# Unrelative Relativity Part 2

Discussion in 'Physics & Math' started by Prosoothus, Jun 12, 2002.

1. ### overdozehumanRegistered Senior Member

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310
Hi Tom

Think about what you're saying. If speed of light changes between reference frames, then laws of optics (for example) will be different in different reference frames. That is clearly not the case. So, there must be still something missing in your picture. Let's see...

Question is, who is measuring the time? If it's the moving object, remember that its clock ticks slower -- so in the end it still gets precisely c as the speed of light. If you are making your measurements from some other FOR, then of course you would have the impression you expressed above. But then, in your reference frame you are still measuring the velocity of light to be c, aren't you? The point is, everybody will measure the speed of light as c, from any inertial reference frame.

That is precisely the opposite of what happened. From the start, Einstein assumed speed of light to be a constant -- c -- for all intertial frames of reference; he was led to such an assumption by consideration of Maxwell's electromagnetic equations describing light. In fact, Einstein was always very fond of Maxwell, and he was quite distressed at conflict between the apparent constancy of lightspeed that came out of Maxwell physics with theoretically variable (zero to infinite) speed of light in Newtonian physics.

Once he made his choice and stuck with Maxwell, he then proceeded to see what the implications would be. To his possible surprise and probable delight, he discovered that the outcome is a self-consistent mathematics that at low-velocity limit reduces to Newtonian. Of course, Lorentz transformations were never just artificially inserted into the theory; on the contrary they are a direct logical consequence of the original premises (lightspeed constancy and special relativity.) I have a book on relativity that derives the Lorentz transformations entirely from those two premises (in fact, the entire theory derives from them.) If you want, I can reproduce that derivation for you.

3. ### allantVersion 1.0Registered Senior Member

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The answer is a if you are an observer moving at the same velocity as the 1 Kg object before the collision, and b if you are an observer moving at the same velocity as the 2 kg object Instead of assuming they are destroyed assume they meld and stay together. What the is the energy foreach observer - the answers given, remember the melded object will still be moving.

There is a third frame you can use, the velocity after the collision, and this is probably what you are thinking of as the "true one and only energy" after the collision, but it aint so.

To return to the original scenario, it is the same if you assume the energy is converted to photons. The are not five and 1 million photons but there is a red/blue shift increasing or decreasing the energy of each photon. So each observer will see a different color, and hence a different total energy. It all comes out in the wash.

5. ### ProsoothusRegistered Senior Member

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1,973
Allant,

Welcome to sciforums!!!

What if the melded object is not moving after the collision? What about the two observers? Both observers came to a stop after the collision, and they are both in the exact same location. Doesn't that mean that both observers are in the same frame of reference? Didn't the frames of reference of both observers merge into one?? If so, how is the energy of the final frame of reference related to the energy of the two frames of reference before the collision.

As you can see, everthing is OK with relativity until two frames of reference merge into one, or vice versa. Everytime this happens, relativity is proved incorrect because there has to be only one result for both observers(since after the collision, they are both in the same frame of reference).

Tom

7. ### allantVersion 1.0Registered Senior Member

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After collision not moving relative to what ?

a) the first object
b) the second
c) something else the just happens to means the result is not moving ?

Two frames colapse into one ? SR applies to inertia frames of reference. Now either the two frames are the same to begin with. or after they merge one or both have been accelerated and are not inertial frames of reference.

P.S thanx for the welcome.

8. ### ProsoothusRegistered Senior Member

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1,973
Allant,

Let's say the Earth, for the sake of argument.

Both frames have deccelerated as a result of the collision. How much energy did the collision produce.

You might say: Well that depends on which observers frame of reference your looking at it from.

Let me remind you, there aren't two observers anymore. The two observers were merged into one observer by the impact, just like the two objects. Technically, the two observers decellerated and came to a stop(relative to the Earth) in the exact same location as the merged object.

Tom

9. ### (Q)Encephaloid MartiniValued Senior Member

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19,125
Let me remind you, there aren't two observers anymore. The two observers were merged into one observer by the impact, just like the two objects. Technically, the two observers decellerated and came to a stop(relative to the Earth) in the exact same location as the merged object.

Technically, the two observers are observing the same impact from different sides since they were viewing each other moving towards one another. Therefore, there should still remain two separate reference frames, not two merged into one.

10. ### FyreStarFaithless since 1980Registered Senior Member

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229

Greetings -

I'm going to throw my hat into the ring here and see if I can help with the clarification of reference frames.

A frame of reference is unlimited in scope. It says "We will mark this point in space and consider everything else in the universe to be moving relative to it". For simplicity's sake, most often the universe is defined as consisting only of two seperate masses. Such masses, when isolated, are still affected by the rest of the universe, but to a degree so small that our conclusions don't change appreciably.

So, we define a point as the center of the frame of reference and then define the rest of the unignored universe in relation to that point. Once that is done, an observer at that point can use equations to model the behavior of other objects. No matter what the other objects do, the defined point (and hence, the observer) never changes position.

I will provide an example. Think of a simple Cartesian coordinate system. For the first frame of reference, I will (arbitrarily, though it does make the math simple) choose the origin as the center point. The 'rest of the universe' consists of three 1kg masses. Mass A is located at (-1,0,0), Mass B at (1,0,0), and Mass C at (0,0,0). From this frame of reference, both Masses A and B are moving towards the observer at a speed of 1 m/s, while Mass C is stationary. After one second, the masses will collide at (0,0,0), at the center of the frame of reference. Afterwards, Masses A and B will be seen by the observer to travel away from him at a speed of 1 m/s.

Now, we perform the same event from Mass A's frame of reference (that is, defining Mass A's position to always occupy (0,0,0) on our Cartesian coordinate system. The observer in this case will see Mass C located at (1,0,0) and Mass B at (2,0,0). Mass C will be travelling at a speed of 1 m/s towards Mass A, and Mass B will be travelling at a speed of 2 m/s towards Mass A. After one second, Masses B and C will reach Mass A and collide, and then proceed to move off at the same respective speeds in the opposite direction. Mass A will have never moved from (0,0,0). (The situation for Mass B is basically the same, just flipflopped over the axis.)

With this in mind, let me address a few of the points you mentioned, Prosoothus.

Regarding the Earth having to 'expand' to satisfy masses being dropped from (for example) each pole: This is a case where you are altering the frame of reference in the middle of the experiment (or simply misdefining it in the first place). A frame of reference can have only one center; also, it can't suddenly change that center in the middle of the experiment without changing the reference frame. You have (0,0,0) set to the rock being dropped on the North Pole. From here, both the Earth and the mass at the South Pole will approach you. I repeat, there can be only one (0,0,0) for each frame of reference. Indeed, the property of residing at (0,0,0) and never leaving it is the definition of a frame of reference. So, clearly, having the Earth approach you from two different directions would directly imply that what you are using is not a frame of reference. A mass can only have one position relative to you, not two.

Regarding the 'proton impacting electron, forming neutron' point: The proton itself is not observer; rather, defining the proton's position to always inhabit (0,0,0) places an 'observer' at the same spot as the proton. I'm going to answer your question in haste, and then explain. You asked "Which frame of reference is the neutron in? The electron's or the proton's?" The answer is: Both. Furthermore, the proton's and electron's continue to 'exist' as long as we want them to. Here's why:

Defining a frame of reference centered (initially!) on the proton, we observe the electron approaching the proton at (0,0,0). When they collide, energy is released and a netron is formed and begins to move slowly away from (0,0,0). This is the point at which frames of reference can become confusing, mostly due to the language we use regarding them. By using the terms 'We', and the personified 'observer', we make it seem as if there is actually a person there 'riding on the proton'. Here's what to keep in mind; from any particular frame of reference, there are infinitely many other frames of reference, all moving relative to the one we're "in". Suppose you were granted the ability to perceive the motion of other frames of reference relative to youself. Standing stationary on a street, you would see these little windows of cartesian coordinate systems flying by you, some accelerating, some not. Sometimes an object will occupy the center of a frame of reference before an event 'dislodges' it. This is what happens with your example. The frame of reference itself continues to exist, unmolested, while the proton we used to find it has been dislodged.

The simplied version of that paragraph is simply that frames of reference exist independantly of the objects you're working with. You can use an object to find a frame of reference, but you can't guarantee that the object will stay at (0,0,0) for eternity.

This explanation turned out to be much longer than I intended. I hope it helps

Thanks,
FyreStar

11. ### James RJust this guy, you know?Staff Member

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33,449
It seems I've missed a few posts in this thread. Most of what I would reply here I have said elsewhere (see, for instance, the "how accurate are atomic clocks?" thread. A few loose ends:

<b>c'est moi:</b>

<i>Further, would you be so kind to define motion.</i>

I agree with Crisp on this. It is a change in the spatial co-ordinate of something, relative to a particular frame of reference.

<i>Moreover, though I have no idea how to do it in practice, but why can't we use light as abs. FOR?</i>

Because all time intervals and lengths of every object would shrink to zero, meaning that everything would be observed to happen at the same place and at the same time. It doesn't tell us anything useful about distinct events.

<i>A new theory is required to perform objective and logical science which both explains and predicts. Physicians should also understand the importance of philosophy for their physics, which most of them don't. Not only the results matter, understanding the results is far more important. Understanding and interpretting them always involves philosophy. Philosophy overrules physics and not the other way around, it's that simple.</i>

Philosophy complements physics, but in no way overrules it. They are different fields of inquiry. Philosophy is based on pure thought. You start from a set of assumptions and see where they lead. Physics is an experimental and observational science, intimately connected to real-world evidence.

<b>Tom</b>:

<i>So how do you know what's the correct model?? The correct model has to pass these three tests:

1) It must be logical.
2) It must overlap reality 100%. If it doesn't, it either isn't correct or it isn't complete.
3) It must be simpler than all other models.</i>

Relativity satisfies (1) and (2). (3) is a silly requirement. Newtonian gravity is simpler than general relativity. Aristotlean physics is simpler than Newton. That doesn't by any stretch mean that Aristotle was right and Newton or Einstein wrong.

<i>If you recall I used the formulas for time dilation and length contraction to prove that light's speed is not c in all frames of reference.</i>

Hopefully, with subsequent explanation, you don't still believe this statement.

<i>If the two of you want to believe that the Earth moves towards the rocks, and that the Sun revolves around the Earth, then I'm happy for you. It's just fortunate that not all scientists, over the years, believed what you do.</i>

All the best scientists over the years have been able to look at things from more than one point of view.

12. ### itchyRegistered Senior Member

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47
Prosoothus,

Do you think the amount of energy caused by the collision is somehow absolute? That there is some absolute reference frame that is the correct one? What is the point of inventing such a absolute frame of reference? What will you use it for? The only interesting frame is the one you are in, that is the one you are measuring the energy from. And depending on the frame you are in you will measure the energy differently because of doppler effects on the light emitted from the collision.

I dont see the contradiction, but perhaps I am missing something.

13. ### c'est moiall is energy and entropyRegistered Senior Member

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583
"""Philosophy complements physics, but in no way overrules it. They are different fields of inquiry. Philosophy is based on pure thought. You start from a set of assumptions and see where they lead. Physics is an experimental and observational science, intimately connected to real-world evidence. """

James R, what the hell are you saying here?

How are you connected to the *real* world around you? How??? Through your thoughts. Tell me, what will influence your sight on that 'real world' of yours? That's right, those same thoughts. Need I say more? This line, "Physics is an experimental and observational science, intimately connected to real-world evidence", is absolutely pure rubbish. It seems philosophy is not a strong point of yours which explains your lack of understanding in some basic matters. You remain arrogant in your "scientific" way of thinking, but you will not learn through that. Do you want to understand the world or do you want to write down abstract formulas and talk about numbers without knowing what influence *your* way of thinking has on them. It's all about points of view, isn't it?

14. ### James RJust this guy, you know?Staff Member

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33,449
It's funny, c'est moi. I'm having a similar discussion in another thread, where I'm being accused of having exactly the opposite opinion that you're accusing me of. It seems to me that somebody here can see both sides of the story, whilst other people are convinced that one side provides all the answers.

15. ### c'est moiall is energy and entropyRegistered Senior Member

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"""It's funny, c'est moi. I'm having a similar discussion in another thread, where I'm being accused of having exactly the opposite opinion that you're accusing me of."""

I'm sorry but I've not been following discussions here for the last month or so. don't know what you're talking about

"""It seems to me that somebody here can see both sides of the story, whilst other people are convinced that one side provides all the answers."""

james R, from all our past discussions I know pretty well how you think and you are not someone who sees both sides of the story which you prooved yet again with you post about philosophy
You have yet to understand why science can fail, even though it seems conducted in a correct way. I'm not telling you this to sound harsh or to start some kind of a vendetta

, I learnd a lot from you here about physics for which I thank you, but you might also learn something from me I think, and that will be more philosophical science than "real" science. I never stop questioning things, I never stop thinking about stuff and I think that's the right kind of attitude and I feel this is commonly lacking in many future and current scientists. The standard theories are there, not to stick to them, but to break them to pieces and to replace them, and then the new standard will have to receive that very same destinity.

THE WORLD <-----> DATA <-----> THOUGHTS - INTERPRETATION & YOUR PHILOSOPHY <----> SCIENTIFIC THEORY

Philosophy, the art of reasoning, does not complement, it is a basic thing in the proces of science. It DOES NOT complement to science.

16. ### James RJust this guy, you know?Staff Member

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33,449
c'est moi,

Take a look at the "Does light have a mass?" thread, around pages 43-44 (in 10 posts per page), then get back to me.

<i>james R, from all our past discussions I know pretty well how you think and you are not someone who sees both sides of the story which you prooved yet again with you post about philosophy.</i>

I don't presume to know how you think, and I suspect that you have no idea how I think.

<i>You have yet to understand why science can fail, even though it seems conducted in a correct way.</i>

Can you give an example in which science has failed when conducted in the correct way?

<i>I'm not telling you this to sound harsh or to start some kind of a vendetta...</i>

I certainly hope not. There's no need to get personal. We're just having a discussion here.

<i>I never stop questioning things, I never stop thinking about stuff and I think that's the right kind of attitude and I feel this is commonly lacking in many future and current scientists.</i>

Why do you think that?

<i>The standard theories are there, not to stick to them, but to break them to pieces and to replace them, and then the new standard will have to receive that very same destinity.</i>

New theories are fine as long as they:
1. explain previous observations as well as the old theories.
2. make new predictions which the old theories do not.
3. are in accordance with the evidence.

<i>Philosophy, the art of reasoning, does not complement, it is a basic thing in the proces of science. It DOES NOT complement to science.</i>

I fail to see how philosophy can be both a "basic thing in the process" and, at the same time "not complement" science.

17. ### FyreStarFaithless since 1980Registered Senior Member

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229
A pyramid of knowledge

Philosophy, and in particular, Epistemology, enable science by defining and developing logic and reason. This is similar to the way science enables technology. Whether he conciously acknowledges it or not, the scientist adopts a very specific philosophy that allows him to proceed through his work. This is not to say that it is required that one study philosophy to partake in science, but that does not mean that the underlying priciples of science are absent.

If you have time, I would recommend reading some of Ayn Rand's epistemological writings.

Thanks,
FyreStar

18. ### c'est moiall is energy and entropyRegistered Senior Member

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""I fail to see how philosophy can be both a "basic thing in the process" and, at the same time "not complement" science.""

by complementing, James R, you mean it is there, but not essential like the fundaments of a house
the rest is just plain boring to start answering

19. ### CrispGone 4everRegistered Senior Member

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1,339
Two can play that philosophy game

Hi c'est moi,

You cannot dismiss James' question by saying it is plain boring to answer to. The part I am refering to is:

I understand what you are referring to, i.e. that correctly conducted science does not necessarily match "reality". Newtonian mechanics can be perfectly good science, but it can give completely wrong predictions. But on the other hand, it was you that pointed out that reality is thightly related to one's point of view.

So doesn't this absolve science from having to be connected to reality to be "good science" ? After all, if reality is only perceived through your own personal looking glass, then there is no place for reality in "good" science, which should be free from all subjectivism, since there can only be one truth (or reality) according to you.

By just looking around you, you should realize that science *is* connected to reality: it is reality that drives scientists to investigate and study. Hence, science appearantly is not objective, and perhaps there is not one reality, but only one's personal perception of what is real.

This somehow reminds me of the discussion we had on relativity, whether time dilatation and lorentz contraction are real or not... Following the reasoning above, they are very real indeed, at least for someone observing them.

Just some thoughts...

Bye!

Crisp (now graduated and absolved of exams

)

20. ### (Q)Encephaloid MartiniValued Senior Member

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19,125
Crisp (now graduated and absolved of exams)

Congratulations !!!

21. ### James RJust this guy, you know?Staff Member

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33,449
c'est moi:

Thankyou for you assessment. Of course, boring doesn't mean I'm wrong. Looks like a cop-out to me.

22. ### c'est moiall is energy and entropyRegistered Senior Member

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"""You cannot dismiss James' question by saying it is plain boring to answer to. The part I am refering to is"""

Yes I can

If I don't feel like answering every bit because it is boring, then this is my right.

quote:
--------------------------------------------------------------------------------

"You have yet to understand why science can fail, even though it seems conducted in a correct way. "

Can you give an example in which science has failed when conducted in the correct way?

--------------------------------------------------------------------------------

"""I understand what you are referring to, i.e. that correctly conducted science does not necessarily match "reality"."""

Indeed, every piece that he just asks 'why' is needless to answer because you know what I mean, hence boring stuff. But since you deepened it out some more, unlike James R, it is interesting again.

"""Newtonian mechanics can be perfectly good science, but it can give completely wrong predictions. But on the other hand, it was you that pointed out that reality is thightly related to one's point of view."""

I guess you're meaning this:

'Do you want to understand the world or do you want to write down abstract formulas and talk about numbers without knowing what influence *your* way of thinking has on them. It's all about points of view, isn't it?'

That last one was actually meant sarcastic, referring to relativity. As for the serious part, yes, everybody seems to live in his or her reality field. For example (this counts for every being on earth), what do you know about the 'reality' of the sun, except for some pictures and seeing this small globe shining in the sky? Nothing. Our reality is all about experience. Experience through our minds (thoughts). There is NO direct connection between 'that' reality out there and ours, which does not mean that there is no such thing as an objective reality.

"""So doesn't this absolve science from having to be connected to reality to be "good science"?"""

I think that here we agree, but may I add that I, as an outsider, am not really interested in a theory, of which Newtonian physics is a good example, of which we know that it mostly works but actually doesn't fit in the "real" world. I want primary understanding. I have no need to be able to predict stuff etc., that's not my job nor responsability. So, if you define 'good' science as being science with which you can predict the stuff you want to, than that's fine for me, but good science for *me* would be the one which stands the closest near that 'real' reality out there, where the line of our thoughts inbetween is the thinnest.

How would you know that the latter stands the closest and not another one? Imagine two theories predicting stuff equally good yet explaining it differently. I think that the one which feels right to most people is the one which stands closest to objective reality. It would give us a mere taste of that real reality, a taste which looses almost all its strengts once it gets mixed with our world of thoughts. You see, the taste of it would appear familiar to everyone, not for a mere 50% of all people, but for 99,9%.

"""After all, if reality is only perceived through your own personal looking glass, then there is no place for reality in "good" science, which should be free from all subjectivism, since there can only be one truth (or reality) according to you."""

See above.

"""By just looking around you, you should realize that science *is* connected to reality"""

"""This somehow reminds me of the discussion we had on relativity, whether time dilatation and lorentz contraction are real or not... Following the reasoning above, they are very real indeed, at least for someone observing them."""

Tricks of a magician look also most real to the unexperienced audience. The thing is, will there still be a line between illusion and 'reality', or is it just an 'illusion' and a barrier created by Language itself? The magician could show the audience afterwards how he created their 'wrong' reality, their illusion. Maybe there is also a way of finding the trick of those dilations and viewing them like the magicians trick, again, but now with understanding. There are some tricks of magicians that, even though you know how they work, you'd still see the same thing happening. Only then, the difference is that you now understand that what you see, isn't actually real. Maybe it is the same with dilations. You'd know about it, but you'd still see the same thing happening.

23. ### CrispGone 4everRegistered Senior Member

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This reminds me of me, 5 years ago

Hi c'est moi,

I can perfectly understand what you are saying. I used to think of science in the same way, but somewhere along the way (in the past 5 years), too much things happened that made me look at life in a more cynical way. People hate me for it, but we'll not go into detail here

. Anyway, my view on science has also changed. I don't want to sound preachy, but you have an idealistic view of science. That is not bad, in fact, it is the most beautiful view on science you can have. Unfortunately I cannot agree with that view, but for some reason I suspect it has to do with other factors (see story about myself above), and not with "experience" or "knowledge" (which after all I don't have, perhaps a tiny bit more than some people, but that's nothing special anyway

). But let's get on with the discussion:

"I have no need to be able to predict stuff etc., that's not my job nor responsability. So, if you define 'good' science as being science with which you can predict the stuff you want to, than that's fine for me, but good science for *me* would be the one which stands the closest near that 'real' reality out there, where the line of our thoughts inbetween is the thinnest."

This is what I meant with an "idealistic view" (and let me remind you that it is not meant in a bad way). Yes, I would also like to have a theory that explains exactly how nature works, and I would also like to be 100% sure that it indeed describes how all those wonderful things happen. Unfortunately, there is no possible way whatsoever of verifying this. The closest you could come is to perform all possible experiments imaginable, and see if your theory holds. I think - and this is even disputable - that a fundamental theory that could explain *all* experiments would indeed be "correct" (i.e. it would describe exactly how nature works).

Practically, it is ofcourse not possible to explain all experiments (the infinite amount of thinkable experiments would require an infinite amount of time to perform). Hence you have to stick with predictions for a finite amount of experiments. What one can say is that a theory that predicts and explains a large amount of experiments could perhaps tell us something of how nature works. Because of this fundamental restriction, I think we cannot define science as "what matches reality" but rather "what matches reality in a close way". But we more or less agree on that, the question ofcourse is ...

"How would you know that the latter stands the closest and not another one?"

"I think that the one which feels right to most people is the one which stands closest to objective reality. It would give us a mere taste of that real reality, a taste which looses almost all its strengths once it gets mixed with our world of thoughts. You see, the taste of it would appear familiar to everyone, not for a mere 50% of all people, but for 99,9%."

I have some problems with the concept "objective reality". This assumes that there is some underlying structure in all matter and energy that is absolute, but is only perceived by us in the wrong way (i.e. through our own world of thoughts). The question here is if this really exists. We'll get back to that in a second.

First, I don't think one can define reality by using "familiarity" or "feelings about it being right". This is very subjective, something we want to avoid at all costs to define the "objective reality" we are talking about. Intuition can really mislead you. I assume you have already met people that at first seemed nice and gentle and then turned out to be straight descendants of the prince of darkness himself. Or, something can look real tasty, but when you actually put it in your mouth all you can think about is the glass of water next to the plate to swallow everything as fast as possible. I have learned not to trust my intuition (really, don't ever try the fried octopus rings

).

To answer the question on whether an underlying reality exists or not... I really liked the analogy with the magician:

"Tricks of a magician look also most real to the unexperienced audience [...] There are some tricks of magicians that, even though you know how they work, you'd still see the same thing happening. Only then, the difference is that you now understand that what you see, isn't actually real."

Yes, but it is an unwritten code between magicians never to reveal their tricks. Then, by logic:
1) You see it happen, i.e. it seems real.
2) There is no possible way of finding out whether what you perceive is real or not.
3) Then there are two possible choices: either it is real, or either it is not.

If it isn't real, then you know you can't trust your perceptions and senses. However, the real "reality" is hidden by your incapability to perceive it. Hence nothing rationally can be said about it, since it is not falsifiable - you can make any claim you want. This is ofcourse not the way science chooses, since it tries to be consistent. Hence from a pure scientific point of view, one is obliged to assume that what one perceives (directly or indirectly) is truelly reality.

Sidenote: by perception I ofcourse mean measurement. We're not talking about distortion effects, but controlled, verifiable measurements by adequate equipment (be it the human eye, or a high-tech particle detector).

I think this is why most scientists are pragmatic: if the theory works, then that is fine for us, there is no science in talking about whether it is real or not. It is also by this kind of reasoning that my view on science has changed

.

Well, is there really an underlying reality ? I hope that you now understand why I think there is no scientific answer to this, but only a philosophical one. I think there is none. If I drop a penny on the floor here, someone on the sun will only know I did that after 8 minutes (once he measured it with his equipment). There seems to be a fundamental constraint in nature that prevents us from perceiving events instantaneously (the finite transmission speed of information). This fundamental limitation leads me to think that the only reality is what one perceives, it is almost if nature wanted it to be that way.

I am waiting to be proven wrong on that one ofcourse (e.g. by explanations of how some of those wierd quantum entanglement experiments can happen), since this could perhaps restore science (for me personally) back to its glorious position it once had

. For now, I stick with my own reality, and to be honest, it works fine that way for me

.

Bye!

Crisp

PS: (Q), thanks!