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A virgin is a woman. Matthew and Paul were using different terms to refer to the same person, Mary, to make different theological points about God's incarnation as Jesus. Different terms in themselves is not grounds for contradiction.
Matthew was quoting Isa 7:14 to make the point that Jesus's birth was another fulfillment of an ancient prophecy. It could have been translated as "young woman" from the underlying the Hebrew but English translations preserve the Septuagint translator's choice of "virgin" especially because the passage also serves to show that the conception was miraculous.
But Paul's point in Gal 4:4 was entirely different. "born of a woman" was meant to highlight that Jesus was God incarnated as a human being to be "one of us", as a member of humanity. Every one of us was born of a woman. The phrasing is about solidarity that shares our being cursed under Adam. Paul's point is that Jesus is acting as a representative (as second Adam) to redeem the whole humanity (v.5) who was "enslaved to the elementary principles of the world" (v.3), upgrading our status from slave to son (by adoption) to heir (v.7).
Another related possible connotation of "born of a woman" that Paul could have tried to communicate to his reader is that the human condition is frail, and it's very gracious for God to suffer this humiliation. This meaning is suggested by a journal article from the January 2022 edition of La Civiltà Cattolica 'Born of a Woman, Born under the Law': Christmas according to Saint Paul, which proposes that the formula "being born of a woman" recalls Job 14:1-2 (ESV):
"Man who is born of a woman
is few of days and full of trouble.
He comes out like a flower and withers;
he flees like a shadow and continues not.
and of Hymns of Qumran where
“born of woman” means “formed of dust,” “creature of clay” (1 QS 11,215; cf. F. García Martínez, Testi di Qumran, Brescia, Paideia, 1996, 95).
This meaning in turn reminds me of the 'dust' in Psalm 103:13-14 (NLT):
The LORD is like a father to his children
tender and compassionate to those who fear him.
For he knows how weak we are;
he remembers we are only dust.
and the 'wildflower' in Psalm 103:15-16 (NLT):
Our days on earth are like grass;
like wildflowers, we bloom and die.
The wind blows, and we are gone—
as though we had never been here.
CONCLUSION: Matthew and Paul tried to communicate many things regarding God's incarnation as Jesus, and in the process used different terms to refer to their Scriptures, our Old Testament. Since the purpose is unified, there is no contradiction.
Upvote:2
The verb in Gal 4:4 is ginomai (Strong 1096) which means 'to come' or 'to become'.
God sent forth his Son, made of a woman, made under the law, [Galatians 4:4 KJV]
ο θεος τον υιον αυτου γενομενον εκ γυναικος γενομενον υπο νομον [TR, undisputed]
The English word 'born' (which is somwewhat ambiguous) is not available in Greek which is more precise and speaks of 'delivery' (tikto, the act of a woman delivering a child) or 'begetting' (gennao, the production of a separate entity into the world after the cord is cut).
Thus there is no contradiction.
What Paul is stating is the manner of the coming of Jesus Christ, that is, by woman. And the condition into which he came : that of an extant legal authority.
Interlinear Greek/English of Galatians 4:4
γίνομαι 'to come' or 'to become' Biblehub/Strong 1096