I have strong feelings both ways, but I find it hard to deny that it probably saved a lot of allied lives. It doesn't mean that it was the "right" choice, but I don't see how it can be possibly dismissed (put yourself in Truman's shoes).
OP2: >> "...saved a lot of allied lives..."
The atomic bombings of Japan had nothing to do with the military campaign in the Pacific theater, which by 1945 had been decisively and indisputably won by the Americans. They had literally run out of targets to bomb in the Japanese home islands. [0-1] An argument could be made that the Atomic bombings saved some American lives by accelerating VJ day and thereby precluding some unpleasant island-hopping, though I've never been terribly convinced.
The atomic bombs were about scaring the Soviets, plain and simple. By '45 the American military was perfectly aware that it had won the Pacific, would soon win the European theater, and had long since turned its attention to preparing for the cold war and plundering German technology and other assets.
[0] See for example Sec. of War Henry Stimson's diary and papers, which are excerpted here among other places: http://www.doug-long.com/stimson5.htm It might help to command-F down to this quote: "I was a little fearful that before we could get ready the Air Force might have Japan so thoroughly bombed out that the new weapon [the atomic bomb] would not have a fair background to show its strength."
[1] https://en.wikipedia.org/wiki/Air_raids_on_Japan#Attacks_on_...
That's true if the Japanese would have fought a land invasion tooth and nail for every inch of ground, civilians charging at GIs with kitchen knives, etc, etc. That's what the Western propaganda of the time predicted; East Asians in general and the Japanese in particular were broadly understood as hive-mind insects yearning to sacrifice their lives and families for the glory of the nation.
The reality is considerably more doubtful, especially now that we know about the government factions that were desperately eager to surrender even before the A-bombs fell. And even if we discount that, there's no excuses made for why the nuclear show of force had to be targeted against civilians, nor why the second bomb was dropped only three days later.
For me, there's a fuzzy line where enough time has passed that actions need to be judged based on what was known, or should have been known, at the time.
The question for me isn't: Did it actually save lives.
The question for me is: Did Truman and Churchill really believe they were saving millions of lives (and, was that a reasonable belief). Which really comes down to: do you believe Truman and Churchill?
My parent said (if I understood correctly) that all the justifications he or she has heard "could not be more unconvincing." That you and I are now talking about how many lives did it really save?, does not change the fact that saving American ( / allied ) lives, is not a wholly unjustifiable (or incomprehensible) thing for a US President to consider. Maybe I was just put off my his or her hyperbole.