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Time Travelers (inference-review.com)
154 points by fern12 on Feb 20, 2018 | hide | past | favorite | 19 comments

This was Gödel's gift to Einstein on his 70th birthday, certainly one of the coolest birthday presents ever.

Closed timelike curves are intrinsic and irreducible features of Gödel space-times.12 If they are possible, so is time travel. And with time travel, certain paradoxes arise. Imagine a traveler arriving in the past and killing his own grandfather.13 Would he survive the encounter if he broke the causal chain leading to his own existence?

There are two popular views about how these kinds of paradox might be managed. The first is committed to an ensemble of equally concrete but different versions of the physical world. Travelers into the past arrive in worlds that are distinct from those that they left. They are free to kill their grandfather secure in the knowledge that their grandfather is not really their grandfather, but something like his counterpart. David Deutsch and Michael Lockwood think that restrictions posed by classical systems on the actions of time travelers imply that time travel must must displace travelers into different worlds.14 It is by no means clear that time travelers under such a scheme are really following a closed timelike curve. Closed timelike curves are paths that return to, or very close to, their own spatiotemporal starting points.15

On quite another view, time travel really does return an agent to his very own past. A temporal rerun is possible only if everything he does in the past is already in place in his history. This means everything. Consistency might be maintained through the most ordinary of physical processes: the gun misfires, or the bullet dribbles out inconclusively, or at the very last moment your grandfather ducks to tie his shoelaces. Your efforts can make the past what it was but they cannot make the past different from what it was.

There is a third, which is that you would arrive in the past only to find that you had no agency at all, no free will. In this formulation you would never attempt to kill your grandfather, and no faulty guns or lucky grandparents are needed. If the present is a hypersurface sweeping through spacetime, leaving the past in its wake, and the past is a set structure... you’re an automaton if you travel to the past. Presumably in this kind of natural order, the future doesn’t exist except as a word and concept. There is the fixed past, and the hypersurface of the present.

Or the future and past are both fixed, and the hypersurface of the present is some strange artifact we perceive. Or... who knows?

> There is a third, which is that you would arrive in the past only to find that you had no agency at all, no free will.

Or perhaps it's only the illusion of agency that is gone.

Good point, and I’ll freely admit that I try not to think about that too much.

That's probably a good strategy. Either it doesn't work that way, and you can go happily about your life, or it does work that way, and you're happier not knowing because knowing only allows for the possibility of negatives. Ignorance is literally bliss.

"and the hypersurface of the present is some strange artifact we perceive"


And that makes it a different present for each and everyone/everything.

And that makes it a different present for each and everyone/everything.

I’ve never thought about it in quite that way before, but yes I suppose so. Each hypothetical observer is going to experience different slices of “present” in something like Block Universe conjecture.

Slightly off-topic but after reading Douglas Hofstadter's books the one thing that I never managed to grok was Godel's incompleteness theorem. It's implications are fascinating but my math background never allowed me to completely understand it's proof.

Pick some formal system X which can handle basic arithmetic. The basic idea is to take the classic paradox, "This sentence is false," and embed it into maths. The sentence actually constructed is something like, "This theorem is not provable within formal system X," and it's phrased as a theorem of formal system X.

The main ingredient we need here is to realize that numbers can represent sentences in formal systems. This should be obvious to computer scientists of today, but it was ground-breaking at the time.

Another thing we'll need is diagonalization. Some folks get hung up on this one; ultimately, you'll have to convince yourself that it's a valid technique.

Hofstadter's explanations, both the long gentle one in GEB and the quick fuzzy one in "I Am A Strange Loop", are much more detailed and correct on this, so I'd advise going back and making sure you understand each step.

Also, something that doesn't get enough airtime is Tarski's Undefinability, which says that a formal system afflicted by incompleteness is not capable of talking about truth in certain basic and important ways which would enable a formal system to fully represent itself.

I'd highly recommend "Gödel's Proof" by Ernest Nagel and James Newman (https://www.amazon.com/G%C3%B6dels-Proof-Ernest-Nagel/dp/081...). It'll probably require a bit of patience to go through, but should be fairly accessible to reader without an extensive mathematics background.

Whole-heartedly seconded. I went through the proof twice, once in a math context and once a philosophical one. This work was referenced in both.

What are the practical implications of the incompleteness theorem?

Mathematics is postmodern; there's no single formal system which is the concrete bedrock of mathematics, and there will always be some true-but-unprovable theorems in consistent formal systems.

Cynics might take this to mean that maths is meaningless. On the contrary, I'd like to suggest that maths is the canonical emergent pattern; any formal system with sufficient complexity to represent itself will automatically exhibit some algebraic structure, and maths emerges naturally from there.

The biggest implication for computer scientists is that, combined with results of Church and Tarski, we should expect that meta-languages are more powerful than object-languages, whether this is by preprocessing and macros like in C/C++ and CL, or by avoiding using meta-features except when necessary like in Python and Ruby.

There will always be more math to be discovered, because there will always be theorems you can state but not prove within the confines of the existing systems.

You cannot ever know for sure that your software is bug free, because no finite system of logic is both true and complete.

At least, that’s my understanding of it.

Not exactly.

For any sufficiently powerful system (Turing completeness is definitely powerful enough) there are statements which are true but cannot be proven by that system. Secondarily, no system can demonstrate its own consistency.

However, if you limit your software enough, you can prove that every possible input is handled correctly. That involves removing Turing completeness. E.g. once you accept a language as a configurator, you can no longer prove that. CSS3 is Turing complete. JSON is not.

One requirement is limiting the size of inputs.

Afaik this is all related to relativity. No good or bad unless unrelated like ++ or -- and we're in a system where, at least humans, only perceive reality as it relates to us. You need air, so do I, together we have a similar need but also together we occupy each other's space and resource

This is a good lecture from CMUs 15-251 class on his two incompleteness theorems https://youtu.be/bmECBK_TOQA

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