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The Renaissance created a number of prominent mathematicians. However, later in the 18th and especially 19th century, Germany and France became the hot centers of mathematical thinking.

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    $\begingroup$ Because England, Germany and France could fund their science much better than Italy could, and science was the driving force of mathematics during that period. $\endgroup$
    – Conifold
    Commented Jun 19, 2017 at 3:04

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Analysis got a head start in the work of Italian indivisibilists Cavalieri, Torricelli, degli Angeli, and others. However, the budding analysis was viewed with disfavor by some in the Catholic hierarchy, including many leading Jesuits. This led to a suppression of active math centers in Italy and a graduate decline of the Italian mathematical tradition. The reasons for the clerical opposition to indivisibles are the subject of dispute among scholars. This recent publication by Sherry attributes this to the association of indivisibles with ideas that were seen as contrary to the Catholic interpretation of the eucharist, while this recent publication by Alexander attributes the opposition to an adherence to a Euclidean ideal of rigor.

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  • $\begingroup$ But what of algebra, geometry, probability or number theory? Why wouldn't Italian mathematicians switch to that? After all, we have Cardano and the Renaissance theoreticians of perspective, but projective geometry owes its birth to Pascal and Desargues, and probability theory to Pascal and Fermat. $\endgroup$
    – Conifold
    Commented Jun 20, 2017 at 23:56
  • $\begingroup$ Well one can offer all sorts of speculations but I would rather not do it here but rather stick to the facts. If you like we can discuss this privately via email. @Conifold $\endgroup$ Commented Jun 21, 2017 at 6:41
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This does not apply only to mathematics, but to all ancient culture. It was almost completely forgotten in Europe, and then it was slowly recalled, in the process which is called Renaissance. So, the natural restatement of your question would be: why did the Renaissance happen in Italy first?

It is hard to name one reason. Probably there were several:

  • First of all, some memories of the ancient culture were certainly preserved in Italy more than in other places in Europe, because Italy was the center of the ancient Roman Empire.

  • Second, and probably more important, was more intensive contacts with the Eastern Roman Empire, which still existed at the time of the Renaissance, and where some ancient heritage was preserved.

  • And, finally, Italy in general had more contact with the rest of the world because the international trade was more developed in Italy. This includes Mediterranean countries and the same Eastern Empire. (After all, Marco Polo and Columbus were Italian, and Fibonacci was a merchant.

To a much smaller extent, one can also see a similar process in the Iberian Peninsula, where Europeans had a long contact with Muslims, who also preserved a part of ancient heritage. However, in Spain and Portugal this process was arrested by religious intolerance after the Reconquista.

So the process seems to start on the cultural border (penetration of "new" ideas) but later it develops mainly in the places where intellectual climate is more more appropriate (Germany, England, France).

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    $\begingroup$ I would also add that famines and wars were very common in Italy, which made people realize praying does not help, and one must do real things to solve real problems. $\endgroup$
    – timur
    Commented Sep 28, 2017 at 3:20
  • $\begingroup$ @Rodrigo de Azevedo: When I was talking on inquisition and the end of reconquista I meant Spain and the end of XV century. This was the time when the climate in Spain changed to hostile to independent thinking. $\endgroup$ Commented Jul 8, 2020 at 17:02
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    $\begingroup$ Rodrigo de Azevedo: Some places in the earlier periods certainly were. Read about Alfonso X of Castile, for example. Also some times and places under Muslim rulers. $\endgroup$ Commented Jul 8, 2020 at 17:17
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    $\begingroup$ @AlexandreEremenko, your answer tries to explain why the Renaissance started earlier in Italy than elsewhere in Europe, but it does not explain why it ended earlier in Italy :-) $\endgroup$ Commented Jun 28, 2023 at 9:54
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    $\begingroup$ @Mikhail Katz: this is briefly addressed in the last paragraph. Trials like those of Galileo and Bruno do not help scientific progress. To be sure such things happened in Germany as well, but unlike Italy, Germany did not have a total power of the Catholic Church, and it was always possible to find places tolerant to intellectual pursuits. A comparison of biographies of Galileo and Kepler shows this clearly. $\endgroup$ Commented Jun 29, 2023 at 6:09
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With the spread of mathematical studies, it is normal for the Italian quota to diminish; for example, in the 20th century, the Soviet Union and the United States were dominant on a global level, yet Russia and the Thirteen Colonies did not make many contributions before the nineteenth century.

Nevertheless, it is not true that the Italian contribution in the 19th century was scarce: Ruffini solved the problem of the algebraic solubility of equations higher than the fourth degree (a general formula does not exist), Beltrami solved the problem of Euclid's fifth postulate (it is independent of the first four), Ricci-Curbastro introduced tensor calculus, Betti extended topological methods from geometry to analysis, Peano solved the problem of the existence of solutions for ordinary differential equations under the hypothesis of continuity (at least one solution exists) and introduced the theory of measure. Additionally, between the end of the 18th century and the beginning of the 19th century, there was also Lagrange (who was from Piedmont, born to Piedmontese parents, raised in Piedmont, and became French through marriage later in life, despite many Anglo-Saxon historians portraying him as a Frenchman born by chance in Turin).

Now, what I sometimes wonder is: why, when talking about the history of non-Euclidean geometry, is everyone mentioned, from Alhazen to Gauss, but not the solver of the problem? Eugenio Beltrami should be a mythical figure in the history of mathematics, yet he is hardly mentioned.

In general, one might question: why are some mythologized, often beyond their merits, while others are unjustly neglected? It is often a matter of subjective tastes (who decides what is important and what is not?), but it is also a political issue. Well, I think I can conclude by saying that Italy has not been able to assert itself culturally in the last two centuries, partly due to a lack of will and partly due to a lack of strength.

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