Cardano, The Gambling Scholar
by
Oystein Ore
Princeton University Press, 1952.
On Renaissance Gambling and the Analysis Thereof
Cardano was a gambler, and an excellent one. Cardano started gambling in his college days, and earned quite a lot by it. Ore includes
(page 109) an anecdote from Cardano's autobiography, in which Cardano brags of gambling with a nobleman every night, taking home
``about one gold piece'' each time, until the nobleman cried quits and made Cardano swear not to come to the nobleman's house to
gamble anymore. Cardano's gambling was probably the impetus for his analysis of games of chance, in two ways. First, as Ore suggests,
Cardano was interested in the mathematical problem of probability, and second, as Bellhouse [1] suggests, Cardano's writing were an
attempt to justify gambling in a society that often condemned it. The Liber de Ludo Aleae is a collection of drafts and
notes that Cardano wrote on gambling over a period of years; these drafts were collected and printed within his Opera Omnia
in 1665. The Liber, published after the work of Fermat and Pascal on probability in 1654, had little impact on the
development of probability.
Ore, in his analysis of the Liber, untangles Cardano's drafts, explicates Cardano's changing conceptions of probability, and constructs an understanding of certain Renaissance gambling practices. Cardano focused on the probabilities involved in dice and card games. A typical dice game was to bet on the number of throws needed to roll a double six, while the card game on which Cardano focused most of his attention was primero, a game whose actual rules are lost to posterity, although Ore reconstructs much of it from Cardano's writings. Cardano's work on dice led him to the correct rules for the probabilities of two dice, although not without some digressions. Cardano's conception of probability changed during the course of writing the Liber. He first used what Ore calls `reasoning on the mean', in which Cardano argues that, if the chance of getting a 6 in one roll of the die is 1/6, then the probability of getting 6 in two rolls of the die should be 2 times 1/6. Cardano realized that this did not lead to the correct results, and shifted to a frequency conception of probability. Ore does an excellent job of making sense of Cardano's writings on this topic. Primero, [2] one of the most popular card games of the sixteenth century, involved bidding, in which players state their hands (or understate/overstate it), and at the end of the hand, if at least two players remain, the player with the best hand who didn't overstate wins the pot. Primero had an interesting provision when there are two players left, in which the player with the lowest bid can offer a settlement to the other, in which the pot will be split according to their rough estimate of the probabilities of each winning. There were standardized settlements, depending on what the bids were, and Cardano analyzed who benefited more from these standardized settlements. He found that the weaker player generally received more from a settlement than from playing out the hand. Bellhouse argues that this argument of Cardano was based on a sense of justice, that gambling is just when precisely when the game is what we would call `fair', and such gambling should not be condemned. This analysis by Bellhouse is one of the ways that new research has complemented the analysis of Ore.
On Cardano, His Life and Character
Cardano was born around 1501 in Milan, possibly illegitimate. His father was a lawyer and a teacher of geometry, although
not at a university. In 1520 Cardano matriculated at the University of Pavia and defended his doctorate in 1526. He was popular
with his fellow students, and footed many of his expenses by gambling. After receiving his doctorate, he applied
for admittance to the College of Physicians in Milan, but was rejected, nominally because of questions about his birth,
but most likely because Cardano was, by all accounts, an overwhelming asshole. Cardano moved to a small town and opened a
practice, but it was not particularly lucrative, and he moved back to Milan (and, for awhile, into the poorhouse) where he
had social connections, both his own from college and the friends of his father, who got him his father's old teaching job.
In 1536, he published the book that launched his reputation: De Malo Recentiorum Medicorum Usu Libellus,
or On the Bad Practices of Medicine in Common Use. This book contained a mixture of good medicine and attacks on
practitioners of bad medicine, a book flamboyant and aggressive in the best Renaissance style. Cardano and his book became very popular, and
in 1539 he was admitted to the College of Physicians in Milan, and published two texts on mathematics. Johannes Petreius of Nuremberg offered
to print any further texts the now well-known Cardano wrote, and so the great Ars Magna made its debut in Nuremberg in 1545.
Cardano was much more a physician than a mathematician, and wrote several more books on medicine. By 1551, as Ore says,
Cardano was one of the best known physicians in Europe, so much so that John Hamilton, archbishop of Scotland,
in 1552 summoned him to Edinburgh to cure Hamilton's asthma. Cardano's trip to Scotland became a sort of triumphal progression through
the capitals of Northern Europe, and his cure of Hamilton's asthma proved to be very lucrative. Cardano's later years were not
so triumphant. In 1560, Cardano's eldest son poisoned his wife, and was executed. In 1570, Cardano was accused of heresy (reportedly
for having cast the horoscope of Jesus) and was briefly imprisoned and his books were put on the Index . He believed, as
did many in the Renaissance, in astrology and divination, but he drew the line at what he considered black magic.
He cleared his name, and died in Rome in 1576.
Cardano was, in all aspects, truly a Renaissance man. The breadth and depth of his interest was astounding, but it was coupled with a prideful, arrogant, violent, humorless character. One of the weaknesses of this book, I feel, is in Ore's trust in Cardano's depictions of himself. Ore draws from Cardano's autobiography [3] , and from anecdotes within Cardano's writings, for his depiction of Cardano. Yet there is a genre of Renaissance autobiography, in which Cardano's writings fall, in which truth and boasting are twined together and difficult to separate. Benvenuto Cellini's Autobiography is a classic of this genre, and notoriously full of misrepresentations. Ore's reliance upon Cardano's writings emerges in his discussion of Cardano's feud with Tartaglia, in which Cardano is assessed as `frank, but somewhat flavored', while Tartaglia is `not always reliable'. In my reading, it seems that Ore admires Cardano, and that Ore's analytic powers are focused on Cardano's mathematics but not upon Cardano's writings.
Ore's account of Cardano's disputes are drawn from primary sources, either letters or books written by the disputants themselves or by contemporaries. He uses these sources to tell a straightforward story, one that research has amplified over the last fifty years. For example, Ore says of Tartaglia ``His command of Latin, for instance, remained insufficient, and so, contrary to the scholastic tradition of his day he preferred to publish books in Italian, actually in Venetian dialect. Tartaglia was an abbaco teacher, not a university professor; he was not working entirely within the scholastic tradition. Furthermore, Latin served a gate-keeping function in the Republic of Letters, in that those who could not express themselves well in Latin were suspect as scholars. The whole question of Tartaglia's grasp of Latin and the response to that of his peers has been embedded, by more recent researchers, in the institutionalization of mathematics within the Latin-speaking universities. There is more context to the role of Latin than Ore was aware of.
Closing
Ore's book is still worth reading. The translation of the Liber de de Ludo Aleae is, as far as I know, the
only English translation, and has apparently stood up well, despite the complete lack of any critical apparatus. Ore's
analysis of the mathematics of the Liber is well done, and his comments on the games of the Renaissance are good.
His stories of Cardano lack the social context that has been provided by later research, and that remains the biggest
mark against the book. It's an excellent read, though, and I heartily recommend it.
249 pages in English, published in hardcover by Princeton University Press 1953.
It was republished by Dover in 1965.