Today, even to become a decent mathematician is not an easy job!
Half a millenium ago, I could probably know everything about the cutting edge of physics, astronomy, math, chemistry and optics in my head at once. 500 years ago, calculus had yet to be invented, fire was still thought to be "somehow" composed of phlogistons, and scientists were jsut coming to grips with the idea that the earth circled the sun.
Even 200 years ago, the knowledge encompassed by science was tiny compared to what we know today.
If you never, in your entire life, have to worry a second if you can feed yourself and your family and keep a roof over your head, you're probably slightly more keen to see what a life of sitting down and wondering might lead to.
As a matter of fact, that was not the case in the Islamic empire - the government had encouraged independent research and learning and funded it quite extensively, which is the biggest reason it was such a scientific boom period. It's quite astonishing how far they've fallen, because at the peak of the European medieval times, the Islamic Empire had been encouraging this sort of scientific research and study like crazy. Perhaps most famous of all are the scientific libraries of Baghdad which once held volumes of literature and research pertaining to medicine, literature, physics, chemistry, and biology; and were made available (mostly) to all.
Invasions and wars in the region, along with the decline & fall of the Islamic Empire brought an end to that golden age of science and knowledge which is really sad because in that short period of time a lot of knowledge was discovered and contributed, lending to the future discoveries by European scientists in Calculus, Chemistry, Medicine, and more.
The Mongol invasion was the beginning of the end, to use a cliche. The Islamic Empire was indeed "borrowing" the talents of the people they conquered, but (the majority of) the accomplishments only took place after the conquest by the Islamic empire. More importantly than who did the thinking (it was basically a mixture of different cultures and races, the majority of which were not native Arabs) is the fact that that sort of scientific expeditioning was not discouraged and that the scientists were given as close to free reign as possible.
Contrasted with the Holy Roman Empire and the Catholic Church's aversion to scientific discovers (Galileo anyone?) you can see that the Islamic Empire played the role of encouraging godfather quite well. Obviously it's not the government nor the religion that makes brilliant people, rather it just sets the stage for the accomplishments to take place.
It's a common misconception that the Islamic Empire == Arabian Empire, since the empire dictated that anyone to enter the folds of Islam would be considered the same "class" of citizen regardless of race or culture (and history can attest to the enactment of these beliefs) - it was the non-muslims that were considered to be second-tier citizens with respect to certain freedoms. Keeping that in mind, Persian or Arab or Kurd doesn't really matter.....
But anyway, here are just a couple of famous Arab-Muslim scholars: Ibn al-nafis, Ibn Khaldun, Ibn Battuta, Al-Kindi, Ibn Al-Haythan, Averroes/Ibn Rushd, Ibn Yunus, Ibn Al-baitar, al-idrisi, hunnayn bin ishaq, jabbir bin hayyan.......
Full list is here, you can click through to see the ethnicity of these scholars: http://en.wikipedia.org/wiki/List_of_Muslim_scientists
Take a look at the creator of cryptanalyis:
http://en.wikipedia.org/wiki/Al-Kindi
He was an arab philosopher, scientist, astrologer, astronomer, cosmologist, chemist, logician, mathematician, musician, physician ...