Madhabs are four schools of Islamic law. The answers of the different legal schools reveal a lot about the different "types" of Islam that flourished. One school states that merely brushing one's hand against a woman is enough to necessitate purification. Another school states that the touch obviously involved "pleasure", perhaps a hearty prolonged embrace or handshake. The school of Abu Hanifa, however, states that "touching a woman" means nothing other than full sexual intercourse. All sorts of other touch can be ignored, and the man can go ahead and pray without doing the ritual washing again. You can see that Abu Hanifa, or the Hanafite school, is most relaxed. This is not by chance. Hanifa was another person of Persian ancestry. His rulings were given in an environment where non-Arabs with their urban customs abounded; indeed in the cities where Muslims lived there were many non-Muslims. Thus his school develops law and interpretation to make allowance for the mentality of the people who will practice these rulings, and not for a Muslim-only tribal or desert society where behaviors are much more conformist and stricter. To this day, the Hanifite school is practiced in areas where the ethnicity is non-Arab: Turkey, India, Central Asia and the Russian Volga region. Again, this shows that Islamic law was adaptive and pluralistic.
What are qiyyas and ijma?
Qiyyas in Islamic law is the process of dedactive analogy in which the teachings of the compared and contrasted with those of the Qur'an. Ijma is an Arabic term referring to the consensus or agreement of the Muslim scholars basically on religious issues.
What role did non-Arabs play in developing Islamic civilization?
Non-Muslims took part in intellectual freedom: Sadya Gaon and Moses Maimonides are two the greatest figures of medieval Jewish philosophy. Many of the core theological and jurisprudential principles were absorbed from the Islamic theological sciences. Indeed, some see Judaism's greatest philosopher, Maimonides, as in effect an Islamic philosopher (the contemporary Jewish expert on Islamic philosophy Oliver Leaman, for one). In addition, the first hospitals were springing up -- places where the new medical knowledge was offered, often for free, to patients. (The Jewish Maimonides was a leading doctor in Cairo during the time of Saladin). This is all part of what is sometimes called the Golden Age of Islam. However, the emphasis here will be more on political developments and further shifts in the nature of Caliphal rule.
What were Muqaffa's and Mawardi's contributions to Islamic political philosophy?
A thinker called al-Muqaffa (756) eventually worked out a political doctrine which put forward a new vision of who the Caliph was. It was neither Shiite, as after all it was difficult to keep on finding Alide heirs, nor Kharijite, whose purity and quarrelsomeness was difficult for an imperial outfit. Muqaffa was himself of Persian ancestry, not an Arab; Baghdad was on the territory of the former Persian empire - and so Muqaffa chose an old Persian template of empire and re-clothed it Arabic, Islamic garb. He advised the Caliph to style himself as a Persian emperor, and to recreate the glory of the Sassanid imperial court. Islam, in that sense, should be a state religion, and the Caliph should be worldly ruler, who nonetheless supported religion. Muqaffa encouraged Caliph Mansur to use his secular power to support and enforce the Mutazilite view on the created nature of the Quran and other rationalist tenets.
A new political scientist arose to theorize and justify was al-Mawardi (1050). He and the Baghdad Caliph were faced with the new situation of the Seljuk Turks. These nomadic people had emerged from north-eastern Eurasia (northern Kazakhstan, Western Siberia), and travelled through Central Asia, emerging now along the Caspian and Black Seas -- on territory that had been incorporated into the Umayyad and then Abassid empires. The best that the Caliph could do was tell these strong warriors that if they wished to continue as Muslims (and like earlier and later nomads they had chosen to become Muslim) they must accept his authority. In effect, then, the Seljuqs conquered territory as they wished, and the Caliph then blessed it and declared it Islamic conquest. He "turned unlawfulness into legality and the forbidden into the legitimate", as Mawardi said, or as we might say, he rubber-stamped the inevitable.