First of all do not play the whole game of enlightened vs non enlightened, it is all romanticist revisionism created and fostered by selective historians who have agendas other than the distribution of truth and universal understanding.
To begin with, the Ottomans while perhaps could have done better in later years, where woefully falling behind and all the while the period ensuring dominance over Europe passed as the the population of Europe and expansion to the New World combined with the eventual rise of secularism and other developments like universal sufferage and end of slavery made the Ottomans backwards in all respects except in Fiqh (lol, they beat Europe in that hehe). Even in earlier Ottoman periods the Ottomans were not any better in agriculture, population, science, etc as Europe and was already falling behind in population and such despite military victory and seeming dominance.
To begin with, I would like to present a fact to you, both the Abbasid and Umayyad failed to last in a real sense longer than 200 years and each time were ripped apart by religious strife, economic hardship and individual nationalism. Regardless of a nations innovation, if it's economics have deteriorated, it's people angry and the those of low income or none willing to die for change, then such a state is destined to fall, no innovation in mathematics can save this inevitability. Keep that in mind before you assume the grandeur and exceptional ability of this state to rule all of Southern Europe, or whatever the conquer, if they get nothing but war right outside Baghdad, how can they expect peace and joy in Italy? It is completely non sensical and the acceptance of whom? Any movement to science in Europe would be then directed to rebellion against this massive foreign entity, not some loving relationship.
Having answered what I suspect to be your question, lest I read wrong, I will just speak my mind on the Abbasid, the supposed golden age, the one that every revisionist and Arab nationalist tells me that we would be swimming in gold and having lunch on the moon had survived instead of those backwards Europeans.
The Abbasid was a period followed by the Umayyad, who had extended the conquests of the Rashidun under the the command of the Prophet (Rasul) to all surrounding areas. The Umayyad period was the height of the Arab warrior and his sword burned with fervor and sought Shahid and the awaited paradise, the Umayyad directed its momentum on all fronts, Baqiyyah Wa-Tatamaddad, through the tribal divisions and decentralization in war towards any opponent outside of the Ummah regardless of creed. The conquests put the Byzantines into a paralyzed state unable to counter and only able to defend with broken hands, and to the East any invasion was foolhardy as the door to Syria far and the armies of the Ummah numerous. However, despite the victory and power on display, the Umayyad with no intervention from the extior began to crumble under the pressure from religious strife and the disenfranchised minorities that made up its empire who stayed at home while the Arab Sunni toiled abroad. The result was the Abbasid revolt, Great Berber revolt by the Khawarij and the revolt of al-Harith Ibn Surayj the Alid. These revolts culminated in the beginning of the Abbasid period in around 750-751.
The great achievement of the Abbasid from the start was not innovation of technology but of the alliance with minorities so to control its vast territory without constant war. This alliance with the learned minorities being Jews, Zoroastrians, Christians, etc where well educated and mastered in the arts of innovation as they were in past times before the rise of Islam. This alliance in support differed with the Umayyad who allied with the tribes of Arabs and the Muhjahid, thus the Abbasid became a more science and culturally oriented state than the war driven Umayyad.
During the course of time as a result, the Abbasid experienced scientific innovation and a movement pertaining mainly to mathematics and star gazing and instituting even more translations of texts into Arabic. In this period however, was driven by a new religious group differing from the previous breakaways (Khawarij, Shi'i and Murji'ah), both from the field of innovation and opium filled palaces was the Mu'Tazila and its subgroups. The sect found that rationality is more important than Imaan (faith) and redefined the religion creating new stipulations, such as:
The Quran was created after the fact and is not eternal.
They said you cannot say Allah exists or non existent, because saying he exists means he is comparing him to creation and if you say he doesn't exist, that is Kufr Akbar.
Various Takfir for silly subjects.
And many others, but this isn't about the Mu'Tazila sect, which I could go over in detail if asked.
The effect of the Mu'Tazila dominance in Baghdad over the supposed righteous and pious Abbasid rulers can be seen evidently. The Mu'Tazila and the Baghdad rulers fearing the power of the Arab tribes and the Muhjahdeen, began bringing in Turkish Mamluks to fulfill roles of Arab warriors and replacing Arabs in areas of power in military matters with these Mamluks. The plan to weaken enemies of the state and limit a harsh view of the in reality, decadent and weak Abbasid rulers. The overall result was the weakening of the Arab martial capability seen during the Umayyad period and the further descent into decadence of the Abbasid rulers who already withheld wealth from the people who where becoming more and more aware of the Kufr of the Abbasid.
The Rise of the revolutionary Shi'i Alid, Yahya ibn Umar in 864 was one of the catalysts for the end of the Abbasid following the series of palace struggles during the Anarchy of Samarra. Yahya Ibn Umar, gained fame through the land for leading a short revolt against the Abbasid in which he was captured and as punishment had his body slowly cut apart in which he refused to submit and was dismembered and burned. The murder of Yahya was met with the cries and anger of the Arabs regardless of creed, with Shi'i saying he was Ahl ul-Bayt and the Khawarij and Sunni seeing him as an Ulema and respected man. This began a series of events (which were already in motion) towards a revolt and series of revolts that would bring the Ummah to its knees...
The trade of slaves from Africa had always been seen and slaves from Uganda, Abyssinia, Zimbabwe and Somalia common place since the days before Islam. However, the condition of slaves, likely pertaining to the decadent and greedy Mu'Tazila faction at Baghdad (who were obsessed with race and the distinctions in them), worsened as they toiled in the plantations and mines of Southern Iraq on constant surveillance from the Abbasid rulers, who administered castration on the men in many cases out of fear of supposed "African fertility". The conditions were grueling and despite the innovations, the foundation of Iraq depended on these slaves producing in South Iraq, thus the conditions continued.
In Iran, the rise of the Khurramiyah, known as the Joyous religion, rebelled and conquered parts of Northern Iran around mountain passes and claimed to undue injustices. The group was founded on a new revolutionary Zoroastrianism focusing on the joy of martyrdom and rebellion upon the enemies of the religion. They would struggle for years on the periphery a constant backdrop that the Abbasid clique attempted to forget. The rebellion lasted until 837, but it's damage to Iran was done and showed the utter dissatisfaction with the rule of the Baghdad clique.
Then came a man named Ali Ibn Muhammad, a revolutionary who oddly seemed to have a single wish, to revolt against the Abbasid with whatever means at his disposal. Initially he claimed to be the successor to the Imam of the Shumaytiyya in Madinah, who claimed to be the Mahdi and was defeated in battle by the Abbasid in Madinah in 815. He claiming this, moved from Iraq to Bahrain and attempted to ferment a revolt amongst the people, claiming to be a descendent of Ali Ibn Talib and the awaited Mahdi, despite his claims he was not taken seriously in Bahrain and was eventually exiled from the area. As this happened, the struggle of the Zanj (Black slaves) remained the same, the toil seemingly endless and salvation non existent (or so it was thought).
Having been rejected by the Shi'i, Ali (referring to Ali Ibn Muhammad for the rest of the post, not Ali Ibn Talib) sought a vector for insurrection. He arrived in Iraq during the rebellion of Yahya Ibn Umar and heard far and wide of his martyrdom and spoke with the Bedoiun and common folk of their suffering at the hands of the Abbasid and the breaking of the religion by the clique at Baghdad that demanded taxation not in the Shariah and sent Mamluks who attacked them and forced them to pay taxes, etc... Ali as well met Khawarij amongst the people who were discontent and worried for the suffering of the people as well as the decadence and Kufr of the Abbasid. They said clearly that the rule was for Allah alone and his Shariah alone and the Abbasid had in their opinion broke this and the Khawarij proclaimed the piety of the poor and disenfranchised in comparison to the oppressive wealthy. The Khawarij in addition to the suffering of the Arab mentioned the Zanj in the marshes of Southern Iraq (Sawad or black land). Ali then went to the Sawad to see for himself.
Then a new sight was seen by the Zanj, an Arab travellor inquiring on the conditions of the people. Ali saw much suffering and despite his initial wish in being the Mahdi and all the power being a Shi'i ruler conferred (look at Is'mail being a living deity), decided that it was through the Khawarij and the revolt of the oppressed and poor that his revolt would gain traction. He then began speaking in the Masjids which now where being filled by slaves and Bedouin as well as hardcore Khawarij revolutionaries. He began to preach, "God is great there is no Hukm except by God and God is Lord, there is no abitration except by God" the call of the Khawarij at the battle of Siffin in revolt against the Ali Ibn Talib, he then began to proclaim with the excitement of his people from many different groups of oppressed disenfranchised people that "By God, the Zanj Slave is equal to the Arab and bloodline mattered not, only the piety of a Mu'min (believer)". After his sermons, the Khawarij in tandem with Ali called the Zanj to break their chains and join in a righteous revolt and the Bedoiun to join and refuse the Zakat and any Muslim upon the truth to join the cause of Allah.
The movement quickly gained fame in its hit and run tactics and its constant manpower by raiding areas and adding its slaves to the army and Bedoiun and Khawarij rebels arriving on droves at the word of the insurrection. The armies armed lightly often with no armor defeated Abbasid armies through sheer will and tactics pushing them across Iraq. Eventually at the height of the revolt Basra was taken and became the capital of the new state.
In the end the Zanj rebellion was quelled at the cost of the Abbasid Khilafah and the complete destruction of Iraq in terms of sustainability. The revolt however was not the last chapter and only the most vicious of the revolts and most widespread close to home to the Abbasid period.
Then arose a new group, the Qarmatians conquered eastern Arabia, sieged Baghdad and destroyed the remnants of the cities in Sawad after the Zanj defeat and picked up on the momentum of the Fatimids in North Africa, proclaimed further revolt and sacked Makkah and desecrated the Kaaba eventually selling the Kaaba back to the weak Abbasids. I could explain more and do the whole story narrative with the Qarmatians and the Fatimids, but I've written too much already lol.
Either way, was algebra and constellations enough to make up for this? Could it save them? What use is a scientist who forgets to look out his window at his people and forgets what gave him power?