What means this word, righteousness? Why, the end, the scope, the well-doing. For what was its design, and what did it enjoin? To be without sin. This then is made good to us (κατώρθωται ἡμἵν) now through Christ. And the making a stand against it, and the getting the better of it, came from Him. But it is for us to enjoy the victory. Then shall we never sin henceforth? We never shall unless we have become exceedingly relaxed and supine. And this is why he added, “to them that walk not after the flesh.” For lest, after hearing that Christ has delivered you from the war of sin, and that the requisition (δικαίωμα) of the Law is fulfilled in you, by sin having been “condemned in the flesh,” you should break up all your defences; therefore, in that place also, after saying, “there is therefore no condemnation,” he added, “to them that walk not after the flesh;” and here also, “that the requisition of the Law might be fulfilled in us,” he proceeds with the very same thing; or rather, not with it only, but even with a much stronger thing. For after saying, “that the righteousness of the Law might be fulfilled in us that walk not after the flesh,” he proceeds, “but after the Spirit.”
So showing, that it is not only binding upon us to keep ourselves from evil deeds, but also to be adorned (κομᾅν) with good. For to give you the crown is His; but it is yours to hold it fast when given. For the righteousness of the Law, that one should not become liable to its curse, Christ has accomplished for you. Be not a traitor then to so great a gift, but keep guarding this goodly treasure. For in this passage he shows that the Font will not suffice to save us, unless, after coming from it, we display a life worthy of the Gift. And so he again advocates the Law in saying what he does. For when we have once become obedient to Christ, we must use all ways and plans so that its righteousness, which Christ fulfilled, may abide in us, and not come to naught.
Ver. 5. “For they that are after the flesh do mind the things of the flesh.”
Yet even this is no disparaging of the flesh. For so long as it keeps its own place, nothing amiss comes to pass. But when we let it have its own will in everything, and it passes over its proper bounds, and rises up against the soul, then it destroys and corrupts everything, yet not owing to its own nature, but to its being out of proportion, and the disorder thereupon ensuing. “But they that are after the Spirit do mind the things of the Spirit.”
Ver. 6. “For to be carnally minded is death.” He does not speak of the nature of the flesh, or the essence of the body, but of being carnally “minded,” which may be set right again, and abolished. And in saying thus, he does not ascribe to the flesh any reasoning power of its own. Far from it. But to set forth the grosser motion of the mind, and giving this a name from the inferior part, and in the same way as he often is in the habit of calling man in his entireness, and viewed as possessed of a soul, flesh. “But to be spiritually minded.” Here again he speaks of the spiritual mind, in the same way as he says further on, “But He that searches the hearts knows what is the mind of the spirit”; and he points out many blessings resulting from this, both in the present life, and in that which is to come. For as the evils which being carnally minded introduces, are far outnumbered by those blessings which a spiritual mind affords. And this he points out in the words “life and peace.” The one is in contraposition to the first— for death is what he says to be carnally minded is. And the other in contraposition to the following. For after mentioning peace, he goes on,
Source: Homilies on Romans (New Advent)