Why do you not see the paradox? When guys on the board say the HK earth based observers saw the Westbound clocks slow down, which is not true, why don’t you correct their errors?[/quote]

The observer on fixed to the earth, not in the plane, saw the clock on the Westbound plane speed up. Anyone who says that all three clocks slowed down relative to each other has stated a paradox.

Actually, nobody that I know or read claimed that the clocks all slow down relative to each other. Find one, I don't think anyone said that exactly. The Westbound clock speeded up relative to the earthbound and Eastward travelling clock. The HK experimental results were consistent with the HK calculations predicted and explicitly shown in the same issue, consistent with the rules stated in the Einstein's 1916 paper, Einstein's 1905 paper, and even "Theory of Electrons" by Lorentz.


The Lorentz book I quote was not just a paper but a 138 page book. [/quote]
You haven't quoted anything in "Theory of Electrons."
Please explain Lorentz's theory to us, especially the math and physics involved, that we may understand.

He imparts the relative motion in Section 4 when he says, ”and if the clock at A is moved with the velocity v along the line AB to B.” He is of course describing the relative motion only from the point of view of K system observers, and this misleads a lot of his readers.

I think you just don’t understand the rules of the theory that he had already written into the theory before Section 4, such as, “The introduction of a ‘luminiferous ether’ will prove to be superfluous inasmuch as the view here to be developed will not require an ‘absolutely stationary space’ provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place,” and, “It is clear that the same results hold good of bodies at rest in the ‘stationary’ system, viewed from a system in uniform motion.”

These two statements make the motion “relative” and not absolute, so the K’ system and its clock are “stationary” to the K’ observers, and the K system and its clock are “stationary” to the K observers. To each system and its observers and its own clock, it is always the “other system” that appears to be the one that “moves” and their own that is “stationary”.

His two Section 4 clocks are in two separate systems as he has already described in the first 3 1/2 sections of the paper. So when he says, ”If at the points A and B of K there are stationary clocks,” the two systems and their two clocks are both stationary relative to each other and they coincide, as in Section 3. You just didn’t notice that “the clock at A” is the K’ system clock. You thought it was a K system clock.

When he says, ”which, viewed in the stationary system, are synchronous;” both systems are stationary relative to each other at this point in time, and both clocks are synchronous in both systems since there is no relative motion going on.

And as I’ve said, if you will get yourself a copy of his 1918 paper you’ll see his diagram of the U1 clock fixed in the K system and the U2 clock fixed in the K’ system, with the two systems and two clocks moving relatively.

See where he says in Section 4, “Further, we imagine one of the clocks which are qualified to mark the time t when at rest relatively to the stationary system, and the time τ when at rest relatively to the moving system, to be located at the origin of the co-ordinates of k, and so adjusted that it marks the time τ.”? The way he wrote the first pc thought experiment in Section 4, there are only two clocks, one in each system, and both are synchronous with each other before the relative motion between the two systems begins. That's why people notice the paradox. In his 1918 diagrams, he does show the U2 clock situated at the origin of the co-ordinates of K’ (which is k in the 1905 paper).

I think you just don’t understand the rules of the theory that he had already written into the theory before Section 4, such as, “The introduction of a ‘luminiferous ether’ will prove to be superfluous inasmuch as the view here to be developed will not require an ‘absolutely stationary space’ provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place,” and, “It is clear that the same results hold good of bodies at rest in the ‘stationary’ system, viewed from a system in uniform motion.”
The stationary system was only an arbitrary inertial frame. He assumed it existed, and defined it as a "system where both the laws of mechanics and the rules of electromagnetism apply." He then shows with equations that if one stationary frame exists, there are an infinite number of systems that also exist where "both the laws of physics and the laws of electromagnetism apply." Thereafter, any one of these inertial frames are equivalent. One could have started with any one inertial and called it the "stationary frame" and then proved the existence of the others, including the one he started with.
This is a typical "uniqueness problem" as usually treated by mathematicians. There is no single "stationary frame" that one is absolutely required to start with. There are an infinite number of inertial frames, and any one will do. However, there are also an infinite number of noninertial frames and if you start your measurements there, you will not be able to use the "laws of mechanics" or the "laws of electromagnetism." The physicists and engineers that use relativity all understand K and K' are different inertial frames, that is how they set up their equations.

I recommend an article by a physicist who has read the 1905 paper and makes the same point that I have been making. The article is:
Hans C. Ohanian, "The role of dynamics in the synchronization problem," American Journal of Physics 72 (2), 141-148 (February 2004). This article discusses the 1905 paper, analyzes the words and equations of the 1905 paper, is written in a refereed publication and is by a physicist. Refereed means it was reviewed by physicists who knew the same technical vocabulary he does, that I do, and that his use of words is not in any way unconventional.

One important point he makes is that the "kinematic part" and the "dynamic part" of special relativity can not logically be separated, at least not completely. The definition of "stationary frame", from which the "kinematic part" is developed, has a piece of dynamics in it. One can not define a unique "synchronization method" in an inertial frame without referring to "the laws of mechanics and the laws of electromagnetism." Otherwise, synchronization has a uniqueness problem.

He does criticize Einstein for not making that point explicitly, and doesn't address the uniqueness problems directly. But he does point out that Einstein had an uncanny talent to find the one and only solution to his word problem without going through all the detailed steps. Basically, he guessed a solution to complex equations, substituted the solution in, and showed that it solved the equation exactly.

H.A. Lorentz solved equations the muscular way, and occassionally made a small error or used an unncessary approximations. Lorentz used 132 pages to derive most, if not all, the results in Einstein's 34 page article. Even if you think that Lorentz basically "got it" you have to admit that Einstein made it much more user friendly.


I have a suspicion that he may have read one of your threads, or even been referring to me in the acknowledgements where he thanks an "unnamed reviewer." Maybe not. There are other articles this year in the "American Journal of Physics" addressing the details of the 1905 paper. However, Ohanian's paper would be a good start.






These two statements make the motion “relative” and not absolute, so the K’ system and its clock are “stationary” to the K’ observers, and the K system and its clock are “stationary” to the K observers. To each system and its observers and its own clock, it is always the “other system” that appears to be the one that “moves” and their own that is “stationary”.
/quote
His two Section 4 clocks are in two separate systems as he has already described in the first 3 1/2 sections of the paper. So when he says, ”If at the points A and B of K there are stationary clocks,” the two systems and their two clocks are both stationary relative to each other and they coincide, as in Section 3. You just didn’t notice that “the clock at A” is the K’ system clock. You thought it was a K system clock.

When he says, ”which, viewed in the stationary system, are synchronous;” both systems are stationary relative to each other at this point in time, and both clocks are synchronous in both systems since there is no relative motion going on.

And as I’ve said, if you will get yourself a copy of his 1918 paper you’ll see his diagram of the U1 clock fixed in the K system and the U2 clock fixed in the K’ system, with the two systems and two clocks moving relatively.

See where he says in Section 4, “Further, we imagine one of the clocks which are qualified to mark the time t when at rest relatively to the stationary system, and the time τ when at rest relatively to the moving system, to be located at the origin of the co-ordinates of k, and so adjusted that it marks the time τ.”? The way he wrote the first pc thought experiment in Section 4, there are only two clocks, one in each system, and both are synchronous with each other before the relative motion between the two systems begins. That's why people notice the paradox. In his 1918 diagrams, he does show the U2 clock situated at the origin of the co-ordinates of K’ (which is k in the 1905 paper).[/quote]

Sorry, I mucked up the last post. I didn't trim the comments at the end, and I forgot to include the point about the vx/c^2 term. Important, the two clocks for K and K' are at x=0. Therefore, they are synchronized regardless of what impart means. Furthermore, the synchronization contains some dynamics in it. Lets start again.

But the two clocks that are coincident on top of each other can't be knocked out of synch by a force "impelling" one, because they are local. You have to look at the full Lorentz equations (ahem), which have a vx/c^2 term. I notice that you never refer to equations at all, do you have a problem with equations? Apply the complete Lorentz equations to the problem. When x=0, vx/c^2 is zero. The two clocks are still synnched.

Again, I refer to Ohanians paper. The idea of synchronization requires force, which is referred to in a simplified but logical manner in the 1905 paper. Ohanian is reading the 1905 paper, and refers to these "paradoxes" as coming long after. However, Einstein could have made the discussion easier without the K and K' reference frames, that is discussed in other AJP articles. Einstein said later that he could have made it still easier. But the 1905 paper was complete itself. Not when artificially chopped in half, however.

Once more, the Lorentz equations are:
x'=g(x-vt)
y'=y
z'=z
t'=g(1-vx/c^2)

g=1/sqrt(1-(v/c)^2)


Not to be confused with the time dilation
dt'=g dt

Warning: the (vx/c)^2 term is important to the synchronization. In a logical argument about relativity, it can't be ignored. In a sense, it eliminates the paradox. I recommend that you discuss it.

Sam5, you're still not playing by the SR rules.
This might be true, but under SR, if you impart a velocity to a system, it becomes a physically different system. So, any measurements done in K' while it is stationary wrt K are useless once you've imparted the velocity to it, and so is the synchronisation! Do us all a favour and check if Einstein uses K' for anything before the velocity change. I'm betting you won't, because he knew as well as we do that unless systems are allowed to accelerate, you cannot compare any values in them from before and after the velocity change, except at the origin. Why do you think he's so specific about stating that the clocks are synchronised in K and not in K'? Rhetorical question - it's because it's absolutely critical to understanding this situation!


Ask yourself which one of the clocks remains in the same system in which the clocks were synchronized.

To use your method, we look at the situation with the clock at A stationary in K' and the clock at B stationary in K, where K and K' are already in motion. What will the situation be once they meet?

If the clocks were synchronised in the K system, then they aren't in K', and A will lag behind B.
If the clocks were synchronised in the K' system, then they aren't in K, and B will lag behind A.

This works regardless of which system you use for your calculations. The system in which the clocks were synchronised attributes the lag to time dilation. The other system attributes it to an initial time difference when the experiment started - time dilation wasn't "enough" to compensate for it, and the amount of lag comes out exactly the same as the other system would have it.


PS: Lorentz, would you pretty please start using the preview button to check your posts beforehand? I'm having a really hard time reading them.

__________________
"We do not require reality to conform to the expectations of the ignorant"

Sam5
Notice, in your description, the two cars are parked side by side, and then they move relatively. That's impossible unless one of them accelerates.
[Sam5]

[Lorentz]
Sure. A velocity was bestowed on frame K. It accelerated a short time during which it accelerated. For that brief time it was not an inertial frame. But after a velocity was bestowed on it, the acceleration stopped. If it hadn't stopped, one couldn't say that "a" velocity was bestowed on it. After the velocity has been bestowed, the acceleration went to zero.

From my Dorset and Barber Unabridged Dictionary (second edition)
1) Impart: to bestow upon another a portion or share of anything.; to give.
[Lorentz]



That means, in your description, that at least one of K and K' is not inertial. That is why you cannot apply the principle of special relativity to that system.
[Sam5]

No, because K' has "a" velocity bestowed upon it. The acceleration has become zero. Both K and K' are now inertial reference frames. A clock can be accelerated to a high velocity, using either a small or large acceleration. During the acceleration, it goes through a number of velocities relative to K. But once it reaches a constant velocity, and the acceleration is zero, it stays at that velocity. Frame K' is moving at a constant velocity relative to K.
Why can't you see that? K' is an inertial frame when the measurements
described are starting. The statement that K' has a constant velocity relative to K is the same as saying K' does NOT accelerate relative to K. The brief period of K' acceleration is ancient history. Acceleration doesn't have to be constant. The clocks stay synchronized at the point x=0 because, independent of v, when x=0 then vx/c^2=0.
0=(v)0/c^2.

It is your mistake when you do so, not Einstein's, or mine, or of what you call phony proofs. It is your error that causes your paradox.[/quote]
[Sam5]
It is your reading comprehension that causes your error, not his.

I tried to turn on the HTML. I changed it to "on" in my profile. However, my HTML is still off.

Make sure your BBcode is on also.

__________________
Hwæt! We Gardena in geardagum,
þeodcyninga, þrym gefrunon,
hu ða æþelingas ellen fremedon.

There is no clock paradox in General Relativity.
[/quote Sam5]


Then the results of Hefele and Keating do not have any paradox. Their predicted results, published in a separate article, are based on general
relativity. What they call special relativity is merely a special case of general relativity, an approximation where the masses of objects are very small. Since they use relativity, and you are comfortable that there are no paradoxes in general relativity, then you have to believe that the results of HK are without paradox. Let me demonstrate by explaining it in general relativity terms.

1) According to general relativity, a gravitational field can cause time for one observer slow down with respect to the other oberserver. Specifically, a clock in one gravitational potential ticks faster than a clock in a lower gravitational potential. If two clocks are near each other but in different gravitational fields, then the one in the stronger gravitational field slows down with regards to the other clock.

2) The equivalence principle: The effects of a gravitation and inertial are indistinguishable. That is, if one sees an effect due to gravitation in an inertial frame, it is locally equivalent to the effect due to inertia in an accelerating frame. There is no way a person using local measurements in an elevator if he is one earth being pulled by the earth's mass or in space in an box that is being forced to accelerate. Every localized gravitational effect has an equivalent inertial effect, regardless if the gravity is caused by a mass or by acceleration.

3) The formula for centripetal acceleration, derived by Newton (or earlier)
is
a=v^2/r


4) The gravitational field from the earth's mass adds to the centripetal acceleration of the earth's rotation to make the effective gravitational field of each clock.

5) The Westbound, fixed ground, and Eastbound clocks are under the same mass generated gravitational field because they are all at the same latitude, throughout the experiment. The only difference in effective gravitational fields comes from the centripetal acceleration.

6) The centripetal acceleration of the Westbound clock is less than the centripetal acceleration of the fixed on earth clock is still less than the centripetal acceleration of the East bound clock. This is because, relative to a hypothetical clock at the center of the earth, the Westbound clock has a smaller velocity than the fixed clock than the Eastbound clock.

7) The Westbound clock has a smaller gravitational potential than the fixed clock which has a smaller gravitational potential than the Eastbound clock.

8) The Westbound clock ticks faster than the fixed ground clock which ticks faster than the Eastbound clock.

9) That is what HK predicted, and the results they found. There is no logical, mathematical, or scientific reason to think there is a contradiction in their results.

10) If they did their experiment correctly, then their results support general relativity under certain conditions.


By general relativity, which has no contradictions according to Sam5, the HK analysis of their data is correct. Or at least, logically consistent.

[Lorentz]

I stand by my argument. A ground-based observer sees clocks flying west run slow compared to a stationary clock on the ground and sees eastbound clocks run slower still. An observer on the westbound plane sees the clock on the ground run slow, but not as slow as an observer on the eastbound plane would.
[Jim]

Sorry, I have to correct you there. The Westbound clock sees the clock that is fixed to the ground tick slower than it. The Eastbound clock ticks still slower. The Westbound clock is the fastest ticker of the three clocks.

This is exactly what Hafele and Keating predicted would happen. H&K predicted that for clocks at the same latitude, the Westbound clock ticks the fastest relative to the ground fixed and the Eastbound plane because it.

The thing is that in the HK experiment, there is no inertial reference frame observer. All three clocks are accelerating outward away from the axis of the earth. However, the Westbound clock accelerates the least. So his clock ticks the fastest. Just like in the twin experiment, the earthbound twin never accelerates at all, while the rocket travel twin has a large acceleration if only for a short period of time according to his watch. The least accelerating observer is has the fastest ticking clock.

Although this formula was derived explicitly after the 1905 paper, I'll write it out anyway. One can derive it from the 1905 formulas using beginning calculus. The formula relating time in an inertial frame, K, to an accelerating frame, K', is:
(dt')=(1+ax/c^2)(dt)/sqrt[1-(v/c)^2]
where "a" is the acceleration of K' and "v" is the velocity of the K' frame. dt' is the time between ticks in the K' frame, x is the distance in the K frame from the origin, and dt is the time between ticks in the K frame. Note that as a increases, the time
This can be derived, easily, from the 1905 paper. However, when using the formula, be very careful about signs. Remember that "a" and "v" can have different values and different sign, and they always have different units.

A geocentric satellite is also not an inertial frame according to SR. It accelerates toward the center of the earth. Angular speed may be constant, the magnitude of velocity is constant, but the direction of velocityis changing all the time. Thus, the acceleration is significant because of the change in direction of velocity.

Not my quote. I was quoting Taibak.

__________________
Never attribute to malice what can be adequately explained by ignorance or stupidity.
Isaac Asimov

I started to read Charles Ives once. The physicist, not the composer. There was a set of volumes in a Duram library dated 1932 that "showed" how Einstein was wrong, and I read some articles written later by Ives. I stopped. He wasn't a real relativity critique, he was just an Einstein basher not a serious scholar. He made some mistakes early on that I caught right away. I decided he was worth any more time after I caught a few big ones. Here are just a few.

1) He starts the first volume, somewhere on the first page: I think that the theory of relativity involves only kinematics, and is not supposed to have forces." He then starts from that premise (which he even from the beginning isn't sure) and starts proving relativity wrong.
Relativity does include forces, and doesn't really stand on kinematics alone.

2) He made an incorrect analysis of a circular Sagnac property. He says that since it is cylindrically symmetric, there should be no beats coming from a circular Sagnac cavity.
Alas, there are no smooth round Sagnac cavities because everything is made of atoms, and even the electrons are descrete. The closest thing to his hypothetic system is a fiber optic Sagnac interferometer, which most assuredly does show beats.

3) He reintroduces forces (ignoring the dynamics Einstein), gets results identical to that of Einstein , and then starts claiming a breakthrough. Because now relativity can not have forces. He became sure of himself on this point after writing about it 200 pages. He didn't start that way.

4) He ignores diffraction optics and surface currents in the circular Sagnac interferometer.

5) He has an obvious chip on his shoulder. He gets very emotional about Einstein's evil plan to hijack science.

Who does that remind me of? Think....think... :roll:

__________________
"If lightspeed has something to do with speed.
how come things can move fast in the dark?"
-James Driscoll (Spaceman), kook, imbecile, idiot.

You might be thinking of Sam5, but from what I've read of his posts he seems to view Einstein as a cult figure, with others to blame for "hijacking science". Now if you really want to see someone who frothed at the mouth and refused to name Einstein except to refer to him as "Albert" or "the inventor of spacetime", check out some of oriel36's posts. Sam5 is a beacon of light and reason compared to oriel36!

__________________
Microsoft is over if you want it.

The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

[Who does that remind me of? Think....think... :roll:[/quote]
You might be thinking of -----, but from what I've read of his posts "hijacking science". [/quote]

Actually, Charles Ives didn't exactly express it that way, and I wasn't going to bring anyone else into this except Yannox, who made that long list of people who were dissatisfied with relativity, or at least Einstein. I just wanted to point out that these people he quotes may not stand up to close examination.
Has anyone else read anything by anybody on this list?

I've looked at Aspden with the intent of critiquing his lecture "Why Einstein is Wrong". I would like to assemble a collection of critiques of some of the work by the people Yannox referred to. Of course if you want to go ahead and do this that's wonderful.

__________________
Microsoft is over if you want it.

The bar has been lowered for the promotion of ATM ideas; the bar for the acceptance of ATM ideas must remain high.

This caused the paradox in the theory. He first said early in his paper, “The introduction of a ‘luminiferous ether’ will prove to be superfluous inasmuch as the view here to be developed will not require an ‘absolutely stationary space’ provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.” But then in Section 3 he introduced “stationary space” in which the K frame was fixed. Thus, he contradicted himself and used the concept of “stationary space,” whereas he had already said, “the view to be developed will not require an ‘absolutely stationary space’.”


I reread that 1905, and some later work. Based on what I read later, especially about the analysis of light polarization, in context he meant a Galilean invariant luminiferous ether would prove superfluous. When he later said that the ether was necessary to get relativity to work, he meant a Lorentz invariant ether.

Thing is, Maxwell equations in vacuum are constrainted by both Galilean and Lorentz invariance. So picturing an aether that has properties analogous to air, with purely classical (i.e., Galilean) properties could be easily done if there were only electromagnetic fields and no other forces in the universe. However, there are other forces in the universe.
A nucleus, or even an electron, could not hold itself together using just electrical forces. Even Heinric A. Lorentz included a nonelectrical force holding the electron together, which was Lorentz invariant. Even simple properties like "transverse polarization only" can't be explained by a straigh tforward ether model.

Konrad Lorenz (these Lorentz/Lorenz guys keep on coming) pointed out that the absense of longitudinal polarization creates real doubts as to the existence of an ether. Again, the context seems to be a Galilean invariant ether. Put enough weird properties to the luminiferous ether, you could get any weird properties you want. But a ether that acts like a normal fluid, or a normal elastic solid, does not. You figure out what type of ether can suppress longitudinal, but not transverse, waves.

Oh just let this thread die!

*Bump*

Agreed :P

__________________
Obedience brings victory
Victory is life
That is the order of things

I think you just don’t understand the rules of the theory that he had already written into the theory before Section 4, such as, “The introduction of a ‘luminiferous ether’ will prove to be superfluous inasmuch as the view here to be developed will not require an ‘absolutely stationary space’ provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place,” and, “It is clear that the same results hold good of bodies at rest in the ‘stationary’ system, viewed from a system in uniform motion.” [/Sam5]

Second Edition of Dorset and Baber's Websters New Universal Unabridged Dictionary, Delux second edition (1979). Definitions of:
Superfluous:
1) More than what is wanted, needed, or useful, surplus, excessive; as, in a composition abounding with superfluous words.
2) Not needed, unneccessary, as in a superfluous remark.


The two statements that you quoted made by Einstein do not logically contradict each other because of the "stationary state" he finally defines does not have "special properties." That is, the stationary state is arbitrarily chosen from an entire ensemble of stationary states with exactly the same special properties. The ether would not be superfluous if there was no system, in uniform motion or not, that had the same properties as the stationary frame.

He later says, "Any reference frame, moving a constant velocity with respect to the stationary frame, has identical properties to the stationary frame initially chosen." He is saying that that uniform frame, moving with a uniform velocity with respect to the stationary frame, has identical properties to the stationary frame. The "reference frame moving at constant velocity" is different than the stationary frame, but has identical properties.

An absolute unique ether is not consistent with two frames having the same properties, even if the two frames are moving at an arbitrary velocity-vector" (please highlight the "vector" part because direction is important).

The ether is "superfluous inasmuch as the view here to be developed does not require an absolute stationary state provide with special properties." However, "special properties" implies that there are only properties that are unique to that one frame.

Einstein clears up the issue of ether mostly in his electrodynamics, not entirely in the kinematics part of the book. Note his example, "nor assign a velocity-vector to a point of empty space where electromagnetic properties take place." Electromagnetic properties mean Maxwell's equations." There is more than one frame where Maxwell's equations hold. That being the case, ether in its "simplest" form is superfluous.

As an example, it can be shown that if Maxwell's equations are valid (the ones written down by Maxwell, ALL of them), electromagnetic radiation in a vacuum moves at speed "c" in all directions. If it doesn't, you need new equations that are other than what Maxwell wrote. The stationary frame represents one possible observer to whom Maxwell's equations and the physical concepts behind them are true.

However, an observer can be moving at a speed "v' in a fixed and unchanging direction with respect to a stationary frame. This observer would be in a different frame. If the ether wasn't "superfluous," then that observer would see the speed of light in one direction different from the speed of light in the other. That would be a proof of a "nonsuperfluous" ether. However, then that observer would have to come to the conclusion that Maxwell's equations are wrong, at least for him. Because if Maxwell equations are right, then light goes at the same speed in all directions. So that for him, who is travelling at a nonzero constant velcoity with respect to the "stationary frame," Maxwell's equations are wrong. However, this has not been satisfactorilly shown by experiment. So Maxwell's equations are either true for all frames, up to what can be shown by experiment, or the ether is "superfluous."

Note that Lorentz believes in an ether, but also refers to the ether as "superfluous," with respect to experimental analysis. The word "superfluous" doesn't mean "nonexistent," it means unnecessary. There is a big difference, which most people can understand, between "necessary" and "sufficient." One can define an ether which is "sufficient" to describe Maxwell's equations being the same in all inertial frames. But it is "unnecessary" because one can always use a different inertial frame. All inertial frames have :identical" properties, there are no "special" properties.


Thus, in the sense that Einstein used it (as does almost everybody else), superfluous means causing a surplus. Superfluous does not mean nonexistent. Whoever told you than Einstein claimed the ether does not exist was wrong. He said that it was "superfluous."[/img]

Lorentz

Sorry, I can't let the subject die. It bothers me not to make a statement with regards to certain types of logical error. I appreciate the BA's patience with me. My remarks may be superfluous to other people, but they are not false.

You quoted and addressed the same post last month though, and Sam5 hasn't posted since. Maybe I better look out for him.