In English translations kata sarka is translated as "according to the flesh" (ESV), "from a human point of view" (NRSV, NLT, NET), and "from a worldly point of view" (NIV) or "a worldly perspective" (CSB).
The issue Martyn faces is if kata sarka should be translated adjectivally or adverbally. If an adjective kata sarka modifies "no one" and "Christ" in the text. In that reading, Paul would be saying he knows no person (or Christ) "in the flesh." If, however, used adverbially kata sarka modifies the verb "to know." In that reading, as Martyn writes, "Paul is speaking about ways of knowing. He now knows no one in a fleshly way; if he once knew Christ in a fleshly way, he knows him in that manner no longer."
To answer the question, Martyn surveys 2 Corinthians 2-6, the context of the text in question, gathering up evidences that Paul saw the Christ-event as the decisive event in cosmic history, a "turning of the ages" from old to new. The Christ-event, then, creates an epistemological crisis, demands a new way of seeing. Consider one of the texts Martyn points to, 2 Corinthians 2.14-17:
But thanks be to God, who in Christ always leads us in triumphal procession, and through us spreads the fragrance of the knowledge of him everywhere. For we are the aroma of Christ to God among those who are being saved and among those who are perishing, to one a fragrance from death to death, to the other a fragrance from life to life. Who is sufficient for these things? For we are not, like so many, peddlers of God's word, but as men of sincerity, as commissioned by God, in the sight of God we speak in Christ.
Notice the point from the last post. The call here isn't for repentance. The focus of action is upon the activity and victory of Christ who "leads us in triumphal procession." But the Christ-event requires seeing properly. Or, rather, smelling properly. For some, Christ smells like death. To others, Christ smells like life. Note the perceptual framing, which is less about repentance than seeing (or smelling!) the new thing God is doing. The fragrance of Christ spreads through the world and to some it smells like rot and to others the sweetest perfume. Two ages, one moving toward life and the other perishing, are now existing side by side and the call is to discern which world or age you belong to.
Following this thread, from chapters 2 to 6, Martyn concludes that kata sarka is being used adverbially in 5.16-17. What Paul is describing in this critical passage concerns epistemology, a way of knowing at the turn of the ages.
