I think I should start by answering the last part of your question about the passages in Qur'an that direct the Muslim women to wear hijab as these passages will definitely answer the first part of the question: Why should women wear it?
And say to the believing women that they should lower their gaze and guard their modesty; that they should not show off their beauty and ornaments except what is (ordinarily) visible thereof; that they should draw their veils over their bosoms and not display their beauty except to their husbands, their fathers, their husband's fathers, their sons, their husbands' sons, their brothers or their brothers' sons, or their sisters' sons, or their women, or the slaves whom their right hands possess, or male servants free of physical needs, or small children who have no sense of the shame of sex; and that they should not strike their feet in order to draw attention to their hidden ornaments. And O ye Believers! turn ye all together towards God, that ye may attain Bliss.
Surah 24 Verse 31
O Prophet! Tell thy wives and daughters, and the believing women, that they should cast their outer garments over their persons (when abroad): that is most convenient, that they should be known (as such) and not molested. And God is Oft-Forgiving, Most Merciful.
Surah 33 Verse 59
The first reason for wearing hijab, then, is that it is a command from God. Religious Muslims - like any other religious people belonging to any revealed religion - don't feel comfortable in deliberately disobeying God! If some Muslims, despite of the semantic clarity of the Qur'anic verses, still think that this ruling is not mandatory, this will not alter the message.
We can have some Muslims who unjustly or ignorantly deny the illegality of drinking alcohol, eating pork or even committing major sins like stealing, fornicating… etc. This does not mean that such things are permitted in Islam! In Islam it is very easy to detect or judge whether a certain ruling is mandatory or not, by going back to the major sources of Islamic legislation. This we technically name as ‘shari'a’. Those sources are the Qur'an, the Holy Book of Muslims, and sunnah, which is the sayings and guiding acts of Prophet Muhammad (peace be upon him), as stated in authentic books. So, simply this is the very same case with hijab.
Notice Carmen that this ruling of hijab, is highly esteemed among Muslim women, despite the big pressure of most regimes in the Muslim majority world to forbid it. Sometimes it is forbidden by force or women are discouraged from wearing it. The issue is seen as a means of secularizing or globalizing the Muslim states! This is, in-fact, except for a few Muslim countries, which encourage their women to observe this ruling or at least "let it go".
The rest are on a savage war against it. Turkey is the outspoken example of such hostility, but the other regimes are not less hostile, albeit silently, towards it. Despite this fact you can find Muslim women insisting on wearing it, even if it costs them their jobs and social status ...
In fact, there is wisdom behind this heavenly command of hijab and behind Muslim women's challenge to wear it. It is that in the Islamic culture - like many other cultures - the idea of women's respectability and virtue is related, among other things, to the propriety of their dress. In Christianity for example, nuns cannot show up without their hijab. Also, regular ladies wear it while attending masses, which is a reflection of self-admission that God wants to see them this way! Also, as far as I know, in Judaism, the most religious faction amongst Jews, namely The Hassidics, have their strict rules about covering women’s bodies. Even in secular societies, some women judges have to cover their heads, during sessions, as a way to show their dignity and self-respect.
Remember that your own puritan American society, before being swept over in the 1950s, by this permissive value system, used to look with embarrassment to any woman going out to the street without her hat! Mini-skirts only appeared recently together with the secular and atheist style of life that some people in the west - and the East as well - chose for themselves.
You can also tell that hijab is the way Islam neutralizes a woman's stereotyped role as a mere female, by inciting the society to deal with her as a human being away from her "extra feminine powers"! No wonder then that the majority of Muslim women refuse to let their ‘mill go with all winds’ and that they insist on keeping their Islamic identity. This is regardless of the consistent unfair criticism and sometimes satire against it. They wear hijab, not only in submission to God's order, but also because their inherent moral code is in full harmony with its philosophy.
Mr. Lamaan Ball, former editor of Ask About Islam, adds:
The term hijab means more than just covering the hair and refers to the general code of modesty outlined in the verses quoted above. To understand what is implied in these verses, consider what it is that women do to show off their beauty. What is it they make a display of to be sexually attractive?
Though to some extent what constitutes showing off of one's beauty depends on the cultural context we live in, there are many parts of the body that are used for showing off beauty across all cultures and these are realities of our human condition not just culture. Hair is recognized by very many of the scholars of Islam, as well as other scholars across the world, as one of them. To be sure that you are not showing off your beauty as a woman you should cover up these areas appropriately.
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