A paradox inside Newtonian world

[Hurkyl]

What's the use of a rigor, if the axioms contradicts each other?

One practical use is to back up your claim that the axioms contradict each other.

If I was merely interested in refuting your original PDF, a perfectly sufficient refutation would be to quote the passage

(*) the center of gravity of anybody cannot be accelerated, unless some external force is applied to it​

and point out that you have not justified it.

Alternatively, I could point out the fact that even if the above statement is true, your scenario involves a configuration of particles, not a body, so (*) doesn't apply.


But if had presented a valid argument with all due rigor, then (presumably) such objections could not be made.



But I'm not interested in refuting you, I'm trying to explain where you went wrong in reacting to the result of your argument. When we boldly venture forth using intuition as our judge of validity, rather than rigor, we have to face the fact we will make mistakes.

For example, we might attempt to apply some concept outside of its domain of validity -- one would typically view your original PDF as a demonstration that that particle configuration is outside of the domain of applicability of (*).


The advantage of rigor, here, is that theorems come with explicit hypotheses, and you can actually check whether a theorem is applicable to a given situation.

I am saying, I see it in ZF+NL.

One of the points I was making earlier is that NL is not a formal system. Newton's laws are conceptual. It really doesn't make sense to ask if NL is self-contradictory: such questions only make sense for particular formalizations of NL.

And although you have not formalized NL, you have (roughly) stated two axioms that you would include (any possible set of values is an admissible configuration of particles, and the center of mass axiom holds for any configuration of particles), neither of which are related to Newton's three laws. So not only have you not shown NL to be contradictory, you haven't even really made progress towards that goal.

 

[Tomaz Kristan]

I don't see, what's your point, Hurkyl?

That we have some mass configurations, which are illegal or non Newtonian - or what?

Newtons laws are ment to hold. No matter how bizarre a chunk of mass or body, or mass point constelation is.

You know that.

 

[Tomaz Kristan]

One of the points I was making earlier is that NL is not a formal system. Newton's laws are conceptual.

What is this? Are you saying, that NL are somehow above the scrutiny for inconsistencies, or what?

 

[reilly]

Physics uses math as a convenient language. As far as we know the nature we observe, here on Earth, is finite. So, the use of infinity in any form in physics is as much a slight of hand as it is simply convenient. Newton, Einstein et all never promised a Rose Garden, never seemed to worry about contradictions in math that are at the margin of physics. From what I know of physics history, physicists have done rather well at pushing the boundaries of mathematics -- normalization of continuous spectra for example, delta functions and distributions -- which the mathematicians later cleaned up in a systematic way.

The problem here is not a physics problem, it is a mathematics problem dressed up to look like a physics problem. As a physicist I say, in spite of my posts, paradox? who cares. The system in question can never by physically implemented, hence we'll never know the right answer as far as physics is concerned.

You don't need experiments in math, so the discussion can go on endlessly.

Regards,
Reilly Atkinson

 

[Hurkyl]

I don't see, what's your point, Hurkyl?

When using informal means, if you derive a paradox, you have not proven your axioms are self-contradictory. In particular, you may have applied your heuristics outside of their domain of applicability. Searching for paradoxes like this is, in fact, a typical way of discovering the limits of an informally expressed idea.

 

[Tomaz Kristan]

When using informal means, if you derive a paradox, you have not proven your axioms are self-contradictory.

Newton Laws ARE an example of a formal system. Not consistent with the math, inside which are presented, as we saw.

 

[Tomaz Kristan]

The problem here is not a physics problem, it is a mathematics problem dressed up to look like a physics problem.

Physics can't have a practical problem. It is, how it is. A physics theory can be flawed.

The pure Newtonism is.

 

[vanesch]

In particular, you may have applied your heuristics outside of their domain of applicability. Searching for paradoxes like this is, in fact, a typical way of discovering the limits of an informally expressed idea.

Yes, that's why I found this example interesting, without being worrysome. Clearly, Newton's laws, with Newton's gravity, cannot be applied to just any mass distribution ; this is what this example demonstrates. However, the question is: how "liberal" can one be with the mass distributions before problems set in...

 

[Tomaz Kristan]

Some people already worry about the so called Seeliger's paradox. An infinite Euclidean universe uniformly filled with mass, leads to the infinite gravity force at every point ... and so forth.

http://www.google.com/search?hl=en&lr=&safe=active&q=seeliger's+paradox&btnG=Search

 

[reilly]

Physics can't have a practical problem. It is, how it is. A physics theory can be flawed.

The pure Newtonism is.

A bit hard to follow. What does your first sentence mean? Strikes me that physics is full of practical problems -- designing and building particle accelerators or electron microscopes, ballistics, ...

Not only can physics theories be flawed, but physicists themselves as well.

But, in spite of any formal problems, Newton and Maxwell seem to work just fine in our everyday world -- airplanes, computers and the internet, radio and TV, highway design, ... That's good enough for most physicists. What's the problem? Does it have bearing on how we use physics in our everyday world, or in getting tenure?

Curious minds want to know?

Regards,
Reilly Atkinson

 

[Tomaz Kristan]

A bit hard to follow. What does your first sentence mean?

Physics in reality, can't harbor a paradox inside. A theory can and is then wrong in a formal sense. And not in accordance with the real world.

 

[Lonewolf]

Physics in reality, can't harbor a paradox inside. A theory can and is then wrong in a formal sense. And not in accordance with the real world.

You do realize that your set up of masses couldn't exist in the "real world", don't you?

 

[Tomaz Kristan]

You do realize that your set up of masses couldn't exist in the "real world", don't you?

Yes, I do. I've mentioned this, too.

 

[reilly]

Physics can't have a practical problem.


The pure Newtonism is.

As the great Count Basie once said: One more once. I don't get it, what kind of practical problems are you talking about? Why can't physics have them?

Please, give us an answer.

Regards,
Reilly Atkinson

 

[Tomaz Kristan]

I don't get it, what kind of practical problems are you talking about? Why can't physics have them?

A problem is always in inadequate theory and only there. I realize, you can't have this construct in practice, something is wrong only inside the Newtonian abstraction of the real world. NOT in the real world.

Okay now?

 

[ObsessiveMathsFreak]

A problem is always in inadequate theory and only there. I realize, you can't have this construct in practice, something is wrong only inside the Newtonian abstraction of the real world. NOT in the real world.

Okay now?

Prove it. You still haven't. All we're getting is handwaving. You haven't even put down one single equation in over 250 posts.

Is it so much to ask that you come up with the equations of motion for your system. Something similar to post 136? Because all we're getting, and all we have been getting since about then from your posts is denial. Please present a valid list of equations(not prose).

 

[Tomaz Kristan]

OMF,

Why should I?

You understand, or you don't. You are on denial, or you aren't.

What would change, a bunch of fancy expressions?

 

[reilly]

A problem is always in inadequate theory and only there. I realize, you can't have this construct in practice, something is wrong only inside the Newtonian abstraction of the real world. NOT in the real world.

Okay now?

No. Why? Bye.
Regards,
Reilly Atkinson

 

[Tomaz Kristan]

No. Why? Bye.

Is this is the way, how the problem will go away?

 

[reilly]

What problem? Oops, my big mouth took over.

If it is such a big and profound problem. write a paper. Or, with all due respect, take yourself less seriously.

 

[Tomaz Kristan]

So you say, there is no problem here?

Or you say there is?

What's your position, really?

 

[George Jones]

I don't know how relevant this is, but the astrophysicist John Barrow, in his book Impossibility: The Limits of Science and the Science of Limits, writew

Time and time again, the development of our most powerful theories has followed this path: ... then something unexpected happens. The theory predicts that it cannot predict: it tells us that there are things that it cannot tell us. Curiously, it is only our most powerful theories that seem to possesses this self-critical feature. ... I believe that we can expect to find more of these deep results which limit what can be known.

He has singularities in general relativity, probabilities in quantum theory, incompleteness in formal systems, and unknown aspects of new theories in mind when he makes these comments.

 

[reilly]

Although I've spelled out my take on the matter above, I'll try once again. The primary issue is convergence of infinite series. As vanesch pointed out, the problem is one of conditional convergence -- Is it a paradox for a series to have different limits? -- Hurkyl has also been right on target. My take involves the difference between open and closed intervals, a distinction not commonly used in physics -- can't find them in nature. And, so far, no one has found a system with an infinite number of particles.

Consider: the boy is a dog. There are at least two meanings to this sentence, a problem characteristic of many languages, including math.

Paradoxes, no doubt, abound in physics. For classical mechanics more fruitful ground might be found in mechanics of continuous media, or in the chaotic difficulties in the mechanics of more than two bodies. Yours is not a paradox.

If you want to convince us, then get to the real issue of convergence -- the normal equal and opposite property of central forces does not hold for the closed interval. Thus the closed interval version has nothing to do with physics.

Regards,
Reilly Atkinson

 

[Tomaz Kristan]

And, so far, no one has found a system with an infinite number of particles.

Of course not. But inside the Newton's abstract world, you have them as many as you wish. Infinity of points is 100% legal.

Paradoxes, no doubt, abound in physics.

They are forbidden. As everywhere. Something may have a "paradox" in name, but no real paradox is allowed.

Yours is not a paradox.

Says you. But have no idea, how the explain it.

If you want to convince us

Well, I don't care for everybody, if she is convinced or nor. I care for a possible hidden error in my line of thinking. No one has been able to show me one. This is a valuable information for me.

the normal equal and opposite property of central forces does not hold for the closed interval.

We have an half open interval here. But this has no role at all.

Thus the closed interval version has nothing to do with physics.

Where did you get this one?

 

[Hurkyl]

Infinity of points is 100% legal.

Prove it.

(and not just that infinitely many points is legal, but any configuration of infinitely many points is legal)

And prove that the center of mass theorem holds for your configuration of points.

 

[Tomaz Kristan]

Hurkyl,

There are NO exceptions. Every finite mass configuration is legal, no matter how bizarre it may looks to someone.

And every finite mass has it's center of gravity.

You think not?

 

[ObsessiveMathsFreak]

There are NO exceptions. Every finite mass configuration is legal, no matter how bizarre it may looks to someone.

And every finite mass has it's center of gravity.

You think not?

You dodged the question. You were asked to prove that an "infinte" configuration of masses is legal. You've only referred to a finite configuration of masses. That doesn't of itself show anything about an infite configuration of masses.

You still haven't proven that an infinite number of particles is a legal configuration. And moreover, given that it is, you still haven't proven that the center of mass calculation is valid either, and even then you still haven't given the motion of the center of mass under the gravity forces you describe.

 

[Tomaz Kristan]

You still haven't proven that an infinite number of particles is a legal configuration.

One ball is legal, isn't it? Divisible more than a finite number of times. Inside Euclidean space, where Mr. Newton operates.

QED.

 

[vanesch]

Of course not. But inside the Newton's abstract world, you have them as many as you wish. Infinity of points is 100% legal.

It is not the infinite number of points in itself that is a problem. It is the fact that your mass density diverges which is the cullprit. And not by itself, but in combination with a force law such as the gravitational law of Newton.
Now, before you say that a mass point has infinite density (which is true), it can often be replaced with a sphere of same mass and of finite density. This can be done each time when other mass points do not come nearer than a distance d in the problem.

The problem with infinite mass density is that the gravitational potential energy can diverge, from which moment onward anything goes. As such, a force law in 1/r^2 is NOT to be had in such a system. For instance, your system wouldn't (I think) give any problem if the interaction law were something like Hooke's law.

So, with a given force law (in casu 1/r^2), certain mass distributions are simply forbidden. In the same way, as with Hooke's law, mass points at divergent distances would be forbidden.

 

[Tomaz Kristan]

In the same way, as with Hooke's law, mass points at divergent distances would be forbidden.

Doe to the fact, the sum of divergent distances is not a finite distance. Can't fit into Euclidean space.

OTOH, 1/r^4 force between points is also legal. Since not illegal.

certain mass distributions are simply forbidden

Where and how and by whom or what?