Did Renaissance mathematicians once consider themselves inferior to the great ancient mathematicians?

Question

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 therefore 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?

Answer 1

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.

Answer 2

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 Nave and to his student Fiore. Tartaglia rediscovered it independently, but knowing that a solution exists is a powerful incentive for looking for it, a nerve-wrecking 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.

Answer 3

The writings of Descartes and Viète do indeed suggest a certain "inferiority complex", believing that their own work was simply rediscovering Greek analytic methods which had been lost by the "barbarians".

From Descartes' Rules for the Direction of the Mind:

But when I afterwards bethought myself how it could be that the earliest pioneers of Philosophy in bygone ages refused to admit to the study of wisdom any one who was not versed in Mathematics, evidently believing that this was the easiest and most indispensable mental exercise and preparation for laying hold of other more important sciences, I was confirmed in my suspicion that they had knowledge of a species of Mathematics very different from that which passes current in our time. I do not indeed imagine that they had a perfect knowledge of it .... But I am convinced that certain primary germs of truth implanted by nature in human minds ... had a very great vitality in that rude and unsophisticated age of the ancient world. ... Indeed I seem to recognize certain traces of this true Mathematics in Pappus and Diophantus, who though not belonging to the earliest age, yet lived many centuries before our own times.

But my opinion is that these writers then with a sort of low cunning, deplorable indeed, suppressed this knowledge. Possibly they acted just as many inventors are known to have done in the case of their discoveries, i.e. they feared that their method being so easy and simple would become cheapened on being divulged, and they preferred to exhibit in its place certain barren truths, deductively demonstrated with show enough of ingenuity, as the results of their art, in order to win from us our admiration for these achievements, rather than to disclose to us that method itself which would have wholly annulled the admiration accorded. Finally there have been certain men of talent who in the present age have tried to revive this same art. For it seems to be precisely that science known by the barbarous name Algebra if only we could extricate it from that vast array of numbers and inexplicable figures by which it is overwhelmed, so that it might display the clearness and simplicity which, we imagine ought to exist in a genuine Mathematics.

Leo Correy, in his recent text A Brief History of Numbers, writes:

Viète, for one thing, saw his own contribution as part of a more general effort to reconstruct the assumed Greek analytic methods that had been ruined by the "barbarians" (namely, the Arabs). In order to stress this attitude as an essential part of his work, he consistently declined to use the term "algebra" in favour of "analysis". He also introduced some new Greek-sounding terms of his own invention such as poristic, exegetic, and zeteic.

Answer 4

Yes, before Tartaglia, 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. Tartaglia was in the period of Renaissance that marked the 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, and so on, discovered by Newton, Descartes, Leibniz and others. This revolution marks that the golden ages of Greek science and knowledge had been completely surpassed.