I was watching the film Minority Report, and it led me and Heschelian into an interesting (though doubtless unoriginal) discussion of time travel-style paradoxes. While the film itself may – or may not -- have managed to avoid paradoxes, one particular scene reminded us both of what’s sometimes called the Knowledge Paradox. I said I believed in the Grandfather Paradox but I didn't think the Knowledge Paradox was a genuine problem. He disagreed.

We were sitting in a restaurant and scribbling on the serviettes, much to the amusement of the rest of our friends.

As we were both philosophy students who had studied logic, we tried to formalise the problem as best we could. Logically, a paradox is a situation where contradictory conclusions can be derived from the same premises.

One of the big objections to time travel is that it can throw up paradoxes. The most famous of these is the Grandfather Paradox, where a person goes back in time to perform an action A change the past in a way that prevents that person from being able or willing to go back and perform action A. Common examples include :

  • Killing an ancestor before you were born
  • Destroying the time machine/inventor etc
  • Preventing bad things (If you succeed, you wouldn’t know to do it..)

Why is this a paradox? Well…..

If you do action A, you won’t be able to do action A.
If you aren’t able to do action A, you won’t do action A

So our time traveller has to both do and not do something simultaneously. That’s not allowed; it breaches the law of Contradiction; it’s a paradox.

JerboaKolinowski tells me I wrongly called a paradox a breach of the excluded middle. He's right. Go look at the differences.

What do you do with paradoxes? Well there a whole bunch of approaches, from denying them to ignoring them to imagining new universes. For examples see almost every time travel film and book. Philosophers sometimes invent entirely new systems of logic without the law of Contradiction. But that’s not what I’m talking about; in the terms of simple logic, the Grandfather Paradox is a paradox.

The Knowledge Paradox goes something like this:

I’m sitting at home one day and I find a plan down the side of my sofa for a Ray Gun. It’s clear, with easy-to-follow instructions. It also contains the following message:

Arieh, this is from your future self. Use it to become rich and famous.”

So I do. The Ari-CorpTM brand Zappomatic becomes the new weapon of choice for all self-respecting criminals and armies. I become very rich, and fund time-travel research. At some point, when I have the time, I travel back to my old house and hide the Ray Gun plans down the side of the sofa.

Now, I've just created something effectively ex nihilo - the Ray Gun. There exists new knowledge in the world that it seems like nobody ever did any work to discover; it just appeared by itself.

(My brother later remarked that he could understand the knowledge going round and round, but he didn't see how did it get there the first time. This is fallacy number one. The answer is that there is no "first time"; it only happens once. It can be a bit hard to grasp this idea, but there's only one me, and I only do the actions in the story once. So it only happens once.)

What's interesting is that this situation doesn't actually appear to be a paradox, at least in the strict logical sense in which we are using the word. No rules of logic are broken.

My friend seemed satisfied with my attack on the Knowledge Paradox, at least in logical terms.

Then, after we had paid our bill and were leaving, we realised a problem.

Me: I've got to give this pen back.
He: Where did you get it?
Me: I just found it at the table.
He: Maybe you travelled back in time and left it there.
Me: No. It would have run out of ink.

He then said Aha!

The problem we had discovered is that of entropy. A blueprint or floppy disk or CD I hide down the sofa would not be the same as the one I find down the sofa. It would be more torn, corrupt or scratched. Then there'a a paradox again; the disk I find will be simultaneously new and old. P and not-P, folks. Paradox city.

But there seem to be a couple of ways out. One is to hide immutable objects -- things that don't change with time. I think an example of this would be an electron, but I might be wrong.

A better and more practical solution is not to take back any objects at all, but rather just the information in your head. Upon arrival in the past, write it on a note with a 'local' pen and paper, and then hide it. Or even tell yourself directly. Then the paradox of aging objects goes away.

N.B. jurph points out that you can take back an object as long as it's not the same object. In his example, I could find my plans or even an actual Ray Gun down the sofa. I could make a duplicate and then time-travel back with it and hide it. That would solve the entropy problem too, in what may be a very neat way.

What if I forget/decide not/can't be bothered to take the knowledge back now I'm rich and famous? Well, that would be a paradox, but it's just a restated version of the Grandfather Paradox (see above).

There actually is a problem with closed loops in standard physics, though. tdent reminded me of something of which I was vaguely aware in another context: Quantum interference. In most time-loop scenarios (the classic is any wormhole), fluctuations in the quantum field will effectively cause positive feedback, adding to themselves up to infinity. This is properly a physical objection, not a logical one, and could be put aside on that basis alone. tdent suggests that actually it's evidence that closed loops are logically impossible under standard physics. This is a stronger line of attack, but still not enough to undermine the basis of my claim. There are potentially ways of dealing with quantum fluctuations to prevent their addition. As long as these methods are mathematically possible, then we can continue.

So far, we've been talking about logical paradoxes only. It's been suggested that there are other paradoxes, perhaps of the metaphysical variety. Maybe the Knowledge Paradox is one of those. That's what my friend suggested, and I didn't argue. It certainly upsets some of our ideas about causality, and the idea of knowledge appearing from nowhere should break some rule, even if it's not entirely clear what rule. But this seems to boil down to saying "Its a paradox because it seems wrong, and it's confusing". General relativity seems wrong and is confusing, but it's accepted as fact now.

I said earlier that there are a number of ways of dealing with logical paradoxes. Some of these are pretty crazy, or very complicated, or both. But it doesn't matter, as it seems we can create knowledge without getting trapped in a logical paradox. We might not like it, but there's nothing irrational about it, at least in the rigorous logical sense.

If you have a technical quibble or a small oversight with anything I've said, /msg me before writing a reply node. I don't want this to go GTKY. I'll credit any questions and try to deal with them. You won't get a writeup, but hey you aren't NFN, are you? If you've got a totally new treatment of the problem, then node away...