See now we're getting metaphysical.
Is an iten cut in two still a whole? or are both now parts of entirely seperate wholes?
s this scientifically impossible?, it can “NEVER” hit the ground…can it?
- Thread starter Antx777
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See now we're getting metaphysical.
Is an iten cut in two still a whole? or are both now parts of entirely seperate wholes?
If you half something an infinite number of times any finite number will reach zero.
Even if the object fell at a constant speed (which is false until it reaches terminal velocity), each time the object has halved the distance to the ground the time left to the ground is also halved. The two converge to zero. So after a certain number of your discrete steps, the object is is immeasurably close to the ground and in an unmeasurable fraction of time will make the gap smaller.
So actually, what happens at the ed happens very quick because the time to half the distance will get vanishingly small.
Once very close, other physical events take over. The object wont actually touch the ground in a physical sense - the electrons will repel each other to make the object float at a small distance.
Even if the object fell at a constant speed (which is false until it reaches terminal velocity), each time the object has halved the distance to the ground the time left to the ground is also halved. The two converge to zero. So after a certain number of your discrete steps, the object is is immeasurably close to the ground and in an unmeasurable fraction of time will make the gap smaller.
So actually, what happens at the ed happens very quick because the time to half the distance will get vanishingly small.
Once very close, other physical events take over. The object wont actually touch the ground in a physical sense - the electrons will repel each other to make the object float at a small distance.
omnipresent Sandwich?
This is Zeno's paradox, which isn't at all paradoxical but can be quite counter-intuitive. Consider travelling 1m, or just "1" for short. Then first we must travel 1/2, and then another 1/2. I.e.
1 = 1/2 + 1/2
Now focus on the last half of our journey. To complete it, we must travel 1/4 then another 1/4. So we have:
1 = 1/2 + 1/2 = 1/2 + 1/4 + 1/4
Now focus on the last quarter of our journey. To complete it, we must first travel 1/8, then another 1/8. So in total we travel:
1 = 1/2 + 1/4 + 1/4 = 1/2 + 1/4 + 1/8 + 1/8
We can keep doing this, so that:
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
The error in Zeno's thinking is he assumes, since there are infinitely many terms in the above sum, we cannot reach our destination in finite time. What Zeno didn't realise is that it's perfectly possible to add up an infinite number of non-zero terms, and get a finite answer.
1 = 1/2 + 1/2
Now focus on the last half of our journey. To complete it, we must travel 1/4 then another 1/4. So we have:
1 = 1/2 + 1/2 = 1/2 + 1/4 + 1/4
Now focus on the last quarter of our journey. To complete it, we must first travel 1/8, then another 1/8. So in total we travel:
1 = 1/2 + 1/4 + 1/4 = 1/2 + 1/4 + 1/8 + 1/8
We can keep doing this, so that:
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
The error in Zeno's thinking is he assumes, since there are infinitely many terms in the above sum, we cannot reach our destination in finite time. What Zeno didn't realise is that it's perfectly possible to add up an infinite number of non-zero terms, and get a finite answer.
I thourght the paradox was not the distance or time but the infinite number of actions involved that was impossible.
Gah need to dig out my old philosophy text book.
Oh no, the dreaded sandwich debate.
I don't know what you mean by "infinite number of actions" I'm afraid!
If you have an infinite number of operations to halve the distance involved then by definition you can never reach the end... your back at the original problem...
I think just as many don't know the full extend of the original paradox.
I was right above, it's the fact it would require an infinite number of actions, which Zeno said was an impossibility.
That is what granularity is about though. Although it is theoretically possible, it is only true when used to prove the paradox. Anything can be broken down into an infinite number of actions, but it's not because that would defeat the point in performing the action as a whole.
Soldato
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The paradox is that an infinite number of events must occur in a finite time interval.
Hmm, I'm afraid I still don't understand (although perhaps we're talking over each other). The argument I presented looks at a journey travelled (i.e. the "1"), and shows it's equivalence to an infinite number of smaller journeys (i.e. the 1/2 + 1/4 + 1/8 + ...). There's no physical "operation" involved.
Unless you're suggesting it should take an infinite amount of time for me to compute the sum 1/2+1/4+1/8+... because there are an infinite number of terms? Is that the case?
The breadcrumb thing is a good illustration actually...
you can keep chopping the breadcrumb down... until chemically its impossible to divide any part and its still chemically a breadcrumb
but you could keep chopping the parts apart if you had a fine enough precision until you've broken it down into the discrete atoms it was made up of... broken down to elementary particles... then what? can you break quarks up? what is the real nature of energy?
you can keep chopping the breadcrumb down... until chemically its impossible to divide any part and its still chemically a breadcrumb
but you could keep chopping the parts apart if you had a fine enough precision until you've broken it down into the discrete atoms it was made up of... broken down to elementary particles... then what? can you break quarks up? what is the real nature of energy?So when you say "operations", it's synonymous with some physical notion of divisibility - whether it be splitting breadcrumbs or measuring and halving distances etc? Then in this case there is not even a whiff of "paradox", because the Planck length will serve as our lower bound (in the distance case). No more splitting, so we only have a finite number of "operations", and all is well.The breadcrumb thing is a good illustration actually...
you can keep chopping the breadcrumb down... until chemically its impossible to divide any part and its still chemically a breadcrumb
Nope.