I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME

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Erasmus

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I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #30 on: January 21, 2006, 08:06:40 PM »
Quote from: "Panda"
Logic is not a descriptive system. In its simplest form, logic is the study of truth preserving arguments.


That's a statement about logic itself, not about the way in which we use logic.  Logic is, as you more or less said, a qualitative study of formal systems and their usage.  But the reason we (scientists) care about logic is that the formal systems in question are used to *describe* the universe.

So for the purpose of discussions on fora such as this, statements like "What people think has no effect on the universe," while true, advance no understanding.  The discussion is about a statement of Zeno's, not about the way the universe works.  So we apply the methods of logical discourse to analyze Zeno's statement.  We find that, as I stated, Zeno's argument is valid but founded on false and incomplete assumptions and is therefore unsound. Since you are clearly formally educated in logic, I will not explain the difference between validity and soundness.

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Zeno's Paradox shows


nothing, as a matter of fact.

-Erasmus
Why did the chicken cross the Möbius strip?

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Cinlef

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I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #31 on: January 21, 2006, 08:19:45 PM »
Sorry wasnt Xenons paradox that its impossible to hit a running man with an arrow? If not then what the hell paradox am I thinking of?
a confused
Cinlef
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Erasmus

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« Reply #32 on: January 21, 2006, 08:45:01 PM »
Quote from: "Cinlef"
Sorry wasnt Xenons paradox that its impossible to hit a running man with an arrow?


Several seemingly paradoxical "observations" have been attributed to Zeno, and while I haven't heard this particular formulation, it smacks of the race between Achilles and the Tortoise.  Achilles gave the tortoise a head start.  He lost because he first had to run the distance of the head start; in that time, the tortoise moved ahead somewhat, effectively starting over the race with a smaller head start.

The error of ignorance of convergent sums is the same.

-Erasmus
Why did the chicken cross the Möbius strip?

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Erasmus

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« Reply #33 on: January 21, 2006, 09:01:39 PM »
Okay, so, wikipedia has an article on Zeno's paradoxes.  One I had not heard before involves an arrow: in any instant (i.e., zero-length interval of time), the arrow does not move at all; thus at any point in time, the movement of the arrow is null, so how can the arrow be moving?  Again, there's a limiting process involved here that Zeno didn't grasp.

Interestingly, the article claims that calculus doesn't resolve Zeno's paradox:

Quote from: "Wikipedia"
It should also be noted that calculus-based solutions that are offered often object to the claim that "it must take an infinite amount of time to traverse an infinite sequence of distances". However, Zeno's paradox doesn't contemplate the time it would take for Achilles to catch the Tortoise; it simply points out that in order for Achilles to catch up with the Tortoise, Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself: time has nothing to do with it. Thus, calculus-based solutions to Zeno's paradoxes often make the paradox into a straw man.


I'd like to focus on
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Achilles must first perform an infinite number of acts, which seems to be impossible in and of itself: time has nothing to do with it.
 I disagree: I think time has *everything* to with it.  The way in which it seems impossible to do an infinite number of things is exactly that it will, intuitively, take forever.  But this intuition is wrong, unless you disbelieve in limiting processes.

[EDIT]  Another problem is the idea of an "act".  Calling the infinite set of things that are happening "acts" is misleading (who's building a straw man now, Wikipedia-article-writer?)  When I run a hundred feet, the impulse for my actions certainly consist of finitely many neurons firing finitely many times.  I can further break down those "acts" into pieces, but eventually, the pieces will not be distinguishable in any meaningful way.

-Erasmus
Why did the chicken cross the Möbius strip?

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Cinlef

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I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #34 on: January 22, 2006, 08:19:49 AM »
Ah thanks
Truth is great and will prevail-Thomas Jefferson

I've said it before and I'll say it again, Cinlef is the bestest!

Melior est sapientia quam vires-Wisdom

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #35 on: January 22, 2006, 11:40:51 AM »
You know, it doesn't seem to matter what one posts around here: it just gets ignored anyway (but thank you, Erasmus, for pointing people in the direction of my argument against Zeno's paradox).

Let's get this straight: there is no paradox in Zeno's argument. He just didn't understand limits very well (as remarked upon previously). The arrow moves, Achilles catches up to the tortoise, and I get from A to B, all with no difficulty. Inifinity and zero are sometimes hard concepts to relate to the universe, because a period of zero time doesn't actually make any sense at all, and infinitely many "intervals" of no time at all makes just as much sense.

I do have a question for Erasmus, which may end up needing another thread: but I don't think one can actually axiomatize the universe. Axioms are unprovable statements about whatever one's looking at which one takes for granted. What would you take for granted in the universe (and not mathematics, please: that's where axioms properly belong!).

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Erasmus

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« Reply #36 on: January 22, 2006, 08:22:34 PM »
Quote from: "Mundi"
I don't think one can actually axiomatize the universe. ... What would you take for granted in the universe


I am so not qualified to answer your question!  Allow me to redirect you to the flying spaghetti monster, who is clearly the authority on such matters.

But how about some wishful thinking: wouldn't it be nice if the universe were indeed a formal system?  What exactly would this mean?

Well, a formal system needs axioms, as you mentioned, and transition rules that can be used to generate new valid theorems based on the axioms.  Basically, think of axioms as "what you start with" and theorems are "what you get".  

In the case of Our Universe, the axioms would be a very general, weak description of a state of the universe that we all agree is a "possible state".  The transition rules are "easy": that's what physics is mostly busy trying to figure out.  Then a "theorem" is another valid possible state of the universe.

So, back to wishful thinking.  First, I'd want the first four postulates of plane geometry (i.e. not the parallel one), but modified for four dimensional spacetime (something like, "At most four lines sharing a point can be perpendicular at the shared point.")  I'd also want something relating mass to distance, since mass bends spacetime.  Then I'd have something about particle trajectories as straight lines through spacetime.  Maybe.

Anyway, I think you're right; this probably belongs in another thread, but I'm happy to talk about it there.

-Erasmus
Why did the chicken cross the Möbius strip?

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #37 on: January 22, 2006, 08:49:01 PM »
And what about the Gödel's Incompleteness Theorem?

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Erasmus

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« Reply #38 on: January 22, 2006, 09:20:13 PM »
Quote from: "Javier"
And what about the Gödel's Incompleteness Theorem?


I elided that for brevity's sake.  But it's an interesting point.  At a conference on string theory last summer, I was sorta convinced that physicists are okay with incompleteness, but want consistency.

You must also ask whether Gödel's Incompleteness Theorem applies to the hypothetical The Theory Of Everything (TOE).  There are some obscure (to me) requirements.  Also, Gödel's theorem is only concerned with the limitations of recursive logics -- logics that are used to discuss themselves.

Basically, I am not sufficiently well versed in mathematical logic to determine in what way, if any, Gödel is relevant for physics.  Maybe unobservable systems (e.g. interiors of black holes, qualia, et al.) are examples of incompleteness?  Thoughts?

-Erasmus
Why did the chicken cross the Möbius strip?

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #39 on: January 24, 2006, 05:42:40 PM »
The incompleteness Godel refers to is a very specific thing, obviously, so I don't think it would necessarily translate very well to other areas. It says, of course, that a system of logic at least as powerful as Principia Mathematica (ie, capable of referring to itself) is necessarily incomplete, in the sense that there are true statements expressable in the system that the system cannot determine whether they are true or false.

Statements like this have already been found: a famous example is the continuum hypothesis (too complicated for an account here).

If black holes were to be incomplete in this sense they would need some way of referring to themselves. Can they?

And weren't we talking about paradoxes? What happened to those?

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Erasmus

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« Reply #40 on: January 24, 2006, 05:54:23 PM »
Quote from: "Mundi"
If black holes were to be incomplete in this sense they would need some way of referring to themselves. Can they?


Well, I don't think black holes themselves would have to refer to themselves.  Gödel doesn't state that only self-referential statements are neither provable nor refutable within the logic.  What has to be self-referential is the logic itself; whether this applies to our universe is not clear to me, but it seems that we can make (arbitrarily accurate) models or simulations of the real universe in the real universe.  That is, for any level of fidelity in simulation < 1, there exists some configuration of the universe so that it contains a simulation of itself of just such a fidelity.  To really show that Gödel might apply to the Universe, we would have to derive some configuration of the universe (possibly involving a simulation of the universe) that would somehow come into contradiction with something we know to be a fact about the universe.

The reason I bring up black holes is that most of their parameters are unknowable, so that any statement you might like to make about what's going on inside can neither be proven nor refuted.

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And weren't we talking about paradoxes? What happened to those?


They have stepped aside... though I think they're being discussed in another thread as well (the omnipotence paradox).

-Erasmus
Why did the chicken cross the Möbius strip?

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #41 on: January 24, 2006, 06:42:25 PM »
This is getting fairly recherche now! Nevertheless: I think if one were to apply something of an analogue of Godel's incompleteness theorem to the self-referential universe that you mentioned, the universe would have to refer to an encoded version of itself, not merely a copy. It has to be the actual thing (albeit in another form). Is a model, of any fidelity at all, actually the real universe, encoded in another way?

And the incompleteness theorem, as I understand it, would not come up with a contradictory statement, if you were somehow to be able to refer to the actual universe rather than a model: it would say, "You're not going to be able to find out whether this is true or false."

This discussion is somewhat moot, though, is it not, for applying theorems of logic to physics is somewhat fraught: the physical world is not defined well enough for theorems of logic to have validity.

I wonder also about whether those quantities inside the black holes really are fundamentally unknowable, or whether we just don't know about them at the moment. That would make them dissimilar to an unprovable statement, which really is fundamentally unknowable (in the sense that we can say it's true or false from what we know already). The irony here is that the statement might well be true, but we'll never know within the confines of the system as it stands.

So: are those quantities inside black holes a priori unknowable, or just unknown for the time being?

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Erasmus

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« Reply #42 on: January 24, 2006, 07:13:25 PM »
Quote from: "Mundi"
This discussion is somewhat moot, though, is it not, for applying theorems of logic to physics is somewhat fraught: the physical world is not defined well enough for theorems of logic to have validity.


I think that's begging the question.  Trying to determine whether the universe is axiomatizable is the same as trying to determine whether the world is "sufficiently well defined" as you put it.  I'm just exploring the possibilities if we assume the physics to be axiomatic.

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So: are those quantities inside black holes a priori unknowable, or just unknown for the time being?


AFAIK they are a priori unknowable from within the system -- i.e. no statement about a measurement done within the universe can be shown to describe quantities inside a black hole.  This is simply the nature of how events in the universe affect each other.  To make a measurement of X, X has to affect the measuring device in some way; no events inside a black hole can affect any measuring device outside.

At least, this is my interpretation of the restrictions relativity puts on information transfer.

Anyway, your otherquestions re: self-embedding of the universe are very interesting but I'm not sure how to address them.  I would only counter with the question: In a self-referential Godel statement, the statement nowhere *contains* itself, but merely something that humans interpret as corresponding to the statement itself.  What is the nature of this binding?  How is it different from an arbitrarily good simulation of the universe made within the universe?

-Erasmus
Why did the chicken cross the Möbius strip?

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #43 on: January 24, 2006, 08:17:52 PM »
I agree about the axiomatization: math is not physics, after all.

Your comments about the black holes make a lot of sense, and I didn't think about it that way. But could a theory based on how a black hole interacts with the universe tell us anything about what's happening inside? Not being a physicist myself, you're going to have to inform me on this one.

You know, although the statements in Godel's theorems which refer to themselves in some way don't contain the statements, they do refer to actual statements rather than approximations to those statements (hence the beauty of the theorem: it wouldn't have worked without this). So a model might be out: it's not the actual universe. However: humans are part of the universe, and are able to refer to the universe itself. Does this mean Godel's theorem can apply through the agency of humans? . . .

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joffenz

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« Reply #44 on: January 29, 2006, 10:46:07 AM »
According to the same logic you can never read this post. Your eyes have to get halfway down the page, then another half, then another half, so you really you can never finish reading this post!

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #45 on: February 03, 2006, 12:14:29 AM »
zeno wrote many paradoxes. zeno understood that their logic was flawed, because they only work if time is a discrete set of points. zeno proved that time is not a discrete set of points by showing that the paradoxes would arise if time was discrete. however, since motion is self-evident, time cannot be a discrete set of points.

it's called reductio ad absurdum.

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #46 on: March 17, 2006, 05:15:28 AM »
Maybe were not home... maybe its just hallucination!:D

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muiny

I CAN PROVE THAT YOU WILL NEVER BE ABLE TO GO HOME
« Reply #47 on: April 06, 2006, 12:52:35 PM »
i can go wherever i want to because wherever i am i am already there where i am going

i i want to go to the refrigerator i make several points to go like halfway there and i am halfwaythere is where ia want to go the half of half and so on and when i am close enough i change where i want to go to where i am