If the only reason to do it is because the people around them will hold their oath in higher regard - shouldn't they swear on a stack of all religious texts, and on all gods?
> Again, you have heard that it was said to the ancients, ‘Do not break your oath, but fulfill your vows to the Lord.’ 34But I tell you not to swear at all: either by heaven, for it is God’s throne; 35or by the earth, for it is His footstool; or by Jerusalem, for it is the city of the great King...
Implying that even in a religious context, one might swear on "one's mother's grave"in the metaphorical sense. So then I think there must be a natural and historical reverence to oaths in general, which sends my mind down more Neil Gaiman-ish paths. I would bet that the significance of an Oath is older than every extant religion.
Furthermore, you can’t prove what happens after you die. I mean you may very well be right, but it’s just your personal belief.
That is a belief in personal responsibility not motivated by fear of punishment by a supreme being.
This quote from Sagan convey's that meaning among others IMO:
It has been said that astronomy is a humbling and character-building experience. There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we've ever known.
Also see
https://en.wikipedia.org/wiki/Lawrence_Kohlberg%27s_stages_o...
Where fear of gods punishement would be at the lowest level.