The lesson that Hawthorne intends to show in his short story "Dr. Heidegger's Experiment" is that if people were ever to be given the opportunity to go back in time, they would also revert back to all the behaviors that characterized them at the time. In essence, people would likely not change. The key reason is simple: people are who they are, no matter in what place or time they exist.
In the story, Dr. Heidegger intends to conduct an experiment where he will give four eccentric old people a youth elixir. He warns them, however, not to allow the feelings of youth to make them go back to committing the same foolish acts that were once dictated by their immaturity and naïveté.
In fact, he encourages the four participants to become beacons of respectability, maybe even inspiring other youths to avoid making bad choices and to start over again with better, respectable lives.
...you should draw up a few general rules for your guidance, in passing a second time through the perils of youth. Think what a sin and shame it would be, if, with your peculiar advantages, you should not become patterns of virtue and wisdom to all the young people of the age!
Clearly, this is easier said than done. Once the participants feel young again, they immediately revert to the exact immature and self-indulgent desires that once marked their lives. They went back to the same issues, same arguments, and same weaknesses.
They learned nothing. Rather than learning from their experiences as older people and applying them to their newfound youth, they instead indulged over and over in past bad choices. In fact, once the elixir wore off, they looked forward to finding the mythical fountain of youth, from which they planned to drink for days on end.
Meanwhile, Dr. Heidegger may have learned that he was right all along: why try to go back in time if it means going back to the same issues that once haunted us? It is no surprise that, when the bottle of the elixir crashes and the liquid spills, Dr. Heidegger does not feel sorry in the least.