David Terr's Website



“Fibonacci Expansions and ‘F-adic’ integers,” The Fibonacci Quarterly, v. 34, 1996

“On the Sums of Digits of Fibonacci Numbers,” The Fibonacci Quarterly, v. 35, 1996..

A Modification of Shanks’ Baby-Step Giant-Step Algorithm,” Mathematics of Computation, v. 69, 2000

Here's a copy of my resume

Time Travel Paradoxes

There are several types of paradoxes associated with backward time travel (into the past). Forward time travel (into the future) does not involve any paradoxes. In fact, we are all already traveling into the future, albeit at a fixed rate. By contrast, the reason why many reputable scientists as well as laypeople dismiss the idea of backward time travel is that it does involve so many paradoxes which defy intuition and common sense, as well as well-established mathematical and physical principles like logic and causality. Nevertheless, there are ways to resolve most of these paradoxes, as we will see. Some scientists do entertain the possibility of backward time travel, and in fact there seems to be evidence that both general relativity and quantum mechanics, the two greatest contributions to 20th century physics, in fact predict some form of backward time travel!

The Grandfather Paradox
Suppose a maniac with a time machine is bent on killing his grandfather in the past, so he steps into his time machine with a gun, goes back in time to when his grandfather was young, and kills him.(The same argument could be applied to the killer's father, but for some reason, his grandfather is usually descibed as the victim.) Then his grandfather won't be able to get married and have children, and thus won't have grandchildren, so the killer will never be born! Then how could he have made the trip to begin with? This involves a logical contradiction, because if time is fixed, then either the killer's grandfather was killed or he wasn't. If he was, then the killer will not be born and if he wasn't, then the killer will be born. There are ways to resolve this paradox. One, which is commonly used in science fiction stories involving time travel, is the notion of alternate timelines, otherwise known as parallel universes. According to this idea, once the killer goes back in time, he creates an alternate timeline in which he kills his grandfather and is not born. In the original timeline, however, his grandfather is not killed and he is born. The notion of parallel universes is appealing for other reasons having to do with quantum mechanics and not time travel per se. Another possible resolution assumes that the timeline is fixed, but that the killer must fail to kill his grandfather (he changes his mind, he has an accident, the gun slips from his hand, etc.), because it never happened. This resolution, while being logically consistent, involves dispensing with free will.

Closed Time Loops (The Chicken and the Egg Paradox)
This paradox is more subtle, but it still presents problems. The best example of this paradox I know of is "Find the Sculptor", a 1946 science fiction short story by Samuel Mines, in which a man builds a time machine and travels 500 years into the future to find a statue of himself in honor of his being the world's first time traveler. He removes it and takes it back home with him, whence it becomes erected in his honor. But when was the statue built? This paradox involves a closed time loop experienced by the statue. Some scientists do entertain the notion of closed time loops, which in fact are predicted from certain spacetime models in general relativity, but the idea is very counterintuitive.

Creating Multiple Copies of Oneself
The idea of alternate timelines resolves the grandfather paradox, but it creates a new one involving the possibility of creating multiple copies of yourself or of anything else you choose to send back in time. Suppose you hop into your time machine, travel back one hour (thus creating an alternate timeline), then meet yourself one hour before you left in the original timeline. Then there would be two of you existing in one time. You could then repeat the trip with your clone, meet another copy of yourself one hour in the past, travel back one hour with him (now three of you), continuing as many times as you like. Although this does not involve any logical contradictions, it clearly defies common sense.

Sexual Paradoxes
These are perhaps the strangest of all time travel paradoxes. One can imagine a scenario involving a closed time loop in which a man travels back in time to when his mother is young and gets her pregnant with himself, thus becoming his own father. Although this is hard to imagine, even stranger such paradoxes are possible. Robert Heinlein wrote a very strange sci-fi short story in 1959 called "All You Zombies" involving a sexual paradox in which a man, having been born a girl and since undergone a sex change operation, goes back in time from 1970 to 1963, has sex with himself as a younger woman, gets her pregnant, has a daugher, and transports the baby girl back to 1945. The baby then grows up to become both the woman and the man, so that this character is both his own father and mother! (Mighty strange, huh?)

No Time Travelers from the Future
If time travel into the past is possible and is discovered someday, then one may ask why we haven't been visited by any time travelers from the future? One would expect there to be large groups of tourists from the future visiting important historic events, such as the birth of Christ or the signing of the Declaration of Independence. There are several possible resolutions to this paradox. For instance, it may be possible to travel back in time, but not to any time before the construction of the time machine built for this purpose. Some such time machines have in fact been imagined which would work according to general relativity. An alarming possibility is that backward time travel is possible, but that mankind will never build a time machine, perhaps because we'll end up destroying ourselves before we get the opportunity to discover time travel.