From: Ste on
On 23 Feb, 08:02, mpalenik <markpale...(a)gmail.com> wrote:
> On Feb 23, 12:16 am, Ste <ste_ro...(a)hotmail.com> wrote:
>
> > > which of these two models is correct or even more
> > > likely, even if you have a hunch or a personal preference for one. But
> > > both the gangster and the mathematician have put forward models with a
> > > testable prediction. In particular, if you tossed it another 100
> > > times, the two theories would have remarkably different predictions,
> > > and the *coin itself* will tell you which of the two models is a
> > > better description of it.
>
> > That's not true. A further 100 tosses would not discern definitively
> > between the two theories, for the outcome would still be technically
> > consistent with either theory.
>
> In fact, though, you can calculate the probability that either
> hypothesis is correct if the coin toss comes out a certain way after
> 100 tosses.  100% heads, for example, would put you well past a 95%
> confidence interval.  And in fact, when experimentalists publish their
> data, they do also publish such confidence intervals.

But even a weighted coin would be unlikely to give 100% heads. In any
event, the point is that there is still no definitive test, and unlike
a simple coin-toss outcome theory, the truth of real-world scientific
theories and experiments are not nearly as easily reducible to this
kind mathematical probability.

And indeed, if the confidence level is 95%, then that doesn't mean
100% of people should hold the weighing-theory to be true (i.e. on the
basis that the most likely explanation is likely to be the correct
one). On the contrary, it would be desirable to have, say, only 95% of
people working within the assumption that the weighting-theory is
true, and the rest working with the assumption that it is not true.



> > > Notice how different this approach is from sitting back and trying to
> > > decide whether mathematicians (or gangsters) are groups of people who
> > > self-select themselves into delusions, and therefore their models are
> > > not to be trusted. Why do that kind of nonsense, when you can simply
> > > ask the coin to show its colors?
>
> > Because in the real world it is not simply a case of flipping the coin
> > an infinite number of times. Let's face it, we both submit to evidence
> > - that cannot be the difference between us.
>
> Actually, you tend to hand-wave away any evidence you don't like as
> 100 years of bad experiments, as in the case of the Michaelson Morely
> experiment and others like it with similar, albiet more refined,
> steups, which you were insisting had problems for a long time.
> Perhaps the difference is, we understand the experimental evidence and
> you don't.

I haven't hand-waved the evidence away at all. If you mean that I've
just dismissed evidence out of hand (presumably because you think I
find it undesirable), then I would challenge you to identify where I
have done this. If you mean something else, then I would ask you to
clarify what you mean when you say I've "hand-waved evidence away".



> > > > > > If you're suggesting that it's improbable that a theory could work not
> > > > > > because its premises were correct, but because it simply promoted
> > > > > > correct behaviours, then wonders why religion has fared so well..
>
> > > > > I don't know of any practical, artificially created devices that are
> > > > > used in everyday life whose designs are based on religious principles.
> > > > > Do you?
>
> > > > But the form of the question is designed to confirm your
> > > > preconception.
>
> > > I don't think so. Perhaps if you gave an example from the most recent
> > > century of physics.
>
> > I'm not arguing that religious principles have been used directly to
> > create "practical, artificial devices". What I did suggest was that a
> > theory can work because it promotes correct behaviours, independent of
> > the truth of its explicit premises.
>
> So, for example, praying to Vishnu is a correct behavior, whether or
> not he exists?

My argument is that it could be in certain circumstances. For example,
if praying provides a context for socialising and socialisation,
thereby forging social links in the community and disseminating a
common ideology, essentially "bringing the community together", then
praying may well be a "correct" behaviour even though its explicit
justification (of appeasing the Gods or whatever) is false.

Just thinking, it's funny how much my views on religion have tempered
over the years.



> > > > If we are to judge theories based on their tendency to bring about
> > > > desirable outcomes (which I think is one of the unspoken assumptions
> > > > in your question) then it does not seem to me that a theory's premises
> > > > need to be factually correct; only that it produces the desired
> > > > outcome. Which is what I said in the first place.
>
> > > Whether a theory's premises are factually correct IS DETERMINED by
> > > whether it produces accurate predictions of measurements. That's how
> > > science makes that determination in the first place.
>
> > Not really. Once you factor in technological constraints, margins of
> > error, empirical constants, domains of applicability, sheer
> > complexity, implict knowledge, and probably innumerable other issues,
> > it becomes very difficult to discern whether the explicit body of
> > knowledge is true or whether it simply produces sufficiently correct
> > behaviours without there being any truth to its explicit form.
>
> And, if you can disprove a current theory, which means
> 1) coming up with a theory that does something the old theory can't
> 2) demonstrate that the new predictions you make are correct
>
> If you can supplant an old theory with a new, better one, you can 1)
> publish it, 2) gain great scientific acclaim, and 3) possibly win a
> lot of money.
>
> If I could disprove Special Relativity, that would be great.  I'd get
> a nobel prize.  But there are over 100 years of experiments that need
> to be explained with any new theory and it needs to have something new
> that relativity doesn't have.

But this mindset seems to be part of the problem. The truth is that
physicists don't even seem to be able to answer some of the most
elementary conceptual questions that are asked here - or indeed even
readily understand the nature of the questions that are asked, let
alone answer them.

And before you jump in and say that this newsgroup is not
representative of physicists, let me be clear that I've read pretty
widely already and this utterly lack of conceptual clarity is by no
means confined to this newsgroup.

I dare say that instead of looking to green-field theories (which you
seem to allude to in terms of talking about "new predictions"),
theoretical physicists would do well to revisit with hammers and tongs
the very conceptual groundwork of relativity.

Incidentally (and I seem to be in a schizophrenic mood, so this may
seem a bit wild at first read, but bear with me), something just
struck me about a problem of frames of reference. We know that the
numbers, the measurements, the times, the distances, all change
depending on the frame of reference. The problem is how to start
routinely expressing essential relationships in a fashion that is
invariant. For example, consider this illustration:

A------B---C

Basically you can express the distance AB with the value x, the
distance BC as value x/2, and the distance AC as value 3x/2. Or you
can express it as AB = 2x, BC = x, and AC = 3x. But this form of
expression always relies on comparison, and if you change the
reference value of X then all the other values change numerically (but
not physically - there is still some essential relationship that is
physically invariant). But how do you iron out the reliance on
comparison? How do you describe something with reference only to the
things in question, and not to a reference standard?

I think accidentally I might have finally articulated what the
essential aspect is of a "physical explanation". It's the invariance
of the relationships between the points, that is invariant no matter
how they are numerically expressed.



> The scientific community and scientific publications aren't afraid of
> publishing new ideas.  Nature even published an article on homeopathy
> once (with a disclaimer at the beginning) because the experimental
> setup seemed to be perfect, and yet, it gave these incredible,
> surprising results, that even after you dilute out all of a solute,
> water still somehow retains the properties of that solute.  This
> prompted other scientists to try to reproduce those results, and when
> none could, the original team was investigated, and it turned out
> there were researchers who were (perhaps by accident) causing errors
> in the experiment.

Lol. Incidentaly, Mitchell and Webb did a brilliant sketch on
homeopathic medicine. You might be able to find it on Youtube.



> The reason the scientists here haven't accepted you with open arms
> isn't because you're preaching something we don't like to hear, it's
> because:
> 1) You claimed that over 100 years of experiments are in error without
> any actual, logical explanation of what that error was

No I haven't claimed 100 years of experiments are in error. You really
must go back and revise what I've actually said.



> 2) You
> 3) You refuse to do any quantitative predictions, which would
> immediately tell you whether or not a given modification to a theory
> is wrong (to use the gravity example, if I said that gravity falls off
> as 1/r instead of 1/r^2, that would have direct, obvious implications
> to the real world that would be easily testable/measurable).

I was using that as a throwaway example Mark, to illustrate a
completely different point. Namely, that a qualatitive understanding
of the solar system is really quite independent of its quantification,
and that I can always adjust the quantification without throwing out
my qualitative understanding (and indeed, I can have a qualitative
understanding without being able to mathematically quantify the
phenomena at all).



> 4) You immediately assume that because certain physics doesn't work
> the way *you personally* want it to, that it is because science is run
> like a religion.

Again, that doesn't capture what I said. I would appreciate it if
you'd start being more sensible about this discussion instead of
constantly implying that my requirements of science are somehow
arbitrary and unreasonable (as against "science" proper, which is
always implicitly held to be fair and objective).



> That *your personal* philosophies are necessarily
> the correct ones, and anything that doesn't fit in with *your personal
> viewpoint* must NECESSARILY mean that science is wrong.

As I've said, there are a few axioms that I hold which are for all
practical purposes irrefutable, and you're right that this leads me to
believe that any part of science that conflicts with them is
necessarily wrong. But what of it? You raise the issue repeatedly in a
tone of emphasis and perhaps outrage, without either acknowledging the
fact and moving on, or else making an explicit point in regard of it.

Indeed, would I be far off the mark if I said that the point you're
really trying to make with these "your personal viewpoint" statements
is really to say that I have only me in support of me, whereas you
have a professional scientific community in support of you. But are
you making these statements as a form of self-comfort, or for the
benefit of the audience? Who?



> 5) It is very clear that you don't have a good understanding of
> relativity or the concepts behind it.

No I admit that I don't.



> Even simply from the fact that
> you're incapable of correctly doing the thought experiments that you
> post here by yourself.

What do you mean?



> It is necessary that you correctly understand
> a theory before you go around pointing out its flaws.

Not if the theory is incomprehensible, and flawed for that reason. And
nor if the theory is so fundamentally flawed that it can be dismissed
having investigated only a small part of it.



> The funny thing is, you accuse so many people of "wrong thinking,"
> because obviously, education must cause wrong thinking, and yet, this
> is all based on the fact that our thinking does not conform to *your
> specific thinking*.

I didn't say "education must cause wrong thinking". You constantly
distort every statement I make - you not only fail to capture the
nuance, but you actually consistently fail to capture the meaning even
in a crude form.



> You never consider that it might be your lack of
> experience, your lack of education, your lack of knowledge, and your
> personal inability to recocile certain concepts that's the problem?
> Instead, it must be that everybody else has the problem. . .

But everybody else doesn't have the problem. Only physicists seem to.
Most people have no idea what physicists are talking about. And indeed
of those who do claim to have some sort of understanding of physics,
they were not the ones who came up with the fundamental theories of
the day in the first place, and the vast majority are not likely to
ever come up with a new mainstream theory or have anything to do with
formulating one. The mere fact that there are a noticable minority of
people who have become skilled in the subject of physics says nothing
to me about its reliability. It counts for nothing. What counts for me
is compelling argument. On any day I would believe a tiny minority
with a compelling argument over a sizable majority with a nonsense
argument. And hence it becomes somewhat irritating to hear appeals to
authority all the time, or to the "principles of science", etc. I've
spoken to enough people over the years about politics to know when I'm
coming up against ideology, and that is what I am most certainly
coming up against here.
From: Jerry on
On Feb 24, 12:31 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote:

> You're acting like using a letter to refer to a velocity is something
> magic.  It's not, it's just a shorthand.

The term "c" has multiple meanings.

There exists a demonstrable maximum possible speed of
communications, designated "c". This constant "c" is a fundamental
parameter in the specification of the geometric properties of
spacetime. Since this constant "c" is finite, absolute time does
not exist, assuming that our universe is correctly described by
this geometric model.

Light happens to travel at a speed which is either equal to "c"
in the sense presented above, or which is so close to "c" that no
measurement has yet established any detectable difference. This
"speed of light" is conventionally ALSO designated by "c", which
represents unfortunate historical baggage reflecting the manner
in which relativity developed.

Should the speed of light ever be discovered not to precisely
equal "c" (and there is some controversial evidence that this
may be the case for high energy gamma rays), this will have NO
CONSEQUENCE WHATSOEVER in regards to the validity of relativity,
since the "c" which is the speed of light is not the same "c"
which determines the properties of spacetime.

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

Here is a parable for you and Bruce to consider. Once upon a time,
in a universe far away, with somewhat different physical rules
than our own universe, lived a race of beings who evolved the
ability to "see" using neutrinos. Their history of science and
technology resembled ours in many ways, except for their use of
neutrinos for many of the purposes to which we put light.

A young xzplinkr clerk, Abixtx Einxtrt, published a paper "On the
Neutrinodynamics of Moving Bodies" that invoked two postulates,
the first stating the principle of relativity, the second
declaring the speed of neutrinos to be a constant. This paper
sparked a revolution in science.

Nearly a century after the publication of Abixtx Einxtrt's paper,
it was discovered that neutrinos have mass. Crackpots all over
the world celebrated the so-called "overthrow" of relativity
implied by this discovery...

Jerry
From: Androcles on

"Jerry" <Cephalobus_alienus(a)comcast.net> wrote in message
news:0baac22d-ad06-4136-b1b2-d7144955080f(a)a18g2000yqc.googlegroups.com...
On Feb 24, 12:31 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote:

> You're acting like using a letter to refer to a velocity is something
> magic. It's not, it's just a shorthand.

The term "c" has multiple meanings.

There exists a demonstrable maximum possible speed of
communications, designated "c".
==============================================
There exists a demonstrable maximum possible speed of
bullshit, designated "Jerry".





From: mpalenik on
On Feb 24, 2:15 am, Ste <ste_ro...(a)hotmail.com> wrote:
> On 23 Feb, 08:02, mpalenik <markpale...(a)gmail.com> wrote:
>
>
>
>
>
> > On Feb 23, 12:16 am, Ste <ste_ro...(a)hotmail.com> wrote:
>
> > > > which of these two models is correct or even more
> > > > likely, even if you have a hunch or a personal preference for one. But
> > > > both the gangster and the mathematician have put forward models with a
> > > > testable prediction. In particular, if you tossed it another 100
> > > > times, the two theories would have remarkably different predictions,
> > > > and the *coin itself* will tell you which of the two models is a
> > > > better description of it.
>
> > > That's not true. A further 100 tosses would not discern definitively
> > > between the two theories, for the outcome would still be technically
> > > consistent with either theory.
>
> > In fact, though, you can calculate the probability that either
> > hypothesis is correct if the coin toss comes out a certain way after
> > 100 tosses.  100% heads, for example, would put you well past a 95%
> > confidence interval.  And in fact, when experimentalists publish their
> > data, they do also publish such confidence intervals.
>
> But even a weighted coin would be unlikely to give 100% heads. In any
> event, the point is that there is still no definitive test, and unlike
> a simple coin-toss outcome theory, the truth of real-world scientific
> theories and experiments are not nearly as easily reducible to this
> kind mathematical probability.

I don't think you understand how statistics works. We can say "a
weighted coin should come up heads > 50% of the time." We can say
"given that we threw the coin n times and got x number of heads, what
is the probability that this is a fair coin." We can then compare the
null hypothesis (that it is a fair coin) to the "heads comes up more"
hypothesis.

>
> And indeed, if the confidence level is 95%, then that doesn't mean
> 100% of people should hold the weighing-theory to be true (i.e. on the
> basis that the most likely explanation is likely to be the correct
> one). On the contrary, it would be desirable to have, say, only 95% of
> people working within the assumption that the weighting-theory is
> true, and the rest working with the assumption that it is not true.
>

First of all, after 100 flips you'd be well above 95%--probably well
above 99%. But regardless, a 95% confidence interval doesn't mean
that 5% of all people should believe that the coin isn't weighted. It
means that everybody should believe that there is a 95% chance that
the coin is weighted. There's a big difference.

>
>
>
>
> > > > Notice how different this approach is from sitting back and trying to
> > > > decide whether mathematicians (or gangsters) are groups of people who
> > > > self-select themselves into delusions, and therefore their models are
> > > > not to be trusted. Why do that kind of nonsense, when you can simply
> > > > ask the coin to show its colors?
>
> > > Because in the real world it is not simply a case of flipping the coin
> > > an infinite number of times. Let's face it, we both submit to evidence
> > > - that cannot be the difference between us.
>
> > Actually, you tend to hand-wave away any evidence you don't like as
> > 100 years of bad experiments, as in the case of the Michaelson Morely
> > experiment and others like it with similar, albiet more refined,
> > steups, which you were insisting had problems for a long time.
> > Perhaps the difference is, we understand the experimental evidence and
> > you don't.
>
> I haven't hand-waved the evidence away at all. If you mean that I've
> just dismissed evidence out of hand (presumably because you think I
> find it undesirable), then I would challenge you to identify where I
> have done this. If you mean something else, then I would ask you to
> clarify what you mean when you say I've "hand-waved evidence away".
>

Ok, let's look at your whole thread about measuring the speed of
light.

You came up with a bunch of hypotheses that have no basis in physical
reality--about how brightness could affect the location of
interference fringes (it doesn't, and I proved it doesn't), about the
speed of light and measured brightness being able to "compensate" for
each other--again none of which have any basis in physical reality--
all because you wanted to dismiss the Michaelson-Morley type
experiments. Again, there was nothing sensible about any of it,
except that it was a way for you to attempt to justify not having to
believe the speed of light isotropy measurements.
>
>
>
>
> > > > > > > If you're suggesting that it's improbable that a theory could work not
> > > > > > > because its premises were correct, but because it simply promoted
> > > > > > > correct behaviours, then wonders why religion has fared so well.
>
> > > > > > I don't know of any practical, artificially created devices that are
> > > > > > used in everyday life whose designs are based on religious principles.
> > > > > > Do you?
>
> > > > > But the form of the question is designed to confirm your
> > > > > preconception.
>
> > > > I don't think so. Perhaps if you gave an example from the most recent
> > > > century of physics.
>
> > > I'm not arguing that religious principles have been used directly to
> > > create "practical, artificial devices". What I did suggest was that a
> > > theory can work because it promotes correct behaviours, independent of
> > > the truth of its explicit premises.
>
> > So, for example, praying to Vishnu is a correct behavior, whether or
> > not he exists?
>
> My argument is that it could be in certain circumstances. For example,
> if praying provides a context for socialising and socialisation,
> thereby forging social links in the community and disseminating a
> common ideology, essentially "bringing the community together", then
> praying may well be a "correct" behaviour even though its explicit
> justification (of appeasing the Gods or whatever) is false.
>
> Just thinking, it's funny how much my views on religion have tempered
> over the years.
>
>
>
>
>
> > > > > If we are to judge theories based on their tendency to bring about
> > > > > desirable outcomes (which I think is one of the unspoken assumptions
> > > > > in your question) then it does not seem to me that a theory's premises
> > > > > need to be factually correct; only that it produces the desired
> > > > > outcome. Which is what I said in the first place.
>
> > > > Whether a theory's premises are factually correct IS DETERMINED by
> > > > whether it produces accurate predictions of measurements. That's how
> > > > science makes that determination in the first place.
>
> > > Not really. Once you factor in technological constraints, margins of
> > > error, empirical constants, domains of applicability, sheer
> > > complexity, implict knowledge, and probably innumerable other issues,
> > > it becomes very difficult to discern whether the explicit body of
> > > knowledge is true or whether it simply produces sufficiently correct
> > > behaviours without there being any truth to its explicit form.
>
> > And, if you can disprove a current theory, which means
> > 1) coming up with a theory that does something the old theory can't
> > 2) demonstrate that the new predictions you make are correct
>
> > If you can supplant an old theory with a new, better one, you can 1)
> > publish it, 2) gain great scientific acclaim, and 3) possibly win a
> > lot of money.
>
> > If I could disprove Special Relativity, that would be great.  I'd get
> > a nobel prize.  But there are over 100 years of experiments that need
> > to be explained with any new theory and it needs to have something new
> > that relativity doesn't have.
>
> But this mindset seems to be part of the problem. The truth is that
> physicists don't even seem to be able to answer some of the most
> elementary conceptual questions that are asked here - or indeed even
> readily understand the nature of the questions that are asked, let
> alone answer them.

We can understand them. The only problem is that you either don't
understand or believe the answers. Mostly the former, which leads to
the latter.

>
> And before you jump in and say that this newsgroup is not
> representative of physicists, let me be clear that I've read pretty
> widely already and this utterly lack of conceptual clarity is by no
> means confined to this newsgroup.

I wasn't going to say that, but what extensive interaction have you
actually had with physicists? And reading "A Breif History of Time"
doesn't count.

>
> I dare say that instead of looking to green-field theories (which you
> seem to allude to in terms of talking about "new predictions"),
> theoretical physicists would do well to revisit with hammers and tongs
> the very conceptual groundwork of relativity.

There is a firm, concise, consistant, logical, and meaningful
framework for relativity. You don't get it--that's fine. But don't
expect everyone else to try to rewrite physics because you don't
understand something.

>
> Incidentally (and I seem to be in a schizophrenic mood, so this may
> seem a bit wild at first read, but bear with me), something just
> struck me about a problem of frames of reference. We know that the
> numbers, the measurements, the times, the distances, all change
> depending on the frame of reference. The problem is how to start
> routinely expressing essential relationships in a fashion that is
> invariant.

This is done through 4-vectors.

> For example, consider this illustration:
>
> A------B---C
>
> Basically you can express the distance AB with the value x, the
> distance BC as value x/2, and the distance AC as value 3x/2. Or you
> can express it as AB = 2x, BC = x, and AC = 3x. But this form of
> expression always relies on comparison, and if you change the
> reference value of X then all the other values change numerically (but
> not physically - there is still some essential relationship that is
> physically invariant).

First of all, physically, if you double the distance between two
objects, that does make a difference. For example, if you bring two
molecules close engouh together, they will start repelling instead of
attracting (this is the principle behind atomic force microscopy). If
you shrank the sun down into a small enough region, it would become a
black hole.

The invariant quantity is the ratio of lengths: 2:1. And yes, this
can be expressed as 1:2.

> But how do you iron out the reliance on
> comparison? How do you describe something with reference only to the
> things in question, and not to a reference standard?

The distance between B and C = 1/2 the distance between A and B.

>
> I think accidentally I might have finally articulated what the
> essential aspect is of a "physical explanation". It's the invariance
> of the relationships between the points, that is invariant no matter
> how they are numerically expressed.

This is what it means to express things in coordinate free geometry,
which can be done in relativity. It still has to be done within the
context of Minkowski spacetime, though. You do not need coordinates
to do geometry, although you *can* use coordinates and get the same
answer.

>
> > The scientific community and scientific publications aren't afraid of
> > publishing new ideas.  Nature even published an article on homeopathy
> > once (with a disclaimer at the beginning) because the experimental
> > setup seemed to be perfect, and yet, it gave these incredible,
> > surprising results, that even after you dilute out all of a solute,
> > water still somehow retains the properties of that solute.  This
> > prompted other scientists to try to reproduce those results, and when
> > none could, the original team was investigated, and it turned out
> > there were researchers who were (perhaps by accident) causing errors
> > in the experiment.
>
> Lol. Incidentaly, Mitchell and Webb did a brilliant sketch on
> homeopathic medicine. You might be able to find it on Youtube.
>
> > The reason the scientists here haven't accepted you with open arms
> > isn't because you're preaching something we don't like to hear, it's
> > because:
> > 1) You claimed that over 100 years of experiments are in error without
> > any actual, logical explanation of what that error was
>
> No I haven't claimed 100 years of experiments are in error. You really
> must go back and revise what I've actually said.
>
> > 2) You
> > 3) You refuse to do any quantitative predictions, which would
> > immediately tell you whether or not a given modification to a theory
> > is wrong (to use the gravity example, if I said that gravity falls off
> > as 1/r instead of 1/r^2, that would have direct, obvious implications
> > to the real world that would be easily testable/measurable).
>
> I was using that as a throwaway example Mark, to illustrate a
> completely different point. Namely, that a qualatitive understanding
> of the solar system is really quite independent of its quantification,
> and that I can always adjust the quantification without throwing out
> my- Hide quoted text -

But again, you're missing the point, that the quantitative explanation
can determine what qualitatively you will see. How do you think
Newton came up with a 1/r^2 law in the first place? It was by
observing eliptical orbits. There was only one quantitative
explanation that can lead to that qualitative behavior (although
coming up with his exact laws required quantitative measurements as
well). The two are not independent.
From: mpalenik on
On Feb 24, 3:30 am, mpalenik <markpale...(a)gmail.com> wrote:

>
> > For example, consider this illustration:
>
> > A------B---C
>
> > Basically you can express the distance AB with the value x, the
> > distance BC as value x/2, and the distance AC as value 3x/2. Or you
> > can express it as AB = 2x, BC = x, and AC = 3x. But this form of
> > expression always relies on comparison, and if you change the
> > reference value of X then all the other values change numerically (but
> > not physically - there is still some essential relationship that is
> > physically invariant).
>
> First of all, physically, if you double the distance between two
> objects, that does make a difference.  For example, if you bring two
> molecules close engouh together, they will start repelling instead of
> attracting (this is the principle behind atomic force microscopy).  If
> you shrank the sun down into a small enough region, it would become a
> black hole.
>
> The invariant quantity is the ratio of lengths: 2:1.  And yes, this
> can be expressed as 1:2.
>

I mean, 3:2 and 2:3, obviously. I dont' know why I thought you had
written X and 2X for some reason.