Time Travel, Paradoxes and Computation
Time travel has been a trope of Science Fiction since its inception as a genre. Perhaps the most famous is H. G. Wells “The Time Machine,” which gives us brief glimpses of the future at several points. The idea is certainly seductive. Who wouldn’t want to be able to whiz off to the future to view the progress humanity has made, or travel to the past and witness historic events?
But whether or not time travel is possible is still an open debate among physicists. In this post I want to discuss some of the paradoxes that would seem to result if time travel is possible, as well as an interesting algorithm for solving NP problems using a time machine.
Some reject the possibility of time travel out of hand. Their arguments rest on the fact that time travel seems to engender several paradoxes.
The most well known paradox is referred to as the “Grandfather Paradox,” which involves an individual who travels back in time and changes history in such a way that negates his own existence, from which it follows that he could never have existed to change history in the first place. It is called the grandfather paradox because it usualy involves our intrepid traveler (whom we shall from here on call Fred) killing his own grandfather before his father is born, thus breaking the sequence of events that lead to Fred’s birth, which is necessary for him to go back in time to kill his ancestor.
While on the face of it, this paradox might seem to indicate that time travel is impossible, it only does so if one is willing to postulate that the time traveller has a sort of radical free will whilst in the “past”, i.e. that nothing conspires to prevent Fred from doing his deed. Of course, the only way to leverage this claim is to suppose that for some reason Fred, while having every intention and means to kill his grandfather, fails to do so. Perhaps he is run over by a car before he can do so. Or perhaps the rifle he is using jams when the trigger is pulled. The obvious problem with this line of reasoning is that we seem to be postulating a number of metaphysical assumptions that seem dubious at best.
Interestingly enough, a scientist calculated the probability that a mass sent back in time could engender a paradoxical situation. He found that though there are many trajectories a mass could take, the probability that the mass would take a trajectory that resulted in a paradox was zero. This led him to postulate what is now known as the Novikov Self-consistency Principle. If correct, this would put the grandfather paradox argument to rest for good, but leave us with the even thornier problem of a reality that seems to “know” that certain events must not be allowed to happen.
Another paradox is called the Ontological Paradox. This paradox revolves around the creation of objects and information. Consider this example: I go to the book store and buy the complete works of Shakespeare. I travel back in time to find a young Shakespeare who has not yet written any plays. I give him the complete works, along with a detailed chronology telling him when to “produce” each work. The question then becomes, who wrote Shakespeare’s plays?
The information in the plays has become a closed causal loop. They had to have been preformed in the past to be written down and known in the future. But they had to be written down in the future in order to be transmitted to the past. The information contained in the plays causes itself to exist, without having been created.
An even stranger version of this paradox involves the time machine itself. Let us say I resolve to create a time machine. I spend months working on it but find some problems insurmountable. One day, an older gentleman appears on my door. He gives me detail schematics for a time machine, as well as a large sum of money to finance its development. I build the machine according to his plans, and using it I get rich playing the stock market. Many years in the future, I travel back in time to visit my younger self, to whom I give the plans for the time machine, as well as the money to build it.
Once again, in this story we have information that has a confused causal history, which makes a loop in time with no discernible start point.
The ability to transmit information into the past would have serious ramifications for computer science and mathematics. With this ability, it is trivial to develop an algorithm that instantly provides the answer to any NP question. Here is how the algorithm works.
For 1 minute, listen for an answer transmitted from the future.
If an answer is received, check to see if it is correct.
If the answer is correct, transmit it to the past, output the answer and halt.
If the answer is incorrect, choose a random answer, and transmit it back in time.
The way this algorithm is structured, the computer (from our point of view) must immediately produce a correct answer, with essentially no computational effort.
It would also be possible to prove that any given proposition has or lacks a finite length proof. Let us suppose we wish to test if some proposition has a non-infinite proof. Proof checking algorithms exist, and are reasonably efficient. So imagine this algorithm:
Wait for a proof from the future.
If no proof is received, generate a random proof and transmit it to the past.
If a proof is received, check to see if the proof has already been verified.
If the proof was verified, transmit it the past, output it and halt.
If the proof was not verified, check the proof for validity.
If the proof is a valid proof of the proposition, transmit the proof to the past with the validity bit set.
If the proof is not a valid proof of the proposition, generate a new random proof and transmit to the past.
There are practical limits to the length of a proof. These limits are fixed by the amount of time it takes to verify the proof, and how large a proof a human can hold in his head. Surely there are proofs out that that are so long as to be basically incomprehensible. We have proof checking algorithms, but if it takes them 1 trillion years to verify a proof, they are not much use to us. A time machine would throw these limits out the window, by making time a reusable commodity. The machine would take 1 trillion years to verify the proof, but we wouldn’t have to wait those years. From our point of view, the correct answer pops out of the machine as soon as we turn it on.
These two algorithms are examples of Time Loop Logic, or Temporal Computing. Note that in both these algorithms, the potential for a paradox exists. If the answer we receive from the future is not the correct one, we transmit a different answer to the past. This is a paradox. But if the self consistency principal is correct, paradoxes cannot happen. So the initial answer we receive from the future must be the correct one.
Perhaps the thorniest problem with those who believe in the possibility of time travel is an analogue of Fermis’ paradox: If there are time travelers, where are they?
Surely in the course of history there must be events that many people would desire to see. The birth of Jesus (or any other major religious figure) for example. And yet we do not have accounts of millions of strangers flooding Bethlehem. This fact would seem to indicate that time travel is never discovered. Or, at least, if it is discovered, it is so far in the future that people no longer care about this time period.
To seek proof of the existence of time travelers, in May of 2005, a student organization at MIT hosted a time travelers convention. Their intention was to broadcast a particular space time location at which time travelers can show up and make themselves known. As mentioned in the article, “The space-time coordinates continue to be publicized prominently and indefinitely, so that future time travelers will be aware and have the opportunity to have attended.” Setting aside the amusement factor of the confusing tense of that sentence, the idea itself seems sound.
Of course, there are people who claim to be time travelers, such as John Titor. Unfortunately, their claims have never stood up to verification. Perhaps if time travelers do exist, they follow a temporal analogue of the environmentalists “Take only pictures, leave only footprints.” After all, it would not be any fun if your vacation to the distant past altered the future you came from from an idyllic paradise to a hell on earth, would it?
In closing, I would like to point out that an anthropic explanation of why time travel is impossible also exists. This line of reasoning states that any universe in which history is mutable is inherently unstable. If we could go back and tinker with the past, eventually one of two situations would occur: Either a history would result in which humans never developed, or a history results where humans do develop but never discover time travel. The first situation is stable because, obviously, we wouldn’t exist to do the tinkering. The second is stable because we would lack the means to tinker.
The second situation has troubling metaphysical implications, however. It postulates that by a simple historical accident, we never discover a technology that we could in fact discover.
If you are interested in pursuing these further, I would suggest obtaining a copy of The Philosophy Of Time, and in particular the chapter titled “The Paradoxes of Time Travel” by David Lewis.
If anyone from the future is reading this, feel free to contact me with verifiable information to back up your claim of being from the future. I do not require much. The winning lottery numbers for the next several years would do nicely. Or perhaps sport scores, or horse race results….