Here at the Nexus, many of us have faced paradoxical circumstances and experiences. Just for fun and discussion, I have started this paradox thread.

Here we can share and discuss well known, or maybe not so well known paradoxes. We can share, discuss and discover paradoxes we may or may not be familiar with. To get the ball rolling, I shall include a few of my favourites:

The Omnipotence Paradox

It is impossible to be able to do anything, even in a universe where anything is possible. If it is claimed that you can do anything; then create me a stone so heavy, not even you can lift it.

If you create it, then lifting the stone is something you can't do. Since you can do anything, you will be able to lift the stone, therefore creating a stone you can't lift is something you can't do.

The Time Traveller's Paradox

If you travelled back in time and killed your grandfather, you would then not exist. Therefore, if you don't exist, there is no-one to kill your grandfather. You would not be able to travel back in time and kill your grandfather, because the very act would cancel itself out.

I guess also there could be further implications. If you attempted it, you could cancel yourself out, because you could change your grandfathers destiny enough that his offspring may be different. I don't think your dad would be too happy about that either.

The Force and Object Paradox

An unstoppable force and an immovable object cannot co-exist in the same universe. What would happen when they meet?

Peace

Macre

The next sentence is true. The previous sentence is a lie.

In the time traveller's paradox, one of the following assumptions is false:

1. Time travel is possible

2. Humans have free will

The other two paradoxes don't really make sense to a modern mind. They're really theological questions which ignore everything we've learned about how the universe works since the birth of Newton.

That is the liar paradox, more succinctly expressed as:

This sentence is false.

There is Zeno's paradox also ( I cut and pasted from an old post of mine, with a hypothetical solution):

An archer shoots an arrow toward a target, and we can say with great authority that in some elapsing of time the arrow will have travelled half the distance to the target. And in another lapse of time, the arrow will have covered half the remaining distance, and in yet another, half that distance. Following this logic, the arrow must ALWAYS follow this pattern, halving the distances progressively, and will thus NEVER hit the target. But as any archer knows (and any ill fated target, for that matter), the arrow does eventually strike. So the mathematics of the infinite seem at best flawed, if not outright false. (I happen to think that the solution to this is that there may actually be discrete units of time, meaning that to say the arrow travels only through space, neglecting time as a factor beyond a measurement of the time required to travel that distance, is to erroneously accord too much importance to space, and not enough to time: with a discrete unit, in other words, the arrow will never actually be caught in the conundrum of halving over and over, but rather at one point the halving will cease and the arrow will jump a discrete unit of time and hit the target. Have I lost ya?)

Creo - you seem like someone who might be current in terms of an actual solution to Zeno's paradox - any ideas?

Cheers,

JBArk

Macre,
With paradoxes 1and 3 there is an assumption that certain absolutes are possible (immovable, unbreakable, etc) where in reality there would be thresholds where this would break down. When examining it from a perspective of using all the available energy in the universe, at one point you would simply have to choose: dedicate the majority of this energy toward trying to move the object, or toward sustaining that immovability by tying up energy in solid matter? At one point the finite nature of the universe places a limit on either the unstoppability or the immovability.

As far as time travel goes: one suggested fix is that the time traveler starts a new time sequence when they step back in time; the existing sequence, with the grandfather living and going on to reproduce, continues to exist, plus a new time sequence where the time traveler rudely interrupts the grandfather's existence. Since this is in fact a new time sequence, there is no paradox.

I'll take this into the realm of hyperspace experience for a sec here I think. One commonly held Newtonian tenet is that two things cannot occupy the same space at the same time, but in hyperspace with multidimensional figures, this seems to not be the case where we have objects folding/unfolding simultaneously, etc...

There are also the paradoxical experiences of feeling hot/cold, big/small, happy/sad all at the same time.

jbark, I think the invention of calculus resolved Zeno's paradox.

But like you, I suspect that space and time are discrete in essence. And that might be another way of resolving Zeno's paradox.

Great input, thank you all:

Creo/Guyomech: Very good points and I am glad you have raised them. You have shown me that my examples are not really paradoxes at all, not in the strictest sense, given the overly hypothetical nature of my examples.

With example one, since the supposed paradox itself expels all that we have currently known of the workings of the universe; then it could be answered with equal disregard in such a way as "They are both possible in a way our human brain can't comprehend".

With example two, Guyomech rightly states that perhaps a new time sequence is started. Again, given the overly hypothetical nature of my examples, this opens the door for equally hypothetical answers.

With example three again Guyomech is correct. Available energy would have to be granted favourably to either force or object. If they were to be absolute, for a third time the example becomes hypothetical. Therefore it could be easily answered with such statements as "The force passes through the object".

Thank you for helping me see the blind spots in my thinking. I'm slightly disappointed in myself for not seeing the examples from this perspective, or not giving them enough forethought to evaluate them as such. Buy hey, this is the Nexus a place to learn and grow.

Peace

Macre

Well, not sure calculus really solved it - I used to be a whiz at calculus, but now I can barely differentiate my integrations. (a lil dmt joke there... Differential Mathematical Troubles?)

After a quick search I found this:



"Imagine you are walking along the number line from 0 to 1 and that you are walking at 1 unit/second, i.e., it should take you 1 second to walk the total distance.

Zeno's paradox says that you shouldn't be able to reach 1, since before getting to 1, you have to first reach the point 1/2, and then the point 3/4, and then the point 7/8, etc., and there are infinitely many points that you have to go through before you reach 1. Surely, it should take an infinite amount of time to pass through these infinitely many points, right?

In fact, it doesn't. First, the amount of time it takes to go from 0 to 1/2 is 1/2 second. Then, to go the additional 1/4 unit of distance to 3/4, it will take 1/4 second. To go the additional 1/8 unit of distance to 7/8, it will take 1/8 second. This continues ad infinitum.

Thus, the total time that has elapsed during the walk to the point 7/8 is (in seconds)
1/2 + 1/4 + 1/8.
In general, you can walk "almost" all the way to 1 by going through n steps of this sort, and the time it will take is
1/2 + 1/4 + 1/8 + ... + 1/2^n

So far, no calculus. Now the reason the paradox seems sound is that the time it takes to go n steps is a sum of n numbers, as in the sum above. You would think that as you sum more and more numbers, the total time will keep increasing and increasing to infinity.

This is where calculus intervenes. In fact, these sums NEVER get bigger than 1. Try it on a calculator. Add up 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/65536, where 65536=2^16 so this is the sum for the first 16 steps of your journey. It is very slightly below 1. In fact, as you add up more and more terms, these "partial" sums of the entire time your walk takes get closer and closer to 1. In calculus, we say that these sums are approaching a LIMIT. This is the fundamental concept upon which all of calculus is based. In general, if some quantity gets closer and closer to some value under some process, we say that the limit of the quantity is that value.

With our example, although it appears that the sum of the times it takes to walk each piece of the journey should be infinite, because there are infinitely many terms, we find that this argument is fallacious because if you actually try adding up a whole bunch of terms, it never exceeds 1. This is quite a surprising result, no? Because of this, we say that the infinite sum

1/2 + 1/4 + 1/8 + ...

with infinitely many terms converges to 1, i.e., we assign a value to what these terms add up to, because if you try adding as many terms as you please, you will find that it just keeps getting closer and closer to 1."

But all this really says is: "if it approaches 1 it must eventually get there", which to me is unsatisfactory. And I am not alone!

Wikipedia expands:

"Some mathematicians and historians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution.[4] Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems."

So there seems to be no consensus between mathematicians and philosophers, or even seemingly between mathematicians!

Cheers,

JBArk

Zen koans often have that paradoxical nature too.

It seems facing paradoxes can lead us into jumping into a higher order where the paradox is resolved when the apparent opposites coalesce into a higher unity.

I think laughter is highly connected to this same aspect of existence, because through humor we put two opposing ideas together and this generates some kind of reaction in the body, as if it needs to release the tension formed from the opposing forces that are explicit or implicit in the joke or humor.

The hidden assumption in the liar paradox is that every statement (that makes sense) must be either true or false.

In fact, there is a third class of 'undecidable' statements.

jbark,

I have always wondered at the discontinuity of Zeno's paradox with reality.

One of my friends (who is not a math type person, but brilliant nonetheless) offered me an idea, and I have not been able to resolve it. He said that there is no infinite anything, that it's a concept made up by us. I am well versed in calculus and differential equations, so my first reaction was, "Nu uh!" since infinity is used all the time in math.

But then I got to thinking about it, and when do we see any examples of the infinite? Is it even possible in this reality? Reading up further in this thread, Guyomech made a good point about there not being an "immovable object" in this reality. The heart of that concept is there is nothing infinite in this reality.

Zeno's paradox works on the assumption that time and space can be divided up an infinite amount of times, but can it be? Perhaps that is why it breaks down when put to practice. I have asked other people, and have not got a definitive answer one way or the other, but I have a question. Is Planck's constant the smallest interval of space/time we can measure, or is it the smallest that can exist?

I think that may be the key. If time itself can only be divided down to one Planck constant long, then Zeno's paradox cannot apply. If time is continuous, then how can I ever get across the room?

Singularities (blackholes, bigbang etc) seem to be real life examples of infinity, no? Even though we don`t see/experience them directly, but that`s what data seems to indicate...

That was my thought, too, but are they really infinite? Black holes have a size, at least in our perception of them. They have a finite gravitational pull, proportional to the amount of matter they contain. On the other side of the event horizon, no one really knows what goes on (although, I just read an article where Stephen Hawking says there may not be an event horizon, here).

So, I guess my question is, are the insides of black holes in our reality/dimension, or are they somewhere else? Perhaps our universe doesn't allow for anything to be infinitely anything, and that's why black holes are the way they are.

I think I'm in way over my head!

Thus my assertion, in the above post, that time must have discreet units.

And furthermore, dividing a space in half has nothing to do with the infinite. But one must admit that at least in the world of space we seem to be able to divide it and subdivide it indefinitely.

(Enter quantum mechanics - when you get down small enough, space itself, and the presence of particles which occupy that space, is mere probability... I think this goes a way to understanding that zeno's paradox is at once unsolvable, yet explainable. (meaning, of course, that it is still a paradox! ))

A barber shaves everyone who does not shave themselves. Does he shave himself?
If he does then he shaves himself... and thus does not shave himself... thus shaves himself... etc.

That's horribly frustrating! ... Curse you O

You forgot the second part:

...and if he doesn't, the first proposition is false. (he does NOT shave everyone who does not shave themselves).

I personally think barbers are special people whose hair is conscious and falls out or grows in shapes according to its own volition. Thus no paradox in the above propositions.

Nice Orion reminds me of the Pinocchio one:

Pinocchio says "My nose is about to grow."

His nose only grows when he tells a lie. If he's lying because he doesn't know his nose is going to grow, then his nose will grow. But if that happens it's not a lie, so his nose won't grow, which makes it a lie again, and so on.

That's just a silly fun paradox, but just for the hell of it I'm going to ruin it and say that it's all about intention. Please no one come forward and say Pinocchio isn't real, that would just be upsetting.

Peace

Macre

Yeah, I realized after I posted that I said nearly the same thing as you. I have a problem with arguing an idea to people and later realizing I was arguing the same point.

I guess my point is that, if time and space are discrete, then zeno's paradox cannot apply to reality. You can dived by half up to the point of 2 Planck constants, divide by half once more, and then you cannot divide (that is, if space and time really are discrete). So the idea of indefinitely (infinitely) dividing the distance in half cannot apply, which is what the paradox is based on.

On a different topic,
Ever hear of "The Game", where you are winning if you are not thinking about playing the game? So, to play the game you have to not play the game. By explaining the game, I am seriously losing the game right now, and so is everyone else reading this. My hope is that some of you are players already, and I have made you lose. I drag anyone down with me when I lose, and I lose a lot.

The game is almost like a computer virus, and I have now infected more of you! Hahahahah! All hail the game.