Pythagoras' Theorem. Evidence of elsewhere or our ingenuity.

[JimC]

Where is the clear, definite, discrete boundary of a planet? Or a star?

In real planetary objects, there will not be the pure mathematical discreteness of a platonic sphere floating in absolute empty space. However, there is a continuous spectrum of "discreteness" from this platonic ideal (never to be seen in its absolute form in a real universe) to the smooth transition shown, for example, when one gaseous compound is in the process of diffusing into a second one.

On that scale, a planet is vastly closer to the discrete end of the spectrum - its tenuous atmosphere is very small beer compared to the surface boundary between it and a dense solid or liquid interior.

That's pretty close to where I'm going with it. But to say that the magnitude of the impact of the atmosphere is small is a subjective measure, I think.

It can be made pretty objective. A tooled up physics team could make a prediction about some aspect of a planets future trajectory with or without an atmosphere. I would bet that any differences would occur deep into the decimal places...

It's horses for courses. The fuzzy boundaries of planets with atmospheres are important (even vital) in some predictive scenarios, but irrelevant in others...

Anyway, consider a rocky asteroid. Sure, there will be a few surface atoms disturbed, moving out and in a little. But it is extremely close to an abstracted platonic example of a discrete object in space.

[Audley Strange]

Since we're talking countability, is discreteness a fundamental feature of anything in existence, or is it projection of how the brain/mind evolved to effectively analyze experience?

I was getting to it. Honest.

Remember a while ago there was some conversation in which I kept persisting that "universe" by definition was a totality, that no matter what you describe, parallel dimensions, the past, phone books, pornographic drawings of Linus from Peanuts, the letter P, all of it is part of one single thing?

Well there is your discreteness. However there is a problem in that the concept of universe as a unit or an object, is that A) we are part of that object
and B) every experience we have of it is dim and internalised. So it definitely is a projection. However is the projection correct, like a torch in a dark forest, or is it just us looking at hallucinations on the cave walls?

So if the totality is a singular unity, where is the discreteness? What's it discrete from?

The void. The Abyss.

Nothing, then?

Well, that remains to be seen. No thing perhaps.

[FBM]

Where is the clear, definite, discrete boundary of a planet? Or a star?

In real planetary objects, there will not be the pure mathematical discreteness of a platonic sphere floating in absolute empty space. However, there is a continuous spectrum of "discreteness" from this platonic ideal (never to be seen in its absolute form in a real universe) to the smooth transition shown, for example, when one gaseous compound is in the process of diffusing into a second one.

On that scale, a planet is vastly closer to the discrete end of the spectrum - its tenuous atmosphere is very small beer compared to the surface boundary between it and a dense solid or liquid interior.

That's pretty close to where I'm going with it. But to say that the magnitude of the impact of the atmosphere is small is a subjective measure, I think.

It can be made pretty objective. A tooled up physics team could make a prediction about some aspect of a planets future trajectory with or without an atmosphere. I would bet that any differences would occur deep into the decimal places...

It's horses for courses. The fuzzy boundaries of planets with atmospheres are important (even vital) in some predictive scenarios, but irrelevant in others...

Anyway, consider a rocky asteroid. Sure, there will be a few surface atoms disturbed, moving out and in a little. But it is extremely close to an abstracted platonic example of a discrete object in space.

OK, but an object's gravitational and electromagnetic fields are integral parts of the object, no?

[JimC]

The surface of the planets crust. Get's a little blured if the planet has an atmosphere, but its not important if its a few hundred kilometres off, when you look at distances of lightminutes, between planets (or lightyears, between stars).

Would airborne soil count as the planet? I'm thinking that the closer you look, the less discreteness you'll actually find. Is the atmosphere part of the planet? It's gravitational and electromagnetic fields? Are they not integral aspects of the planet's being? And a star...where's its clear cut-off point? You know much better than I that when you get down to the atomic level, things are quite fuzzy indeed. I'm sort of the impression that we make mental models of these things AS IF they had clear, definable boundaries, but that doesn't mean that they actually do.

Yes, there is a fuzziness, but it is very minor in its nature. A clear lack of discreteness would see an objects density change smoothly from centre to an infinite distance. However, in fact we see massive, sudden changes in density, if not to zero, then a very tiny figure indeed. The discontinuities are not absolute in a mathematical sense of some arbitrary figure for density changing to zero instantly, but that is not a requirement to treat them as discrete objects in many useful ways. Depending on the prediction required, one may or may not need to take into account the minor discrepancies that derive from the solar wind, magnetic fields etc.

I wouldn't disagree with any of that, but treating them as if they were discrete is not quite proving that they actually are so, is it?

It depends whether you require the meaning of the word "discrete" to be absolute. I would argue that this requirement is impossible for any descriptor (such as round, of straight, or flat, or discrete) of any object in the real physical world; it applies only to idealised platonic objects. Any object in the real world, when seen at a small enough scale, exhibits a quantum fuzziness at its boundaries. So, proof is going to be impossible if you insist on an absolute definition of physical descriptors. Instead, we need to assess the degree to which a given object approaches the platonic ideal one is considering.

[Audley Strange]

I do appreciate all this.

Re: Fuzziness, it would be because each component is still part of a larger single deterministic process right?

[FBM]

@ Jim: OK. Sounds good to me.

[Audley Strange]

The idea of a universal set is a handy way to think about things but doesn't work logically. Runs into several paradoxes.

Does the set of all sets contain itself, is that one of them?

[JimC]

FBM wrote:

OK, but an object's gravitational and electromagnetic fields are integral parts of the object, no?

Firstly, an uncharged object, containing no moving electrical charges, may exhibit no electric or magnetic fields at all (other than the extremely localised ones of its atoms)

Secondly, they may be integral to it in one sense of the word, but unimportant in others. The fields around an object will not do the same things as the object itself - they will not stop another object, they are not opaque to radiation, etc. The difference between the inside and the outside of the object is real and significant - it is not like the difference between a thick area of fog, and a slightly thinner patch, with an even gradient between them...

[Audley Strange]

JimC is on fire!

[FBM]

FBM wrote:

OK, but an object's gravitational and electromagnetic fields are integral parts of the object, no?

Firstly, an uncharged object, containing no moving electrical charges, may exhibit no electric or magnetic fields at all (other than the extremely localised ones of its atoms)

Secondly, they may be integral to it in one sense of the word, but unimportant in others. The fields around an object will not do the same things as the object itself - they will not stop another object, they are not opaque to radiation, etc. The difference between the inside and the outside of the object is real and significant - it is not like the difference between a thick area of fog, and a slightly thinner patch, with an even gradient between them...

Ah. My understanding was that atoms are 99.9999999999999% empty and that it is only their fields that repel other "solid" objects. Am I getting my facts out of whack? This obviously isn't my field. But it seems to me that electron shells are pretty foggy, once you abandon the outdated planetary model of the atom.

[MiM]

OK, but an object's gravitational and electromagnetic fields are integral parts of the object, no?

I think that is a very interesting question in itself, but in this context it can be avoided by restricting the discussion to discrete bodies of mass, with the additional points made by Jim.

[JimC]

I do appreciate all this.

Re: Fuzziness, it would be because each component is still part of a larger single deterministic process right?

Actually, I think not. I was talking of fuzziness at a quantum level; fundamentally, this is down to the fact that the physical parameters of a system such as the position of an electron at any given time are not absolute in a classical sense, but determined by a probability distribution. Thus, when you get to a sufficiently small scale, there is no absolute boundary to any object, which is only a property of an idealised platonic object.

The key point, however, that we can predict many things in the macroscopic universe to strikingly high accuracy levels without worrying about this level at all...

[FBM]

Well, I'm aware that I can annoy the fuck out of people with my questions, so I'll bow out and let you guys enjoy the discussion.

[JimC]

FBM wrote:

OK, but an object's gravitational and electromagnetic fields are integral parts of the object, no?

Firstly, an uncharged object, containing no moving electrical charges, may exhibit no electric or magnetic fields at all (other than the extremely localised ones of its atoms)

Secondly, they may be integral to it in one sense of the word, but unimportant in others. The fields around an object will not do the same things as the object itself - they will not stop another object, they are not opaque to radiation, etc. The difference between the inside and the outside of the object is real and significant - it is not like the difference between a thick area of fog, and a slightly thinner patch, with an even gradient between them...

Ah. My understanding was that atoms are 99.9999999999999% empty and that it is only their fields that repel other "solid" objects. Am I getting my facts out of whack? This obviously isn't my field. But it seems to me that electron shells are pretty foggy, once you abandon the outdated planetary model of the atom.

Sure, when you get down to that scale, they are indeed insubstantial. However, the outer boundaries of the electrons orbitals still represent fairy rapid discontinuities - there may be an extremely minute probability of an electron being outside its "typical" boundaries, but the probability distribution falls away very rapidly. It is not a smooth, "even" fuzziness, involving steady changes in probability density. The discontinuities are very real indeed....

[MiM]

I do appreciate all this.

Re: Fuzziness, it would be because each component is still part of a larger single deterministic process right?

I don't believe in determinism, and I have argued about it here before. Quantum randomness ultimately destroys determinism on all levels. I think I used the examples of radiation induced cancer and atomic bombs, but it also would make something like the exact moment for a star to go supernova impossible to predict. Add chaos theory to that, and the universe will never be like God imagined at the outset.