(#3) A doctor, a gambler, a mathematician and lesser known founder of Theory of Probability & Complex Numbers

A true desire to gamble and win led to the discovery of probability

¹(reading time: 15 mins)

“I prefer solitude to companions, since there are so few men who are trustworthy, and almost none truly learned. I do not say this because I demand scholarship in all men -- although the sum total of men's learning is small enough; but I question whether we should allow anyone to waste our time. The wasting of time is an abomination.”

- Girolamo Cardano (1501 - 1576)

Golden words from Girolamo Cardano. A forgotten hero, the father of Probability, and Complex Numbers deserve our attention and gratitude. This is my attempt to weave an interesting story on his life while trying to uncover his mathematical genius without ignoring the struggles and achievements in his life.

Early Years

Girolamo Cardano was the illegitimate child of Fazio Cardano and Chiara Micheria. His father was a lawyer in Milan but his expertise in mathematics was such that he was consulted by Leonardo da Vinci on questions of geometry. In addition to his law practice, Fazio lectured on geometry at the University of Pavia. When he was in his fifties, Fazio met Chiara Micheria, who was a young widow in her thirties, struggling to raise three children.

Chiara became pregnant but, before she was due to give birth, the plague hit Milan and she was persuaded to leave the city for the relative safety of nearby Pavia to stay with wealthy friends of Fazio. Thus Cardano was born in Pavia but his mother's joy was short lived when she received news that her first three children had died of the plague in Milan. Chiara lived apart from Fazio for many years but, later in life, they did marry.

Cardano at first became his father's assistant but he was a sickly child and Fazio had to get help from two nephews when the work became too much for Cardano. However, Cardano began to wish for greater things than an assistant to his father. Fazio had taught his son mathematics and Cardano began to think of an academic career. After an argument, Fazio allowed Cardano to go university and he entered Pavia University, where his father had studied, to read medicine despite his father's wish that he should study law.

When war broke out, the university was forced to close and Cardano moved to the University of Padua to complete his studies. Shortly after this move, his father died but by this time Cardano was in the middle of a campaign to become rector of the university. 

His campaign for rector was successful since he beat his rival by a single vote.

World of Gamble

Cardano squandered the small bequest from his father and turned to gambling to boost his finances. Card games, dice and chess were the methods he used to make a living. Cardano's understanding of probability meant he had an advantage over his opponents and, in general, he won more than he lost. He had to keep dubious company for his gambling. 

Once he lost money at an alarming rate in a gambling game. When he finally discovered that the cards were marked he did not hesitate to draw his dagger.

(Turns out honesty was not a virtue in those days - as depicted by below painting - Two cheats and one dupe)

The Cardsharps c. 1594 by Caravaggio

Two cheats and one dupe. One cheat, who concealed extra cards behind his back, plays with the unworldly boy while his accomplice peeps at the victim’s hand and signals with his fingers. This was Cardano’s world.

He stabbed the cheat in the face and forced his way out of the gambling den into the narrow streets of Venice, recovering his money on the way. Running for his life in complete darkness he slipped and plunged into muddy waters of a canal. Not the best place to be, especially if you cannot swim. It was sheer luck that he managed, somehow, to grab the side of a passing boat and get himself lifted to safety by a helpful hand.

However, once he regained his posture on the board, Cardano found himself facing the man with a bandaged face, the same who had cheated him at the gambling table few hours previously. Perhaps it was the chill of the night that cooled the tempers or perhaps neither of the two wanted trouble with the notoriously strict Venetian authorities, the fact is, there was no brawl. Instead, Cardano was given clothing and travelled back to Padua in an agreeable company of his fellow gambler.

Early years of ‘Probability’

We do not know for sure, but it could have been the Venetian incident that prompted Cardano to write notes on probability. He knew that cheating at cards and dice was a risky endeavor, so he learned to win “honestly” by applying his discoveries concerning probabilities.

His Liber de Ludo Aleae (The book on games of chance), is a compilation of his scattered writings on the subject, some of them written as early as 1525, some of them later, around 1565 or so.

He started with the notion of fairness or, as he put it, “equal conditions”:

The most fundamental principle of all in gambling is simply equal conditions, e.g. of opponents, of bystanders, of money, of situation, of the dice box and of the die itself. To the extent to which you depart from that equity, if it is in your opponent’s favour, you are a fool, and if in your own, you are unjust.

He correctly enumerated the various possible throws, i.e. 6 for one die, 6 ×6 for two dice, and 6 ×6×6 for three dice. For example, when discussing the case of rolling two symmetric dice he wrote ...

there are six throws with like faces, and fifteen combinations with unlike faces, which when doubled gives thirty, so that there are thirty-six throws in all,...

Trivial? Perhaps, but, for the time, Cardano showed remarkable understanding that the outcomes for two rolls should be taken to be the 36 ordered pairs rather than the 21 unordered pairs.

Cardano’s careful enumerations provided, at the very least, good explanations why certain numbers of points were more advantageous than others. This was something many dice-players had known from their experience, and even though they could relate it to the number of ways the throws can come out their counting was very problematic.

For example, it was known, and regarded as puzzling , that in a throw of three dice the sum of points is more likely to be 10 than 9, even though there are six ways in which the sum can be nine

1 + 2 + 6, 1 + 3 + 5, 1 + 4 + 4, 2 + 2 + 5, 2 + 3 + 4, 3 + 3 + 3,

and there are also six ways for the sum to be ten,

1 + 4 + 5, 1 + 3 + 6, 2 + 4 + 4, 2 + 2 + 6, 2 + 3 + 5, 3 + 3 + 4.

The fact that the outcomes should be taken to be ordered triples (27 of which sum up to ten but only 25 to nine) was not well understood.

Thus even if Cardano’s discussion had been limited to calculating the correct chances on dice and cards, it could have been regarded as a great achievement, but he went further than that.

He made several insightful general statements about the nature of probability. For example, he realized that when the probability of an event is p, then by a large number n of repetitions the number of times the event will occur is not far from np.

All this was written more than a century before Chevalier de Mere, an expert gambler, consulted Blaise Pascal (1623–1662) on some “curious problems” in games of chance. Pascal wrote to his older colleague Pierre de Fermat (1601–1665), and it was through their correspondence, as we are often told, the rules of probability were derived. The fact is, Liber de Ludo Aleae appeared in print over eighty years after Cardano’s death and about nine years after Pascal’s first letter. Thus, it is reasonable to assume that it had no impact on the subsequent development of the subject.

However, in all fairness, one should recognize the fact that Cardano was the first to calculate probabilities correctly and the first to attempt to write down the laws of chance.

Solution to the enigmatic cubic equation

In 1526 Cardano was awarded his doctorate in medicine. The first few years of his medical practice, in a small village near Padua, were difficult, but eventually fortune smiled on him and he was appointed a public lecturer in mathematics in Milan. His interesting and entertaining lectures attracted large audiences.

Married, with three children, he settled to a relatively comfortable life. He gradually established a new medical practice and acquired influential patrons and patients. In 1536, a fifteen year-old Lodovico Ferrari entered Cardano’s service as an errand boy. Cardano soon realized that he had acquired an exceptional servant. For all his irascible temper, Lodovico was a mathematical prodigy and before long he became Cardano’s most loyal friend and disciple. The two became very close collaborators and went on to work out a general algebraic solution to the cubic and quartic equations, probably the most important mathematical achievement of the 16th century.

Although solutions to some particular cubics had been known for some time it was a general algebraic solution that still eluded the best minds of the Renaissance. Then, sometime around 1515, Scipione del Ferro (1465–1526), a professor of mathematics in Bologna, found a general rule for solving a specific cubic equation of the form x3 + cx = d, with c and d positive. It was a real breakthrough, but del Ferro kept the solution secret. Why?

Because, many mathematical tools were treated like trade secrets at that time. It may sound strange in our publish-or-perish age, but keeping some mathematical discoveries secret was quite common at the time. After all, del Ferro and his colleagues made living by offering their services to whoever offered the best pay. Patronage was hard to come by, there was no tenure, university positions were few and held by virtue of eminence and reputation, and challenges could come at any time. Scholars were often involved in animated public debates that attracted large crowds. Students, merchants, noblemen and all kinds of spectators would gather in public squares or churches to watch the spectacle. The basic rule of combat was that no one should propose a problem that he himself could not solve. Reputations, jobs and salaries were at stake.

We do not know whether del Ferro ever used his result in a mathematical contest, but he was aware of its value, and shortly before his death in 1526, he passed the secret to his son-in-law, Annibale della Nave, and to one of his students, Antonio Maria Fiore. Although neither man published the solution, rumors began to spread that the cubic equations had been solved.

In particular Niccol’o Fontana of Brescia (1499 or 1500–1557), better known under his nickname Tartaglia boasted that he had discovered the solution to cubics of the form x^3 + bx^2 = d (again b and d positive).

At the time Tartaglia was a teacher of mathematics in Venice and Fiore, a native of Venice, was very keen on a good teaching job in his hometown. Confident in his mathematical abilities, Fiore challenged Tartaglia to a public contest. It was a bad idea. Tartaglia was a much better mathematician and, as it happened, the night before the contest on the 13th of February 1535, he had figured out del Ferro’s solution.

The contest, held in Venice, was a humiliating defeat for Fiore and a great victory for Tartaglia. Overnight, an unknown teacher of mathematics from Venice became a nationwide celebrity. His star was rising. However, it ain’t over ’til the fat lady sings.

Enter Cardano.

When he heard of Tartaglia’s triumph, Cardano asked him to reveal the secret and to give him permission to include the solution to cubic equations in Practica arithmeticae generalis, a book which he was preparing for publication. He promised to give full credit to Tartaglia, but Tartaglia categorically refused, stating that in due time he himself would write a book on the subject.

When? He would not say, as he was preoccupied with his work on ballistics and translating Euclid’s Elements into Italian.

Cardano did not give up. After several exchanges of letters Tartaglia accepted an invitation to visit Milan, possibly in the hope that through Cardano’s connections he could secure a lucrative job with the Spanish governor in Milan. The visit took place on the 25th of March 1539. This much we know for sure, but what exactly happened during the visit is not clear. We have two contradicting stories, one by Tartaglia and one by Ferrari.

Tartaglia claimed he divulged the secret to Cardano in the form of an enigmatic poem but only after Cardano had taken a solemn oath to keep the solution secret.

Ferrari, who was present at the meeting, swore that Cardano took no oath of secrecy.

The word of one man stands against that of the other.

One way or another, in May 1539 Practica arithmeticae generalis appeared without Tartaglia’s solution.

However, Ferrari and Cardano were working hard and managed to extend Tartaglia’s method to cover the most general case of the cubic equation. Ferrari went even further and worked out solutions to quartic equations. Meanwhile, Tartaglia still had not published anything on the cubics. In 1543, following rumors about the original discovery by del Ferro, Cardano and Ferrari travelled to Bologna to meet Annibale della Nave. After examining del Ferro’s papers they found a clear evidence that twenty years earlier he indeed discovered the same solution as Tartaglia. Thus, even if Cardano had been sworn to secrecy the oath was no longer valid.

In 1545 Cardano’s Ars Magna was published. It was a breakthrough in mathematics, a masterpiece comparable in its impact only to De revolutionibus orbium coelestium by Copernicus and De humani corporis fabrica by Vesalius, both published two years earlier.

In the book Cardano explores in detail the cubic and quartic equations and their solutions. He demonstrates for the first time that solutions can be negative, irrational, and in some cases may involve square roots of negative numbers

After the publication of Ars Magna Tartaglia flew into a wild rage and started a campaign of public abuse directed at Cardano and Ferrari. He published his own work New Problems and Inventions which included correspondence with Cardano and what he maintained were word-by-word accounts of their meetings.

They can hardly be regarded as objective, in fact, they read like a public rebuke.

However, Cardano, who was now regarded as the world’s leading mathematician, couldn’t care less. He did not pick up the fight, letting his loyal secretary, Ferrari, deal with it. And so Ferrari did. He wrote a cartello to Tartaglia, with copies to fifty Italian mathematicians, challenging him to a public contest. Tartaglia however did not consider Ferrari as worthy of debate - he was after Cardano. Ferrari and Tartaglia wrote letters, trading insults for over a year until 1548 when Tartaglia received an offer of a good teaching job in his native town of Brescia. Most likely, in order to establish his credentials for the post, he was asked to take part in the debate with Ferrari.

Tartaglia was an experienced debater and expected to win. The contest took place in the Church of Santa Maria del Giardino in Milan, on the 10th of August 1548. The place was packed with curious Milanesi, and the governor himself was presiding. Ferrari arrived with a large entourage of supporters, Tartaglia only with his own brother; Cardano was, conveniently, out of town. There are no accounts of the debate but we know that Tartaglia decided to flee Milan that night.

Ferrari was declared an undisputed winner.

‘Complex Numbers’ gets legitimacy

Back in the Renaissance negative numbers were treated with a bit of suspicion, so taking roots of the suspicious numbers must have been almost heretical. After all solving equations meant solving specific mercantile or geometric problems. Thus “things” were measurable entities and whenever solving the quadratic equations, such as x^2 + 1 = 0, led to the square root of a negative number it was assumed that the problem was meaningless with no solutions

Cubic equations were different. Some of them had perfectly respectable solutions, which could be easily guessed, and yet the square roots of negative numbers popped up halfway through, in the derivations of these solutions, and there was no way to avoid or to ignore them. This, to say the least, was puzzling. The general solution to the depressed cubic reads

where

With their confusing notation and their reluctance to accept negative numbers the Renaissance mathematicians initially failed to grasp that this is indeed the general formula, which solves all cubics not just some specific cases.

Cardano discusses a problem of finding two numbers which sum to 10 and such that their product is 40.

The solution is, of course, 5± √ −15.

Finding it difficult to make sense out of such “numbers” Cardano took a purely instrumental approach. He noticed that if you are prepared to ignore the question of what the square root of minus fifteen meant, and just pretend it worked like any other square root, then you could check that these mathematical entities actually fit the equation.

He wrote:

“Putting aside the mental tortures involved, multiply 5+√ −15 by 5− √ −15, making 25 − (−15) which is +15. Hence this product is 40.”

In his book, Ars Magna, Cardano remarks that √ −9 is neither +3 nor −3 but some “obscure third sort of thing”.

This is how complex numbers were announced to the world.

Today, these “useless” discoveries are indispensable to all practicing physicists and mathematicians. Indeed, complex numbers and probabilities underpin the best framework theory we have today - a superb description of the inner working of the whole physical world - the quantum theory that I describe further in my post here.

Epilogue

Let us get back to 1548.

What happened after the contest in the Church of Santa Maria del Giardino in Milan? Tartaglia’s appointment in Brescia was not renewed. He taught there for about a year but his stipend was not paid. After many lawsuits, he returned, seriously out of pocket, to his previous job in Venice. He spent much of the rest of his life plotting and collecting a dossier against Cardano.

Tartaglia died in poverty in Venice on the 13th of December 1557.

Ferrari became famous. He was appointed a tax assessor to the governor of Milan, and soon after retired as a young and rich man. He moved back to his hometown - Bologna - where he lived with his widowed sister Maddalena. In 1565 he was offered a professorship in mathematics at the university but, unfortunately, the same year Ferrari died of arsenic poisoning, most likely administered by his sister.

Maddalena did not grieve much at his funeral and having inherited his fortune, remarried two weeks later.

Her new husband promptly left her, taking with him all her dowry.

Maddalena died in poverty.

Cardano outlived both Tartaglia and Ferrari. In 1546, fifteen years into his marriage, his wife died leaving Cardano the sole caretaker of his three children. This did not seem to affect him that much. He remained in Milan, lecturing in geometry at the University of Pavia, making money both as a physician and as a writer. From the day he published Ars Magna till about 1560 he had everything he could possibly wish for - fame, position, money, respect. These were his golden years, but then, as it often happens, came difficult times.

His eldest and most beloved son, Giambatista, married a woman, Brandonia di Seroni, by all accounts, a despicable character. A woman whom Cardano described:

a worthless, shameless woman

Cardano continued to support his son financially and the young couple moved in with Brandonia's parents. However, the di Seroni’s were only interested in what they could extort from Giambatista and his wealthy father, whilst Brandonia publicly mocked her husband for not being the father of their three children.

Publicly mocked and taunted about the paternity of his children he reached his limits of sanity and poisoned her.

He was arrested.

Cardano did everything he could for his son. He hired the best lawyers and paid all the expenses. Five doctors were brought in who stated that Giambatista’s wife had not been poisoned, or at least, had not received a fatal dose.

The trial judge decreed that to save his son's life, Cardano must come to terms with the di Seronis. They demanded a sum which Cardano could never have found.

Giambatista was tortured in jail, his left hand was cut off and, on 13 April 1560, he was beheaded.

It was a real turning point in Cardano’s life; he never recovered from his grief. He gave up his lucrative medical practice in Milan and moved to Pavia, where he became pathologically obsessed about his own safety. He reported a number of intrigues, attempts on his life and malicious accusations of professional incompetence and sexual perversion.

In 1562, forced to resign his position in Pavia, Cardano, with some help from the influential Borromeo family, secured transfer to a professorship of medicine in Bologna. For a while Cardano’s life resumed its old order; he was happy to touch base with his old friend Ferrari.

However, problems with his children continued.

Cardano’s daughter, Chiara, died of syphilis, contracted as a result of her prostitution. His second son, Aldo, a perpetual thief, moved with him to Bologna but did nothing but drink and gamble. On a number of occasions frustrated Cardano had to bail him out of the staggering debts. Finally, when in 1569 Aldo gambled away all of his personal possessions and was caught stealing a large amount of cash and jewelry from his father, Cardano had him banished from Bologna.

In 1570, Cardano himself was imprisoned for a few months by the Inquisition in Bologna. The charges against Cardano are not known. Some point to Cardano’s connections with Andreas Osiander, one of the leaders of the German Reformation, as a possible cause of the involvement of the Inquisition. By the time one of his most vicious enemies, Tartaglia, was already dead and it is very unlikely he had anything to do with it.

On his release from prison, Cardano was sworn to secrecy about the whole proceedings and forbidden to hold a university post. Helped by another of his loyal pupils, Rodolfo Silvestri, he moved to Rome and appealed to the Pope, Gregory XIII, who granted him a small pension and allowed him a limited practice in medicine.

It was in Rome he started writing his autobiography De vita propia liber.

Cardano died on the 21st of September 1576.

Some say that Cardano predicted the exact date of his own death, some say that he starved himself to death to make this prediction true, we will never know.

Sources:

1. https://www.azquotes.com/quote/727271

http://www.arturekert.org/Site/Varia_files/wCardano.pdf

https://mathshistory.st-andrews.ac.uk/Biogrhttps://www.azquotes.com/quote/727271aphies/Cardan/#:~:text=He%20had%20to%20keep%20dubious,valuable%20time%2C%20money%20and%20reputation.