Keith’s Conundrums: Three Time Puzzles

Keith Douglas

What, then, is time? If no one asks me, I know; if I wish to explain to him who asks, I know not.
— Augustine of Hippo

A big ball of wibbly wobbly, timey wimey stuff.
— The Doctor

Last time we discussed at length the classic liar’s paradox and its surrounding complexities. I have no view on how to solve this family of puzzles, other than to note that it is incredibly hard to do any work here. I do regard it as an almost absolute constraint that classical mathematics be “recoverable.” I also think refiguring some of classical analysis first and then rejigging the official logic(s) would be nice.

There are people who would find “Gibbs is brown” to be socially sanctioned shorthand for “Gibbs is brown in such and such contexts to humans, etc.,” with a specific Gibbs in mind. This is important because some really pedantic people would say the original has no truth value because of this. I am not one here. I think one can attribute truth values to ordinary uses of predicates, at least sometimes.

Three Time Puzzles

Onto new stuff — or is it old stuff? I cannot claim to have originated these time puzzles, but I can give them new twists. The first is on the biology of time travel; we follow that with a puzzle about continuity in time (or so it might appear) and the nature of infinity. The third is the most challenging. At least to my mind — it is also about continuity and also about scientific realism, mathematized modelling of the world, etc. Onto the show — literally, since our source is the great science fiction cartoon comedy Futurama.

1. Fry’s grandfather. Futurama made fun of and invoked many staples of science fiction, fantasy, and even philosophy. In “Roswell That Ends Well” (spoilers ahead!), the hapless 31st century Planet Express crew discover that they are the source of the UFO crash in Roswell, New Mexico, in 1947. (You know the show is fiction when we all know it was actually time-travelling Ferengi!)

However, since one of their number is, to put it kindly, a bit clueless sometimes about interactions with others, he must desperately avoid a grandfather paradox. The usual grandfather paradox — Can a time traveller kill his own ancestor? If so, what happens? — is alluded to, but Fry unfortunately engages in an activity that seems at first glance to create another paradox: He discovers to his horror that he will be/is/was his own grandfather.

Your puzzle is thus: What proportion of Fry’s father’s paternal DNA is shared with that of (a) Fry and (b) Fry’s grandfather. In particular, does Fry have any paternal DNA mutations relative to his father? Is this paradoxical? I encourage you to use this exercise to explore “possibility” as some fans of the show think that there are other paradoxes lurking in the situation described. (Spoilers finished.)

2. Thomson’s lamp. Unlike the first puzzle, this one is actually from the philosophy literature directly: a 1954 article in the famous journal Analysis. Imagine someone who performs a series of tasks as follows. At one minute from our start, she turns on a lamp. Thirty seconds later she turns it off again. Fifteen seconds after that she turns it on again, and so on. What is the state (on or off) of the lamp after two minutes?

Clearly this is a situation that is nomologically impossible. For example, as far as we know, electricity does not flow infinitely fast, so there’s always a delay on the lamp changing state. But at least some philosophers like to worry about more general notions of impossibility. I have mentioned that logical impossibility applies to propositions or statements, not to events. However, speaking loosely: Is this situation a logically impossible one?

Where does the contradiction lie? That’s the hard part: showing that there is in fact a contradiction. Try to reduce the problem to showing a suitable description, such as “The lamp is on and the lamp is off.” Alternatively, pick another contradiction.

3. Velocity. This one is the hardest of all; I personally still wrestle with what to say about it. It requires some familiarity with calculus, so if you do not have that background please bear with us until next time — or go and learn!

I owe my knowledge of this particular puzzle to the 1903 text of Principles of Mathematics by Bertrand Russell. At this point in his long life and career, Russell was getting out there in mathematics but was not the popular public gadfly he was to become, nor had he done much work in pure philosophy. Nevertheless, the logicist thesis (that mathematics and logic are identical) is presented and a large amount of material on the foundations of mathematics and indeed also in mathematicized physics is presented.

Consider the definition of velocity in classical mechanics: v = ds/dt. Substitute definitions (of derivative, continuity, limit, etc.) until one has phrased everything in terms of the definition of limit. See here to follow along. Look carefully at the use of those absolute value bars. They concern the value of a function’s “neighbourhood.” Imagine calculating the velocity of a particle at time a using the function f, from the referenced page, as a position function (let x be the time variable).

This requires the position function to extend before and after the point as depicted, because we “imagine” adding and subtracting delta. If a is “now,” then it seems that the particle, in order to have a velocity, must in some way already “know” its future position. If it “jumps” too sharply, then it has no velocity at all (the derivative is undefined).

Does this entail that the future is in some sense “already there”? Is one taking the notion of limit too literally? Does physics end here, while the rest is mathematics? Should one use mathematics with another notion of limit (or continuity)? Is the definition of velocity using calculus itself wrong?