In anthropology, religion is considered one of the "five basic institutions" that all societies have and have had as far back as we can go in human history. The other four are family, political, economic, and educational. There is an interrelationship between these five in all societies, so religion does play a role in each of them to varying degrees.
The Golden Age of Islam very much had an interrelationship with the development and manifestation of the educational systems in many countries, and Islam still plays a role in modern Islamic societies of course.
True.
But, I would like to point out, for paarsurrey's sake more than for yours, that the development of "Islamic" science come from educational institution, and not from religious institution (Islam).
So really, I am addressing this reply to paarsurrey, because I think you are already aware of this.
Islam, their Qur'an, their prophet (Muhammad) and their deity (Allah) have nothing to do with the "Islamic" science.
Though, in the west, science and education were being lost, or at least seriously downgraded, due to the Dark Ages, starting in the 2nd half of the 5th century CE, the eastern civilizations, including the Byzantine (or Eastern Roman) empire and the Sassanian (Persian) empire, didn't have this Dark Ages, therefore they didn't lose their science.
When the Muslim Arabs began their conquests in Syria, Persia and Egypt, they were taking lands where people already have science. Just because people in those lands converted, the science didn't originate with Islam. And it wasn't Islam (or the Qur'an) that contribute to science, but the Muslims who learn from the existing science that was already in place. People who already have, either Greek or Roman (or both), and Persian educations in scientific fields.
Sure, Muslim scientists did expand and made some good improvements to the existing science by the 9th century, but that's what call progress.
By conquering those territories belonging to the Byzantine and Persian empires, the Arabic-ethnic Muslims were able to reap the rewards of education that they lacked, education like in science and mathematics and engineering and architecture.
Architecture is another area that the Arabs of Muhammad's time, lacked. When the Arabs conquered Syrian and Egypt and Persia, they gained access to architecture that the Arabs had never possessed. All along the North African coast which the Arab army conquered, were Roman cities and towns, and therefore Roman commissioned building. When the Arabs crossed the Gibraltar into the Spanish peninsula, in 712, were more Roman cities, like Hispalis, now called Seville.
Muslims of today, tends to overlook that science from the Golden Age of Islam didn't just appear out of nowhere. The Arabs took whatever science that was already out there, from the people they had conquered and converted. They owed their science to the Greek, Roman and Persian, more so than the Arabs.