiment many times.The rolling of two dice, of one die any number of times, etc.  By showing us how to properly discover the elementary sample space, he pioneered a significant piece of the statistical approach to scientific experimental design and has quickened the pace and reliability of science itself.

Google AdSense a poor Judge

Google recently sent me $50 worth of vouchers on AdSense; thanks.  I signed the Persistent Struggle up to a campaign, only to receive an email from AdSense reviewers stating that I couldn't use it since they considered the blog contentless except for adverts.  I queried this decision on several occasions; they reviewed it and uphold their opinion.  Now they've disabled the very possibility any monetisation at all, even independent of AdSense, on the basis that it has adult content.

Virtually every single link I have in there is either to Wikipedia or academic books for sale in Amazon, about which I think my blogs clearly add content.  My guess is some prude at Google doesn't like the accompanying images to my blog posts and has decided to treat my blog in the same category as adult porn.

Shame on you Google.

Monday, 13 June 2011

Every which way

I've having a great time reading up on the history of probability right now.  I'm coming to the end of a stint trying to get to the bottom of what the Italian Cardano actually contributed to the history of probability (separate blog post soon to come), and have found a connection with an earlier man, a Spaniard called Ramon Lull which gets to the essence of what Cardano is supposed to contributed  - combinatorics.

Both men seem to have been somewhat obsessed with sex. Now it is clearly the case that sex and gambling have always been popular activities of the rich and powerful, with no doubt the evolutionary biologist offering up the conspicuous spend implied in gambling as a form of cultural peacock feather.  But perhaps even the men who invented the subject were likewise disposed.

Cardano, one of the first men to write his own detailed autobiography, gives the following description of himself: "I was ever hot-tempered, single-minded and given to women.  From these cardinal tendencies there proceeded truculence of temper, wrangling, obstinacy, rudeness of carriage, anger, and an inordinate desire for revenge in respect of any wrong done to me." (FN David, p42).  David also tells if how Cardano's day job as a medical doctor was frustrated by the refusal of the College of Physicians of Milan to allow him to join their ranks  "..possible because of his reputation as a gambler and his licentious conduct..."( FN David, p44)  When he was a student there was apparently gossip about  his predilection for "singing boys" , which according to F.N. David became louder after his wife dies, and his university wanted to sack him due to the ".. number and character of young men who frequented his house".  His daughter eventually wrote him a letter saying she was  ashamed to be related to him.

It is somewhat ironic that his main contribution to probability was his insistence on visiting all combinations of the possible when examining simple randomisation machines.  Perhaps these complete samplings bore a similar motivation to that more famous libertine the Marquis De Sade, whose descriptions of sexual couplings was exhaustively complete.  The other ironic element of Cardano's life  was his persecution complex , which is a kind of re-telling of the more or less random facts of his life in a way which banishes randomness and replaces it with an authoritarian determinism.    He then finally loses his job by doing the very opposite to religion - he publishes a horoscope of Jesus - thereby applying the everyday use of spiritual/divinatory randomness to a religious figurehead.  In one last overturning of randomness and determinism, he is said to have forecast his own death date by divination, then proceeded to make the prediction come true by starving himself.  A weird kind of performativity. 

In "The Drunkard's Walk" (written nearly 50 years later, in 2008), we hear that the official reason for dismissal was "sodomy and incest".  We hear that his mother was promiscuous, that his daughter was promiscuous, and at 16 seduced her brother.    This son, Bernstein (p46) is described as "deaf in his right ear, ..he had two toes on his left foot, his third and fourth counting the great toe ... were joined by one membrane.  His back was slightly hunched..".  Perhaps incestuous combinations had been going on for generations, given this list of low grade genetic abnormalities.    Cardano is the first mathematician supposedly to describe so called Cardanic cycloids the geometry of circles within circles.  But these diagrams seem to me quite related to the work of early logician Raymond Lull.  He used circles within circles to enumerate many possibilities in a logical and consistent way.  His main purpose was religious, but that is hardly surprising for a 13th Century thinker.  "The beauty of women, oh Lord, has been a plague and tribulation to my eyes for because of the beauty of women I have been forgetful of thy great goodness and the beauty of thy works" Logic Machines and Diagrams (p3, Gardner).

Is it too fanciful to suggest that both men transformed a passion for exhaustive sexual coupling into the beginnings of combinatorial mathematics?  Lull's conversion happened while riding a horse - he spotted a woman he'd been  trying to fuck for some time enter a cathedral.  Still mounted, he rode into the cathedral in pursuit of her.  She arranged to meet him in a private room, whereupon she showed him her breasts - but they were riddled with cancer.  Take a good look at the foulness of this body you've been lusting after.  You would have been better advised to pursue the Christian God than my body.  Lull went on to create elaborate and exhaustive rules of combining ideas and words - partly a proto-automatic writing prop in the style of Burroughs, partly one of the second millennium's first logic machines.

In another post, I'll look at the ideas of combinations and permutations from a mathematical point of view, as they appeared in Lull and Cardano.  The kind of abstract transformation embodied in Lull and Cardano's stories reminds me of Duchamp's The bride stripped bare by her bachelors, even