The word Paul used that's translated "completeness" is:
1) brought to its end, finished
2) wanting nothing necessary to completeness
4) that which is perfect
4a) consummate human integrity and virtue
4b) of men
4b1) full grown, adult, of full age, mature
To as student of philosophy, as I believe both Paul and the Corinthians were, the word naturally would be connected to teleology, the study of the final cause or purpose of things. Paul contrasts it with the "partial" (meros <3313>), which will pass away. According to the Theological Dictionary of the New Testament, Volume 1, there's a philosophical angle to μέρος as well:
The LXX does not use it for human or animal parts of the body, but Philo has it in this sense, as well as for parts of the soul or the world. Accepting the unity of the cosmos, Philo teaches the Stoic harmony of the parts. The parts share in the totality only for the sake of the totality, and the truly perfect good is a whole. As a part, man cannot be the direct image of God, only an image of the logos.—p. 585
Chapter 13 is sandwiched between two parts of Paul's instructions on spiritual gifts. Chapter 12 addresses the problem that different people get different gifts, which apparently caused dissension among the believers in Corinth. Paul pulls out one of his favorite analogies: the church as a body with a variety of members. He transitions to his definition of love:
But earnestly desire the higher gifts.
And I will show you a still more excellent way.—1st Corinthians 12:31 (ESV)
The "still more excellent way", therefore is agape love. It is a power that brings the parts together and provides the ultimate purpose for the spiritual gifts. So Paul follows Philo's argument that as individual parts, we aren't yet direct images of God. So, when will that happen? When will we see "face to face"?
The passage lists several things that will cease:
- knowledge (gnōsis <1108>)
The easiest reading is that knowledge will pass away at some point in the future when we have direct communication with God. Prophesies and tongues as specialized gifts provided to communicate with other believers would also be unnecessary at that point in the future. Therefore, the cessationist, must argue that somehow knowledge has also come to an end.
Since I can't think of a plausible argument for why knowledge might continue after other spiritual gifts have ceased, I believe this passage must be looking forward to the future when Jesus will return and the church will meet Him in person.