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Frank Wappler said some stuff about [...]    It would certainly help if we could at least agree on terminology.  I use the notion of accuracy as given in Bevington/Robinson   Data Reduction and Error Analysis for the Physical Sciences , p. 2:     The accuracy of an experiment [or estimate] is a measure of how close   the result of the experiment [or estimate] is to the true value.    and I understand the true value of a physical quantity in any  particular trial to be the value that would be obtained, at least  in principle, by conducting in this trial the reproducible  measurement procedure through which this physical quantity is defined.           Bevington is not a book of philosophy. The concept of true value doesn't imply that one exists which is free from the context of how a theory defines it. The only philosophical aspect I can recall seeing in bevington, was in regard to misguided attempts to facillitate finding true values by discarding outliers and some admonishments against data smoothing as ill-conceived attempts to find more or less truth in a theoretical prediction by removing or smoothing over the the points one knows to require more care .         If you are describing a sphere, then you aren't going to find some means to discover whther or not your choice of origin for theta,phi is truer than just defining it. More likely, attempts to force the sphere to yield to insisting its really a prolate ellipsoid, will produce ways to show exactly that.         The only real difference between the truth of special relativity and say, the truth of galilean relativity is the willingness and ability of each theory's proponens to meet the requirements of self- consistency. With special relativity, little effort is required to be consistent with observation. With galilean relativity, I doubt there exists a person that both knows how to fix all of the problems through new features and unintuitive paramaterizations that would also be stupid enough to do so.         Science typically implies that true mean something along the lines of irreducible representation . Nature doesn't have to agree with science other than being self-consistent. With respect to special relativity, why should parameters which are not independent, lead one to expect otherwise? If you actually consider what you have to measure c, I don't think you'll be able to point to a velocity ruler.         It's also not necessary. All that is required to distinguish between theories A & B is that both agree on measuring the relevant values the same way (even if a hypothetical theory, C, would consider this pathetic). If A and B agree on what outcome should distinguish between them for any experiment using measurents both agree upon, then that's all there is. Until one has a theory C that allows for comparing measurements to select between it and both A and B such that the result is self-consistent, that's all you get. There is no tautology involved in choosing to define a quantity which is not fundamental to the theory in the way the theory uses it. In relativity, not only is the value of c not fundamental, since it acts as a scale, it's not really crucial that light propagate at this velocity unless it leads to some other contradiction that the prospect of massive photon causes.         There exist any number of ways to determine whether special relativity or something else gives a correct prediction without finding the one true way to measure and calibrate the world. Fortunately. For example, on may distinguish, without ambiguity, whether galilean or special relativity predicts the correct outcome of an experiment without an elaborate, if not dubios scheme to discover the hypotheical truth of anything. Galilean relativity allows me to use any clock. That means it's totally unnecessary to so fret over the details of synchronizing _frame_s. If any snchronizing is required, special relativity beats galilean relativity. This is easy to determine. I can weigh out 2 - 1 kg samples of a radioactive isotope and fly one around the world. I can determine the activity both before and after using the same equipment. As long as a statistically sufficient number of decays have occured to give meaning to the difference, then I can say that one of the two is wrong. Determining how wrong is meaningless and doesn't make the other more correct. One has successfully eliminated either A or B using measurements both agree upon to predict mutually exclusive outcomes. If you want to do better, you need another hypothesis to compare. The only tautology occurs when trying use a theory to prove itself because you find something to dislike about self-consistency.

I am sure you can find many references on the web that explain how the age of the universe is determined. I have found, and read, several such pages myself. Did you find among them at least one that defined age of the universe in terms of a reproducible measurement procedure, relative to which the accuracy of various other estimation methods (for instance those offered at Ned Wright's site) might be determined? I could _define_ the age of the universe in seconds as the number of periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the caesium-133 atom since the universe began divided by 9,192,631,770. That would be the 'true value' as discussed above. All practical measurement procedures would then have some accuracy in relation to that definition.

[...] For example, on may distinguish, without ambiguity, whether galilean or special relativity predicts the correct outcome of an experiment Given the experimental specifications about the counts and even about the trajectories of either sample (as far as they are definite), which correct and unique outcome (in terms of the actual, true, correct and unique counts) do you suggest is predicted by SR or Galileian relativity? Could the predicted and/or actual outcome depend on _anything that's not_ explicitly part of the experimental pre_script_ion, such as about the distribution of (gravitational) potential difference, neutron flux density, or mean temperature in the experimental region? Can the principle of stationary action be used to derive predictions about the most probable distribution of potential differences from outcomes and coordinate relations measured in past trials? Regards,     Frank  W ~@) R p.s. In responding to your post I've written direct replies only to some of your statements. Should you find that I had failed to address particular important points of your post, please remind me on them.

Frank Wappler said some stuff about    That's ignoring their decisive difference:         No. I claim that I can insure there is no difference. That was the point. I can always add a new free parameter to a theory to make it agree with all of the data.  SR is _base_d on reproducible definitions, namely on Einstein's  calibration procedure and the associated distance definition;  the Lorentz coordinate transformations _follow_ from those  measurement procedures, in terms of the physical quantities   t and x which are _defined_ through those measurement procedures  and last but not least in terms of v which can be defined as  d/dt( x( t ) ) in terms of the already defined quantities.    OTOH, I don't know of any reproducible measurement procedure  by which to define the symbols that appear in Galileian  transformation equations (except approximately as quantities  that are defined through the SR procedures, in the limit of  v << c, where v is defined/measured through the SR procedures  in the first place).         I simply add a correction to galilean relativity. for example:         E = p^2/2m + f(p,m,c)         After adjustments, I get the same result relativity gets. The only criteria that prevent me from doing that ad nauseum is a simplicity and self-consistency requirement. Simplicity is philosophical. Self-consistency basically causes anyone not convinced by simplicity to simply give up once maintaining the facade becomes too difficult. That doesn't mean one can't do it.     All that is required to distinguish between theories A & B is   that both agree on measuring the relevant values the same way    Precisely. This implies:  The measurement procedures must be defined a priori,  and theories are distinguished by predicting distinct results.  Depending on the results, the theories can be divided into those  that are already falsified, and those that are still viable;  but the preselected measurement procedures simply remain in any case.    Notice that theories and (systems of) measurement procedures are  quite distinct concepts.         No. They are not. Suppose you define c = constant and get the usual L'= L [1-b^2]^{1/2}, b = v/c. My theory defines L to be a constant but c to vary. Then I have: c = v/[1-(L'/L)^2]^{1/2}. I meaasure L in a moving _frame_ fixing defining the frequency of the light to be related to wavelength and velocity such that the wavelength is defined to be the same in both _frame_s. We now have measurement procedures that depend upon the theory, since length doesn't mean the the same thing.   I can weigh out 2 - 1 kg samples of a radioactive isotope    Pending the de_script_ions of procedures you mean by weighing out  and for identifying radioactive isotopes         I actually don't have to weigh. It's redundant. I can use the same measuring devices used for the rest of the experiment to make the same measurements prior to the experiment and determine that both have the same decay rates. Since Both are in the same rest_frame_, and both theories agree on the physics there, decay rate means the same thing. At least for identical isotopes. Both theories agree on what isotopes are.  (which would have to be the same reproducible procedures _both_  in SR as well as Galileian relativity),         The experiments are identical. The only difference is that galilean relativity states that any clock is correct while relativity states that the correct clock is the one in the rest_frame_ of the clock. All that matters is that the clocks start out the same and end differently. How differently doesn't matter so long as both theories have no overlap withing the experimental uncertainty. That rules out one of the two as a viable candidate.    let me assume that the experimenters involved can _count_  (actually, truely, correctly), at least in principle,    and let me consider samples of constituents  for which it is expected (_base_d on previous trials) that  any subsequent count finds fewer than (or at most as many as)  any previous count (that's assuming each particular experimenter  can _order_ the own counts into those obtained earlier and  those obtained later).    Having made those assumptions, one may even require that (at least)  two sorts of constituents, counts X and counts Y, be distinguished  in each sample, such that the relation    ln( X_later / X_init ) =approx=             ln( Y_later / Y_init ) *                     ln( X_then / X_init ) / ln( Y_then / Y_init )    is expected to hold usually (within deviations of order  +/- sqrt( X ) and +/- sqrt( Y )) for any three sets of  counts init , then , and later ;  let me then consider one such sample separated into two equals:    X_init_sample1 = X_init_sample2, Y_init_sample1 = Y_init_sample2.     and fly one around the world.    Using the SR measurement procedure, the corresponding coordinate  relations can certainly be defined and measured very specificly;  for instance that both samples moved on the same circle a plane  perpendicular to the worlds rotational axis, with angular  velocities that are different in magnitude and/or in direction .    OTOH, pending a reproducible procedure of Galileian relativity for  defining and determining whether someone flew around the world ,  let me assume that Galileian relativity allows at least to define  and to distinguish whether two (e.g. the two samples) are separated  or whether they are together, meeting each other.         How many measurements are you making? You make a SINGLE measurement. You place _ONE_ sample to leave and return. Both galilean and special relativity have a clear concept of going away and returning. They disagree only on what the clocks read. I don't fly one sample and call that a relativistic measurement and the repeat the experiment and call it a galilean measurement. That's stupid. I fly one sample and call it a measuremnt that both agree upon in all details other than than what the clock reads when they open the airplane door.   I can determine the activity both before and after   using the same equipment.    Pending the de_script_ion of how to define and determine activity ,  which would have to be the same reproducible procedure _both_  in SR as well as Galileian relativity,         Do they differ?  If so, then then the experiment can't work. This is a direct contradiction to your assertion that measurements are not theory dependent and supports my claim that measurements are theory dependent, and comparisons can only be made with measurements both theories agree on how to perform. There is no _object_ive absolute.  one may continue to assume that the constituents in either sample  (individually) can be counted. Further, in order to calibrate  which, if any, pairs of counts sould be _compared_ to each other,  besides the initial counts,  let's assume that the counts when the samples meet again  are relevant and determined equally in both SR as well as  Galileian relativity.         There is no calibration necessary. Have you never made a comparison where the calibration drops out as a common factor? These are usually the types of experiments one bends over backwards to perform, not avoid.    One would find    X_meet_sample1 =< X_init_sample1, Y_meet_sample1 =< Y_init_sample1,  X_meet_sample2 =< X_init_sample2, Y_meet_sample2 =< Y_init_sample2,         Why are there 2 samples. I only need the first to determine the outcome. The second is superfluous and has no advantage even statistically than just adding both samples together and performing a single trial.  and    ln( X_meet / X_init ) =approx=             ln( Y_meet / Y_init ) *                     ln( X_then / X_init ) / ln( Y_then / Y_init )    for each sample inividually, and for any particular count obtained   then , between the counts obtained at init and at meet ;    that's as specified above, else this particular experimental trial  would have to be discarded.    However, in general, the counts _might be_ significantly different  between the samples         I'm afraid I have no idea what you are trying to do. This is a very simple measurement, which has no ambiguity as it was given. [...]    since those differences were not specified by the experimental  procedure. Consequently one can call those differences the  nontrivial result of the experiment.     As long as a statistically sufficient number of decays have occured   to give meaning to the difference,    ... let's assume that;  e.g. by considering only those trials in which    X_init_sample1 - X_meet_sample1 sqrt( X_init_sample1 ), and    sqrt( X_meet_sample1 ) 1, etc. ...         You can't assume this by simply pulling the out of the air. However it's not difficult to figure out what requires. Relativity predicts: dN/N  = -gammalambda dt Galilean R predicts: dN/N' = -lambda dt (N-N'/)N_0 = exp(-lambda(gamma - 1)t) choose N-N' to get the statistics you want.     then I can say that one of the two is wrong.    Which of the two is wrong, and why?         If the sample remaining behind == sample travelling, relativity is         wrong.         If the sample remaining behind undergoes more decays, then galilean         relativity is wrong.         Reason? Because that was the premise that both theories agreed upon. If you have a _different_ theory, then the experiment doesn't apply to you. If you think galilean relativity states something different, then we have different theories, and again, the experiment doesn't apply to you. I've ruled out _A Theory_. Doing so is meaningful, regardless of whether or not it satisfied your plans for the final, unambiguous, all-encompassing experiment to determine _the_answer_ for all time. I propose that no such experiment is possible. I can always create a new theory that is in every way identical to your correct one, using variables to mean anything at all provided I introduce whatever parameters are required to make it work. Nothing but a philosophical
... wiêcej »

In fact it would be more rigorous to define D as the limit as the size of the individual d measurements tends towards zero and the number of steps increases accordingly. That's just as well; but keep in mind that in order to take such a limit of an _expression_, the corresponding _expression_ must be defined in the first place, i.e. involving general nonzero values of x . IIUC, you suggest to determine D_{ Us_now }( Us, G ) as limit_{ n