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Thomas Aquinas discusses Mary as the sinless Mother of God in the Summa Theologica (Third Part, Question 27, Article 4), writing that Mary
would not have been worthy to be the Mother of God, if she had ever sinned. First, because the honor of the parents reflects on the child, according to Prov. 17:6: "The glory of children are their fathers": and consequently, on the other hand, the Mother's shame would have reflected on her Son. Secondly, because of the singular affinity between her and Christ, who took flesh from her: and it is written ( 2 Cor. 6:15): "What concord hath Christ with Belial?" Thirdly, because of the singular manner in which the Son of God, who is the "Divine Wisdom" (1 Cor. 1:24) dwelt in her, not only in her soul but in her womb. And it is written (Wis. 1:4): "Wisdom will not enter into a malicious soul, nor dwell in a body subject to sins." We must therefore confess simply that the Blessed Virgin committed no actual sin, neither mortal nor venial.
Does Catholicism apply all these reasons to the mother of the "Mother of God"? If not, why not? And if yes, what about the Grandmother of the "Mother of God", and so on backwards to Eve?