Robert Heinlein's 1941 story "By His Bootstraps" begins with the narrator writing in a philosophy thesis that time travel is impossible because time, in Immanuel Kant's terms, is only empirically real and does not exist independently among things in themselves. The narrator is then suddenly surprised to find two different versions of himself arriving from the future, with conflicting warnings and promises about what he can do. Traveling to the future, he meets an older man who repeats the promises, but whom he ends up distrusting. After some confusion, back in the present, he obtains some supplies and returns to the future to a period significantly earlier than when he would met the older man, intending to contest the future with him. Eventually, however, it turns out that he himself is the older man and his future is in fact, pace Immanuel Kant, secured.
A paradox of time travel arises in relation to this story. The narrator does indeed set himself up "by his bootstraps"--his present and future selves all interact with each other to produce the events. The paradoxical nature of this comes down to the case of a notebook that was provided to the narrator by the older man in the future. It contained a vocabulary of the language that was spoken by people in the future. The narrator learns the language and, as the book wears out over the years, copies it over into a notebook he had fetched from the present. This notebook, as it happens, is the very one he, as the older man, then provides to his other self. He is therefore the same person who both learns the knowledge from the notebook and put the knowledge into the notebook in the first place. The vocabulary as a certain list of items arranged in a certain way was thus complied by no one whatsoever. The knowledge exists in a closed temporal loop and is in an important sense uncaused or uncreated. The narrator himself notes that there is something peculiar about this.
Peculiar indeed. A very similar paradox, allowed by the possibility of the same kind of temporal loop, can become a reductio ad absurdum for time travel. We see just such an paradox in the 1980 movie Somewhere in Time, staring Christopher Reeve and Jane Seymour. As a young man, Reeve encounters an old woman who gives him a watch. Later he becomes obsessed with the painting of a woman in an old 19th century hotel. He decides that he must meet that woman, and he thinks it is possible because of the theory of a professor he had for physics. The professor thinks that it is possible to will one's self back in time, as long as what one carries along is not anachronistic for that time.
Reeve outfits himself for the 19th century and actually succeeds in willing himself back into it. He meets the woman in the picture, played by Jane Seymour, and he is able to win her heart, so that she returns the love he felt ever since seeing her painting. He gives her the watch that he had acquired many years before from the old woman. Then, as their mutual happiness seems assured, Reeve discovers a penny from the 20th century in his suit, and the anachronism vaults him back into the present. He is unable to endure separation from his beloved, starves himself to death in his hotel room, and, apparently, is reunited with her in the Hereafter.
The old woman who gave him the watch in his youth was, of course, Jane Seymour's character, lived to a ripe old age just to see him again. The watch, therefore, was obtained by Reeve from Seymour and was obtained by Seymour from Reeve. In a closed temporal loop, like the knowledge in the notebook in Heinlein's story, the watch is uncreated. But this is impossible. The watch is an impossible object. It violates the Second Law of Thermodynamics, the Law of Entropy. If time travel makes that watch possible, then time travel itself is impossible.
The watch, indeed, must be absolutely identical to itself in the 19th and 20th centuries, since Reeve carries it with him from the future instantaneously into the past and bestows it on Seymour. The watch, however, cannot be identical to itself, since all the years in which it is in the possession of Seymour and then Reeve it will wear in the normal manner. It's entropy will increase. The watch carried back by Reeve will be more worn that the watch that would have been acquired by Seymour.
The reductio ad absurdum created by the watch can be fixed up in a couple of ways. First, we might think that entropy could be reversed by time travel, so that forms of matter would be restored to that state they would have been at the earlier period. But this will not do, since Reeve himself would then be restored to the state his matter was in in the 19th century, which, whatever it was, would not be in the form of Christopher Reeve.
Second, we might think that time travel puts one in an alternative universe. In some universe, the watch is manufactured and bought in the ordinary way, and then the older Jane Seymour, for whatever reason, gives it to the young Christopher Reeve. He goes back in time, to an alternative universe where Seymour did not acquire a manufactured watch, and gives her his. Then she gives it to him later; and he returns to a different universe, where Seymour does not buy a watch but acquires a somewhat more worn watch from him. The temporal loop thus generates a spiral of alternative universes. Unfortunately, it would require a spiral of an infinite number of alternative universes, as each watch in a particular universe is returned to a new universe where it can exist in its increasingly worn state. In some universe, the watch would disintegrate while in Seymour's or Reeve's keeping and need to be discarded; but Reeve would keep returning to the past, unless the watch turned out to be some causal factor in his falling in love with the picture..
Every instance of time travel generating an infinite number of alternative universes might be thought to violate Ockham's Razor, especially since the idea that an alternative universe could be generated in the first place has disturbing consequences for the metaphysics of identity. What does it mean if there are an infinite number of each of the characters, all facing a universe slightly different? Simplicity and common sense rebel against such principles. But without them, time travel, that would allow for the sort of temporal loop in which the paradoxical and impossible watch of Somewhere in Time becomes possible, is itself impossible.
Kant's theory of time may go unrefuted after all.