Could Paul have argued against a Jewish tradition in I Corinthians 11:1-16?

Question:

While looking around for a Jewish tradition from the first century I'd read some years before that was almost word for word what Paul addressed in the Corinthian letter, I ran across your page.  I'm not sure the point you were making in "Images of Head Coverings During Worship".  Anyway, as long as I was passing by I thought I'd send you a copy of something I'd sent my father on the same subject.

Dad,

This is the division I referred to in my message to Al Maxey concerning his Head Covering article you sent, which I replied briefly on.  When you look at the passage this way, it seems too obvious.  Sure would be nice to have the original copy of the letter to see any demarkations between parts that may have existed, also nice to have the original letter although not necessary since many of the questions can be guessed by the answers given.  This, however, seems to be a specific doctrine among the Jews which Paul felt constrained to repeat before addressing it.

1 Corinthians 11

Paul's Prologue:
1 Be ye followers of me, even as I also am of Christ. 2 Now I praise you, brethren, that ye remember me in all things, and keep the ordinances, as I delivered them to you. 3 But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.

The Jewish tradition stated:
(also found in ancient Jewish writings)

4 Every man praying or prophesying, having his head covered, dishonoureth his head. 5 But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head: for that is even all one as if she were shaven. 6 For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered. 7 For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man. 8 For the man is not of the woman: but the woman of the man. 9 Neither was the man created for the woman; but the woman for the man. 10 For this cause ought the woman to have power on her head because of the angels.

Paul answers the tradition:
11 Nevertheless neither is the man without the woman, neither the woman without the man, in the Lord. 12 For as the woman is of the man, even so is the man also by the woman; but all things of God. 13 Judge in yourselves: is it comely that a woman pray unto God uncovered? 14 Doth not even nature itself teach you, that, if a man have long hair, it is a shame unto him? 15 But if a woman have long hair, it is a glory to her: for her hair is given her for a covering. 16 But if any man seem to be contentious, we have no such custom, neither the churches of God.