How often we have heard the expression, "I will forgive, but I will not forget." We can be very grateful that God does not think that way. Not only does he forgive our sins, but the Bible says repeatedly that he chooses not to remember them. In fact, it is part of the new covenant that, "I will forgive their sins and I will no longer remember their wrongs." (Jeremiah 31:34).
Does this actually mean that God is forgetful? Of course not. I believe God uses this terminology to give us the clear message that he does not keep accounts against us. In (Isaiah 43:25) he uses an expression that indicates that he erases any record he had written down against us: "I am He who blots out your transgressions for my own sake, and I will not remember your sins." (RSV)
Paul's way of saying it in verse 19 in the passage above is to use a word that was part of bookkeeping language, "God did not keep an account of their sins." More than that, Paul puts it in a context where he is speaking about our own calling to reconciliation. As God has done, so are we to do. As he has been reconciled to us by removing any record of our sins against him, so should we also do in our efforts to be reconciled. "The old is gone."
I may not be capable of actually forgetting something that was said about me or done against me. But am I able to remove that event completely from my records? I know people who claim to have forgiven but, if you should ever do something to hurt them today, you will hear them say, "You have done that again." "Again" means they still have the old ones on their books. They do not just remember mentally, they still hold it against you. They have not actually forgiven you.
If I hold on to those old hurts, I will never be able to be truly reconciled with that person. If I have really forgiven them, then I have nothing on record against them. Now reconciliation can begin.