Pannenkoek’s Time Logic

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I have this one surreal memory from Christmas a few years ago. My brother was excited to tell me about this thought experiment he came up with. He took out a white board and started giving me a lecture on something like backwards causality or a simplified version of time travel ― not in a physics sense, but in a mathematical or philosophical sense that dealt with navigating the paradoxes. I’d never heard anything like this before. He used this simple but very foreign reasoning which seemed to be logically consistent.

At first I was able to follow along. But after a few minutes, it started to go over my head.

After the lecture, he never mentioned it again. To the point where I almost thought I dreamt the whole thing.

Years later, I asked him if he would try to explain it to me again. At first he didn’t even know what I was talking about. It turns out he’d almost completely forgotten about it. I told him the bits and pieces I remembered and that jogged his memory and he was able to rederive it. He wrote it out this time and gave it to me as a Christmas present.

The following are the first few paragraphs.

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[Pannenkoek] “A scientist discovered a way to send information back in time a couple seconds. His experimental setup was as follows:

  • At time 0, the scientist turns on a machine.
  • At time 5, a message from the future is received.
  • At time 7, the machine sends a message back in time 2 seconds.

The scientist wanted to test what would happen if he caused a time paradox to occur, namely by turning off the machine at time 6, which was after the message was received but before the message was sent. So he rigged the machine so that at time 6, the machine would experience a 50% chance of turning off, which would make it unable to send the message at time 7. After several trials, the scientist observed that in the cases when the machine turned if at time 6, it had not received a message at time 5. In other words, whether it received a message at time 5 depended on what would happen at time 6.

The scientist then modified the setup so that the 50% chance of turning off would only occur if the machine did receive the message at time 5. Now it was impossible for the machine to not receive a message at time 5, since if it didn’t receive the message at time 5, then the machine wouldn’t turn off at time 6, and so it would send the message at time 7, causing a paradox (since it would be sending a message that wasn’t received). Thus, the only possible variation in this setup was whether the 50% chance of turning off at time 6 would succeed or fail. If the 50% chance of turning off failed, then the message would be received and sent in a logically consistent manner. But if the 50% chance of turning if succeeded, then the machine would have received a message that was never sent, which is a paradox. The scientist ran this test 1,000,000 times, and he found that the 50% chance had failed every single time. More abstractly, since the scientist had created a scenario where the 50% chance succeeding would cause a paradox, reality had made it so that the 50% chance would always fail.

Thus, the scientist concluded that it must be impossible to cause a time paradox. This meant that if information was received from the future, then it was inevitable that this information would be sent back in time to that point. With this knowledge, the scientist got to work developing a machine that would help him achieve his true goal ― to live forever…”

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“What about this situation?

  • At time 5, a message is received or not.
  • At time 6, if a message was received, shut off the machine.
  • At time 7, send a message back 2 units of time.

Once you remove the chance for a paradox not to happen, what’s the result?” [1]

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[Pannenkoek] “He’d probably have a heart attack before shutting off the machine. In my thought experiment, I was careful always to give the universe an ‘out’ to prevent any dangerous situations like that.”

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There are two more eponymous laws that are relevant.

“According to the Novikov self-consistency principle, if an event exists that would cause a paradox or any ‘change’ to the past whatsoever, then the probability of that event is zero.” [2]

“According to Niven’s Law, if a given universe permits the possibility of time travel and of changing the past, then no time machine will be invented in that universe.” [2]

[Pannenkoek] “That last phrase of Niven’s law ― ‘then no time machine will be invented in that universe’ ― doesn’t seem strong enough. Someone could invent one and just not use it or use it to go forward in time.”

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A few years later, I stumbled upon Wikipedia’s Time-Loop Logic article. I’ve been wondering if it’s same thing and he independently discovered it or some part of it.

Time-loop logic is a hypothetical system of computation that exploits the Novikov self-consistency principle to compute answers much faster than possible with the standard model of computational complexity. In this system, a computer sends a result of a computation backwards through time and relies upon the self-consistency principle to force the sent result to be correct, provided the machine can reliably receive information from the future.” [3]

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“A distinguishing feature between different types of time travel in fiction is whether the time traveler was there when history happened the first time around. Is there a kind of self-consistency? Or does the very act of time-traveling to the past change what happened and force the universe onto a different trajectory of history from the one experienced prior to traveling?” [4]

The Legend of Zelda: Ocarina of Time contains both kinds of time travel. The part with the Song of Storms is the ‘Harry Potter type’ with no branching paths, while the parts with the magic bean platforms are the ‘Bill and Ted type’ where there are branching paths.”i[5]

The Song of Storms is an example of “an entity tracing out a circular path in time, looping around on itself so that it has no beginning and no end. A famous example of this can be found in the 1980 movie, Somewhere in Time. The main character is given a gold watch by an old woman. He then travels back in time and gives the watch to the same woman when she was in her youth ― the very watch that she’ll, some decades later, give to him. How did this watch come into being?” [5] I think these count as strange loops.

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“As far as we know, time only moves in one direction ― forward ― but if you could loop back to visit the universe at an earlier point in time, a famous paradox arises. What if you killed your grandfather when he was a child? Then your father or mother wouldn’t have been born, so you wouldn’t have been born, so you wouldn’t have been able to go back in time to kill your grandfather in the first place. Paradox.

But let’s just follow the paradoxical timeline through beyond its paradoxical conclusion. You go back in time, kill your grandfather, thus you aren’t born so you can’t go back in time, thus your grandfather isn’t killed, thus you are born, so you go back in time and kill your grandfather, and so on… I’m describing this as a linear series of events but really it’s two entangled histories happening in parallel. And so, at least from a logical perspective, this looping timeline is entirely consistent. The main point is that sometimes we think a situation creates a paradox when it doesn’t.” [4]

[Pannenkoek] “I’ve always wondered why it’s called the ‘grandfather paradox’. Wouldn’t just ‘father’ suffice?”

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“Regaining Lost Time”

[6]

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Citations

  1. April Shaw
  2. Wikipedia’s List of Eponymous Laws article
  3. Wikipedia’s Novikov Self-Consistency Principle article
  4. ~minutephysics
  5. (source lost)
  6. my concept, drawn by ~Pharaoh

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