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The renaissance created a number of prominent mathematicians during that era. But 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 Jun 19 '17 at 3:04
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This does not only apply 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 is why renaissance happened 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 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 Italians, and Fibonacci was a merchant.

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

So the process seems to start on the cultural boarder (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 Sep 28 '17 at 3:20
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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 Jun 20 '17 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 $\endgroup$ – Mikhail Katz Jun 21 '17 at 6:41

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