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anonguest

Help Me Solve This Simple 'logic' Puzzle

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I know that the premise is wrong, for obvious reasons. Maybe it's just Monday, and I am still sleepy, but feel really dopey for not being able to readily put my finger on where the flaw in the 'logic' of the premise below is.....

Bullet is fired at person/object that is running/moving away from the gun.

Bullet travels at steady and unchanging 100 metres/sec. Object moving away from gun travels at steady and unchanging 1 metre/second.

Receding object is exactly 100 metres away from gun barrel muzzle when bullet exits the barrel.

Thus....

After precisely 1 second has elapsed the separation between the front of the bullet and the receding object will now be only 1 metre - since in 1 second the bullet has covered the initial 100 metre separation but the object has itself moved onwards 1 metre.

After another 1/100th of a second has elapsed the bullet has now covered that extra 1 metre, thus having travelled a total of 101 metres. BUT in that same extra 1/100th second the object too has continued to move away - and is now only 1cm away from the bullet.

This remaining 1cm gap between bullet and object is now covered by the bullet in only 1/10,000th of a second! But again the object continues to move away, and now remains only 1/1000th mm away.....

and so on....

The 'logic' being applied effectively claiming that, although the time 'steps' get progressively smaller and the spatial separation gets smaller down to atomic and sub-atomic length scales and beyond, it is impossible for the bullet to ever reach and hit the the object??

So.....what am I missing?

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So.....what am I missing?

Time won't keep slowing down as the bullet keeps getting closer :)

After 2 seconds the target person is 102m from the gun man, and the bullet would be 200m away, except.it isn't because it hit the poor sap. He should have run sideways!

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I liked best:

Peter Lynds has argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.[25][26][27] Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes.

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Guest eight

So.....what am I missing?

Whatever you're shooting at.

Have you tried it with arrows and tortoises?

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I know that the premise is wrong, for obvious reasons. Maybe it's just Monday, and I am still sleepy, but feel really dopey for not being able to readily put my finger on where the flaw in the 'logic' of the premise below is.....

Bullet is fired at person/object that is running/moving away from the gun.

Bullet travels at steady and unchanging 100 metres/sec. Object moving away from gun travels at steady and unchanging 1 metre/second.

Receding object is exactly 100 metres away from gun barrel muzzle when bullet exits the barrel.

Thus....

After precisely 1 second has elapsed the separation between the front of the bullet and the receding object will now be only 1 metre - since in 1 second the bullet has covered the initial 100 metre separation but the object has itself moved onwards 1 metre.

After another 1/100th of a second has elapsed the bullet has now covered that extra 1 metre, thus having travelled a total of 101 metres. BUT in that same extra 1/100th second the object too has continued to move away - and is now only 1cm away from the bullet.

This remaining 1cm gap between bullet and object is now covered by the bullet in only 1/10,000th of a second! But again the object continues to move away, and now remains only 1/1000th mm away.....

and so on....

The 'logic' being applied effectively claiming that, although the time 'steps' get progressively smaller and the spatial separation gets smaller down to atomic and sub-atomic length scales and beyond, it is impossible for the bullet to ever reach and hit the the object??

So.....what am I missing?

That it's complete ******?

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That it's complete ******?

It's not. As referred to earlier, these are valid paradoxes and really troubled the Greeks.

How to do an infinite number of infinitessimal things in a finite time envelope?

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It's not. As referred to earlier, these are valid paradoxes and really troubled the Greeks.

How to do an infinite number of infinitessimal things in a finite time envelope?

Do the sums. The maths actually work out that in this case the infinite number of ever-smaller times actually add up to a finite time.

Weird, I was talking about this with someone earlier today.

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It's not. As referred to earlier, these are valid paradoxes and really troubled the Greeks.

How to do an infinite number of infinitessimal things in a finite time envelope?

You only have to advance the time by a tiny amount to demonstrate that it's wrong as the bullet emerges from the running man's stomach, so BLT appears accurate in his assessment.

It demonstrates excellently the value of practical application over theory, especially if the running man happened to be Blair.

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Hmmmmm.....interesting. Thanks!

As you imply in your original post, I don't think this is about denying that the bullet hits the target. It's a thought experiment with whether assumptions about how the universe works reconcile with actual outcomes or not.

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Do the sums. The maths actually work out that in this case the infinite number of ever-smaller times actually add up to a finite time.

Weird, I was talking about this with someone earlier today.

That's not in doubt, we already know the time is finite because the arrow (or in this case the bullet) reaches the target.

The paradox arises from the infinite number of steps you've done in a finite time. How can that be?

I think there are various sensible ways around the paradox, but as mullets said, it can be a way of challenging your pre-conceptions about how the universe works.

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That's not in doubt, we already know the time is finite because the arrow (or in this case the bullet) reaches the target.

The paradox arises from the infinite number of steps you've done in a finite time. How can that be?

I think there are various sensible ways around the paradox, but as mullets said, it can be a way of challenging your pre-conceptions about how the universe works.

It's about as useful as philosophy.

Would the bullet ripping the running man's intestines out make a sound if he had no ears and no-one else was listening?

Yawn.

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For me this seems pretty simple to answer:

The bullet will hit the object at about 1.01 seconds.

What you are doing is looking at points in time between 1 second and 1.01 seconds. The point in time is always an exponentially smaller step forward than the last so it makes sense that you will never hit 1.01 seconds.

Just as if you take 100 seconds and continuously divide it by two you will never reach zero.

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That's not in doubt, we already know the time is finite because the arrow (or in this case the bullet) reaches the target.

The paradox arises from the infinite number of steps you've done in a finite time. How can that be?

I think there are various sensible ways around the paradox, but as mullets said, it can be a way of challenging your pre-conceptions about how the universe works.

That infinite number of terms is the maths I was talking about, using an infinite sum to calculate the answer, not just a little bit of fiddling around solving simultaneous equations of motion. That you can have a sum of infinite terms which produces a finite result is mathematically demonstratable (and happens all the time in integral calculus).

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For me this seems pretty simple to answer:

The bullet will hit the object at about 1.01 seconds.

What you are doing is looking at points in time between 1 second and 1.01 seconds. The point in time is always an exponentially smaller step forward than the last so it makes sense that you will never hit 1.01 seconds.

Just as if you take 100 seconds and continuously divide it by two you will never reach zero.

How do we ever get to tomorrow? The second hand can get closer and closer to midnight but how does it ever get past 11:59.999 recurring?

Interestingly enough, you can prove that 0.999 recurring is the same as 1. Which rather neatly, logically and mathematically sorts out the paradox.

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Just as if you take 100 seconds and continuously divide it by two you will never reach zero.

Which is where the paradox comes in. You can divide by two an infinite number of times to get an infinite number of positive real times and never reach zero. Instincitvely a sum of infinite positive real values should be infinite, although it plainly isn't in reality in cases like this (and can be demonstrated mathematically that it isn't.

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How do we ever get to tomorrow? The second hand can get closer and closer to midnight but how does it ever get past 11:59.999 recurring?

Because each second is the same as the last. Not an exponentially smaller number than the last.

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Guest eight

That infinite number of terms is the maths I was talking about, using an infinite sum to calculate the answer, not just a little bit of fiddling around solving simultaneous equations of motion. That you can have a sum of infinite terms which produces a finite result is mathematically demonstratable (and happens all the time in integral calculus).

Remind me never to get stuck next to you at a dinner party.

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I know that the premise is wrong, for obvious reasons. Maybe it's just Monday, and I am still sleepy, but feel really dopey for not being able to readily put my finger on where the flaw in the 'logic' of the premise below is.....

Bullet is fired at person/object that is running/moving away from the gun.

Bullet travels at steady and unchanging 100 metres/sec. Object moving away from gun travels at steady and unchanging 1 metre/second.

Receding object is exactly 100 metres away from gun barrel muzzle when bullet exits the barrel.

Thus....

After precisely 1 second has elapsed the separation between the front of the bullet and the receding object will now be only 1 metre - since in 1 second the bullet has covered the initial 100 metre separation but the object has itself moved onwards 1 metre.

After another 1/100th of a second has elapsed the bullet has now covered that extra 1 metre, thus having travelled a total of 101 metres. BUT in that same extra 1/100th second the object too has continued to move away - and is now only 1cm away from the bullet.

This remaining 1cm gap between bullet and object is now covered by the bullet in only 1/10,000th of a second! But again the object continues to move away, and now remains only 1/1000th mm away.....

and so on....

The 'logic' being applied effectively claiming that, although the time 'steps' get progressively smaller and the spatial separation gets smaller down to atomic and sub-atomic length scales and beyond, it is impossible for the bullet to ever reach and hit the the object??

So.....what am I missing?

The world is not infinitely divisible. Nice summary here: http://www.rphv.net/blog/zenos-paradox-and-planck-units/

A variation of this argument uses a footrace between Achilles and the Tortoise to illustrate a similar point about relative motion. Douglas Hofstadter offers a wonderful, witty dialog describing this contest in his staggeringly thought-provoking book Gödel, Escher, Bach.

It took a solid two thousand years for mathematics to catch up to Zeno’s road-crossing chicken. In the 1600s, Isaac Newton and Gottfried Wilhelm Leibniz independently devised a clever system for dealing with the limits and infinite series implicit in Zeno’s Dichotomy; today, this branch of mathematics is called calculus, and it’s one of mankind’s greatest (and most useful) intellectual achievements.

To some, though, purely mathematical explanations aren’t always intuitively satisfying. It’s nice that math acknowledges that a chicken CAN cross the road… but HOW? And what does it mean? What exactly happens when things move?

Viewed from this standpoint, Zeno’s paradox isn’t really about math at all. Instead, it’s more closely related to a question about the fundamental nature of Life, the Universe, and Everything, namely:

Are time and space continuous or discrete?

Put another way, this question asks, “Can space and time be infinitely divided – is there always a way to split things in half, no matter how small… or is there some ‘smallest interval’ of space/time that can’t be chopped apart?”

Quantum theory provides one possible resolution to Zeno’s Dichotomy. Developed around the turn of the 20th century by Boltzmann, Planck, Einstein, Heisenberg, and a host of others as a way to describe the behavior of matter at subatomic levels, quantum mechanics works – numerous experiments have demonstrated quantum theory’s ability to predict observable phenomena.

One of the basic tenets of quantum theory is the notion of quanta. Essentially, a quantum (plural: quanta) is a “smallest amount” – a quantity which cannot be divided. In physics, a quantum of length is about 1.6 × 10−35 meters, and is known as a Planck length after the physicist Max Planck. This is an unimaginably short distance, but it implies that a road-crossing chicken can only travel “half the remaining distance” 116 or so times (because 2-116 ≈ 10−35) before he must either “go all the way” and travel an entire Planck length, or remain where he is and go nowhere.

It seems, then, that quantum mechanics has resolved Zeno’s Dichotomy paradox by demonstrating that space is discrete, and motion is achieved by traveling many, many Planck lengths one after another.

Unfortunately, it’s not quite that simple.

In fact, another of Zeno’s ideas – he was quite thorough! – disabuses this notion as well. This paradox, known as the Arrow, uses the relationship between time and distance to show that an entirely discrete universe is impossible as well. Though Zeno probably knew very little about the formal link between time and space, Einstein’s theory of relativity establishes a clear interrelationship between space and time via the speed of light c, which represents the maximum speed at which ANYTHING can move: 299792458 meters per second. Thus, a quantum of time is the amount of time it takes light to travel one Planck length: about 5.4 × 10−44 seconds.

Zeno’s Arrow paradox goes something like this:

Consider an arrow in flight. If time consists of indivisible instants, then nothing can happen “in between” units of time, and an arrow in motion is, at each smallest moment, stationary. The arrow can’t be moving, since that would imply some degree of motion “in between” the indivisible instants of time – which would imply an amount of time smaller than the indivisible unit. Thus, at any given instant, an arrow in motion is equivalent to an arrow at rest that was placed, motionless, in the same position. However, it is contradictory to say that an arrow in motion is EVER identical to an arrow at rest, because one is moving and the other is not.

Thus, like the wave-particle duality of matter posited by quantum theory, it seems that space and time are neither fundamentally discrete nor continuous, but are instead – somehow – both.

This suggests some natural questions. If the universe is both discrete and continuous, is it possible to measure and quantify the “discrete-ness” or “continuity” of a particular part of the universe? How much “discrete-ness” does a subatomic particle have? How much “continuity” is possessed by an arrow in flight? And what is it about these objects that determines these properties? Finally, and perhaps most importantly to the chicken… why?!

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How to do an infinite number of infinitessimal things in a finite time envelope?

I can't now recall where, but I once read/heard the Mandelbrot set described as a finite area enclosed by a perimeter of infinite length. Not that the Mandelbrot set is unique in this, I assume--what with Koch's snowflakes and Wossname's gaskets--but, at the time, the notion interested me.

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As you imply in your original post, I don't think this is about denying that the bullet hits the target. It's a thought experiment with whether assumptions about how the universe works reconcile with actual outcomes or not.

In the thought experiment, the rate of time passing is slowed...

this clearly isnt reflected in reality.

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