All content copyright © 2010-2019 Frank Revelo
Mother is the animal self or soul. She is often represented as a Virgin because she is to be considered as pure and without blemish, like everything in the material world. That is, we must accept matter for what it is, without criticism as to its essence. We can no more complain about our animal self than we can complain about the law of gravity—he who blasphemes against the Holy Ghost (the Virgin in the New Testament) can never be forgiven. Similarly, he who kills his mother (denies his animal self) will be persecuted forever by the furies (suffer mental illness).
Father is the moral code of society, the "law", the "word", the "seed", creator of the Spirit. To kill the Father is dangerous, but a necessary step on the path to enlightenment. Thus Queen of Swords in the Tarot is depicted with the head of the Father dripping blood. To kill the father is to free oneself from the tyranny of excessive programming, such that the individual will is repressed in favor of the will of the society.
Son is the seed of the Father, or word, implanted in the Virgin and become flesh, spirit crucified upon the cross of matter. Son rather than daughter because we humans are primarily spirit, with only a small residual animal nature. That is, happiness in humans is mostly dervived from our spiritual nature, or mind, rather than the body and soul.
The Father (and also the Devil, the Father's opponent or alternate face, as the case may be) should not be depicted by graven images, since "moral code" is an abstraction and to represent abstractions with concrete images is inherently misleading. However, this rule is often violated, mainly because low intelligence types simply can't cope with abstraction. Much of the stupidity in religious discussions (God as a bearded old man flying around in the sky, Devil as a creature with horns and clove feet, etc) is due to violating this rule on graven images. By contrast, both Mother and Son can be represented by graven images without causing confusion. Mother represents the soul, which is tightly connected to the concrete body. Son represents the total self, including the concrete body.
Trinitarianism is something of a compromise by the establishment priesthood, who represent the will of society, since it acknowledges the existence of the individual and thus the possibility of conflict between individual and society. Pure monotheism denies this potential conflict and admits the exitence of one thing and one thing only: the will of society, expressed through the societal moral code. Pure monotheism typically must be enforced with physical violence, but that is expensive. Practically speaking, "you can catch more flies with honey than vinegar", which is why the Catholic church has usually tolerated Virgin worship. Similarly, the Hindu establishment tolerates Kali/Durga worship and even pantheism, though they would certainly prefer monotheism. The Protestants eliminated the Virgin and merged the Son back with the Father, and so recreated monotheism. On the other hand, the Protestants (and Moslems to some extent) permit the laity to act as priests (to question the societal moral code and push to have it changed) whereas the Catholics do not, so in terms of oppressiveness to the individual, these various religions are probably equivalent.