Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: McKay on 04/08/2012 10:13:43
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So I found this video, in witch it was said and explained that all movement should be impossible .. it was explained somehow like this (too bad i can not find the video now) : imagine a arrow (or bullet.. or anything else) shooting/ moving from start to its target - in order to get to the target, the object has to reach the trajectories/ paths mid-point. To reach this mid-point, it has to reach the mid-point between start and first mid-point. To reach this new mid point, it has to reach the mid point between start and mid points mid point.. and so on to infinity.
So, according to this, all movement should be impossible - it would take infinite amount of time to traverse any infinitely small distance.
Even although the time to traverse each sub-mid-point would decrease, no matter how small the time gets, it would have to be multiplied by infinity, as there would be infinite .. steps.
That is .. weird.
It got me thinking and what I thought up was - space must be quantized. In that case, there would be a finite amount of steps and time required would be finite.
.. but that got me thinking even further - if space is quantized, then there is a actual limit of how round a circle can be - with the smoothness increasing with size! (basic geometry)
And so, to conclude, if no circle can be a perfect circle and the roundness (for the lack of a better word) is limited, then the constant pi (3.14159...) does not have to be irrational and it can be finite..
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So I found this video, in witch it was said and explained that all movement should be impossible ...
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#The_dichotomy_paradox
Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy (https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#The_dichotomy_paradox)), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Proposed_solutions
... if no circle can be a perfect circle and the roundness (for the lack of a better word) is limited, then the constant pi (3.14159...) does not have to be irrational and it can be finite..
Pi is derived from mathematics which has ideal perfect shapes, not real-world objects made from lumpy atoms.
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In the original version of Zeno's paradox (pre-quantum theory), there was a flaw in the logic: It is possible to add up the infinite series of 1/2+1/4+1/8+....; the answer is finite and equals 1. This matches our experience that an arrow can reach the target.
Just because it takes an infinite amount of time to write the equation in full does not mean that it takes forever for the arrow to get there! Of course, mathematicians get bored writing infinite things out in full, and so have a notation to write this infinite series in 1 short line.
In the quantum version of Zeno's paradox, the Heisenberg uncertainty principle tells us that you cannot measure the precise position of any object, and the hypothesised Plank length is so short, that even macroscopic objects should be able to "quantum tunnel" over such a small distance. So motion to hit a target is still possible under quantum theory.
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Zeno's paradox originates in ancient Greece and is essentially a verbal paradox demonstrating the limits of everyday language to describe some things. It can in a simple sense become a mathematical paradox of infinity in everyday mathematics. However as pointed out it can be clearly dealt with mathematically given the correct way of expressing the problem it can also be cleared up in everyday life by observing that things do move.
Zeno's original way of expressing the problem was in fact from the opposite direction because the no movement concept would have been refuted straight away.
Zeno described a race between Achilles and a tortoise and showed that even though Achilles was running faster than the tortoise he would never overtake it.
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We have a clash of opinion here - one is that there is no problem and that there can be an infininte number of steps between two points in space, and the other is that there can't be - that the universe must be granular. Anyone who thinks that there can be an infinite number of consecutive steps simply hasn't understood the point of the Zeno's Arrow paradox: with an infinite number of consecutive steps where each step has a duration of any size, it will necessarily take forever to get from A to B. If the steps have no duration at all, then there can be no such thing as time - all events would play through in a single instant (of no length whatsoever), including the entire history and future of the universe.
With Zeno's Hare and Tortoise Paradox the solution is again to have a granular universe. Movements are all made in jumps and we have to focus on how long things are stationary. The wait-times for the hare are shorter than the wait-times for the tortoise, so it gets more turns to move and is able to catch the tortoise when the tortoise is not moving and leave it behind in the same manner.
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In the original version of Zeno's paradox (pre-quantum theory), there was a flaw in the logic: It is possible to add up the infinite series of 1/2+1/4+1/8+....; the answer is finite and equals 1. This matches our experience that an arrow can reach the target.
Put another way, let's say it takes 1 second to traverse the distance. If you chop it in half, it would take 1/2 second to traverse each half, which adds up to 1 second total. If you chop it in half again, you'd end up with 1/4 second to traverse each of 4 segments, which would add up to 1 second total. No matter how you chop it up, 8ths, 16ths, 2nths, it still takes 1 second because you're adding together many very small times, which can still add up to 1. Zeno didn't know the proper way to deal with this mathematically when you allow it to be chopped up endlessly, but modern calculus introduced the idea of a limit, which shows that no matter how small you chop these segments, the total time taken is still 1 second.
The question of what happens in reality on the quantum scale is another thing altogether, but the original paradox was resolved by calculus.
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Calculus didn't solve the paradox at all - it simply provided a way of calculating the exact answer which no amount of further chopping can take things beyond - it provides no mechanism for making infinity times zero equal to one.
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Calculus didn't solve the paradox at all - it simply provided a way of calculating the exact answer which no amount of further chopping can take things beyond - it provides no mechanism for making infinity times zero equal to one.
David, you're mistaking infinity for a number. You can't multiply by infinity--it isn't a real number. You can't add up an infinite number of things--infinity isn't a number. What you can do is to allow the number of steps to increase without limit, and the step size to decrease proportionally without limit. This is how calculus resolved the paradox.
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Zeno was fully aware that you could increase the number of steps without limit and that the step size would decrease proportionally without limit, but he was also able to see that it did not resolve his paradox. You either have to stop at some point and accept that the universe is granular or you have to take it all the way to the point where an infinity is introduced, at which point the maths breaks down. Zeno's paradox tells us that the universe is granular.
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No, David, you don't have to introduce infinity as a number in this limiting process. That's the point of calculus.
Talking about "granularity" of the universe is beside the point. As I said, the quantum question is another thing entirely. Calculus resolved the classical Zeno paradox.
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Yes you do, JP - that's the whole point of the paradox. Calculus doesn't do anything to address the issue - all you're doing is making an assertion that it solves a problem which it manifestly doesn't. Anything less than going to the point where an infinity is brought in simply gives you a set of tiny chunks which can be cut in half over and over again forever, and that is the infinity coming in again whether you like it or not. If there's a finite number of chunks, it's still granular, and that's the case all the way down until you make the jump to infinity. The idea that calculus has anything new to say about the paradox is just wishful thinking.
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Calculus tells you why trying to shoehorn "infinity" into the problem as if it's a number as you're doing is wrong.
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Calculus didn't solve the paradox at all - it simply provided a way of calculating the exact answer which no amount of further chopping can take things beyond - it provides no mechanism for making infinity times zero equal to one.
David, you're mistaking infinity for a number. You can't multiply by infinity--it isn't a real number. You can't add up an infinite number of things--infinity isn't a number. What you can do is to allow the number of steps to increase without limit, and the step size to decrease proportionally without limit. This is how calculus resolved the paradox.
Infinity can be a real number, as a human with a designated life span, you will just never be able to count to it. Or count a set number of objects, within that time frame. So any math performed on that un- knowable amount, within a humans lifetime, would be a guesstimate.
Infinity just means outside the realm of exacting mathematics. To say that by completing steps one two and three, and proving it, that step infinity will follow at a later time in some predefined order, is not scientific. No one has ever counted the particles of electricity that exist in any common object at any given time, so to say we have counted or weighed known objects, is a certain lie. It is a guesstimate that we live with.
Look at it this way, a particle of electricity is so small, moving so fast, that no one can ever see it.
Not that anyone sees any matter. Rather they see what other particles of light, bring to their eyes, about the object we claim to see. That is why no one will ever see a particle of electricity. Or ever know if the many particles of light needed to make our eyes or cameras work, are only representing one particle of electricity we think we are seeing or an infinite number. That is why for many, many, years, a particle of electricity has been considered, infinitely small. And to me will stay that way.
My point being since we are not going to be able to prove something infinite as being something finite, we should probably not worry about complex formulas that may only be exaggerating the known error. Not really aiming this at you in particular.
Sincerely,
William McCormick
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Calculus tells you why trying to shoehorn "infinity" into the problem as if it's a number as you're doing is wrong.
Calculus does nothing of the kind. Zeno clearly understood this issue better than the calculus priests. Think about it carefully for a moment. How many numbers are there in the series 1, 2, 3, 4, 5, etc. if you count them all up. You get infinity: that's just well-established mathematics, so there's no dispute about that. What happens if you use a different series like 1, 2, 4, 8, 16, etc.? You get infinity again (and although it may be a different kind of infinity, it's still an infinity). Now do the same with the series 1/1, 1/2, 1/4, 1/8, etc. - how many of them are there? Infinity again. Dispute that if you like, but you're moving away from established mathematics if you try. So, what is it that makes you imagine that the idea of calculus can override that?
If you can go on counting up the items in a series forever, which in the case of Zeno's arrow paradox we can, then that necessarily gives us an infinite number of steps whether you're prepared to admit it or not, and that's the whole point of Zeno's arrow paradox - to ban us from considering the case where the infinity is involved, you aren't addressing the paradox at all, but merely avoiding the issue. The paradox is very specifically about the infinity, and calculus doesn't let you off the hook in that regard at all. If the infinite steps have a duration, the action can never complete, and if the infinite steps have no duration, the scale is lost and they don't add up to the required answer.
What is calculus? It's a mathematical trick get around the infinity problem for calculating exact values in awkward cases where you can go on working with smaller and smaller pieces forever, but that is all it is - it is not a mechanism to make the impossible possible.
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which in the case of Zeno's arrow paradox we can,
I don't think we can. The logic in the version of the Paradox of the Arrow that I know is faulty, so it's not really a paradox at all. I would not be too surprised to learn that Zeno understood that and decided to pull a fast one on us.
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Zeno was fully aware that you could increase the number of steps without limit and that the step size would decrease proportionally without limit, but he was also able to see that it did not resolve his paradox. You either have to stop at some point and accept that the universe is granular or you have to take it all the way to the point where an infinity is introduced, at which point the maths breaks down. Zeno's paradox tells us that the universe is granular.
Yes, and the time interval also decreases proportionally without limit. You will end up with an arbitrarily large number divided by the same arbitrarily large number, which, if I'm not mistaken, is equal to one. Doesn't strike me as strong evidence that the Universe is granular.
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Calculus tells you why trying to shoehorn "infinity" into the problem as if it's a number as you're doing is wrong.
Calculus does nothing of the kind. Zeno clearly understood this issue better than the calculus priests. Think about it carefully for a moment. How many numbers are there in the series 1, 2, 3, 4, 5, etc. if you count them all up. You get infinity: that's just well-established mathematics, so there's no dispute about that. What happens if you use a different series like 1, 2, 4, 8, 16, etc.? You get infinity again (and although it may be a different kind of infinity, it's still an infinity). Now do the same with the series 1/1, 1/2, 1/4, 1/8, etc. - how many of them are there? Infinity again. Dispute that if you like, but you're moving away from established mathematics if you try. So, what is it that makes you imagine that the idea of calculus can override that?
Those sets are the same kind of infinity in size. They're countably infinite. That also has nothing to do with what we're discussing. It's completely different from saying that infinity is a real number that you can add or multiply by.
Look, you can argue that you're not making a mistake by treating infinity this way all you want, and throw around pejoratives like "calculus priests," but the fact is that you're wrong. You argued above that if you can multiply infinity by zero which is wrong--that expression doesn't even make sense in math since infinity can't be multiplied: it's not a number. Then you argue that adding infinite numbers of things together leads to an infinite result. That's also wrong, and you can prove that various infinite series converge using calculus/analysis.
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In the original version of Zeno's paradox (pre-quantum theory), there was a flaw in the logic: It is possible to add up the infinite series of 1/2+1/4+1/8+....; the answer is finite and equals 1. This matches our experience that an arrow can reach the target.
Put another way, let's say it takes 1 second to traverse the distance. If you chop it in half, it would take 1/2 second to traverse each half, which adds up to 1 second total. If you chop it in half again, you'd end up with 1/4 second to traverse each of 4 segments, which would add up to 1 second total. No matter how you chop it up, 8ths, 16ths, 2nths, it still takes 1 second because you're adding together many very small times, which can still add up to 1. Zeno didn't know the proper way to deal with this mathematically when you allow it to be chopped up endlessly, but modern calculus introduced the idea of a limit, which shows that no matter how small you chop these segments, the total time taken is still 1 second.
The question of what happens in reality on the quantum scale is another thing altogether, but the original paradox was resolved by calculus.
I think I see how you mean there JP, that it mathematically can't be a 'infinity', as adding it up again always will give you the same 'finite' answer. Or am I reading you wrong here? On the other hand? What stops me from splitting that second infinitely? One can argue that this second must be the finite answer to all splitting but there is still a 'infinity' represented in the 'instant' it takes, as it seems to me?
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Zeno was fully aware that you could increase the number of steps without limit and that the step size would decrease proportionally without limit, but he was also able to see that it did not resolve his paradox. You either have to stop at some point and accept that the universe is granular or you have to take it all the way to the point where an infinity is introduced, at which point the maths breaks down. Zeno's paradox tells us that the universe is granular.
Yes, and the time interval also decreases proportionally without limit. You will end up with an arbitrarily large number divided by the same arbitrarily large number, which, if I'm not mistaken, is equal to one. Doesn't strike me as strong evidence that the Universe is granular.
The time interval also decreases proportionally, but the whole point of the paradox is that when you take it to infinity in order to get rid of any possible granularity, the duration of the infinite chunks is zero, at which point the whole thing loses track of the total duration. If you don't take it that far, you're still dealing with granular solutions, just with smaller and smaller grain at every step.
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Those sets are the same kind of infinity in size. They're countably infinite. That also has nothing to do with what we're discussing.
The idea that they might count as different kinds of infinities was just mentioned in passing, and I suspect you're wrong about them being the same kind - maybe you aren't up to speed with the latest developments in mathematics in that area.
It's completely different from saying that infinity is a real number that you can add or multiply by.
It has everything to do with what we're discussing - it's the entire point of the paradox, and your proposed solution (which isn't actually yours, but is being pushed by the priests and must therefore be correct no matter how wrong it is) is to ban the issue from being addressed and claim that avoiding the issue equates to a solution.
Look, you can argue that you're not making a mistake by treating infinity this way all you want, and throw around pejoratives like "calculus priests," but the fact is that you're wrong.
Well, obviously you're right because you're simply right, and never mind what correctly applied reason has to say on the matter.
You argued above that if you can multiply infinity by zero which is wrong--that expression doesn't even make sense in math since infinity can't be multiplied: it's not a number.
If you don't take it to the point where there are an infinite number of chunks, you're actually pushing a granular solution, so you're not succeeding in solving the paradox with a non-granular solution. If you attempt to consider an infinite number of chunks, the maths breaks down and you fail. That's the whole point! You are pushing an infinite number of granular solutions and claiming they aren't granular.
Then you argue that adding infinite numbers of things together leads to an infinite result. That's also wrong, and you can prove that various infinite series converge using calculus/analysis.
No, I point out that any attempt to add up the infinite number of chunks goes wrong because if they're all zero in duration, there's nothing to add up, and if they're all greater than zero you can only get an infinite result (and this is not incorrect maths) - neither are mathematically useful as the whole scale has been lost. The only solutions are granular, and the tricks of calculus have nothing profound to say on the matter whatsoever.
If when you count up the items in a series of numbers such as 1, 2, 3, 4, etc. you get an infinity, what sense does it make to say that if you add up 1 + 1 + 1 + 1 + etc. forever you don't also get an infinity? That is identical to counting the items in the series. You are simply not entitled to pick and choose in such a way as to decide that one of these is valid and the other invalid when they are identical processes.
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In the original version of Zeno's paradox (pre-quantum theory), there was a flaw in the logic: It is possible to add up the infinite series of 1/2+1/4+1/8+....; the answer is finite and equals 1. This matches our experience that an arrow can reach the target.
Put another way, let's say it takes 1 second to traverse the distance. If you chop it in half, it would take 1/2 second to traverse each half, which adds up to 1 second total. If you chop it in half again, you'd end up with 1/4 second to traverse each of 4 segments, which would add up to 1 second total. No matter how you chop it up, 8ths, 16ths, 2nths, it still takes 1 second because you're adding together many very small times, which can still add up to 1. Zeno didn't know the proper way to deal with this mathematically when you allow it to be chopped up endlessly, but modern calculus introduced the idea of a limit, which shows that no matter how small you chop these segments, the total time taken is still 1 second.
The question of what happens in reality on the quantum scale is another thing altogether, but the original paradox was resolved by calculus.
I think I see how you mean there JP, that it mathematically can't be a 'infinity', as adding it up again always will give you the same 'finite' answer. Or am I reading you wrong here? On the other hand? What stops me from splitting that second infinitely? One can argue that this second must be the finite answer to all splitting but there is still a 'infinity' represented in the 'instant' it takes, as it seems to me?
The problem I'm pointing out is that you can't treat "infinity" as a number. You have to be very careful when saying you're taking an infinite number of things or making things infinitely large or small. For example, infinity cannot be treated as a real number. Zeno's paradox essentially argues that if you divide the interval into an infinite number of steps, each one still takes a tiny amount of time to cross, call this time t. The argument goes that the total time to cross N of these steps is N times t therefore the total time to cross an infinite number of these steps is infinity times t, which must be infinity!
The problem is that you have to treat the idea of infinity in a mathematically correct way. It isn't a number like 1 or 10 or Pi or square root of 2. It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite.
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The problem is that you have to treat the idea of infinity in a mathematically correct way. It isn't a number like 1 or 10 or Pi or square root of 2. It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite.
The time is finite, but the amount of chopping you can do is infinite. Every solution short of that infinity is a granular solution. You cannot escape from a granular solution without directly addressing the infinity, and any attempt to handle that infinity will self-destruct.
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David, I'm not sure how to respond to you. Similar to your "Einstein was wrong" thread, your main arguments against the mathematical resolution of the paradox are insults aimed at mathematicians and error-filled proofs of your ideas. I've pointed out where you're going wrong in your proofs and you respond with more insults at the "high priests of mathematics." If you don't want to discuss mainstream mathematics or science, why are you posting on a mainstream science Q&A forum?
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David, I'm not sure how to respond to you. Similar to your "Einstein was wrong" thread, your main arguments against the mathematical resolution of the paradox are insults aimed at mathematicians and error-filled proofs of your ideas.
Everything I said in the thread about Einstein's philosophical interpretation of relativity was correct - I've just started a new thread about it because nothing's been done about the previous one which was wrongly hacked out of a thread and shunted into "new theories" where it clearly doesn't belong.
I've pointed out where you're going wrong in your proofs and you respond with more insults at the "high priests of mathematics."
No, you've simply made claims about mathematics which don't stack up. The priests I refer to are not all mathematicians, but to specific ones who push faulty philosopy as fact and misinform the public, shutting down their thinking in the process. Calculus does not solve Zeno's arrow paradox - claims that it does are simply false.
If you don't want to discuss mainstream mathematics or science, why are you posting on a mainstream science Q&A forum?
When I see people pushing ideas that are wrong on the basis of authority, that is sufficient justification for putting across an alternative view. You push other people's assertions while I push mine, and the people who read the arguments put forward to back the assertions can make up their own minds. History is littered with cases of false beliefs holding sway, and when they're eventually overturned everyone looks back and says how silly people must have been to believe whatever it was - the same will happen again over many things that are mainstream and wrong. My objective here is simply to open people's minds to the possibility that certain beliefs are wrong, or at the very least go beyond the competence of the people who came up with them.
In this argument here, we have had a claim that 1 + 1 + 1 + 1 + etc. doesn't add up to infinity, whereas if you count all the numbers in the series 1, 2, 3, 4, etc. you get an infinite number. The former is the very process by which the latter is done. You can't just ban things from being infinities on an arbitrary basis - that is neither mathematics nor science.
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JP knows his math David :) I'm pretty sure on that one, doesn't mean that one can't have a different opinion though, just as I'm trying to see where that limit is thought as. To me the reason could be that a second is a limit of sorts whereas a infinity? On the other hand "It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite."
So i split a second in 2, then 2 again, then 2 again. then 2 ... ad infinitum, but how and where do I reach that limit? What in calculus demand that I can't just keep doing so for ever JP? There has to be some hidden logic to this reasoning that I'm missing.
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It's really quite simple: at every level of chopping, you have a granular solution, each one with a finer grain than the one before it, but because there are an infinite number of these granular solutions, the solution isn't granular. In other words, it's turtles all the way down.
The infinity is the elephant in the room - it's there, but people don't want to see it and have trained themselves not to do so in order to maintain their belief that calculus solves a problem which it actually fails to address. We all know that the time is finite and that things can actually move - Zeno understood that fully well. His paradox is focused squarely on the infinity issue, and calculus doesn't engage with that. Calculus gives you exact answers to questions like "what would the answer be if you could add everything up infinitely", but it simply does it through a trick.
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If you are really going to get to some reality or truth about this subject, consider the actuality, the paradox does not.
An arrow does not just move ahead. Any object in this universe made of matter, is expanding and contracting, constantly, at some frequency. When you get down to the increments, that are less then the movements of the arrows expansion and contraction, you will find that your timing of the arrows contraction and expansion, will be more important then the increment it moved.
That is why we use infinity, to describe just such a proposed experiment. The variables become to great. To silly to try to guess at, or confirm.
Sincerely,
William McCormick
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David,
I think you better set down precisely the version of the paradox that you are referring to so that we can test the logic to see if it even is a paradox, and, unless you have certain mystical powers, it might be best if you stop claiming that you know what Zeno was thinking.
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JP knows his math David :) I'm pretty sure on that one, doesn't mean that one can't have a different opinion though, just as I'm trying to see where that limit is thought as. To me the reason could be that a second is a limit of sorts whereas a infinity? On the other hand "It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite."
So i split a second in 2, then 2 again, then 2 again. then 2 ... ad infinitum, but how and where do I reach that limit? What in calculus demand that I can't just keep doing so for ever JP? There has to be some hidden logic to this reasoning that I'm missing.
In calculus, you can. That's why I was careful to say above that I'm not talking about quantum effects. In classical physics and calculus, there is no issue in chopping up time and distance into as small a chunk that you want. There is a problem with saying you have an infinitely small segment unless you're precise about what infinity means. Infinity is not a number like 1, sqrt(-1) or Pi. Infinity has to be used with care as a concept for very large things. It does have meaning, but since it's not a number, it doesn't make sense if you try to add, divide or multiply by it. You can (and physicsts often do) use it similarly to a number to mean things can increase forever or decrease forever without stopping, but that's a concept, not a single number.
Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want. If you do that, each segment is of finite time. But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite. The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross. For example, if you chop it in half, each segment takes 1/2th of a second to cross. 2*1/2=1 second. If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second. If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1. Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit. Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.
Zeno's paradox is a bit more sophisticated, since it's basically chopping the segment in half, then that half in half, then the next half in half, etc., so it's not as obvious that it has to sum to 1, but it's a similar proof in calculus. You can write the sum of a finite series and calculate the sum as you chop it up without limit. (The answer was actually found before calculus--thanks to Matthew for pointing that out to me--but calculus really made it rigorous and showed how to treat it within a firm mathematical framework.)
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JP knows his math David :) I'm pretty sure on that one, doesn't mean that one can't have a different opinion though, just as I'm trying to see where that limit is thought as. To me the reason could be that a second is a limit of sorts whereas a infinity? On the other hand "It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite."
So i split a second in 2, then 2 again, then 2 again. then 2 ... ad infinitum, but how and where do I reach that limit? What in calculus demand that I can't just keep doing so for ever JP? There has to be some hidden logic to this reasoning that I'm missing.
In calculus, you can. That's why I was careful to say above that I'm not talking about quantum effects. In classical physics and calculus, there is no issue in chopping up time and distance into as small a chunk that you want. There is a problem with saying you have an infinitely small segment unless you're precise about what infinity means. Infinity is not a number like 1, sqrt(-1) or Pi. Infinity has to be used with care as a concept for very large things. It does have meaning, but since it's not a number, it doesn't make sense if you try to add, divide or multiply by it. You can (and physicsts often do) use it similarly to a number to mean things can increase forever or decrease forever without stopping, but that's a concept, not a single number.
Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want. If you do that, each segment is of finite time. But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite. The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross. For example, if you chop it in half, each segment takes 1/2th of a second to cross. 2*1/2=1 second. If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second. If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1. Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit. Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.
Zeno's paradox is a bit more sophisticated, since it's basically chopping the segment in half, then that half in half, then the next half in half, etc., so it's not as obvious that it has to sum to 1, but it's a similar proof in calculus. You can write the sum of a finite series and calculate the sum as you chop it up without limit. (The answer was actually found before calculus--thanks to Matthew for pointing that out to me--but calculus really made it rigorous and showed how to treat it within a firm mathematical framework.)
Due to the reality that exists, isn't the paradox then as stated, just a mathemtical exercise, admittedly or obviously not taking into consideration any actuality or reality?
Sincerely,
William McCormick
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Ahh, cool JP :) Now I see how you meant. And it's a basic principle of calculus as I remember, very vaguely though :) n*1/n=1 I mean.
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William, to me it's logic and it is correct, at least for calculus. And as we use that kind of math for all sort of daily needs and constructions, and find it working? But ask JP of what he didn't discuss instead :)
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Due to the reality that exists, isn't the paradox then as stated, just a mathemtical exercise, admittedly or obviously not taking into consideration any actuality or reality?
There are various versions of the paradox and there is no definitive Zeno version. That was lost to antiquity. As David is using it to claim that Calculus is somehow defective and that space is granular, I think it's important that he specifies his particular version of it.
However, the version I'm aware says something like "at an instant in time, an arrow.........." The counterargument is, because time flows, there are no "instants in time", and if you use instants of time you have just assumed that time is granular which will lead to the inevitable conclusion that space is also granular.
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David, you might be interested in Georg Cantor (http://en.wikipedia.org/wiki/Infinity) where it's mentioned:
Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.
There is a particular limit problem I'm working on at the moment that goes like this:
limit as x->1 of 1/lnx - 1/(x-1)
As x approaches one, both 1/lnx and 1/(x-1) approach infinity. So you could say in this case "infinity" minus "infinity" equals 1/2, which is what the value this expression takes on at exactly one. Clearly, these are different "sized" infinities, or I believe the technical term is these are infinities of different cardinality?
Anyway, I'm pretty sure the correct way of treating infinity is as a tendency, rather than strictly a number. It is the tendency for a measure to grow without bound. It isn't any particular number. This implies that even if you were to define a symbol for some sort of discreet infinity, something that represented more a hard boundary than a tendency, the infinity we're used to would still grow without bound past that boundary, because...that's what it's defined to do!
And of course if it grows without bound, it naturally represents an infinite family of numbers, which is precisely why you can't treat the symbol for infinity, itself, as a number: the symbol takes on meaning relative to expression in which it is used. Do correct me if I'm wrong, that's as far as I understand at the moment.
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William, to me it's logic and it is correct, at least for calculus. And as we use that kind of math for all sort of daily needs and constructions, and find it working? But ask JP of what he didn't discuss instead :)
But the tip of the arrow is going to be found going backwards, or at least at different velocities, at certain cycles of the arrows, expansion and contraction in the real world, once the increments of the arrows movement, get very small. All depending on the arrows veocity.
So the whole theory is void once you pass a certain point of decreasing increments. The arrow at the midway point in time, may not be at the midway point. Because the arrow itself is expanding and contracting at its own rate based on the material it is made from. By increasing the level of accuracy you uncover variables not considered before.
So if you were to call the first half of the arrows journey n, and the second half of the arrows journey n, then n*1/n may not equal 1 Ha-ha.
By claiming you split the arrows journey in half you declared both halves equal. There should be no problem with that equation but in real life there is.
That would be the case in the real world, using very exacting measurements. For most non exacting purposes it would be true.
That was the purpose of infinity, if you don't know, and you cannot measure it in this lifetime, or you do not wish to waste a lifetime, call it infinity. If someone wants to go there let them, you already turned down the job.
Sincerely,
William McCormick
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Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want. If you do that, each segment is of finite time. But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite. The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross. For example, if you chop it in half, each segment takes 1/2th of a second to cross. 2*1/2=1 second. If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second. If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1. Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit. Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.
All you're really saying is that calculus includes as one of its principles a rejection of a particular idea of infinity, so it isn't calculus that's solving the paradox at all, but a flat rejection of a particular idea of infinity is by itself being used to reject the arrow paradox on the basis that it involves a faulty kind of infinity. Calculus simply doesn't engage with the kind of infinity which Zeno is using.
The issue then is about the validity of the kind of infinity in the paradox. Now we're getting somewhere at last - I can actually see now that you're rejecting Zeno's kind of infinity on rational grounds and that the success of calculus appears to give you a justification for doing so, so you attribute everything to calculus. Your argument is essentially that there is no such thing as infinity in the sense in which it is used in the paradox, so the paradox is not valid on that basis: there can be no such thing as the kind of infinity which Zeno had in mind because it can never be reached by anything - there is no forever that's long enough to make one, and if you're cutting somthing in half, then that half in half, then the resulting quarter in half, and so on forever, there is again no possible way of ever getting to an infinitieth, so the concept of infinity which Zeno was using is a kind of infinity which can never be realised. That is good thinking. I like it.
But there's a problem with rejecting Zeno's kind of infinity - his kind of infinity still has to apply to the inherent nature of a universe where you can chop things into smaller and smaller chunks forever. If it doesn't, you're going to have a universe where a finite amount of chopping must be capable of hitting the grain. If there is no grain, you have necessarily introduced Zeno's kind of infinity and made his paradox valid. Either way, the universe must be granular to solve the paradox.
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David, you might be interested in Georg Cantor (http://en.wikipedia.org/wiki/Infinity) where it's mentioned:
Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite.
That's the kind of thing I was referring to, though I've only heard about it through radio programmes which I can't link to, but from what I've heard there are people who are now doing useful new kinds of maths with different kinds of infinities. At the bottom of this page - [link removed as an experiment to find out why this post is being rejected for using a blacklisted term] - it says:-
There is a whole theory about the sizes of infinite sets, much of which was developed by the mathematician Georg Cantor. It is a very deep theory, much of which depends on several things that are far beyond the scope of hub pages. If you are interested and have a little bit of mathematics training, I would recommend visiting here http://people.umass.edu/gmhwww/382/pdf/09-infinite%20sizes.pdf. You will learn many things such as the nice theorem "there are an infinite amount of sizes of infinity".
[http://nauticaricky.h u b p a g e s.com/hub/Different-Sizes-of-Infinity - the only way to get there (for anyone who feels the desire to) is to copy that, remove the spaces and paste into address bar.]
The reason I brought up the idea of different kinds of infinity here was simply to discourage anyone from wasting time by going down that route as an objection to my argument on the basis that cutting things in half, then the half in half, then the quarter in half, etc. is the wrong kind of infinity - I wrongly thought it probably was a different kind of infinity, and had that been the case there was a an easy way to show that it wasn't relevant. Having looked back at JP's comment about it, I realise now that I misinterpreted his reply and thought he was rejecting the idea that there were different kinds of infinity, but he was actually just saying that they were the same kind.
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And of course if it grows without bound, it naturally represents an infinite family of numbers, which is precisely why you can't treat the symbol for infinity, itself, as a number: the symbol takes on meaning relative to expression in which it is used. Do correct me if I'm wrong, that's as far as I understand at the moment.
I think you're right. I think the kind of infinity Zeno's paradox involves cannot exist at all, and the consequence of that is that our view of the nature of the universe as something that could be chopped up forever without hitting the grain is wrong. There is no finite number in an infinite series that can't be reached by going through that series, and all numbers in an infinite series are finite. The only way the universe can have the property that you can't hit the grain is if it's built on infinities of the kind Zeno was using, completely beyond the reach of finite values.
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Zeno's paradox requires you make some assumptions about the mathematical model you're using. It's pointless to say you're using the "real model of the universe," since there is no such thing that we've discovered yet.
I was perfectly clear that I was talking about the classical Zeno's paradox in which it is assumed that the universe can be chopped up into arbitrarily small pieces. Calculus resolves that, as it (and limits) were invented to deal with things being chopped up into arbitrarily small (or large) pieces.
I also noted that there are good reasons to believe that you do hit some fundamental limits when chopping things up due to quantum effects, and that version of Zeno's paradox is not resolved by calculus and limits.
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A little off the main line of this discussion, but I would submit that we have very good evidence that space cannot be anything but infinitely divisible -- that is, that it cannot be pixellated. Why not? Because there is a relatively small and finite number of ways in which uniform elements may be stacked in an infinite array. They correspond to the 400-odd crystallographic space groups, each with its own particular symmetry.
The point is that none of them can possibly be isotropic, (that is have perfect spherical symmetry), and so with any stacking of finite elements in three-dimensional space there must be preferred directions, that would readily show up in a macroscopic observation. The preferred directions would even have to be a convenient several degrees apart.
The fact that we have no observations that indicate any absolute or preferred directions in space seems to me to be a very solid indication that there is no pixellation, at any degree of fineness, of real space.
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I'm a slow learner Damocles.
Are you suggesting that if uniform elements only can be stacked certain ways, but not other, that this stacking also should be expressed by matters grouping in space? The 'preferred directions' I can see but for that to define the grouping of galaxies for example? Surely interesting, so you better go deeper there :)
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Assuming I correctly understand your post Damocles, rather than zero pixellation at all degrees of fineness, couldn't another possible reason be that space wasn't "stacked" piece by piece to begin with? But that all the 'pieces' of space were laid in place instantaneously?
I think the only reason I somewhat understood your post was that this "preferred direction" you mentioned showed up while programming the area in a 2d-neural network I worked on as a hobby sometime back. There was some sort of bias in the direction that the algorithms were working because I unwittingly laid out the neurons and the connections between them in some predefined way.
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Suppose that space is 'pixellated' in some way, and that there are finite sized chunks of distance as a result.
Premises:
(1) That space is 'full'. It has to be for the concept of space to make any sense -- else the question of what is between the finite atoms of space becomes a meaningful one.
(2) That the geometry of space is locally Euclidean. I suspect that this stipulation is one of convenience rather than necessity.
It follows that:
(1) We cannot inquire into the shapes of the pixels or atoms of space -- spatoms -- because the idea of measurement of distance on a scale smaller than these supposed entities has no meaning.
(2) Any unit of distance must be an integral number of spatoms
(3) Spatoms must be distributed to fill the whole of an Euclidean 3-dimensional space.
The most isotropic tiling of three dimensional space that is possible has the symmetry of a cubic close-packed crystal lattice. There are 2-dimensional sections of this lattice that are triangular close-packed.
Imagine a measurement made of a large scale object along an edge of a triangle in this lattice. It will involve, say, N spatoms. Now rotate the object through 30° without changing the plane. It now involves 2N/√3 spatoms -- roughly a 20% increase.
We do not observe a 20% variation between a minimum and maximum distance measurement when we rotate a real object in any direction.
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Make the 20% of the last posting a 15% -- more accurate, but it is immaterial to the thrust of the argument anyway. (Moderator, my communication seems to freeze when I try the 'modify' link on my original posting).
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the first step, which is half the distance,can also be considered a distance in and of itself?
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the first step, which is half the distance,can also be considered a distance in and of itself?
That is true of a crystal lattice, but not of an atomically pixellated space. The first step, like any of the other steps, must take the integer value 1.
It is a bit like the original naive concept of an indivisible atom -- you can conceive of half an atom in such a world, but it cannot exist.
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Damocles, I agree with you that there can't be a lattice of space pixels. But it could also be the case that there is a smallest meaningful or measurable distance (or time), but that these distances aren't located at fixed positions in space. In other words, our ideas of space and time might break down at a small enough distance, which doesn't require that these "smallest" distances arrange themselves in to a lattice.
From what I understand, this is what some quantum theories of gravity propose, but I'm out of my depth in trying to understand the details.
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Quite interesting, and I will have to reread you Damocles. As for JP:s suggestion it also make me think of the observer dependencies noted in relativity, relative 'a absolute globally true distance' there. As in the often cited example of the infalling muon.
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Still interesting Damocles, I'm revisiting it, and a use of latices I haven't seen before. I adhere to the idea of a 'flow' myself, or 'no pixelation' if one like. If we assume this as a fact then what does it make of particles of rest-mass, can we assume a pixelation there? And space, what does it make of a vacuum. Something without structure? Distance exist, as well as 'motion', both related to the observer. And with matter a property containing a lot of space.
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I have difficulty understanding why we cannot have a physical space that is continuous and infinitely divisible. It is surely the case with the (one dimensional) space of real numbers. I am not used to dealing with spaces that take into account general relativity with high gravitational fields, large distances, or relative speeds close to c0.
It seems to me that there is nothing in quantum mechanics or quantum electrodynamics -- a relativistic theory with virtual pair creation/destruction etc., that has been proved empirically to be extremely accurate on a local scale -- that says that space is not infinitely divisible or that space without some matter in it is impossible (not counting the "vacuum field" as "matter", of course).
It is a mathematical fact, and not too difficult to prove, I should imagine, that any locally Euclidean space of dimensionality greater than 1 cannot be both isotropic (uniform in every direction) and pixellated (completely filled with a stacked set of subspaces with similar geometry to each other).
Aristotle and Plato believed that space had to be filled with some matter, in order to hold things apart. Aristotle considered that the presence of nothingness was an oxymoron, and that if it were not for the presence of the "lumeniferous aether", material things as we know them would all collapse together. Zeno devised his several paradoxes to prompt closer attention to the supposed impossibility of motion that he considered a concomitant of current views of the world. Leucippus and Democritus (pre Aristotle) considered that the world was full of just two types of stuff -- substance, and void. The matter was in indivisible lumps (a tomos = indivisible) of various shapes and sizes, which were kept apart by their continuing motion, and by continually bouncing off each other, and that the different properties of different material arose from different shapes of the "atoms" of substance. The void was a continuous (and presumably infinitely divisible) backdrop to these "atoms".
I envisage the world as lumpy matter in continuous motion in an infinitely divisible void. This is much the way that Leucippus and Democritus saw it 2500 years ago. There is nothing in such a view that is problematic for any of the science that I do. But then I am only a chemist, not inquiring closely into substructures of protons, neutrons, and electrons. Things may or may not be different for a particle physicist or a theoretical cosmologist.
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Damocles, I don't think there is any reason why space can't be continuous and infinitely divisible. The only major issue is that we know our theories break down at small enough scales (the Planck time and length). The breakdown of our theories doesn't mean that suddenly space and time become discretized. It just means that we need a new theory.
There is a good reason to think that this new theory might involve space-time doing some funny things, however. General relativity describes gravity in terms of the geometry of space-time. Quantum mechanics describes all other forces in terms of quantized particle interactions. For consistency, it's natural to expect that on a small enough scale, gravity should behave like particles. Since the fabric of space-time is so intimately connected to gravity, it's natural to expect it might take on quantized, particle-like properties.
And of course this quantization happens on scales so small that we haven't even been able to detect it in particle accelerators, which are capable of breaking apart even protons and neutrons.
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Lovely discussion people. Just wanted to try and boil it down to words I understand. :)
so...
This Zeno paradox (as a pure mathematical construct and not in any way an observable phenomenon) is resolved because although it plays the "infinity" card (infinite amount of steps), it also implies another "infinity" (namely the smallness of those steps) which cancels it out. I guess if you look at it another way and say that the first step is an infintely small step, you can't then turn around and say it is a finite time, because then you could use the opposite argument and say that if a journey is taken in infinitely small time steps, but in each time step there is a finite distance travelled, nothing takes any time.
So based on the Zeno paradox the universe could be granular or infinite and it wouldn't change the results, as long as your were consistant with how you applied it.
I lost some of the argument but that's kind of where I ended up with it. Too simplistic?
Also, in terms of real stuff, I was led to believe that when you get down to fundamental particles, they exist in a kind of probability field. Wouldn't this resolve the granular universe by saying that each particle could exist anywhere, in infinite detail?
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Sorry, Bizerl, I was not meaning to get over-technical and leave you behind. There are some very deep philosophical issues involved, and I guess my own interest was in putting some of them before the theoretical physicists who participate in this forum to see how they would react to some of the ideas in my own mind.
You are quite right. The Greek philosophers and mathematicians had a significant problem with the concept of infinity, and lacked the mathematical tools to deal with it. That is another aspect of Zeno's paradoxes. With the mathematical approaches we have been taught, especially the inclusion of differential and integral calculus, we have effective and consistent ways of dealing with the infinite and infinitesimal. That does not address philosophical questions of whether or not we should consider the infinite and the infinitesimal as existing in the real world, but as far as science is concerned it does provide us with reliable terms for observation and description of the real world in a physical model.
From here on, I am getting to a personal, and perhaps slightly tongue-in-cheek viewpoint. There are huge differences between the ways that chemists and physicists look at things. There are also huge differences between the work that theoretical physicists actually do, and the way that they present their findings to a lay audience. When they are actually working, researching, and discovering things, they are manipulating high-powered and completely inaccessible mathematical expressions as they develop and explore mathematical models in an attempt to describe the physical phenomena that they are trying to address. When they try to report their finding to a lay audience, they often use analogies and make philosophical pronouncements that their work really does not warrant, but they find themselves unable to communicate in lay-accessible terms in any other way.
Now physicists like aesthetic simplicity in their models, and often this search for aesthetic simplicity is very productive of new results. For example, chemists recognize and work with three types of sub-atomic particle -- protons, neutrons, and electrons. They are very happy with that picture because it represents a simplification of their previous conception of 80 or 90 individual elements with puzzling patterns and differences in their properties -- a chaotic picture indeed! But theoretical physicists were intrigued by the fact that protons and neutrons had nearly the same mass, but different electrical charge, and that isolated protons were stable whereas isolated neutrons underwent radioactive decay. So they worked hard to produce a mathematical model where protons and neutrons were symmetrically equivalent -- where a proton could emerge as a neutron under some sort of symmetry operation. When this model was developed, it suggested that there should be a number of other sub-atomic particles, some of which were known from previous experiments, and some of which were discovered when they were specifically looked for. These other particles had approximately the right masses and lifetimes that the mathematics had suggested. The theory was very fruitful.
From bizerl:
Also, in terms of real stuff, I was led to believe that when you get down to fundamental particles, they exist in a kind of probability field. Wouldn't this resolve the granular universe by saying that each particle could exist anywhere, in infinite detail?
You certainly have reproduced pretty much what a physicist would probably tell you in the first sentence. The problem is in the last three words of your post. The Copenhagen interpretation of quantum mechanics, which seems to be the only viable one although many scientists are trying to come up with alternatives, says that not only is there an unavoidable minimum uncertainty in any measurement you make on a particle (well, most measurements), but that this uncertainty is a property of the particle, and not just a failure of the measurement method. The problem is that if you try to look at the detail of a particle -- for example to examine its shape or its parts -- then the uncertainty in the measurement of each part is greater than the uncertainty in the measurement of the whole: that is that the parts of a particle are blurred or defocussed over a larger amount of space than the particle itself occupies. That is certainly the sort of thing that the mathematics says; I am not at all sure of my translation of it back into the real world! But those last three words are the essence of the problem.
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I fully enjoyed your eh, wiev look back? tillbakablick as we say in Swedish. Furthermore, to ignore what people thought before us isn't that clever, it speaks more of hubris than understanding to me. Although we have come quite a bit on the way to getting those ideas into some proper reference frame, possibly :)
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Sorry, Bizerl, I was not meaning to get over-technical and leave you behind.
It doesn't take much to leave me behind! And thank you for the consideration and the great reply.
This may be the right thread for another mathematical anomaly from the depths of my highschool classes. I seem to remember a technique to convert a repeating decimal to a fraction (see here) (http://www.basic-mathematics.com/converting-repeating-decimals-to-fractions.html). When applied to 0.99999..., this result = 1. Is there a difference between 0.99999... and 1? I've always thought of 1-0.99999... as infinitesimal. Is that right?