16th Century Italian Mathematics (it's more interesting than you think)


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This kind of thing went on all of the time in 16th century Italian mathematics, but I'm going to focus on one particular instance: the solution of the cubic equation.

For those of you who are not familiar with what a "cubic" or "cubic equation" is, it's an equation involving one term with x^3. For example, 4x^3+2x^2+x+9=0 is an example of a cubic. Mathematicians had struggled with finding a solution to an arbitrary cubic for over a thousand years before this story took place. If you look at the cubic formula, you can see why.

It is the year 1525 and the mathematical scene in Italy has never been so intense. Mathematicians would challenge each other to "math duels" regularly, in order to prove their superiority and higher knowledge. Mathematicians who won math duels won exorbitant prizes and fame. Those who lost were shamed, disgraced, and most importantly lost their teaching jobs for the rest of their lives. A math duel worked as such: each mathematician would give the other problems to solve that they deemed impossibly difficult that they knew how to solve. Whichever mathematician could solve his rival's problems in a set amount of time (usually 30 days) won. These were commonplace, and since the stakes were so high mathematicians often held findings secret so that they could win in one of these duels.

Let's first look at an aging mathematician named Scipione del Ferro. He has worked for his entire adult life attempting to solve problems such as these, and he believes he has a solution to a certain type of cubic called the "depressed cubic". He witheld his findings from the public because he was terrified of being challenged to a math duel. Many mathematicians had this sort of "ace up their sleeve" to prepare themselves for math duels. Ferro would up never publishing his results, and he died in 1526. The only way we know of his solutions is that his student, Antonio Fiore, was given them.

Now, let's look at Tartaglia. Tartaglia was a young mathematician who wished to rise to fame and fortune. He had found a method to solve all cubics, not just the depressed cubics. He decided that with his new method, we would challenge the most prominent mathematician of the time to a math duel and gain national fame. Since Ferro, the leading mathematician of the time, had just died, Tartaglia chose to challenge his understudy Fiore to a math duel. Tartaglia with his new method won handily and rose to the spotlight of the mathematical world.

In the midst of Tartaglia's success, he was contacted by a man known as Gerolamo Cardano. Cardano was a publisher, and he wished to publish Tartaglia's secret in his textbooks and make millions. Tartaglia, also wishing to make millions, was extremely hesitant with giving his secret away. He also might have been challenged to another math duel, to which his secret was his protection. Tartaglia decided that he would publish his own work in a paper of his own and receive credit for what he had done before allowing Cardano to publish his work.

Cardano gave Tartaglia a deal. Cardano would agree not to publish Tartaglia's work until Tartaglia's paper was out, so long as Tartaglia agreed not to go to any other publishers. They agreed, and Tartaglia told Cardano his method and Cardano promised not to publish Tartaglia's work without his consent.

Cardano, however, still wishing to make millions, went on a quest of his own. He searched through all of the available mathematical literature and found several unpublished works by the mathematician we talked about earlier, Scipione del Ferro. He found that in Ferro's work was the foundation for Tartaglia's method. Ferro had all of the pieces he needed in order to find a solution, but never put them together in the right way. Cardano, having seen the solution Tartaglia had, was able to fix Ferro's work into the solution for the general cubic. And he did it without ever once directly using Tartaglia's work, thereby keeping to his deal! Now Cardano could publish Tartaglia's work and reap all of the benefits.

So he did. And Tartaglia was pissed.

Tartaglia, in retaliation, challenged Cardano to (what else?) a math duel. Cardano deflected the duel to his student, Lodovico Ferrari. Ferrari, an aspiring young mathematician, had discovered a solution to the quartic (even more difficult than the cubic). He beat Tartaglia in the duel. Tartaglia lost his job and his fame. He died penniless.

Cardano became rich off of his publications. Ferrari became rich and got a prestigious teaching job at Bolonga (he would eventually, sadly, be murdered by his sister over monetary issues).


The above is a true story. If only mathematics was as awesome as that today.
Who would've thought of math being a life/death duel (well, at least in terms of fame). But yeah, an interesting read for sure, and here I was thinking it would be a boring topic.
Why couldn't we be alive in those days, high-stakes battles of logic would be very interesting. We don't get anything that cool :(
Mathematicians were like the samurais of italy. I think I might challenge someone to a math duel one day.
Have you heard of the "Ruffini method"? It's pretty damn awesome, since it kets you resolve equations of practically any degree (sorry if my terms are off, I'm literally translating the italian definitions). Every time you repeat the process you reduce the degree by one, while multipling the equation by (x - y). Therefore you can simply repeat it until you have the equation of second degree, which can be resolved normally


Ce soir, on va danser.
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fucking ITALY man, that country has been nuts for thousands of years

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