Not aesthetic grounds-- the grounds would be the definition of what a theory is.Hurkyl said:
There have to be some defining assumptions that theories make, such as objectivity and repeatability. These can never be proven, only falsified. That is the important kind, the "bridge-building" kind, of predictions that theories must make. These are of the "weather prediction" kind, and are the useful predictions, the bridge-building predictions, that science makes. To make predictions of that nature, there is no need to pretend theories are things that they are not.
That is circular reasoning, you simply define straight that way. All we can say is their accelerometers read zero, if we want to think of that as special that's up to us-- there's no need to go and build physics around it.
At last we see the appearance of the word "orthonormal", which I've been hammering for awhile now.
I remain unconvinced of that, and this is an important purpose of the thread. The key thing I have maintained is not that SR makes false predictions for quantitative measurements within the regime where it has been tested, nor that it is unable to predict the dynamics of any particle with a known proper acceleration that satisfies certain other assumptions (as are necessary in either classical physics or Dirac's formulation of quantum mechanics). Rather, its problems are pedagogical, in that it may make unnecessary guesses that could prove to be false in future experiments outside the realm where it has been tested. Such false "predictions" are not an important part of any theory, just as it was not an important part of Newton's laws that they work to arbitrary speeds (and the fact that they don't has in no way compromised their use in situations where they are warranted).
Yes, but all that goes right into the definition of a theory, as I alluded to above. We do not need to add special postulates to handle that, it is in all scientific theories from the start. This is my point, the importance of understanding what aspects of our theory are there because that's how we define scientific theories, what aspects are there because they unify existing observations, what parts are extensions that we are curious about testing and have no idea if they will work or not (like Newton and arbitrary speed), and what parts are just pure fantasy (like MWI) that we have no reason whatsoever to ever pass a falsifiable test.Tomorrow is a new regime too.
But I still don't know which of the two versions of "confidence" you mean. I would say the confidence afforded to us by the scientific method is of the first kind I listed, but you seem to be talking about the second situation.
It wasn't, any more than it was "scientifically correct" to think Newton's laws would extend to arbitrary speed, or that Ptolemy's model would hold up to more precise observations. The only things that are scientifically correct are to expect predictions "within the box" of the current dataset to work, that's like predicting the weather or building a bridge. Other types of predictions are called "guesses", and are not scientifically correct to expect to work (a point history has been rather clear on, especially once you bear in mind that "the winners write the history").
No, it had no empirical support (even in the absence of gravity), as it was only formulated and tested for inertial observers. Indeed, it breaks down when you leave that observational regime, as is not untypical of phyical theories.
If by that you mean that "global Minkowski is known to be wrong", I agree.
I thought you were pointing out that -+++ is the same as +---. What is written is merely a re-affirmation of what I've been saying all along-- that the Minkowski metric is invariant only under the transformations of the Poincare group (and is not invariant under arbitrary coordinate transformations or changes of observer, though its signature is).
If you want to use calculus to integrate the metric between events from the perspective of a constantly accelerated observer, you need to integrate the Rindler metric, not the Minkowski metric. The latter gives you the wrong answer, that's the point.
That depends on what you mean by "Minkowski space", this is very much the point here. What many (most) mean by that phrase is, "a metric space ruled by the Minkowksi metric", but if you take that meaning, your statement is wrong. It is only right if you take the more general meaning of a "space governed by Minkowski geometry, constrained by metrics of Minkowski signature and the tensorial transformation rules between them." More to the point, you seem to be imagining that events themselves are members of a vector space, but they are not-- that stucture must be imposed on them by choosing basis vectors, i.e., by the coordinates. Hence, changing observer/coordinates is indeed a mapping of the vector space into itself, a mapping that if it leaves the events unchanged it changes the vectors that are associated with them, or if it leaves the vectors unchanged then it changes the events associated with them. I think one is the covariant picture, the other the contravariant. I believe the "dual space" is what you get if you make the opposite choice.Hurkyl said:
As I said, if the events are taken to be invariant, then the vectors have changed. As in your example.
I thought this is what you are imagining, but I think you are incorrect. The events are one thing, the points in Minkowksi space are another, and the connection is made via the coordinatization. You are imagining that they both stay the same when we change oberver/coordinates, but they do not-- one or the other must change.
Translations of the origin are of no interest to me, we are talking about changing reference frame. In other words, we talking about the vectors that connect events, not the vectors that connect events to an origin.
Rest assured, that is not what I'm trying to do. The fact that an observer is on a worldline is very much my concern with the standard formulation of special relativity-- it not only requires the worldline be inertial, it further require the worldline has always been inertial and always will be. That's terribly over-restrictive, and there's just no need for it.
Again, this is not the standard formulation, involving the Minkowski metric.
Yes, this is what I'm saying, except I'm saying that the red herring is the Minkowski metric. We don't need a symmetry group that limits our postulates, indeed general relativity figures out how to do it with no such limitation.
That's what I've been saying, as I recall right about the time I was "boring" and "exasperating" DaleSpam right out of the thread.
Mathematically, that definition is: (given an ambient formal logic and formal language)Ken G said:
There is always a symmetry group, whether you state it explicitly or not.
e.g. general relativity is symmetric under any isometry of differential manifolds, and the frame bundle is symmetric under global and local Lorentz transformations.
*shrug* I guess there's nothing left to say but "you're wrong". (And similarly for many of the points in your previous posts)
Minkowski space does not have an origin. It is not a vector space. (Just like Euclidean space)
If you don't have empirical evidence that Newton's laws shouldn't extend to arbitrary speeds, then you don't have any scientific grounds for expecting them to fail for arbitrary speeds. (I'm assuming you meant to make a sensical statement -- we knew even before special relativity that notions of absolute velocity had no physical meaning)
As this is a physics forum, I would have preferred the scientific definition. For that, I think Wiki (http://en.wikipedia.org/wiki/Theory) does fine:Hurkyl said:
My point is that the symmetry group is all there is-- there is nothing special about inertial observers simply because they exhibit that symmetry. Arbitrarily accelerated observers exhibit other symmetries, what we need is not a way to flag the inertial ones, but rather a prescription for matching the symmetry to the observer.
You said that when you defended the idea that the Minkowski metric was invariant to any coordinate change, but now you are recognizing orthonormal transformations, and when you claimed that changing an observer was "just a coordinate change, not a mapping from spacetime into itself", a claim you have also apparently backed off on.
Metrics apply to vector spaces, so once again, if "Minkowski space" takes on its usual meaning as "the spacetime vector space with the Minkowski metric", then it is a vector space.
I'm confused what confuses you about this perfectly natural statement about the normal formulation of special relativity (and I remind you of the John Baez quote I posted earlier in the thread).
You may indeed assume I was making a sensible statement there. Furthermore, if you have no reason to expect they do extend to arbitrary (relative-- obviously) speeds, then why do you think it is "scientifically correct" to expect they will? This is precisely what I am saying is not scientifically correct, as history has shown many times. It is scientifically correct to form no opinion in advance of the observation.
Be aware that many people (even technical people!) often make statements with an implicit assumption of nontriviality -- I would be extremely hesitant to take such an common-language heuristic explanation as being accurate on such a detail. (in fact, I would even expect experts to disagree on that detail)Ken G said:
You seem to read that statement much differently -- I would read it as, for example, a theory of fluid motion is not expected to make predictions about photonics.
The twin (pseudo)paradox is, by definition of 'paradox', an example of fallacious reasoning. The resolutions of the twin paradox are meant specifically to identify the flaw in the reasoning and explain why it is a flaw.
Yes, the metric on Minkowski space is invariant under coordinate change. Yes, what you described as 'changing an observer' appeared to be nothing more than a coordinate change. Yes, for an actual (affine) transformation of Minkowski space itself to respect the metric, it must be Poincaré.
Yes, vector spaces may have metrics1. So can affine spaces. And pseudo-Riemannian manifolds by definition have a metric. Minkowski space, like Euclidean space, is not a vector space.
Your assertion that all worldlines are inertial is patently false. And note that the Baez quote doesn't say anything about observers or worldlines.
I wouldn't.
In fact those reasons are still applicable today.Indeed, and I am suspecting that the right way to do special relatilvity is similar. By "right", I mean the way that carries no unnecessary concepts that are included purely for convenience and familiarity, but which ultimately replace the actual point of what has been discovered about reality with pictures (like the isotropic speed of light or the Einstein simultaneity convention) that are useful in practice but of deceptive physical content. They are fine for doing calculations, but may not be the best way to unify speical relativity with other advances in physics.Hurkyl said:
That's exactly why I would not count it a theory, nor the goal of science.Hurkyl said:
Correct, I read it differently. I read it that, for example, a theory of particle dynamics that explains ideal gases would not be expected to describe the motions of those same particles when confined to the scales inside atoms.
I agree that it is quite difficult to say categorically what is a difference "in principle", but nevertheless this is the charge that is put to science-- when you are building a bridge, for example, you face that charge all the time.
The only paradox there stems from the different sounding explanations. That problem would be avoided in the approach I'm advocating.
(shrug)-- that is simply wrong, what more can I say. We were making progress when we established that the invariant was only the signature of the metric, and that the manifold was Lorentzian as a result. Don't backslide now.
Equivocation. We have always been talking about the standard way special relativity is described and axiomatized, right from the start of the thread. As such, "the metric on a metric space" is just what we've been talking about. What we discovered of importance, in my view, is that it is not the metric at all that generates gravity-free dynamics, it is the Lorentzian geometry of the manifold, which is connected to the signature of the metric. If you want a true coordinate-free invariant, you must work with the covariant/contravariant dual spaces, as neither the covariant metric tensor, nor the contravariant metric tensor, is by itself invariant to the kinds of transformations of spacetime into itself that we need to do physics from the perspective of different observers.
A pretty good indicator that I never made, or even thought, any such assertion.
Nevertheless, what it did say is something you have simply ignored.
I haven't the vaguest idea what you are trying to say here, because taking the literal meaning is obviously "patently false".But we do have reasons to expect Newtonian mechanics to work for arbitrary relative speed: all of that pesky empirical evidence supporting Newtonian mechanics.
Students have to learn coordinates -- they are so computationally useful that it would be harmful to deny them that knowledge. And while students learn coordinates, many will make mistakes, and some will rediscover the twin pseudoparadox. Showing them an unrelated derivation of the same quantity does not help -- it does not fix their misunderstanding of coordinates.Ken G said:
Wrong. See:
So, why did you say:
I meant exactly what I said -- this very day, we have reasons to believe Newtonian mechanics works for arbitrary relative velocities, and that is perfectly consistent with the scientific conclusion that Newtonian mechanics doesn't work for arbitrary relative velocities. I'm trying to provoke thought about how empiricism works.
But I'm not talking about the students who have made a mistake, I'm talking about those who did everything correctly and can't figure out why the two observers cannot agree on the reason that one is younger. That's a problem. The standard answer is "the reason is itself coordinate dependent", and I used to accept that answer. Now, I see it as flawed-- it misses the whole point of relativity. If the point of relativity is that different observers can use the same physics, then they should also be able to do it in a way that finds the same answer to the same question. It really isn't that hard to do relativity that way, it would just use coordinates as a convenience rather than as something seen as an integral part of the physics.Hurkyl said:
No, my approach is not at all useless. Let's look at how my approach applies to the centrifugal force example. I say, "why do I weigh less at the equator", and someone else says "because centrifugal force counteracts some of gravity". Someone else says "no, that's wrong, there's no such thing as centrifugal force to counteract gravity, it is because I simply require less force from the scale to move in a circle at the equator". Two different answers to the "why" question, which is the only sense to which the twin problem is a "paradox" (once we've recognized failures of our daily intuition does not count as paradoxical). In first-year physics, the first answer is "wrong" and the second is "right", but later when you learn general relativity, either answer is just as good.
I do not know what you mean by that statement. To me, if someone says "the metric is invariant", they mean the action of the metric on all vector pairs v_1,v_2, selected from a vector space, is invariant under any coordinate change (i.e, homeomorphisms) of the vector space. Since the action of the metric is not invariant under any homeomorphism that is not orthonormal, that contradicts your statement. Obviously you mean something different by it, but the real issue here is, if you pick up a relativity textbook and start reading about the "Minkowski metric", are you reading about the actual thing that defines the structure of Lorentzian manifolds, or are you reading about some specially coordinatized version of that true structure? I say the latter, and that's the problem with it.
Then it is not invariant under arbitrary transformations. You have not established why there is anything to "confuse" here. Note that we are not discussing whether or not mathematicians know what a Minkowski space is, they invented it, nor are we discussing whether or not dynamics on a pseudo-Riemannian manifold are locally that of a Lorentzian manifold, we know that they are. We are talking about whether or not we should treat the Minkowski metric as something special, or as just one from a whole class of metrics with the same signature that are all equally physically "real" and equally selected from the class of metrics important for understanding spacetime from the perspective of any observer.
That quote does not claim worldlines are inertial, it says that noninertial observers also have worldlines and should be able to coordinatize spacetime using constantly changing basis vectors. For example, Einstein's simultaneity convention is a lot different for such an observer (allowing time to be perceived as going backward, for example).
I'm not getting the connection, it just sounds like a contradiction. Why would we care if we have reasons to believe Newtonian physics works for arbitrary v if we know it doesn't?I meant exactly what I said -- this very day, we have reasons to believe Newtonian mechanics works for arbitrary relative velocities, and that is perfectly consistent with the scientific conclusion that Newtonian mechanics doesn't work for arbitrary relative velocities. I'm trying to provoke thought about how empiricism works.
Ken G said:
A cute cartoon: http://www.xkcd.com/123/Ken G said:
On the contrary, it's a classic example of understanding coordinate representations, but not what is being represented by those coordinates.
Your example wasn't a paradox -- it was two ways of deriving the same thing. A paradox is when you have two (valid) arguments that arrive at contradictory conclusions.
There is no such thing as a "best" answer. The critera for judging the 'goodness' an answer depend on the circumstances, and generally speaking, different answers will be better in different situations.
Right there -- that's your problem. You have confused the notion of a 'coordinate change' with the notion of a 'homeomorphism'.
Anyone can coordinatize spacetime in any way they please. (I don't think that 'constantly changing basis vectors' has any literal meaning, but I think I know what you mean) Coordinate charts have nothing to do with observers.
Because we want to do science correctly -- in particular, we don't want to make actual mistakes, nor do we want to force science to conform to our a priori biases.
(There are ways to accommodate our a priori biases without hacking the philosophy of empiricism to pieces)But what you haven't answered is, why should we need to "expect" anything at all? If we already have observations that we think are relevant to the new ones, then it is the behavior in the old observations, not any theories we built from them, that cause us to "expect" something in the new observations. That's natural-- if previous results relate to new ones, we use them, that's the bridge building kind of predictions that science makes-- the important kind. But again, in that situation it is the existing observations that creates the cause for new expectations-- theories are nothing but a way to unify the information in the existing observations. When we forget that, we land in all kinds of hot water, throughout history.dx said:
But on what basis do you say that it is equally likely? On the basis of experience, of prior observations of symmetric objects. If you hand someone a shoe, should they also expect it is equally likely to end up on the sole or the top, or would it be natural to adopt no expectation at all until some experience was built up around objects of that shape? Aren't you simply using what you know about symmetries to build your expectation, not any kind of "null hypothesis"?
Why not? What if it was a shoe instead of a coin?
When we have information, i.e., past experience that seems relevant, yes. It is a very difficult issue to decide what constitutes "relevant past experience", yet we have to do just that every time we build a new bridge or a new airplane, so it's not a new problem to identify when we are really probing a new regime. You might say "but the aprpropriate equations for plane flight are known", but that really isn't true-- there's no such thing as "the appropriate equations" in an absolute sense, paradigm choices always have to be made, based on experience.
But again, note that I am distinguishing two types of prediction here, one is the type that says a certain drug therapy might cure a disease in an individual even though no testing was ever done of that drug on that individual. That's the important kind of predictions that science makes, "inside the box" of what we have experience with, and we need to be able to expect them to be right to gain the value of science. But predictons made "outside the box" are something very different, and have a far spottier record in science-- such as predicting that the drug will also work on other diseases that bear some resemblance but which we have no data for that drug. More to the point, the usefulness of predictions like that is very different, they are only there to guide new hypotheses and new experiments-- there is no need to "expect" them to be right (and any practicing physician who "expected" such predictions to be right could make significant errors in judgement).
Well this is exactly "the rub", when can you tell if your theory should be expected to work or not. Some Earthlike physics works on Mars, and some doesn't, pure and simple. There's no reason to expect that it either will or it won't, except as guided by past experience around extrapolating a particular theory in that way.
We can certainly agree on that, the issue is, did we have any right to be surprised that it didn't? I say, no, that is false surprise, engendered by a fallacious idea about what physical theories really are-- a fallacious idea that we seem to be even more likely to fall into in modern areas like interpretations of quantum mechanics or the "landscape" in fundamental particle theory. Basically, Einstein got away with telling reality that it ought to bend starlight passing a massive body, and that made us forget, for the umpteenth time, that the winners write the history.
Right, that is the need for understanding tensor quantities, and how various quantities transform. Those are clear enough to mathematicians, but it is the physicist's problem to try to use that in their interpretation of reality. Are they doing a good job, or mistaking computational conveniences for statements of what is real? Is there any difference? Mathematics doesn't answer those questions, they are metaphysical.Hurkyl said:
Then you can answer, is the centrifugal force something real?
There is no such paradox in the twin "paradox", for anyone who can do relativity, just like there's no paradox in saying 1+1 is either 1 or 2 for someone who can add. So to call it a paradox is indeed to embrace the type I was talking about-- two different sounding answers to the same question, that are both right, yet seem like they contradict. We tolerate that situation in the current formulation of relativity, and that is what I am suggesting is a weak pedagogy to accept.
But that's easy, it's no more interesting than someone learning to add for the first time, encountering "paradoxes" because they are simply doing something wrong. I'm talking about the aspects of that paradox that survive a fully correct treatment in some axiomatic system, yielding contradictory sounding descriptions of the reality of what happened.
Of course. Yet it falls to us to make that call anyway, constantly, both as teachers and as we ourselves try to obtain the most facile understanding of how reality works.
You said that before, but I'm claiming that when the "coordinate change" corresponds to the way a different observer measures reality, we are indeed talking about a homeomorphism-- that the physical approach makes that the required picture. In mathematics, you have more freedom to decide if you want to imagine that a change of coordinates either simply relabeled the same vectors with new names, or if it mapped the old vector space onto a new one where the structure is the stucture of the names. You make that choice when, for example, you plot a trajectory in polar coordinates. Do you write rectangular axes labeled theta and r, and plot curvy paths on them to show inertial motion, or do you draw little circles cut by radial wedges, and plot straight lines? These are mathematically equivalent, so I don't see why you are saying that one must make a distinction and cannot see a coordinate change as a homeomorphism.
It is trivial to create an automorphism from that by inverting the first coordinate function (it is invertible), and applying the second coordinate function. That is a perfectly valid association of a coordinate change with an automorphism on X, it seems to me, and indeed it is just what is often done in physics (as in using centrifugal forces in finding a Roche lobe, for example, where certain equations are applied prior to the final mapping back to X).
But they are not very different ideas, expressly because we are dealing entirely with coordinate homeomorphisms here. Hence, automorphisms of R^n extend trivially by the action of the coordinate function to automorphisms of X. This is crucial, the structure of X is preserved on R^n, so there is not the distinction you describe.
If that were true, then they would have no physical meaning and would be purely abstract mathematical concepts. We must have a way to connect observations to coordinates. Saying I can coordinatize spacetime any way I want is like saying I can name events "Tom", "Dick", and "Harry" if I want-- but I'm not doing physics unless I can connect these names to a ruler and a clock somehow.
I don't see your view that empiricism is some kind of "weighing" of pro and con evidence. It is generally accepted that no theory can be proven true by any number of successes, but can be proven to need modification by any single significant failure. That stems from the need for a theory to unify, not replace, data. Of course we use "false" theories all the time, but again that is based on our experience with using them, not any kind of fundamental theoretical stance about how reality must work. All you seem to be saying here is that if we find a regime where a theory breaks down, we will still use it in regimes where it does not, but that merely serves to underscore what I mean by the reliable type of predictions that science makes-- in contrast with the guesses masquerading as predictions.
My end goal would be to not confuse predictions with hypotheses, to avoid mistaking the various purposes for which we have theories in the first place.
I can either win the lottery, or I can not win. There is also "two sides" to that issue. If I have no idea what my chances of winning are, shall I use that as a reason to conclude my chances are 50-50 in lieu of new information? Or shall I simply say I have no idea what the chances are, and the fact that there are only two possibilities gives me no help whatsoever, because I have no reasonable basis for using that information?granpa said:
Call it what you like, but it still doesn't mean much of anything.
That is true in all situations and is unrelated to the issue of the likelihood of the two events. You said "on what basis do you say that it is equally likely? on the basis that there are 2 sides." I assumed the "it" was which side will occur, not what is the frequency that you can be right. Even if there are 99 sides, I can be right 50% of the time by randomly selecting between 1 or 2-99 to bet on. Or I can predict a coin flip 33% of the time by choosing randomly between heads, tails, and that it will end up on its side. None of that has anything to do with "the number of sides" that can occur, it is just a way to manipulate a winning chance via a betting strategy. What's the relevance?granpa said:
Ken G said:
Ken G said:
Ken G said:
Ken G said:
Actually, it's not that simple; the problem of a priori probabilities is a significant philosophical issue -- this is a common and often useful convention, but it's far from clear that it would be the "most logical expectation". And it's not strictly necessary anyways -- one can view the scientific process as merely determining which theories have the stronger Bayes factors, rather than trying to determine which theories are most probable. i.e. science seeks to accumulate evidence, not to uncover truth.dx said:
Are any forces 'real'? What about resolving forces into components, or decomposing it into contributions from different sources?Ken G said:
That's why it's more properly called the twin pseudoparadox -- there is a flaw in the reasoning that leads to the contradiction. If you are talking about anything other than the (fallacious) lines of reasoning that lead to the conclusion that each twin sees the other one age less, then you are not talking about the twin paradox; you're talking about something else.
I can't decipher any content in this. Anyways, the thing I've learned both from experts and my own experience is that for any subject, it is best to understand it from many different points of view. This way, we can select a point of view most suited to the problem of interest -- and even better, we can transfer between different points of view, so as to apply a wider variety of methods to the problem.
I'm not.
That's called a "coordinate chart". And anyone can use whatever coordinate chart they want, consider things relative to more than one chart, or even not use a coordinate chart at all.
Such a thing cannot be proven for exactly the same reason that a theory cannot be proven true. And even a single significant failure usually isn't enough to yield convincing evidence that a modification of a theory. For example, equipment failure or improper experimental procedure are usually more likely 'explanations' of a significant failure.
granpa said:
Which does it need to do, predict within the well tested regime, or beyond it? If I have a theory that predicted in the tested regime, then I can build bridges with it. Exactly why do I need to be able to extend my predictions beyond that regime? I would call that nothing but the formation of a hypothesis, and we don't need a theory to do that, we can try anything we like.dx said:
Exactly my point-- the pattern is in the data, so it lives in the tested regime. Extensions outside that regime are either part of that pattern, in which case they are not outside that regime, or they don't, in which case they are. We don't know in advance what the regime is, but we still have no need to make any guesses about what the regime is. There's simply no need for it, you are never going to use the theory to do anything in a regime in which it has not been tested (other than to form a hypothesis, but then there is no need to "expect" anything, you are just deciding what experiment you think should be done. Indeed, "expecting" results tends to lead you to not bother with the experiment.)
But that's the definition of the Fibonacci series. It's only a theory if you don't know where that series comes from (this is the problem we have in physics). If the series is coming from observations of some kind where you are not just getting out what you put in, then you have no reason to expect the form will continue indefinitely. If it worked for a million terms, it seems probable it will work for many thousands more, because why should they be special, but what about a billion more? No reason to expect that, unless you are just getting out what you put in (as in the Fibonacci series itself).
It is to avoid fooling oneself. Feynman has a great quote that science is about learning how to not fool yourself, given that you are the easiest person you can fool. I see science falling into that same trap, it is not taking its own principles far enough if it keeps causing us to fool ourselves into false expectations, and then having "revolutions" later on.
I would say that is exactly the way science should be done, yes. Granted, it is no simple matter to define what is meant by an "untested regime", but for now we'll simply allow there is such a concept even if we can't be terribly precise about what it is.
I agree, except I would simply call that the great discovery of quantum mechanics. That is was also a revolution was all our own fault. Columbus "discovered" the New World for Europe, its existence was not a "revolution". The ancient Greeks knew there was some 15,000 miles of ocean out there, so for anyone to "expect" it to be empty would just be guessing.
But it was very much a comeuppance, indeed that is the main reason Poincare did not discover it himself. He saw it as some kind of mathematical trick, he couldn't believe that it was a description of how reality actually worked. That's what I mean about putting the "cart" of expectations before the "horse" of observing reality.
I claim your analysis is using the symmetry of the coin, and that's why it seems "arbitrary" to do anything else. But if you know the coin has a symmetry, you are indeed using knowledge of the coin. Write that same argument but for a conical hat.
True, but that would not lead me to expect a 50-50 chance, it would lead me to simply say I have no meaningful way to assess the probability. Probability requires a great deal of knowledge about what variables are outside your control-- if you don't even know that, it is a meaningless concept.
I don't agree, all experiments can have two possible outcomes-- a particular one, and anything else. Shall we start with the assumption, then, that any outcome you can name has a 50-50 chance of happening, on the grounds that we have no other information about the probabilities of "all other outcomes"? We always have to group outcomes, there's no absolute sense of "the possible outcomes of an experiment". Even if you are flipping a coin, there is the location of every other particle involved in that experiment. You can say you don't care about them, so you are grouping outcomes. So am I in the above.
The point there is that at first glance, we may think we are saying something fundamental about the theory gravity that it works on Earth and on Mars. But we are not, we are saying something fundamental about the observations we already had that we used the theory of gravity to unify. We observed what aspects of a planet control its gravity, and built a theory that reflected that. So when we look at other planets and find the theory works, it is because we put "other planets" right into the theory. If the "other planet" is a neutron star, we get a breakdown, and if it's like the planets we built the theory for, we don't.
phyti said:
Exactly. My question to you came in response to your claim that centrifugal force confusions arise from understaning how to do coordinates, but not understanding what they represent. What do they represent? They are means of manipulating quantitative information, how can you do the manipulation correctly and still be "missing something", as you appear to suggest?Hurkyl said:
Correct, I"m talking about something else-- something that remains an issue even after you know how to do relativity (what else would be interesting?). That "paradox" is that the two observers can do everything right and still come up with a very different answer as to the "cause" of the age difference. That's normally an accepted aspect of relativity, but my point is, it doesn't need to be so.
Quite so. Then from that perspective, you may interpret this entire thread as asking the quesiton, "what is the formulation of relativity that uses only postulates that we have really established observationally, i.e., postulates that are not subject to being overturned with new observations unless we somehow did the existing observations wrong."
Right-- the "most suited" aspect of this approach is that it is most suited to not making claims on reality that we haven't the vaguest idea are true (nor have any "Bayes factors" to apply).
So you are contesting my mathematical proof that coordinate automorphisms on R^n extend trivially to topolological automorphisms on X, and therefore the "distinction" you draw does not really exist?
All you are doing is naming the action, but any such naming doesn't change the point that a coordinate chart means nothing in physics unless you can make a connection with an observable. So no, anyone cannot use any coordinate chart they want-- they must be able to describe its connection with clocks and rulers, or other measuring devices, or it just isn't meaningful physics. That's why a coordinatization is a homeomorphism on the topological space that comes complete with automorphisms onto other coordinatizations, all of which extend trivially to automorphisms on the topological space.
Although that's all true in principle, in practice that isn't the way we conceptualize our art. Although it can be argued that the Michelson-Morely experiment was of no significance until it was reproduced, for just those reasons, the way we describe the progress of science is quite different. Take it up with the legacy of Einstein, as it includes his famous quote "No amount of experimentation can ever prove me right; a single experiment can prove me wrong." The literal truth of this is not really the point, I would say the significance of the remark is that physics lives in little boxes called "appropriate regimes", so you can have a hundred experiments in one regime and learn nothing about some other one, until a single experiment is done in that other regime. My goal is to recognize this right up front in how we postulate our science, basically so that we can really keep better track of what we are actually doing-- thereby eliminating the problem of "revolutions" (in the sense I'm using it not dx's more general meaning of any significantly new discovery).
You may be correct that probability is different from expectation (I only know the latter as a result-weighted version of the former), but what point are you making about observations in physics?granpa said:
How does a theory show something is real? I thought observations were the only things capable of that.Chatman said: