How can one respond to those who cite the hadith of Fatimah bint Qays in the Two Sahihs as evidence for the permissibility of intermingling, specifically from two aspects: first, the entry of the Companions to Umm Sharik's house, and second, Fatimah's observing her 'iddah in the house of Ibn Umm Maktum?

The hadith (definition) of Fatimah bint Qays is not abrogated. Her observing the 'iddah (definition) in the house of Ibn Umm Maktum was out of necessity because her husband had divorced her thrice. The Prophet (peace be upon him) saw that her observing the 'iddah with Umm Sharik would cause hardship, so he ordered her to observe it with Ibn Umm Maktum because he was blind. This falls under the principle of choosing the lesser of two evils.

As for the Companions entering upon Umm Sharik, it was due to their frequent visits to her because of her righteousness. This is permissible and there is no problem with it, because what is forbidden to prevent a means (sadd al-dhari'ah) is permitted for an overriding benefit (maslahah rajihah). With the safety from seclusion (khalwah) and temptation (fitnah), the prohibition of mixing (ikhtilat) is negated. This cannot be compared to the calls for permissiveness and mixing in contemporary times.

Similarly, it is permissible to look at a beardless youth (amrad) for teaching Islamic sciences and necessary crafts. It is not piety to abandon teaching, even if it requires a gaze from which no feared harm is anticipated. This is what Ibn Hajar al-Haytami answered, and the matter revolves around what is in the heart and what the circumstantial evidence of the soul testifies to.

The consideration in mixing is the fear of temptation or the existence of its likelihood, because the prohibition of mixing is not for its own sake but to prevent the means to temptation. The blameworthy mixing is that which involves a gathering between non-mahram (definition) men and women without need, or involves soft speech or uncovering of 'awrah. As for the permissible mixing, it is that which is safe from temptation.

The sacred texts warn against the temptation of women, for they are the greatest temptation for men. Whoever seeks safety in his religion and his worldly life must keep away from them as much as possible.