I can’t wait, although I often wonder and am rarely answered: on what is your non-religious morality based?
EMPATHY.
Empathy is a sensation, not a reason; I feel it, and am a happy atheist; you feel it, and we could be neighbors in a lovely secular country. But what about people who don’t feel it? Not sociopaths, mind you: the many people whose ‘circle of empathy’ (to borrow Carl Sagan’s wonderful phrase) is narrow?
I can’t stay someone’s hand before violence by advising, “Feel empathy, damn you!” I believe in neither damnation nor coercion of feeling. This is the reality of the world; we are not all empathetic.
Those are two different problems. The first is your question, why be (non-religiously) moral? RJS answered but did not elaborate a morality based on empathy, although many have done so throughout the ages (see Kant in particular). The idea that we feel someone else’s pain as our own has been central to liberal philosophy since before the Enlightenment.
The second problem is in your response which suggests that the test by which we measure morality must be that which convinces others not to act in a certain way (in this case violently). But all systems of morality apply only to the individuals who subscribe to them. The test, properly understood, is whether or not that morality convinces me to stop acting violently. In this case, empathy could surely convince me not to act violently towards others.
Modern philosophies of morality have become quite sophisticated, but still retain their roots in some notion of human dignity. A brilliant theory is Amartya Sen’s (in economics) and Martha Nussbaum’s capabilities approach.