This is one of those questions that gets asked a lot, though often as a result of some misunderstandings of terminology. Often this kind of question comes from a critic of Calvinism…or perhaps more specifically, from a critic of the Calvinistic doctrine of Total Depravity. They see the words, “Totally Depraved,” and assume that what the Calvinist means is that human beings are as bad as they possibly can be. Yet, a brief survey of the world around us will illustrate that even unbelievers do noble things and the worst of mankind could act worse than they do. So, the critic says, “See, the world around you disproves your doctrine.”
Yet, the argument is more of a straw man than anything else. All that the world around us proves is that the misunderstanding or misrepresentation of the doctrine that the critic holds is untrue, but that says nothing about the doctrine itself. The doctrine itself simply speaks to the fact that the whole person has been affected or tainted by sin and thus the whole person (mind, body, soul, actions, etc…) needs redeeming. Yes, we could be worse, though God restrains our sin lest this earth become little more than a shadow of hell.
The other question raised by statements like this has to do with the definition of the word, “good.” Certainly Jesus teaches that only God is good (Luke 18:19). And so, the quick answer is, “No, no one can do anything Good because God alone is good.” Yet, that seems to beg the question because the person asking is not asking about good in the absolute sense, but simply about the things that we might appreciate as good, noble, or perhaps as meritorious.
Of course, the answer is still, “no.” The reason for this is that God considers anything that is not done “in faith” to be sin (Romans 14:23). Further, Paul writes to the Colossians that everything they do should be done in the name of Jesus Christ giving thanks to the Father through Him (Colossians 3:17). And so, if the wicked have rejected God (which they have) and do not honor the Son, Jesus Christ (which they do not), then it is impossible for them to do anything good in God’s eyes no matter how noble we might view an action in our own eyes.
And so the Heidelberg Catechism makes it clear in question number eight, that unless we are born again by God (thus having faith and doing all in thanksgiving to God through Christ) then we can do nothing good. This is both a matter of Total Depravity and of Intentionality. Lots of pagans do things that are noble on a human level, but in God’s eternal economy they fall short.