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In the book "What Do You Care What Other People Think?", Feynman talks about how in the 16th century Niccolo Tartaglia discovered a solution to cubic equations. He says while this was not a major discovery, it was a boost to the confidence of Mathematicians of the day, since not even the great Greek mathematicians had been able to solve cubic equations. In Feynman's words:

They were very upset when I said that the development of greatest importance to mathematics in Europe was the discovery by Tartaglia that you can solve a cubic equation: although it is of very little use in itself, the discovery must have been a psychologically wonderful because it showed that a modern man could do something no ancient Greek could do. It therefor helped in the Renaissance, which was the freeing of man from the intimidation of the ancients.

Is this true? i.e did Mathematicians in Europe, before Tartagalia consider themselves inferior to greeks, and did Tartaglias's discovery help dispel this inferiority complex?

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Feynman is being... liberally creative. What he says is his own interpolation that "makes sense" from the perch of today. "Must have been psychologically wonderful", perhaps, but "freeing of man from the intimidation of the ancients" is not how the men of Renaissance generally felt. The intimidation they sought the freeing from was not of the ancients, but of religious and scholastic orthodoxy, to which "the wisdom of the ancients" was something of a cultural ground and counter. The feelings toward the ancients were more along the lines of awe and admiration. Ancient works were avidly sought after, translated, transmitted, and enthusiastically built upon. The Renaissance does translate as re-birth, it is not liberation from the ancients that it championed, but their revival, and validation by them.

As for the specifics. First, the discovery of the solution to the cubic was made 30 years before Tartaglia, by del Ferro, and Tartaglia knew about it. Del Ferro did not make it public, but entrusted the secret to his son-in-law and successor Fiore. Tartaglia rediscovered it independently, but knowing that a solution exists is a powerful incentive for looking for it, a nerve-recking experience as we know from Tartaglia's own description, and not giving up. In fact, he rediscovered it while preparing for a "mathematical duel", a common practice at the time, he was called to by Fiore, see Why is “Cardano's Formula” (wrongly) attributed to him? and Saiber's Niccolò Tartaglia's poetic solution to the cubic equation for more on the story.

Second, it is not clear that the men of Renaissance believed that the methods they discovered were unknown to the ancients, I am not aware of Tartaglia, or any of his contemporaries, expressing a Feynman-like sentiment. Instead, there was at the time a popular "conspiracy theory" that the ancients deliberately hid some of their "secrets of the art". Around the time of Tartaglia, Copernicus, looked for affirmation of his heliocentric system in the writings of ancient Pythagoreans, like Philolaus and Ecphantus, and his contemporaries saw it (mistakenly) as a revival of the Pythagorean system. In the preface to De Revolutionibus, he also writes of the reluctance of the Pythagoreans to share their discoveries with the unlearned commoners, see SEP, The Renaissance: Ficino, Pico, Reuchlin, Copernicus and Kepler. Even later, when it came to the calculus methods, Torricelli wrote:

"I would not dare to affirm that this new Geometry of indivisibles is a totally new discovery. I could more easily believe that the ancient Geometers have made use of this method of discovery of the most difficult theorems, though they preferred another way in demonstrations, either to conceal the secret of the art, or not to give any occasion of contradiction to their jealous detractors".

Wallis and others expressed similar opinions. After quoting them in Cavalieri and Euclid, see Revolution and Continuity, p. 159, de Gandt further comments:"By claiming that the ancients possessed analogous instruments, the creators felt justified in adopting new ways. Their invention was a restitution of the ancient skill...". Solution to the cubic was, no doubt, inspiring, but not in the way Feynman imagined.

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  • $\begingroup$ To clarify the context of Feynman, he was talking in reference to a trip to Athens, where modern Greeks were essentially engaging in ancestor worship, and much of that same feeling of holding the ancients in high regard above all. Feynman was using the achievment of Tartaglia is a counter-point to this $\endgroup$ – Steve Sether Oct 29 at 17:54
  • $\begingroup$ @SteveSether Such stories often have a rhetorical motivation, and I am not even sure that people telling them feel they are saying something factual and are obligated to be accurate. Changing or adding details and emphasis, embellishing, sometimes even inventing are taken as a kind of fine pedagogical touch. It wouldn't be a problem if the audience took it in kind, but the catch is that inspirational stories only have the desired psychological force when they are taken as true. A make believe with a moral does not impact people as much. $\endgroup$ – Conifold Oct 30 at 23:42
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The answer to your question in the title is definitely yes. For example, Fermat wrote in a letter: "Perhaps, posterity will thank me for having shown that the ancients did not know everything". Fermat lived a century after Tartaglia, and the general opinion was still that "the ancients knew everything". This opinion began to change only after the invention of Calculus.

But what Feynman says is not quite true. Tartaglia's discovery (as well as some earlier works, like those of Fibonacci) was not so significant to change the general opinion of superiority of the ancient mathematics. In fact, the top achievements of the ancients were not fully understood and incorporated to contemporary mathematics until the 19th century.

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  • $\begingroup$ I think Feynman wasn't saying this was some the achievement that finally convinced Mathematicians of the time that they could be as good as the ancients, but merely this was one piece of the puzzle. Now... why Feynman chose Tartaglia is an interesting question. You'd think there must have been some other, earlier problem Renaissance mathematicians solved before cubic equations? $\endgroup$ – Steve Sether Oct 29 at 18:01
  • $\begingroup$ Yes, Tartaglia was one of the first people who solved a problem whose solution was not known in antiquity. But mathematics consists not only of solving problems. Some of the important advances before Tartaglia are due to Fibonacci and Oresme, mathoverflow.net/questions/269893/…, for example. $\endgroup$ – Alexandre Eremenko Oct 30 at 14:07
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Yes, before Tartagalia, Europe had experienced a dark age in culture and science (the Middle Ages) over a thousand years. So it was natural for scholars at that time to think they were inferior to Greeks. Tartagalia was in the period of Renaissance that marked resurrection of art and scientific activities from the golden ages of Greeks science and knowledge led by Plato, Aristotle and Archimedes.

The Renaissance marks the fundamental and comprehensive changes of European culture during the 15th and 16th centuries, which led to the demise of the Middle Ages, and for the first time ever, embodied the values of the modern world. Achievements from polymath artists led by Da Vinci and scientists such as Copernicus, Galileo, Kepler, and so on, proclaimed that their age had progressed beyond the intellectual decline of the Middle Ages and accomplished resurrection from the ancient Greek and Roman civilizations.

The scientific revolution started in the Renaissance continued into the 17 century and led to the discovery of new scientific fields like Analytic Geometry, Calculus, Mechanics by Newton, Descartes, Leibniz and others. This revolution marks that the golden ages of Greek science and knowledge had been completely surpassed.

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