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Part one:

An important thing to remember when considering infinity is that the word is used in different ways by different people. It is very difficult to make any progress in discussing infinity unless one is sure that all those concerned are using infinity in the same way. We will start by looking at some of the ways in which people in different disciplines use infinity.

Mathematical infinities are in some respects the easiest to deal with, but in other ways are the most complex. Mathematicians tended to avoid infinities until Georg Cantor tackled them and tamed them. Cantor discovered a richness and variety in the infinite that surprised, and to some extent disconcerted his contemporaries. He succeeded in bringing infinities into mathematics.

Perhaps the most surprising thing, at least from the point of view of the non-mathematician, was that Cantor designated some infinities as “countable”. By this he did not mean that anyone could count to infinity, but simply that such an infinity could be placed in a one-to-one relation with the natural numbers. (The term for this one-to-one correspondence is a “bijection”) Two examples of countable infinities might be the odd and even numbers, as either of these sequences can be put into a one-to-one correspondence with the natural numbers.

Uncountable infinities are those that cannot be put in a one-to-one relationship with the natural numbers; such as the apparently infinite sequence of real numbers between, for example, 0 and 1, or 1 and 2.

Cantor’s infinities are mathematical “truths”, in other words, he was able to derive them from a set of non-contradictory axioms which maintained logical self-consistency. Such infinities, therefore, “exist” in the mathematical sense, but do not necessarily have any counterpart in the physical world.

Countable and uncountable infinities exist only in the minds of those mathematicians or scientists who are using them. In the physical realm it is impossible to produce an infinite number of material objects. The uncountable infinities may, at first sight, seem to offer a more concrete example, but consider the numbers between 1 and 2, in principle, it seems possible to continue producing smaller and smaller fractions for ever, but no finite, physical quantity could be divided infinitely. The object would have to be infinite to start with, which begs the question, and provides no real concrete example.

In mathematics it is acceptable to refer to a large number as “approaching infinity”. In fact this is not possible, because however large a number might become, it is still infinitely far from becoming infinite.

What about “physicists’ infinities”? This is an area in which the hitch-hiker is likely to find much confusion. To the physicist infinity is context related. For example; the context for an infinite universe would be that of confinement to the three dimensions of space and one of time that we experience. On the other hand, the context for a finite universe would be that of a universe from which it would be possible to escape. In quantum mechanics, entanglement is seen as a way of placing particles outside the Universe. Thus it can be “proved” that the Universe is both finite and infinite at the same time, depending on context. Perhaps, even more confusing for the non-cosmologist, is the fact that in this context the Universe might be considered as being infinite in one direction, but not in another.

Once one has grasped the various ways in which the term “infinite” is used it becomes possible to work with it in much the same was that many scientists work with quantum theory, simply by accepting that it works, without asking how, or why. Infinity becomes a tool which can be used to express ideas and concepts. The physicist can then say “Beyond this, we are not dealing with science”.


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First query that comes to mind:
A fundamental constituent of matter (quark or electron) would seem to be a force in a spatial probability distribution. May it not be true that while no meaningful direct measurement is possible below the Planck length, spacetime is not granular and divisibility of those spatial regions is actually infinite?


"Time is what prevents everything from happening at once" - John Wheeler
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The possibility of the infinite division of spacetime is very real, as you say, it may not be granular.

I would offer the following thoughts on that subject:

1. There seems to be increasing reason to believe that spacetime might be granular, so that possibility has to be taken seriously.

2. QM appears to be the “theory of the very small”; why should spacetime be exempted from its division into quanta?

3. If, below the Planck length, measurement is no longer meaningful, does this not take sub-Planck measurement out of the realm of science?

BTW, in Part 2, I propose to look a little more at the scientist’s idea of infinity, then, perhaps, venture into the realm of the philosopher.


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And Bill S that I would say is a fairly accurate portial of that argument ... thats gets my vote.

When we go the other way and look at the expanding universe thats where it all gets interesting :-)


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If Orac approves, I'm obviously not being controversial enough. I shall have to do something about that. smile


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You really want that "prize" don't you :-)

Last edited by Orac; 11/02/11 02:40 AM.

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I think that the subject the BS has selected has a great significance in physics. We have to distinguish between mathematical and physical infinities. So I will ask the same question that I have asked in another thread. Can there be an infinite number of finite things? Can a finite thing be divided into infinite pieces? I think both are impossible.

If space-time is granular, then it should be finite and also it will not be possible to divide it into infinite no of pieces (it is just granular). If it is not granular, then it will be infinite and can be divided into infinite number of pieces.

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Originally Posted By: F
If it is not granular, then it will be infinite and can be divided into infinite number of pieces.


That's the bit I have trouble with.


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Originally Posted By: O
You really want that "prize" don't you


Any prizes going; I'm up for them. smile

Part 2 is almost ready, not very controversial, I'm saving that for part 3. laugh


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Part 2

Trying to look at infinity from the point of view of a physicist, it becomes clear that it is something that needs to be worked with. One way of doing that is to introduce the idea of sets. In set theory it seems to be permissible to accept that there could be an infinite set which contains all possible finite sets. Given this scenario, it becomes possible to work with any of the finite sets, acknowledge that each is part of an infinite whole, but not actually have to include the infinite in any calculations or conclusions.

Philosophers have wrangled for many centuries, not just about the nature of the infinite, but also about whether or not such a thing existed. Giordano Bruno (1548 – 1600) was a Dominican Priest. Although he was neither a scientist nor a mathematician, he firmly believed that the Universe was infinite. He was sufficiently unwise to commit his beliefs to writing, which brought about his torture and death. Possibly the scientific world owes more to Bruno than is at first obvious. Galileo was acutely aware of the treatment of Bruno by the Inquisition. Had he not been, he might have been less cautious and could have suffered an earlier and more severe fate.

Galileo said: “…we attempt, with our finite minds, to discuss the infinite, assigning to it properties which we give to the finite and limited; but I think this is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.

Cantor disagreed; he developed a theory of different sizes of infinity. All countable infinities were the same size, and represented the smallest infinity, designated by the Hebrew letter “aleph” with a subscript zero (said: aleph-nought). The uncountable infinities, on the other hand were not just numerous, there was an infinite number of them.

Thomas Aquinas, in the 13th century, seems to have specialised in “proofs” of all kinds of things, usually of a theological nature. However, his “proof” of the unreality of an infinity of material objects, which sounds like an early version of set theory, held its ground until Cantor took it in hand.

Aquinas said: “…any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.

Immanuel Kant did not like the idea of an actual infinity. He said: “….in order to conceive the world, which fills all space, as a whole, the successive synthesis of the parts would have to be looked upon as completed; that is, an infinite time would have to be looked upon as elapsed, during the enumeration of all coexisting things.

Aristotle had argued against the actual infinite, and had replaced it with the “potential infinite”. It says something for Aristotle’s influence that this idea dominated thinking for two thousand years, and can still be found in some circles today. However, although Aquinas had used the idea of sets to argue against actual infinity, it was the defining of a set that actual made infinity a reality. That came about more or less like this: it had been argued that the integers were not infinite because they could never be presented as anything other than a finite quantity. The definition of a set, however, presented the integers as a single entity – a set – which, as such could be considered infinite. Thus it was deemed possible to have an infinite set of finite sets or objects. Of course, such an infinite set is still only a mathematical infinity, which has existence only in the mind.

Ironically, it was Cantor who cast some doubt on this idea of an infinite set. He discovered that infinities are insuperable. He worked out that a never-ending ascending hierarchy of infinities must exist. There was no overarching infinite set of infinities. Although this does not prove that an infinite set of finite objects cannot exist, it must ask a question.


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Originally Posted By: F
If it is not granular, then it will be infinite and can be divided into infinite number of pieces.


When you say “infinite” I assume you mean “smooth”, or perhaps you mean smooth and infinite.

If it is smooth and infinite, there would seem to be no limit to the number of times it could be divided; but the same considerations would apply to this as to Rede’s query, above.


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I am going to call this old school scientific rejigging of infinity is discussed in the article below.

The title is wrong "The hocus pocus that made quantum theory work" it should be "The hocus pocus that made QED theory work" as QED is a small historic part of QM but thats a small issue.

http://www.newscientist.com/blogs/culturelab/2011/11/how-we-travelled-beyond-infinity.html

As per our other thread this is back in 70's physics when we had finiters solid little partciles whizzing around a nucleus view of the atom.

The silly part was they even realized the problem look at the backdrops of there own comments.

(http://en.wikipedia.org/wiki/Quantum_electrodynamics)

Quote:

In time this problem was "fixed" by the technique of renormalization (see below and the article on mass renormalization). However, Feynman himself remained unhappy about it, calling it a "dippy process".


Quote:

Conclusions

Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as the anomalous magnetic dipole moment. However, as Feynman points out, it fails totally to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem."


But it gave brilliantly predictable results so those little solid partciles grew a following and were taught to people including finiter :-)

It still stuns me that the idea of a solid particle was somehow so believable given all the problems with it.

I ask myself over and over how did science convince itself that particles were real and physical what is so "believable about them" that people will invent anything to make them so and theories work.

In the end I am forced to conceed we covet to our senses and we desperately want to be standing on "solid land" rather than standing on a solid surface that is actually a sea of microscopic waves. Solid is solid ... liquid is liquid our senses override our logic. I suspect thats why those supergels and "oobleck" fascinates kids (http://www.instructables.com/id/Oobleck/).

So I guess what you are asking Bill S is how did science come to renormalize infinity and the answer is not comforting.

Modern QM probably takes a much stricter stance with infinity and only allows it with context. A physical infinity can not exist.

Edit: I have been reading whats in the public domain about this, What I find most daming is in the Attitudes and interpretation section of (http://en.wikipedia.org/wiki/Renormalization). Dirac, Feynmann all understood the problem how could science teach the solid particle view of an atom it staggers me.

Edit: I am sure Bill S would have loved Diarc

Quote:

I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it!


Edit: I like and embrace this statement "If a theory featuring renormalization (e.g. QED) can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems", it is on those grounds we kill finiters little solid particles :-)

Last edited by Orac; 11/03/11 10:07 AM.

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For those interested in my view .. I firmly backed into the Nobel Laureat Steven Weinberg corner and was there before he won the prize :-)

To my mind he has the most concise and consistant view of things and its best to let him do the speaking for himself. Trust me if you have the time and inclination this is worth the read and is layman enough you should be able to follow.

http://www.nybooks.com/articles/archives/2011/oct/27/symmetry-key-natures-secrets/?pagination=false

The key points he picks up to do with this discussion are

Quote:

The origin of accidental symmetries lies in the fact that acceptable theories of elementary particles tend to be of a particularly simple type. The reason has to do with avoidance of the nonsensical infinities I mentioned at the outset. In theories that are sufficiently simple these infinities can be canceled by a mathematical process called “renormalization.” In this process, certain physical constants, like masses and charges, are carefully redefined so that the infinite terms are canceled out, without affecting the results of the theory. In these simple theories, known as “renormalizable” theories, only a small number of particles can interact at any given location and time, and then the energy of interaction can depend in only a simple way on how the particles are moving and spinning.

For a long time many of us thought that to avoid intractable infinities, these renormalizable theories were the only ones physically possible. This posed a serious problem, because Einstein’s successful theory of gravitation, the General Theory of Relativity, is not a renormalizable theory; the fundamental symmetry of the theory, known as general covariance (which says that the equations have the same form whatever coordinates we use to describe events in space and time), does not allow any sufficiently simple interactions. In the 1970s it became clear that there are circumstances in which nonrenormalizable theories are allowed without incurring nonsensical infinities, but that the relatively complicated interactions that make these theories nonrenormalizable are expected, under normal circumstances, to be so weak that physicists can usually ignore them and still get reliable approximate results.

This is a good thing. It means that to a good approximation there are only a few kinds of renormalizable theories that we need to consider as possible descriptions of nature.


And that is the context and key to your problems with infinity.

Last edited by Orac; 11/03/11 02:19 PM.

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Originally Posted By: O
So I guess what you are asking Bill S is how did science come to renormalize infinity and the answer is not comforting.


You're ahead of me there; renormalisation should rear its head in Part 3, if I get that far. As for how science came to renormalise infinity; it will be interesting to compare our answers.


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Quote:
Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it!


This raises an interesting question. Is there, in reality, any difference between infinitely small and infinitely great?


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Originally Posted By: Bill S.

This raises an interesting question. Is there, in reality, any difference between infinitely small and infinitely great?

Mathematicaly, there is difference. The space is infinitely small indicates there is no space; and space is infinitely great means the space is infinite.

IMO, space without matter represents the physical reality of nothingness. With or without matter, space is infinitely large, and not infinitely small. So an 'infinitely large space' is a physical reality. The same can be said about time. Both these are non-quantized, ie, space and time are are not grainy. But matter is grainy and made up of particles which are real in all respects. The number of particles can never be infinite because (physically) we cannot have an infinite number of finite particles. Thus it is the graininess or the quantum nature that decides whether a physical entity is finite or not (and we have only three entities).

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And therefore according to QM your theory is not the theory of everything as we state

"If a theory featuring renormalization can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems"

You require renormalization so therefore even if you were right you must by definition be only an approximation.


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Originally Posted By: Bill S.

This raises an interesting question. Is there, in reality, any difference between infinitely small and infinitely great?


Yes at science.

Infinitely small can hit a cut off boundary such as surface infinitely large implies there is no cutoff boundary and so they technically differ at science.

Dirac talked about that you can have something so infinitely small that you can throw it away and have a consistant theory and Plank distance is the obvious example.

But you cant have an infinitely big distance cutoff for example because you can't throw it away saying it has no effect.


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Originally Posted By: F
The space is infinitely small indicates there is no space; and space is infinitely great means the space is infinite.


This equates to answering the question "what's the difference between white and black?" with: "White is white because it shows no colour; black is black because it is black."

That's the "physicist's" answer isn't it; absolutely right, but completely useless. smile

However, let’s take a closer look at your logic.

The space is infinitely small indicates there is no space.

Space is infinitely great means the space is infinite.

We cannot have an infinite number of finite particles.

Space without matter represents the physical reality of nothingness.

Nothingness is the only thing of which we can have an infinite amount.

An infinitely large amount of nothingness = nothing.

An infinitely small amount of nothingness = nothing.

Ergo, infinitely small = infinitely large.


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Originally Posted By: O
Infinitely small can hit a cut off boundary such as surface


If it hits a cut-off boundary, how can it be infinite?

Quote:
you can have something so infinitely small that you can throw it away


Surely this is infinitesimal, not infinite.


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Originally Posted By: Bill S.
This equates to answering the question "what's the difference between white and black?" with: "White is white because it shows no colour; black is black because it is black."

That's the "physicist's" answer isn't it; absolutely right, but completely useless. smile


It's only apparently useless to you :-)

For science it answers our questions we need answered some of what you are seeking is outside the domain of science.

That which is outside science we care not about.


Originally Posted By: Bill S.

However, let’s take a closer look at your logic.
The space is infinitely small indicates there is no space.


Again we don't have a problem with no space. Its stuff thats smaller than plank distance it doesn't exist in our realm.

Again I think you are talking about well what is the stuff thats smaller than plank distance and we go you may never know why get hung up over it, it's not in our space we cant access it.


Originally Posted By: Bill S.

Space is infinitely great means the space is infinite.
We cannot have an infinite number of finite particles.


Again at science I have told you consistantly from QM we agree with you on that.

Space is only infinite to us in terms of observation and travelling ... physically it is most certainly finite.


Originally Posted By: Bill S.

Space without matter represents the physical reality of nothingness.


We disagree totally.

If you read the articles I linked it is quiet clear from QM eventually all matter must disolve based on radioactive decay. All matter seems to have a decay constant it is just very very large but not infinite.

That does not mean the universe ceases to exist just because matter doesn't exist it simply becomes a universe devoid of matter.

So if no forces conspire to kill the universe inbetween time QM would say the universe will continue until it dissolves into a empty matterless place.

What happens then or whats governs things from there at this stage that is outside the realm of science as we have doubts we can have physical access to test that realm.


Originally Posted By: Bill S.

Nothingness is the only thing of which we can have an infinite amount.

An infinitely large amount of nothingness = nothing.

An infinitely small amount of nothingness = nothing.

Ergo, infinitely small = infinitely large.




Thats all an esoteric argument.

The same argument exists in mathematics ... prove to me you have zero.

The argument goes you have +5 and -5 or you have +10 and -10 etc how can you prove zero.

Zero can occur by balancing infinite negative numbers versus infinite positive numbers.

So zero mathematically is also an enigma it can never be proved to exist as you don't have zero of anything.

In the end this all falls to occum's razor (http://en.wikipedia.org/wiki/Occam's_razor)

What you say may be true Bill S but it is unimportant from empircal studies point.

At science we apply the razor and simply say what you may well be true it is totally unimportant for the same reason as the mathematical use of the razor on defining zero, your argument leads nowhere and provides no workable framework.

You may find that uncomfortable ... so be it ... there are anti-razor groups.

Last edited by Orac; 11/04/11 03:07 AM.

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Originally Posted By: O
Again I think you are talking about well what is the stuff thats smaller than plank distance and we go you may never know why get hung up over it, it's not in our space we cant access it.


Planck distance is a finite distance. I'm not asking what is smaller than Planck distance. All I am saying is that if something is infinitely small it does not exist in the material world.

Quote:
Space is only infinite to us in terms of observation and travelling ... physically it is most certainly finite.


The first part requires a very specific definition of "infinite", which I would not argue with, as long as we are clear about the context. I see no reason to argue with the second part.

Quote:
That does not mean the universe ceases to exist just because matter doesn't exist it simply becomes a universe devoid of matter.


Presumably this universe is not devoid of energy, so it is not nothingness.

Quote:
Zero can occur by balancing infinite negative numbers versus infinite positive numbers.


Undoubtedly this is mathematically true, but since you can never demonstrate that you have a positive or negative infinite number of anything, the argument is pointless in reality.


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Originally Posted By: Bill S.

Planck distance is a finite distance. I'm not asking what is smaller than Planck distance. All I am saying is that if something is infinitely small it does not exist in the material world.


Then you agree with science .. plank distance is a bounding condition for us.


Quote:

Presumably this universe is not devoid of energy, so it is not nothingness.


Correct at science we would simply a universe devoid of matter and full of energy and possibly dark energy and matter if such things exist.


Quote:
Undoubtedly this is mathematically true, but since you can never demonstrate that you have a positive or negative infinite number of anything, the argument is pointless in reality.


But I can have +5 and -5 and +10 and -10.

I urge you to read (http://en.wikipedia.org/wiki/0_(number))

It evokes the same problem

Quote:

Records show that the ancient Greeks seemed unsure about the status of zero as a number. They asked themselves, "How can nothing be something?", leading to philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum


See I assure you it is exactly the same argument ... infact I did this exact dialog with my mate socratus (Israel Sadovnik Socratus).

I have seen this exact argument many times and it dates back to the medieval times.

Infinity in many ways is just a more modern version of the same dance over zero.

Hence the solution for science is that infinity can only be valid with a given context and zero falls to occum's razor it means the lack of something.

Last edited by Orac; 11/04/11 09:22 AM.

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Originally Posted By: Orac
And therefore according to QM your theory is not the theory of everything as we state

"If a theory featuring renormalization can only be sensibly interpreted as an effective field theory, i.e. as an approximation reflecting human ignorance about the workings of nature, then the problem remains of discovering a more accurate theory that does not have these renormalization problems"

You require renormalization so therefore even if you were right you must by definition be only an approximation.


My theory does not require re-normalization. The physical world comes into the picture only when matter is present. For matter, everything is finite. Even the number of universes is finite, but with an arbitrary limit (may be one or very very large number). Thus basically, my theory qualifies as a candidate for the theory of everything.

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Originally Posted By: Bill S.

That's the "physicist's" answer isn't it; absolutely right, but completely useless. smile
....
Ergo, infinitely small = infinitely large.

I agree with you that the answer is useless as far a physical world is concerned. Whether the space without matter is infinitely large or infinitely small, it is physically nothing and so of no use to us. Whether both are equal or not is a question that has no logical answer. Science requires logical answers, and so science cannot answer that question. We have to leave it to the philosophers.

However, once we have matter (which is grainy), a physical world just exists; everything connected with the physical world is finite. Science is expected to answer the structure and the interactions (of this physical world) at all levels, starting from the quantum level to the cosmic level. How space, time and matter (with their respective qualities) came into existence is beyond the scope of scientific explanation. That is, there will always be something arbitrary, and you have to say 'white is white and black is black' because they are so. This arbitrariness will be the minimum in a real 'theory of everything in physics'.

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Originally Posted By: finiter

My theory does not require re-normalization. The physical world comes into the picture only when matter is present. For matter, everything is finite. Even the number of universes is finite, but with an arbitrary limit (may be one or very very large number). Thus basically, my theory qualifies as a candidate for the theory of everything.


It does you just don't realize it if your using standard unlike charge attraction theory or do you have a does your theory have a different way on how that works?
.

Last edited by Orac; 11/05/11 12:26 PM.

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Part 3

So far we have looked very briefly at infinity as it is used by mathematicians, scientists and philosophers. For the present, at least, I propose to let the philosophers’ views take a back seat, and try to concentrate on the more scientific aspects of the infinite.

In general, scientists seem to be quite comfortable with the ways in which they use infinity, and if these definitions serve the purpose of making scientific calculations and theorising easier and more straightforward, as long as long as they do not lead to errors, I am very happy to leave them alone. There is no need to look for any reality that might underlie QM in order to use it; similarly, there is no need to look for deeper meanings of infinity if the “shut up and work” ethic is all that is needed in a given situation. However, for those of us who neither work in science, nor shut up readily, questions must remain.

When considering infinity, one of the major questions that must be asked relates to the nature of the Universe. Is the Universe finite, or infinite? I have already raised the possibility that it might be both, but this possibility needs to be examined rather more closely. If we accept the Big Bang theory, it looks as though the Universe must be finite; it had a beginning when it was infinitesimally small, but it was, never-the-less, finite. However much it grows, it can never become infinite. Would that it were that simple! There are those in scientific circles who propose that the Universe was infinite, even when it was small. On the face of it this seems self-contradictory, but we must remember that, to the physicist, infinite may simply mean that there is nothing outside it, or even that there is no limit to the extent to which it can grow. However, outside that specialised usage, we have to ask if that is what infinity really means.

In visualising a Universe that grows from a miniscule speck to the vastness that we observe around us it is easy, especially for the non-scientist, to picture a sphere with a clear boundary. Today, most cosmologists accept that the Universe has no boundary. Does this mean that they think it is infinite? Apparently it does not. Current thinking is that the Universe is a hypersphere with a very slight (perhaps imperceptibly slight) positive curvature. If this is the case, it could be finite, but have no boundary. Scientifically, there is no reason to argue that a finite space must have a boundary.

What is a hypersphere? The simplest explanation is that it is the 4-dimensional analogue of our familiar 3-D sphere. The surface of a 3-D sphere is often cited as an example of a finite surface that has no boundaries. This is fine as an example of something that is finite and boundaryless, but, unfortunately, many popular science books describe this as infinite; which, manifestly, it is not. The surface of a sphere is a 2-D entity which requires a 3-D object for its existence. It follows that the surface of a hypersphere must be a 3-D entity that relies on a 4-D object for its existence. So, we are asked to accept a fourth dimension of space if the Universe is a hypersphere, but that may be the lesser of two evils, as we are about to consider.

Next we need to ask what the situation would be if the Universe were flat – absolutely flat. Would this mean that either it had a boundary, or it was infinite? Is there any way in which a zero-curvature, boundaryless space could be contained in a finite volume? Is it logically or mathematically possible? Unintuitive as the answer may seem; it is “yes”. This is because cosmology is a mathematical science, and space is represented by a mathematical concept called a differentiable manifold which does not have to be surrounded by a higher dimensional space. A manifold is a topological space that equates closely to Euclidian space when considered on a sufficiently small scale. A manifold is said to be differentiable when, on a sufficiently small scale, it is close enough to vector space to permit the application of calculus. Space simply has to look right in any location you might choose, and the locations have to be smoothly compatible where they join. By way of example; take a square sheet of paper; mark North and South at top and bottom, and East and West on either side. If you now glue the North and South edges together, and the East and West edges together in such a way as to form a torus, you have no boundaries. You might also object that your paper is no longer flat, but cosmologists assure us that there is no need to perform this difficult gluing job; all you have to do is specify the neighbourhoods at the East and West edges as if they were joined, and do the same for the North and South edges, and you will have a flat space which is finite, but has no boundaries. If you think this is taxing the credulity overmuch, you are in good company. It appears that very few cosmologists are working on this line of reasoning, and most of those who are are trying to establish that it is not representative of our Universe.

What about a universe that is both finite and infinite, depending on the point of view of the observer. As mentioned above, this would have to be a context related universe, which, generally speaking, is what we would be dealing with if we were talking to a physicist. The context may be existence within the Universe. The Universe is expanding, but we have no evidence that it is expanding into anything. As far as we can assess our Universe is all there is. We could never visit all parts of it, nor could we travel all round it, because of its vast extent, and the speed at which it is expanding. For all practical and observational purposes it might as well be infinite. Physicists are content to refer to this as being infinite. Once again, as long as we are aware of the context in which we are using the term, that is a legitimate use of “infinite”.

The same physicist might also describe the Universe as finite if he/she is discussing a topic in which something outside the Universe designates the context. The concept of the multiverse, for example, might provide such a context, as might certain concepts in QM. Again, provided the context is clear, this seems reasonable, but we are still left with the question as to whether this is all there is to infinity.


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Bill, in your first post you say,
Originally Posted By: Bill S.
Part one:... Once one has grasped the various ways in which the term “infinite” is used it becomes possible to work with it in much the same was that many scientists work with quantum theory, simply by accepting that it works ...
This prompts me to ask the following:
Various ways? You mean used in mathematics? And in physics?
Used? By whom?
Who are some of the "many scientists" who simply accept--I presume you mean blindly?--that quantum theory works?

You comment, "without asking how, or why." Why not? What is wrong with asking: How? And Why?

Bill, you add,
Quote:
Infinity becomes a tool which can be used to express ideas and concepts.
What kind of ideas and concepts?

You also add, "The physicist can then say,
Quote:
Beyond this, we are not dealing with science.
So then, what do you suggest? Bring in the clowns? Or is there room for philosophers and theologians?--some of whom may also be careful scientists.


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Originally Posted By: Rev
This prompts me to ask the following:
Various ways? You mean used in mathematics? And in physics?
Used? By whom?
Who are some of the "many scientists" who simply accept--I presume you mean blindly?--that quantum theory works?


"various ways….by whom?":
In mathematics; e.g. look at the work of Cantor.
In physics; look at Orac's contributions to this thread and in Is the physical world real?
It should not be necessary for me to point you to examples of the use of infinity by philosophers and theologians; you are probably better acquainted with those than I.

Quote:
I presume you mean blindly?


I think, if I had meant “blindly”, I would have said “blindly”.

Quote:
You comment, "without asking how, or why." Why not? What is wrong with asking: How? And Why?


What gives you the impression I was saying that anything was wrong with asking how and why?

Quote:
What kind of ideas and concepts?


How about singularities, infinite curvature, infinite speed, infinitely small particles and infinitely large space for a start?

Quote:
So then, what do you suggest? Bring in the clowns? Or is there room for philosophers and theologians?--some of whom may also be careful scientists.


Just in the two threads I mentioned, Orac has protested that aspects were “not science”. In your opinion, does that put him among the “clowns”?

There is plenty of room for philosophers and theologians, but simply because I acknowledge that scientists may express opinions about what is or is not, in their view, science, does not mean I am trying to exclude anyone.


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Originally Posted By: Bill S.
....
Just in the two threads I mentioned, Orac has protested that aspects were “not science”. In your opinion, does that put him among the “clowns”?

There is plenty of room for philosophers and theologians, but simply because I acknowledge that scientists may express opinions about what is or is not, in their view, science, does not mean I am trying to exclude anyone.
Bill S:
Please accept my apology if in my questioning I came across like as if I think of myself as a prosecuting attorney. My intention was to get my own understanding and own thinking clear, not to question the sincere thoughts of others.

It is my sincere hope that the day is not all that far off when the borders between body/mind/spirit--matter, thought and will--will no longer be there; that we will soon be able to travel from matter, to mind, to spirit like we can now travel from water, to land, to air and even to moon.


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I am happy to be a clown whistle

I am a QM scientist these days we have thick hides ... goes with the territory.


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Absolutely no need for an apology, Rev, a lively verbal exchange tends to clear cobwebs from places where cobwebs shouldn't be.


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Part 4

It is easy to think of renormalisation as just a device for getting rid of troublesome infinities in QM, but, as usual in modern science it is not as straightforward as that. In classical electrodynamics a problem occurred because – for example – the electron was considered as a point particle, but the electromagnetic field associated with it had to be described by a more continuum-like field theory. Essentially, this problem was overcome in QED, because both could be treated as quantum fields. However, a related problem remained, because if calculations try to take into account a particle’s reaction to its own EM field, run-away solutions are encountered which go to infinity. These infinities are embarrassing because they make nonsense of the calculations. Renormalisation involves some complex calculations and clever mathematics which effectively replace the infinities with finite physical values for the quantities involved. Although, in some scientific circles, renormalisation is regarded with considerable suspicion, it seems to be an essential process if physicists are to make any progress with important calculations in QED. Practically the process seems to work and has generated predictions that have been confirmed to a high level of accuracy. It may be just a mathematical device, but some wiggle-room has to be allowed, in the same way that it is considered acceptable to invoke extra dimensions in order to make theories work. The important lesson to take from this is that infinities that are not constrained by specific contexts, or limited by mathematics do not fit into finite, physical calculations. Every infinity that finds its way into scientific endeavour must be, in one way or another, “renormalised”.

Have we reached a point where we have to accept that any non-renormalised infinity is outside the scope of science? I think not. A final point we have to consider before plunging into the complexities of the infinite is the nature of nothing. It is often said that the Universe came from nothing, but when one looks more closely at this “nothing” it inevitably begins to look very much like “something”. Like infinity, nothing seems to be quite a flexible term in science, especially in cosmology. For the sake of clarity, I must stress that when I refer to nothing I do not mean some quantum fluctuation of the vacuum, or anything else that might turn out to be something; I mean absolutely nothing.

I am going to start with an assumption. Obviously any assumption must be open to challenge, but without making some basic assumptions scientific progress would stagnate. The assumption is this:

There can never have been a time when there was nothing; otherwise there would still be nothing now.

In other words, something has always existed. Asking what that something might be is probably a shortcut into philosophy, but there is no need, at least at this juncture, to ask that question. All we need to do is to accept that something has always existed – something is eternal. We need to start by looking at the nature of eternity and how we might try to understand it. The popular image of eternity is that it stretches infinitely far into the past, and infinitely far into the future, but can this really be how it is? It is often said that in eternity, anything that can happen will happen, an infinite number of times. Let’s try to equate that with our image of eternity. Eternity stretches infinitely into the past; so an infinite amount of it is behind us. During that past time, everything that can happen will have happened, an infinite number of times. If everything that can happen has already happened an infinite number of times, how can these things happen again? How can things happen more than an infinite number of times? Here we have a paradox. If we regard eternity as linear, things have already happened an infinite number of times. An infinite number must, surely, be unsurpassable, yet there is still an infinite number to come. Undoubtedly there would be a way round this if we were dealing with mathematical infinities, but here we have a physical infinity. If we need a context for it, that must be the absolute need for something to be eternal. Am I confusing things by using “eternal” and “infinite” as though they were interchangeable? Is something that is infinite necessarily eternal? Until someone can demonstrate a mechanism by which something finite can become infinite, we are forced to conclude that anything infinite must always have been infinite, and must, therefore, always have existed. This brings us to a logical conclusion:

Nothing can become infinite; therefore any infinite thing must always have existed, and always been infinite.

Can there be change in infinity? As long as we regard eternity as being a beginningless, endless procession of time it is quite reasonable to accept that change can occur within “subsets” of that eternal time. However, we have already looked at a major problem involved in regarding eternity that way. The problem does not go away if you try to divide eternity into subsets. Because there must already have been an infinite number of subsets, and every change that can happen in every one of those must already have happened, an infinite number of times, again we are faced with the problem of having to add to that unsurpassable infinity. The only reason that time seems to us to have any connection with eternity is that we are creatures of time, and the only way in which we can come near to visualising eternity is to imaging it as an endless expanse of time. We coined the word “eternity” to describe that limitless expanse of time, but our understanding needs to evolve. A similar thing has happened with the word “Void”, which originally meant a vast expanse of nothing, but is now understood as a seething mass of energy and virtual particles. We are unable to imagine non-linear “time”, and seem to have become the more so as our science, technology and sophistication have progressed.

In the same way that it makes no sense to try to divide eternity into portions of time, so it makes no sense to try to divide infinity into a collection of finite objects. At first sight, it seems logical to suggest that infinity might consist of an infinite number of these finite objects. However, as we considered earlier, it cannot have become infinite, so it must always have been an infinite number of finite things. We need to build change into our model, so finite, material objects must be subject to change. If this infinite collection of objects has always existed, then every possible change must have happened an infinite number of times, so we are back with the old familiar problem. We seem to have reached another logical conclusion:

Eternity is not a vast expanse of time. Infinity is not an extremely large number.

It is probably time to return to the question I side-stepped earlier on the grounds that it might involve philosophy, rather than science. The question was asking what the something might be that has always existed. The non-speculative answer must be “everything that exists now”.

This leads to the question: Have I “painted” myself into a corner? I am proposing an infinite cosmos in which there can be no passage of time, and therefore no change; yet we live in a finite Universe in which we observe constant change. I suspect that there are those who would object that any speculation along these lines would be outside science, and such may be the case, but having made a proposal it seems necessary to defend it. That means there will have to be a “Part 5”, but, hopefully, I can keep that short.


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Two new readings for you

Rev's going to love the first

“God exists since mathematics is consistent, and the devil exists since we cannot prove it.”
(http://steiner.math.nthu.edu.tw/disk5/js/history/infinity.pdf)


And finally .. this is another Bill but for the last name it could be you :-)
(http://scienceblogs.com/stoat/2011/09/galileo_on_infinity.php)

Last edited by Orac; 11/08/11 07:02 AM.

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Originally Posted By: Bill S.
If everything that can happen has already happened an infinite number of times, how can these things happen again? How can things happen more than an infinite number of times? Here we have a paradox.

Not really a paradox, Bill. That infinite number isn't a fixed number. You can add to it an infinite number of times.

http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

"These cases demonstrate the 'paradox', by which we mean not that it is contradictory, but rather that a counter-intuitive result is provably true: The situations "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.

Some find this state of affairs profoundly counterintuitive."

Originally Posted By: Bill S.
Am I confusing things by using “eternal” and “infinite” as though they were interchangeable?

No confusion, Bill. There are two dictionary definitions of the word eternal:
(1) time without beginning and without end
(2) timeless.
Clearly, we're discussing (1)


"Time is what prevents everything from happening at once" - John Wheeler
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Originally Posted By: Rede
That infinite number isn't a fixed number.


I have to go a step further and say: That infinite number is not a number at all. If you regard it as a number, it is only a mathematical infinity. If it were truly an infinite number, it would have to encompass all the numbers that could exist, which is manifestly nonsense. Outside of mathematics, infinity is not a number.

Hilbert’s Hotel is something I have given quite a lot of thought to. It is clever, but it works only if you apply mathematical principles to infinity. You have to treat infinity as though it were a very large, flexible number, which it is not.

Quote:
(1) time without beginning and without end (2) timeless. Clearly, we're discussing (1)


There are two definitions because eternity is used in two distinct ways.
(1) is equivalent to mathematical infinities.
(2) is the non-mathematical definition and is equivalent to real infinity.


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You know you would have been burned at the stake if rede and I had our way.

http://www.youtube.com/watch?v=yp_l5ntikaU


BEDEVERE:
Tell me. What do you do with witches?
VILLAGER #2:
Burn!
VILLAGER #1:
Burn!
CROWD:
Burn! Burn them up! Burn!...
BEDEVERE:
And what do you burn apart from witches?
VILLAGER #1:
More witches!
VILLAGER #3:
Shh!
VILLAGER #2:
Wood!
BEDEVERE:
So, why do witches burn?
[pause]
VILLAGER #3:
B--... 'cause they're made of... wood?
BEDEVERE:
Good! Heh heh.
CROWD:
Oh, yeah. Oh.
BEDEVERE:
So, how do we tell whether Bill S is made of wood?
VILLAGER #1:
Build a bridge out of him.
BEDEVERE:
Ah, but can you not also make bridges out of stone?
VILLAGER #1:
Oh, yeah.
RANDOM:
Oh, yeah. True. Uhh...
BEDEVERE:
Does wood sink in water?
VILLAGER #1:
No. No.
VILLAGER #2:
No, it floats! It floats!
VILLAGER #1:
Throw Bill S into the pond!
CROWD:
The pond! Throw Bill S into the pond!
BEDEVERE:
What also floats in water?
VILLAGER #1:
Bread!
VILLAGER #2:
Apples!
VILLAGER #3:
Uh, very small rocks!
VILLAGER #1:
Cider!
VILLAGER #2:
Uh, gra-- gravy!
VILLAGER #1:
Cherries!
VILLAGER #2:
Mud!
VILLAGER #3:
Uh, churches! Churches!
VILLAGER #2:
Lead! Lead!
ARTHUR:
A duck!
CROWD:
Oooh.
BEDEVERE:
Exactly. So, logically...
VILLAGER #1:
If... Bill S... weighs... the same as a duck,... Bill S is made of wood.
BEDEVERE:
And therefore?
VILLAGER #2:
A witch!
VILLAGER #1:
A witch!
CROWD:
A witch! A witch!...
VILLAGER #4:
Here is a duck. Use this duck.
[quack quack quack]
BEDEVERE:
Very good. We shall use my largest scales.
CROWD:
Ohh! Ohh! Burn the witch! Burn the witch! Burn him! Burn him! Burn him! Burn him! Burn him! Burn him! Burn him! Ahh! Ahh...
BEDEVERE:
Right. Remove the supports!
[whop]
[clunk]
[creak]
CROWD:
A witch! A witch! A witch!
BILL S:
It's a fair cop.
VILLAGER #3:
Burn him!
CROWD:
Burn him! Burn him! Burn him! Burn! Burn!...
BEDEVERE:
Who are you who are so wise in the ways of science?
ORAC AND REDE:
We are Orac and Rede, keepers of the sacred infinity.


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Looks like you've been weighed and found wanting to be a duck Bill (hehe, sorry, couldn't resist that).


"Time is what prevents everything from happening at once" - John Wheeler
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My wife says: if I weigh the same as wood, does that mean I'm a block-head?

I just shouted "Duck!!!"


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Orac, that's the best link you have posted; the science is impeccable!


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BTW; how many examples can you find of scientific concepts for which people were burned at the stake that didn't turn out to be right. smile


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Well now. Bill S. is a duck. Bill S. is wood. Therefore Bill S. is decoy duck. Wooden decoys can be valuable. We need to see how much we can sell him for on EBay.

Bill Gill


C is not the speed of light in a vacuum.
C is the universal speed limit.
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Quote:
Wooden decoys can be valuable.


Therefore Bill S. could be valuable; best not burn him at the stake.


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Originally Posted By: Bill S.
Part 4
There can never have been a time when there was nothing; otherwise there would still be nothing now.

Nothing can become infinite; therefore any infinite thing must always have existed, and always been infinite.

Eternity is not a vast expanse of time. Infinity is not an extremely large number.

The question was asking what the something might be that has always existed. The non-speculative answer must be “everything that exists now”.

I am proposing an infinite cosmos in which there can be no passage of time, and therefore no change; yet we live in a finite Universe in which we observe constant change.

I have been busy for a few days, and so did not respond to your posting. I do, in fact, agree with all the above statements that I have quoted from you.

IMO, there is nothing unscientific in your assumptions. The 'infinite' part may be left to the philosophers to speculate, and the finite part, to the scientists to explain logically.

Adding a little more (based on my theory): The infinite space and the infinite time remain as the arena. A 'very large number' of universes exist in the infinite space. Taken as a whole, the 'Ensemble', containing 'the billions of universes' does not change with time. Each universe is a single grain of matter; it is finite in space and remains pulsating (has a finite period of pulsation). During pulsation, changes happen within the universe, but it always remains as a 'system of masses'. During expansion, masses change from hot to cold (I propose cold energy), and during contraction, the masses change from cold to hot.

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Originally Posted By: F
IMO, there is nothing unscientific in your assumptions. The 'infinite' part may be left to the philosophers to speculate, and the finite part, to the scientists to explain logically.


I like the first sentence. smile

I have a few problems with the second bit. If they are not addressed in "Part 5", we can come back to them.

I said I would try to keep Part 5 short - some hopes - but with any luck it will not be infinite!


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Part 5

We have to consider that if we divide or multiply infinity by anything, or if we add anything to it, or subtract anything from it, it still remains infinite. This line of reasoning may seem to militate against anything I have said about the division of infinity leading to the production of finite parts. However, all it really does is establish that infinity cannot be subjected to mathematical processes. Of course, it can be argued that mathematical infinities are quite at home in mathematical processes, but I would contend that this just serves to underline the profound difference between mathematical infinities and physical infinity.

We need to look at the idea of performing any mathematical functions on a physical infinity. If it is truly infinite it is the sum total of all that exists. We cannot add anything to it, because there is nothing to add. We cannot subtract anything from it, because there is nowhere outside it that we could take anything. To talk of multiplying or dividing such an infinity is simply a conceptual exercise that has no real meaning.

Another question presents itself here: Outside of mathematics, does it make sense to talk of an infinity of any specific thing? Elsewhere I have reasoned that it makes no sense to talk of an infinite number of finite objects. What about infinite space, infinite time, infinite matter or infinite energy? It might seem reasonable to argue that it is not possible to have any one without all the others. For example, it would not be possible to have infinite matter without infinite energy (because E=mc^2), then you would need infinite space in which to accommodate it, and infinite time in which it would have to exist, since it must always have been. Perhaps Galileo was right, perhaps our finite minds are poorly equipped to grasp the concepts of the infinite. However, this should not stop us from trying.

Because Hilbert’s Hotel is a firm favourite with those who seek to gain a hold on infinity, or just to perform mathematical “tricks” with infinity, it needs a brief mention. Hilbert’s Hotel is a hypothetical hotel with infinitely many rooms, all of which are occupied; obviously by infinitely many guests. Another guest arrives looking for a room. A typical example of the way in which this guest is accommodated is found at: http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel.

“Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.”

This is little more than a verbal “conjuring trick”, if n = infinity, Hilbert’s trick works because we never have to consider what happens to the nth guest who is moved out of the nth room. Obviously there cannot be an nth room containing an nth guest, because we can never count to infinity, nor can there be a room beyond the nth room, so the whole thing is nonsense. If there were a last room into which the last guest could be moved, the new guest could be put in that room and the infinite shuffling would be totally unnecessary. It might be argued that this is a thought experiment, so it does not need to be practically possible. Even if one accepts Terry Pratchett’s (1989) definition of a thought experiment, that it is: “One that you can’t do, and which won’t work”, it has to bear some relevance to the real world. Hilbert’s Hotel falls far short of that. By what lunacy would anyone suggest that an infinite hotel could fit into a finite universe? What is an infinite number of people? If you already had an infinite number of people in the hotel, where would another one come from? (Forget mathematics; think about a real situation).
(BTW, Pratchett’s explanation, in this same book, of the “Schrödinger's cat” thought experiment is a “must read” for anyone with an interest in quantum physics, and a sense of humour).

The concept of infinity could never be kept permanently out of the realms of mathematics, and rightly so. I often think it is a shame that they didn’t change its name when they turned it into something that could be used in calculations. As we saw when considering renormalisation, anything other than a “doctored” infinity makes nonsense of mathematical calculations.

Two of the eminent mathematicians who, before Cantor, attempted to tame the infinite were Galileo and Leibniz. A good comparison of their respective approaches can be found at:
(http://steiner.math.nthu.edu.tw/disk5/js/history/infinity.pdf)
so there is no need for extensive repetition here. However, a few relevant extracts may help to clarify certain points. Galileo was quite clear that one could not progress from finite to infinite. “The transition cannot be realized step by step by continued divisions of the divisible, because such successive divisions do not lead to a last division.”

Regarding the nature of this transition Galileo held that: “When a terminated quantity proceeds to the infinite, ‘it meets with an infinite difference, what is more, with a vast alteration and change of character’. And indeed: The corresponding notions result from each other by logical negations: finite becomes infinite, divisible becomes indivisible, quanta become non-quanta”.

Significantly, Leibniz considered “the identification of an infinite number – on condition that there is such a number – with zero, and not, as Galileo had done, with unity”. Identification of infinity with unity would “assume the existence of an actually given infinite number”, which would be absurd.

A major advance arising out of Leibniz’s work was that he facilitated inclusion of indivisibles in mathematics. The way he did that is illustrated in his definition of straight lines as rectangles for the purpose of calculations of space. He lays “….the foundations of the method of indivisibles in the soundest way possible, and it supplies the proof of the method of indivisibles which enables us, ….. to find the areas of spaces by means of ‘sums of lines’. He explained these sums as sums of rectangles with equal breadths of indefinite smallness”. Obviously this is a sleight of hand trick which, while it works perfectly well in its appropriate context, did nothing to change Leibniz’s “lifelong rejection of the actual infinite in mathematics”.

A significant passage, relating to Leibniz’s work, from the article linked above is the following. “The demonstrations of the forty- five theorems that follow are based on this concept of infinitely small and infinite quantities, that is, quantities which are, according to his definition, positive, but smaller than any given quantity or larger than any given quantity. The decisive aspect is that they are quantities, fictive ones certainly, because they were introduced by a fiction, but quantities nevertheless. It does not matter whether they appear in nature or not, because they allow abbreviations for speaking, for thinking, for discovering, and for proving.”

It is not my intention to attempt to diminish the importance of Leibniz’s work. I have neither the mathematical skill, nor the wish to do that. I seek simply to show that infinitely small and infinitely large “quantities” are mathematical tools, and that Leibniz was well aware of that fact. His own assertion that: “We need not take here the infinite in the strict sense of the word” is probably all the testimony we need in that regard.

The way in which circuitous arguments can develop when trying to merge infinity and mathematics is illustrated in this quote from the linked article: “Hence a contradiction with the Euclidean axiom on the part and on the whole seems to emerge, because a part equals the whole. This result, however, was absurd in Leibniz’s opinion; it came about only because the consideration proceeds from a last abscissa which does not exist in reality, or in other words, because the indivisible, the point, is identified with an infinitely small quantity, even though both notions differ basically from each other”.

One thing emerges with certainty; infinity “in the strict sense of the word”, cannot be manipulated mathematically. This leads to a startling realisation. No part of infinity can actually be distinguished from any other. This is tantamount to saying that no part is different from any other. In other words; every part of infinity – if there could be said to be parts – would be the entirety.

The part must be the whole.


1989. Pratchett Terry & Jolliffe Gray. The Unadulterated Cat.


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Originally Posted By: Bill S.
Part 5 ...
One thing emerges with certainty; infinity “in the strict sense of the word”, cannot be manipulated mathematically. This leads to a startling realisation.

No part of infinity can actually be distinguished from any other. This is tantamount to saying that no part is different from any other. In other words; every part of infinity – if there could be said to be parts – would be the entirety.

The part must be the whole.

1989. Pratchett Terry & Jolliffe Gray. The Unadulterated Cat.
Sounds like what I have in mind when I speak of immanent and transcendent Being, G.O.D.

=====================================
Earlier, Orac said
Originally Posted By: Orac
Two new readings for you

Rev's going to love the first:

“God exists since mathematics is consistent, and the devil exists since we cannot prove it.”
(http://steiner.math.nthu.edu.tw/disk5/js/history/infinity.pdf)

And finally ..
(http://scienceblogs.com/stoat/2011/09/galileo_on_infinity.php)

=============================================
THE BEGINNING OF INFINITY, by David Deutsch
=========================
http://www.brainpickings.org/index.php/2011/07/27/the-beginning-of-infinity-david-deutsch/
____________________________________
Like DD, I do not believe in a god. I simple say that I know and experience that G.O.D. is. My role, as a bit of infinity, is to willingly choose to tune into and connect with the entirety. BTW, "connect with" is the meaning behind the Aramaic (common language in Jesus' day) word, slaha, "to pray".


G~O~D--Now & ForeverIS:Nature, Nurture & PNEUMA-ture, Thanks to Warren Farr&ME AT www.unitheist.org
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Yes I know and have read the original book by David Deutsch the fabric of reality.

I did get a laugh out of one of the book reviews from Peter Forbes of the independant.

Quote:

He endorses Stephen Hawking’s view that we would be wise to colonise space because the asteroid that will certainly come one day might be beyond even the capacity of our nuclear weapons.


What hawkings said was “The threat of the Earth being hit by an asteroid is increasingly being accepted as the single greatest natural disaster hazard faced by humanity”

He is a phsicist he is not stupid enough to think you can blow an asteroid up with a nuclear weapon.

To explain Rev there is no atmosphere on an asteroid most likely, therefore no big shockwave and huge bang just a bright flash and lots of radiation which would look remarkably similar to a small sun :-)

Only in the movies do you blow asteroids up with nuclear weapons!


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Quote:
blow an asteroid up with a nuclear weapon.


Even if you could blow up an astreoid, by the time it was close enough for you to reach it, wouldn't the resulting shower of bits be as destructive as the whole asteroid?


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Originally Posted By: Bill S.

One thing emerges with certainty; infinity “in the strict sense of the word”, cannot be manipulated mathematically. This leads to a startling realisation. No part of infinity can actually be distinguished from any other. This is tantamount to saying that no part is different from any other. In other words; every part of infinity – if there could be said to be parts – would be the entirety.

The part must be the whole.

I do not agree with the term 'cannot be manipulated mathematically'. The relation between zero and infinity is aesthetically beautiful, and mathematically logical. The whole of mathematics is conceptual and not real. The concepts of negative numbers, imaginary numbers, etc. are aesthetically as beautiful as the concept of infinity. Actually, zero is not the other end of infinity, it is 'negative infinity' that lies at the other end. Mathematically, infinity does not represent the whole, it represents the concept of limitlessness. Mathematically, part cannot be the whole, we can take parts from the whole, and the whole gets reduced by that much.

However, I think that 'infinity cannot be manipulated physically'. Physically, infinity represents the whole. So we can leave the infinity, as I have suggested earlier, to the philosophers to speculate (because it cannot be manipulated logically). Physically, the part has a representative nature of the whole; and explaining the part tantamount to the explaining the whole, and that is what is expected from the scientists. The concept that 'the part is the whole' comes under philosophy; we can say either 'yes' or 'no' and argue, and never arrive at any logical conclusion; in the end we will have to accept that it is just a 'matter of belief'.

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Finiter, to some extent we seem to be saying the same thing here, but arriving at different conclusions.

You say that “Mathematically, infinity does not represent the whole, it represents the concept of limitlessness”. This is precisely what I have been arguing on this forum, and in other places, for a long time. Such an “infinity” can, of course be manipulated mathematically, it is just limitless, a mathematical “infinity”.

You say that “The whole of mathematics is conceptual and not real.” This is true, but mathematics has been described as the language of nature. As a vehicle “for speaking, for thinking, for discovering, and for proving” it is unsurpassed. Mathematics can be applied to the 4-dimensional (or 3-D if you prefer) reality we observe, and can be extrapolated to the multi-dimensional “realities” we can imagine. Without some sort of renormalisation, or replacement by the concept of limitlessness, though, infinity is closed to mathematics, and, as you say, to physical manipulation.

I completely agree with you that “zero is not the other end of infinity”. In fact, I don’t believe that infinity is at one end of any sort of continuum.

Negative infinity? I think that must be a concept for the philosophers. smile Infinity, on the other hand, needs to be understood if we are to have any real grasp of the origin and nature of the Universe.


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BS,you regard yourself as a hitchhiker, and I am just a layman interested in physics. We agree on many things, and that is very interesting.

You said, "Mathematics is the language of nature".

I think you have slightly exaggerated the role. Language is used to describe. So you are suggesting that mathematics can describe the physical world. I would suggest a lesser role; it is just a tool to extract results. The physical world should be explained physically, and for that verbal explanation will be enough. However, all physical laws should be mathematically valid. Thus, mathematics has a role in proving whether the physical model is correct or not. The role of mathematics as a vehicle for 'speaking, thinking and discovering' is minimal. The line of thought should be like this: visualize a physical model, then verify whether it is mathematically feasible; if not feasible try another physical model and so on. It should not be in the reverse order: that is, you should not impose a mathematical model (even if it agrees with the observations) on the physical world. The reason is that all mathematically valid explanations are not valid physically, whereas all physical models should be mathematically valid.

So there is a need to distinguish between mathematical and physical infinities as suggested by you. I agree that infinity is not an end. However, zero may be an end or a beginning. Negative infinity is a mathematical concept. It can have some role in philosophy. I think philosophy is linked to other branches like this: philosophy - mathematics - science - philosophy, thus forming a cycle. Mathematics is conceptual, science is real, and philosophy, both conceptual and real.

You said, "Infinity, on the other hand, needs to be understood if we are to have any real grasp of the origin and nature of the Universe."

As you said initially, 'something cannot come out of nothing'. So there is no origin; or, origin will always remain a mystery. In explaining the nature of the universe, infinity has no real role. That is, if a physical theory leads to infinity (other than the basic assumptions that space and time are infinite, and for which there are no explanations), the theory is wrong; there should not be any re-normalizations.

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Finiter, I think we could agree on more points than may be immediately obvious.

For example: I said that mathematics has been described as the language of nature, not that I thought it was the language of nature. I agree that words can do a very good job where actual description is concerned, but as one who has greater facility with words than with figures, I suppose I would.

You say: “…all mathematically valid explanations are not valid physically”. I agree absolutely, and think there are very few areas in which this is more obviously true than in dealing with infinity.

Where we do seem to differ is in our respective attitudes to the role of infinity in explaining the origin of the Universe. We can say the “origin will always remain a mystery”, and that is a very reasonable position to take. On the other hand, we can explore the whole question to see if there is a logical and scientific link between the infinite and the finite. Such an endeavour might involve making some basic assumptions, but as long as one remembers that they are just assumptions and tries to verify them scientifically, that seems a legitimate way to go.

“…there should not be any re-normalizations.”

I think renormalisation is a reasonable mathematical tool in that it seems to produce results that agree well with observation. Ancient peoples managed to fit supernatural explanations to physical observations, such explanations remained “valid” until better, more scientific, explanations were found, often these explanations were superseded. Possible, renormalisation will be of value until something better is discovered. I would say, keep it, use it, but don’t “deify” it.


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Originally Posted By: Bill S.
Even if you could blow up an astreoid, by the time it was close enough for you to reach it, wouldn't the resulting shower of bits be as destructive as the whole asteroid?


In the movies the small bits burn up harmlessly in the atmosphere .... I mean those nuclear bombs are powerful apparently they blow the meteor to bits apparently :-)

But yes Bill S I will leave the offical calculations to NASA and they put the limit at 400 meters. Anything bigger than 400m could probably not be nullified by any amount of nuclear weapons at best you would probably break it into big pieces and those are still going to smash into earth.

2005 YU55 which whistled past this month was of this sort of size.


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Thread drift is a wondrous thing! How did we get from infinity to blowing up asteroids in the movies?

Someone must have some strong objections to the infinite stuff. Where is Kallog when you need him?


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Originally Posted By: Bill S.
Where we do seem to differ is in our respective attitudes to the role of infinity in explaining the origin of the Universe. We can say the “origin will always remain a mystery”, and that is a very reasonable position to take. On the other hand, we can explore the whole question to see if there is a logical and scientific link between the infinite and the finite. Such an endeavour might involve making some basic assumptions, but as long as one remembers that they are just assumptions and tries to verify them scientifically, that seems a legitimate way to go.

“…there should not be any re-normalizations.”

I think renormalisation is a reasonable mathematical tool in that it seems to produce results that agree well with observation. Ancient peoples managed to fit supernatural explanations to physical observations, such explanations remained “valid” until better, more scientific, explanations were found, often these explanations were superseded. Possible, renormalisation will be of value until something better is discovered. I would say, keep it, use it, but don’t “deify” it.


I think our our differences are narrowing further. Is there a logical and scientific link between the infinite and the finite?

Here, we have to make clear the meaning of 'assumption'. Without going into the dictionary-defined meaning, I would say that an 'assumption' has no proof; ie, it cannot be logically proved. And, a logical proof, in the context of physical world, is a mathematical proof. So an 'assumption' has no mathematical proof and so any assumption is not part of physics in the true sense.

However, we can have 'logical assumptions'. We can use a 'logical assumption' as a starting point when we propose a theory. But, before concluding, we will have to logically prove the assumption. So a 'logical assumption' is provisional, and we are liable to prove it before we say 'it is finished'.

Now, coming to the point, your statement 'Such an endeavour might involve making some basic assumptions ........ a legitimate way to go' is right. You are suggesting that we have to make a 'logical assumption', and later try to prove it. However, in my opinion, only assumptions (which cannot be logically proved) are possible; I am ruling out the possibility of a 'logical assumption' that can link infinity with finiteness. I think this is the main difference between our views.

Re-normalization is a mathematical tool. Though the process is complex, the logic behind it is simple: it can extract results. As such, it is useful and I am not at all questioning its validity. I am only saying that to explain the basics of this physical world, neither 'the probability equations' nor 'the re-normalizations' are required. The reason is that the physical world we confront (ie, our universe) is deterministic and finite.

Based on my theory, the 'Finiteness theory' (sorry, I have to refer to my theory just to make it clear that my argument is based on an alternate theory), mass, length and time (MLT) are the only basic parameters. To explain the fundamentals of this physical world, we require only these parameters. These parameters are deterministic and do not lead to any infinities. However, the permutations/ combinations of these parameters and their squares,cubes, etc., will create a very complex state (like that of a living being like us interacting with its surroundings)that we have to use 'probability equations' and 're-normalizations' to extract results. But basically the physical world is deterministic, and that is why the probability equations and the re-normalizations give requied results.


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Finiter, I like your distinction between "assumptions" and "logial assumptions".

As far as the possibility of a logical assumption that can link infinity with finiteness is concerned; I suspect there should be a logical link somewhere, but it obviously needs more thought.


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Just come back from walking dog. A thought struck me while out. A long time ago I reached the conclusion that nothing finite could became infinite. The obvious corollary, that nothing infinite could become finite, had somehow not registered. There has to be more to be said about that. smile


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IMO, your line of thinking is correct. I will give examples to clarify my view. Suppose matter has a fundamental particle having finite mass and finite volume (matter is grainy), then the universe made up of matter will never be infinite, it will always be finite.

Now taking the corollary: Suppose the space is infinite, then you can never have a finite space. That is, the space cannot be grainy or there is no quantized space. From zero space(a dimensionless point) to infinity (having no limit) the space is continuous. That is, if starting from a point you increase the space by making it an expanding sphere (space is three dimensional and so we have to expand it in all directions) the space enclosed can take any value (not restricted to a subset of values). However the finiteness of each value is just conceptual, not real.

Time is also like space; it is not grainy, but it is only one dimensional. So expanding it from a point of zero time we get a line extending in both directions infinitely. For any point on the line we get a finite value, which is conceptual, but not real.

If there is only space and time and no matter, then it represents a physical 'nothingness', but it is real, consisting of infinite real space and infinite real time. Here, it may appear that physical infinity and physical zero becomes equal. Therefore, do you think that we can conclude that physically infinity and zero are equal? I have never thought of such a possibility.

However, that may be philosophical statement, implying that the real physical world is finite. Such a philosophical statement, I think, does not rule out the possibility of more than one finite physical worlds.

Can that be a logical statement (something that can be proved mathematically) ? I have to think about it.

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Originally Posted By: F
....physically infinity and zero are equal? I have never thought of such a possibility.


I toyed with the idea of infinite = zero, which progressed to infinite = zero = the present.
Any thoughts?


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Physical infinity and physical zero can be argued to be the same, implying that 'any physical thing should be finite'. In that context, ie, when we consider only that two, it has no past or future as nothing would have existed in the past and nothing will exit in the future. So it is always the present. However, you are not suggesting that, I think.

You may be suggesting that the present cosmos is 'zero' and 'infinite' at the same time. And, so it is always 'the present' for the cosmos. If this is not a mathematical statement, but a physical statement, then 'equal to' becomes 'equivalent in a physical sense'. This can mean that the 'infinite cosmos' does not change with time and so it is not possible to differentiate it from a 'zero cosmos'.

If you are thinking on the above lines, then our ideas converge to that extent. However, the cosmos by virtue of its 'the present-status' denies any role for an observer, and hence denies any 'logical assumption'. But the 'finite universes in the cosmos' (proposed by me) are free from that 'infinite-zero syndrome', and offer a role for the observer and a possibility of 'logical assumptions'. So my argument is that the role of science is restricted to the finite universe of ours. The rest (including other universes if any and the cosmos as a whole) has to be left to the philosophers.

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Finiter, you may have already answered these questions, but shortage of time makes it more practical to ask (again?) rather than go hunting.

In your theory can:
1. an infinite "something" be composed of finite objects?
2. an infinite "something" contain finite objects?

You have developed the infinity/zero/present idea on more fronts than I had. All I had done, so far, was to think that the present becomes more illusive, the closer we try to look at it. Ultimately, we have to say either that it has extent, Planck time perhaps, or it does not have extent; in which case, it would seem to be possible to describe it as "infinitely small". This seems to leave us with three possibilities:
1. Time is quantised.
2. Time is not quantised.
3. Time is not real, it is just a conventional way of recording change.

I know where you stand on 1 & 2, but what about 3?


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Regarding the first part: In my theory, infinite space (I will not call it 'infinite something' because it will create confusion)is real, but without matter it is physically 'nothing ', and represents the limit of reality (zero of reality). So when I say the infinite space contains a very large number of finite universes, it does not mean that the infinite space is composed of finite things. The space is composed of space only and not matter. An infinite space (because it is space) can accommodate(not exactly contain) finite objects in it. Simply put, space is real in itself, matter just exists in the three dimensional space, and there is no inherent relation between space and matter.

Regarding the second part: Time is real and infinite like space, but without matter, it is physically nothing. However, it is not just something connected with change; it is real in itself, whether anything changes or not.

Thus in my theory, for a physical world to exist there should be matter, and the number of matter particles in the cosmos will be finite (even though metaphorically we can say it is infinite). For the three dimensional matter to exist, there should be a three dimensional space. Thus, in addition to the infinite space, we have to deal with the 'space connected with matter'. This 'space dimension' of matter changes with time. So matter has a 'time dimension' also. Though the infinite space and infinite time are non-quantized, the 'space and time associated with matter' are quantized.

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Finiter won't like it but to be released tomorrow

http://www.physorg.com/news/2011-11-scientists-vacuum.html

The vast infinite vacuum of space is indeed popping in and out of existance.

I can hear it now but wait there has to be another explaination (waves hands) :-)

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I don't see why Finiter would necessarily dislike this. His theory differentiates between infinite nothingness and 'space connected with matter'. Perhaps I am misunderstanding his theory, but I would have thought that something might exist in the 'space connected with matter', and that this "something" could be converted into particles.


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Originally Posted By: F
Time is real and infinite like space, but without matter, it is physically nothing.


Just trying to get my head round this one. Are you saying that time is a real entity, but it exists only if/where matter exists, and that without matter it does not exist?

If this is so; in what sense is it a real entity?

"What then is time? If no one asks me, I know;
if I wish to explain it to someone who asks, I know not.
" St. Augustine.


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Finiter, your infinite nothingness prompts a thought. In the sentence: "nothing can be infinite" the meaning varies depending on where you place the stress. which, if any of the following would you agree with.

1. “Nothing can be infinite”: meaning, it is not possible for anything to be infinite.

2. “Nothing can be infinite”: meaning, it is possible for nothing (= the absence of anything) to be infinite.

3. “Nothing can be infinite”: having more or less the same meaning as the first statement, but implying that it could be finite.

Orac will probably say I'm playing word games, but Rev should be proud of me. laugh


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Originally Posted By: Bill S.
Perhaps I am misunderstanding his theory, but I would have thought that something might exist in the 'space connected with matter', and that this "something" could be converted into particles.

In my theory, the particles cannot get converted into 'space' or 'energy'. The 'space connected with matter' implies only that matter has volume (that two particles cannot remain at the same position) and the position of the particle can change (the particle can move).

Physical world is real. Real things can have only positive values greater than zero for their mass, volume, energy, etc. Zero is the limit of physical reality (it cannot go to the negative domain). When the mass, volume, energy, etc. become zero, there is no matter; ie, there is 'physically nothing'. That 'physical nothing' means there is the three dimensional space which remains forever without any change. That 'physical nothing' is real; ie, infinite space and infinite time are real. We can say that these remain as the arena which does not change (this has been the earliest or classical concept about space and time).

Regarding 'nothing can become infinite', I follow your third alternative. Here, 'nothing' refers to physical things, and the statement implies that physical things should be finite. Your second alternative is also correct; 'physical nothing' can be infinite (like space and time without matter)

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Originally Posted By: Orac
Finiter won't like it but to be released tomorrow
The vast infinite vacuum of space is indeed popping in and out of existance.
I can hear it now but wait there has to be another explaination (waves hands) :-)

I went through the link that you have given. It is interesting. The net effect is that microwaves were emitted during their experiment. They explain it as particles popping out of vacuum (or field). Obviously, I will propose an alternate explanation. Let us wait till we get the details.

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Originally Posted By: O
The infinity discussion is half mumbo jumbo


This is the sort of thing I'm looking for. It's the bits that are mumbo jumbo, and why, that I need to be able to get to grips with.


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Originally Posted By: Bill S.
This is the sort of thing I'm looking for. It's the bits that are mumbo jumbo, and why, that I need to be able to get to grips with.

To understand the physical world, you have to start from space and time; here, infinity matters. So any discussion regarding infinity is not just mambo-jumbo. We will just continue.

Now I will ask you a few questions.
1.Can a finite mass have infinite gravity?
2.For the gravitational force to be zero do two finite masses have to remain at infinity?
3. Is the quantum information infinite?

I will answer 'NO' to these questions. What is your opinion.

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I guess you are not looking for yes/no answers. If you are I ask for your patience as I have been known to “go on a bit”. smile

1. If the concept of a singularity is correct, then the answer must be “yes”. However, in the case of a black hole; the matter of which the original star is composed bust be finite, therefore its gravity must be finite. On the basis that nothing finite can become infinite; the idea that a singularity can develop, and that this will develop infinite gravity, must be inaccurate in some way. Furthermore, if gravity is a result of the presence of matter, gravity could be infinite only if matter were infinite. It is difficult to see how matter, in our Universe could be infinite, since, as far as we are aware, the Universe is finite.

This does leave the possibility that infinite matter exists in an infinite cosmos, in which case, presumably, there would be infinite gravity. However, this does seem to differ from the infinite gravity = infinite curvature of spacetime proposed for a singularity.

It looks as though we agree; the answer to Q1 is “no”.

2. Two finite masses at an infinite distance from each other will remain at this infinite distance, however they travel relative to each other. The infinite distance can never become finite. If they are an infinite distance apart, they can never have been at a finite distance, so they cannot be aware of each other, nor can one influence the other. The question might have some relevance in a philosophical context, but physically it makes no sense.

3. I don’t know enough about quantum information to give an off-the-cuff answer to this one. My initial thoughts would go something like this: If quantum information relates only to the Universe, it is finite. On the other hand, if it extends to an infinite cosmos, then it is infinite. Orac will undoubtedly put me right on this, but my feeling is that quantum information might be infinite, but we have no way to provide physical proof of that.

The short answers would, therefore, be: 1. No. 2. N/A. 3. Probably infinite.


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The second question is actually related to the first question if viewed at a different angle. When will the force between two finite masses become zero? I think that the force between two finite masses will be equal to zero at a certain finite distance. That means beyond a certain distance there will be no force between them (theoretically also). Or the field should also be finite.

What is 'quantum information'? In classical physics, matter has properties. But, in QM, they refer to it as 'quantum information'. That is, they think that we cannot know 'the properties' at the quantum level; however, there is something that we just do not understand, and so call it 'quantum information'. I think the term is now used as a 'jargon' to indicate that there is something 'sacred' there at the quantum level, which we can never hope to understand.

However, I think the only information that can be present at the quantum level is restricted to 'mass, space and time'. If we think the 'physical something' is finite, then the 'quantum information' contains just 'three bits' of information. With this 'three bits', it should be possible to describe the whole universe.

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Originally Posted By: F
I think that the force between two finite masses will be equal to zero at a certain finite distance.


How could you determine this distance? Wouldn’t the decrease in the force between two finite objects approach zero as a hyperbolic curve approaches a straight line? Only if you accept that the force is actually quantised could you reach zero.


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I'm sure Orac will have something to say about the quantum information, (if he is still posting in this thread) and whatever it is it will be more insightful than anything I could say on the subject.

As I recall, your theory includes an infinite "cosmos" (not your term, I know). How do you relate finite quantum information to that?


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Quote:
If we think the 'physical something' is finite, then the 'quantum information' contains just 'three bits' of information.


Presumably there would have to be a difference between the quantum information needed to cope with finite and infinite "somethings", but would the difference be one of kind or just of quantity?


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Just ordered a new book Bill S based upon recommendation of Jester from resonnaces. Interesting title let you know how it pans out.

http://www.amazon.com/Infinity-Puzzle-Quantum-Orderly-Universe/dp/0465021441


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Everything associated with matter is quantised. So as you have expected, in my opinion, the force is also quantised. The distance, at which the force is zero, depends upon their individual velocities.

'Finite quantum information' relates to 'finite universes' in the infinite cosmos. The infinity of the cosmos is just a quality of the space, which is not quantised and so has no quantum information.

As pointed out earlier, the 'infinite something' is physically 'nothing'. It is not quantised (any quantised entity will be finite) and so does not require any quantum information.

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I had a look at The Infinity Puzzle on Amazon a while ago and was not sure about it. I shall be very interested to know what you think.


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Finiter, I follow the logic of your finite “something”/infinite “nothing”, and as long as you restrict your concept of infinity to this definition there is no problem. Let’s look a little further, though.

There was never a time when there was “nothing”.
The Universe is finite, so it had an origin.
The cosmos is infinite, so it has always been.
Something finite cannot become infinite; nor vice versa.
The only logical origin for the Universe is the cosmos.
“Nothing” cannot become “something”.
The cosmos must, therefore, be “something”, otherwise there would be no Universe.


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I got your point. You are arguing that since the universe is finite, it should exist for a finite time. Is it? That is, you are saying that a finite thing cannot exist for an infinite time; or, the concept of 'finite/infinite' should be applied to both time and space in the same manner.

My counter argument is that time and space are not exactly similar. So the concept of finiteness/infinity cannot be applied in the same way to both. Whether it is a 'physical something' or 'physical nothing', both can exist, because both are real. They are not interchangeable; ie, 'physical something' cannot change into 'physical nothing', and vice-verse. So both should exist for an infinite time and both have no origin.

So in my opinion, the cosmos as well as the finite universes exist for ever. The cosmos without any change, and each universe undergoing a cycle of changes. The period of the cycle (time required) is finite and is the same for all universes.

Thus, in a way the time factor of the universe is quantised and finite. That is, the changes inside the universe does not proceed in any direction for an infinite time. After a finite time the changes reverse and the universe oscillates between two states. This oscillation or pulsation repeats infinitely.

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Originally Posted By: F
This oscillation or pulsation repeats infinitely.


No time for a proper reply at the moment, but I have to note the "infinite series", always trouble!


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Originally Posted By: F
My counter argument is that time and space are not exactly similar. So the concept of finiteness/infinity cannot be applied in the same way to both.



This does not necessarily follow logically. Matter and energy are not “exactly similar”, but the concept of finiteness/infinity is applied in the same way to both.

Would I be right in thinking that in your theory each universe is finite, but is eternal?


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Originally Posted By: Bill S.
This does not necessarily follow logically. Matter and energy are not “exactly similar”, but the concept of finiteness/infinity is applied in the same way to both.

Would I be right in thinking that in your theory each universe is finite, but is eternal?

I agree. But, consider that infinite cosmos is physically 'some thing', then our universe can be infinite or there will be infinite universes. Thus in our argument we reach were we started. We will have to conclude that there can be an infinite number of apples. However, we think it is not possible to have an infinite number of apples. So I think considering a 'physical something' to be eternal is logical (in the sense that the argument sounds logical though we cannot prove it). (In my theory, energy is not a separate entity, it is just a quality of matter, like mass and volume.)

In my theory, each universe is finite but eternal. Suppose the infinite cosmos contains only say 100 universes; these should always be there. The infinity of the cosmos is physically 'nothing'; it is just space and time. So these universes cannot come from that infinity or disappear into that infinity; so these are eternal. The infinity has no effect on them and so these are not 'parts' of infinity. The infinite nothing just provides the environment (space and time) required for the universes to remain.

Anything inside the universe is part of the universe. The universe has an influence over them. Universe being finite, everything inside it has a relative beginning and a relative end. The number of events possible in it is finite; and no event inside it will proceed to infinity. So within a finite period of time everything 'that should have happened' would have happened. After that it will be repetition. So every object inside the universe is reborn infinitely.

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Originally Posted By: F
So within a finite period of time everything 'that should have happened' would have happened. After that it will be repetition. So every object inside the universe is reborn infinitely.


By "repetition" do you mean that each time a universe is reborn it goes through the same sequence of events; or are events different each time?


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Quote:
In my theory, each universe is finite but eternal.


An eternal series is just an infinite series with a slightly different name.

In eternity, everything that can happen will happen an infinite number of times.

If each succession of universes is eternal; at any given point it has already existed for an infinite time, so everything that can happen will already have happened, an infinite number of times.

I know we have been at this point before, but I’m still not clear as to how you resolve this.


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The universe goes through the same sequence of events if we take an overall view (for example, for every human being the sequence is the same; born, lives for some time, and then dies; however what he does during life may be different). Similarly, in the details, the the things may be different, however the direction in which things happen will be the same.

Yes, everything that can happen would already have happened an 'infinite number of times' at any point in time. It is impossible to resolve this. However, strictly speaking, the term 'infinite number of times' is not correct. We have to say that the 'number of times' has no limit if we go backwards or forwards in time. As the physical something cannot disappear, this is the only possibility.

That is, we can remove the spatial infinity, by taking the 'physical something' to be always finite. But the infinity in time cannot be removed completely; but can be restricted by assuming the cycle of events to be finite in time. The cycle of events is always there as a 'physical action' of the 'physical something'.

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From your first paragraph, I assume that each cycle of an individual universe contains different people, doing different things, from those in preceding and following cycles.

We seem to agree that "the term 'infinite number of times' is not correct", but for different reasons.

If my understanding is correct, you regard the past and future of eternity as limitless. You seem to have done your own bit of "renormalisation" here, effectively getting rid of the inevitable infinity and replacing it with two limitless periods of time.

I would maintain that the term 'infinite number of times' is not correct because it implies that infinity/eternity can be measured in terms of time.

You say "the infinity in time cannot be removed completely", as though the addition of the word "completely" means that you have removed enough of it to be able to discount it. (more renormalisation) I would maintain that the infinity must either be in your calculations, or not. You can't do away with part of it and ignore the rest.


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Originally Posted By: Bill S.
From your first paragraph, I assume that each cycle of an individual universe contains different people, doing different things, from those in preceding and following cycles.


Exactly; a 'Bill S' and a 'finiter' discussing infinity would never have happend before, and will never again happen, though humans 'will be' present in all cycles.

Originally Posted By: Bill S.
I would maintain that the term 'infinite number of times' is not correct because it implies that infinity/eternity can be measured in terms of time.

You say "the infinity in time cannot be removed completely", as though the addition of the word "completely" means that you have removed enough of it to be able to discount it. (more renormalisation) I would maintain that the infinity must either be in your calculations, or not. You can't do away with part of it and ignore the rest.

My reasoning is also the same I think. Eternity cannot be measured in quanta of time. The 'finite time required for a single cycle of events' is not exactly a constituent of eternity. The time is continuous, not quantized, and always moves forward.

The infinity never comes in the calculations because all calculations are restricted to the finite universe. The 'infinite' part remains in the assumed part of the picture for which there is no logical (mathematical) proof. Science or physics can only explain the 'finite' part, and for this there should always be a logical (mathematical) proof.

In short, in my opinion, our picture of the cosmos will always consist of two parts: the 'assumed' part and the 'logically assumed' part. The former has no explanations regarding why it is so, we can say 'it is so' because 'it is so'. For the latter, we will have an answer for all questions regarding why it is so. The latter part comes under science. However, it cannot exist without the former, which is the foundation. The former part, though assumed and unexplainable, should be in agreement with the latter, and should be as minimum as possible. Then only will our picture of the cosmos be logical.

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Originally Posted By: F
So these universes cannot come from that infinity


Quote:
However, it [the finite universe] cannot exist without the former [the infinite cosmos], which is the foundation.


Rev says that having your cake, and eating it, seems like a good idea. This looks like what you are trying to do here. I’m not sure it works in the world of science.


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Originally Posted By: F
The 'finite time required for a single cycle of events' is not exactly a constituent of eternity. The time is continuous, not quantized, and always moves forward.


Is the second sentence intended to explain the first?

I don’t see how time being continuous, quantised or even being just a convenient means of measuring change would prevent it from being a constituent of eternity, if your theory is correct.


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Originally Posted By: Bill S.
Rev says that having your cake, and eating it, seems like a good idea. This looks like what you are trying to do here. I’m not sure it works in the world of science.

I am sorry that my explanation lacked clarity. It should be read as: However it (the logical assumption) cannot exist without the former (simple assumption), which is the foundation. I was referring to 'the two types of explanations'.

However, the finite universe cannot exist without the infinite cosmos; the cosmos serves as 'the arena' and 'not the foundation'.

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Originally Posted By: Bill S.
Originally Posted By: F
The 'finite time required for a single cycle of events' is not exactly a constituent of eternity. The time is continuous, not quantized, and always moves forward.


Is the second sentence intended to explain the first?

I don’t see how time being continuous, quantised or even being just a convenient means of measuring change would prevent it from being a constituent of eternity, if your theory is correct.

Yes. By the term 'constituent', I meant 'the part' with which the whole is created. If time was qunatized, then each quantum would be a constituent of time, and time itself would have been finite. As time is continuous, it is infinite or eternal.

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Originally Posted By: F
If time was qunatized, then each quantum would be a constituent of time, and time itself would have been finite. As time is continuous, it is infinite or eternal.


You seem to be saying that if time were quantised, it would have to be finite; yet your cosmos is composed of finite universes, but is infinite. You might reason that your cosmos is not composed of universes, it simply contains them, but I have the impression that your cosmos would be "nothing" if it did not contain universes.


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Yes. The cosmos, if it does not contain any universes, would be physically nothing; it would represent the reality 'nothingness' (a physical zero); it would be simply 'space and time' without anything in it, and 'space and time' alone cannot create any thing physical out of it; or, any 'physical thing' is composed of matter.

That is, the cosmos is simply inert and helpless to interfere with any physical thing that may be present in it. Anything that happens to the physical thing is due to its action; or it is an action-reaction syndrome. Physically, action is energy and reaction is force; quantitatively both are equal.

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This seems to make sense, but there is still the problem of the infinite series. Being a bit short of time for checking back through past posts, I shall have to ask you to correct me if I am wrong, but did we not agree that one could not have a physically infinite number of finite objects?


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Yes, we agreed that there cannot be an infinite number of finite objects. The only point where we disagree is about the infinite number of 'finite events'.

I think we have come to the end of discussion. The discussion was 'very useful as far as I am concerned'. Though my theory remains the same, my insight into it has increased. I have not considered one important point: that I have used different standards for avoiding infinity in respect of space and time. Is there any justification for that? (one is three-dimensional and the other is one-dimensional; the former can be closed and the latter will always be open; I will start thinking in that direction).

If you think that we can close this thread, then we can close it. If you think there are still other points, we will continue. A closely related subject (to infinity) is the question, 'Where should the line between metaphysics and physics be drawn?'. If you are interested, we can start a new discussion on that.

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Why should you be able to have an infinite number of finite events, if you cannot have an infinite number of finite objects? This sounds a little like semantics.


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As far as my theory is concerned, it is like that. I came to be aware of it only when you brought up that point for discussion (that was what I said in the last post).

Now I have to search for any possible reasons. A 'thing' fundamentally occupies space. An event, on the other hand, 'occupies' time. Space is three-dimensional and so can be closed. Time being one dimensional can never be closed. From this, can we come to the conclusion that the number of things will be closed or finite, whereas the number of events will be open or infinite (or the same event can go on happening infinitely)?

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Quote:
Space is three-dimensional and so can be closed. Time being one dimensional can never be closed.


I feel sure you must have reached this conclusion via a logical process, but it eludes me.


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Quote:
Space is three-dimensional and so can be closed. Time being one dimensional can never be closed.


I feel sure you must have reached this conclusion via a logical process, but it eludes me.


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Sorry, I'm not intentionally repeating myself. For reasons best known to itself, my computer is getting into duplications. Like me, it's getting on a bit. smile


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Let’s look at wave/particle duality and in particular the extensive nature of the wave. I believe that in QM the full wave description of all the waves goes on to infinity in space and time. Those who are familiar with my views on infinity will know that I have problems with finite things going to infinity. However, one has to consider the possibility that the wave function of any particle might not be finite at any stage of its existence.

Physicists can create Bose Einstein condensates that are of a finite and observable size, but in which the atoms exist together and jointly within this space and it is not even theoretically possible to distinguish between them or where they are individually. Effectively, every atom in the condensate occupies every part of the finite space occupied by the whole condensate.

Extend this idea to the “infinity of space and time” occupied by every wave function of every particle. We now have an infinity in which every part is equal to the whole. The entire concept of parts, or divisions, becomes just a function of our restricted view of eternity/reality.

It might be argued that temporal divisions must exist in order to accommodate the changes between waves and particles, but this is only a perceptual change. A particle may be no more than a contracted description of the full wave; a sort of “Fourier transform”. The distinction between waves and particles may be no more than perceptual distinctions that are necessary for us to make sense of our 4D Universe.

Thus, QM becomes a window on infinity which only the “shut up and calculate” mentality prevents us form looking through.


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Lack of response to the last post suggests that fellow posters might consider this thread as deceased, perhaps with some justification. However, I still have the idea going round in my head, so a further attempt at “exorcism” might be worth a try. smile

Let's return to the wave/particle duality. My understanding is that QM is consistent with the idea that particles may not actually exist as physical entities. The more precisely one can locate a wave, the more it resembles a particle. This might lead to the conclusion that a particle is nothing more than a precisely located wave.

The next step is to consider that the wave, in its own frame of reference, is still an expanded wave, going to infinity and occupying the entire Universe. Only in the frame of reference of an observer might the wave be considered as a particle, having location and mass.

The idea that every wave occupies the entire Universe, and might continue to do so, even if an observer perceives it as being localised, may be counter-intuitive, but that certainly does not necessarily mean it cannot be the case. QM tells us that our reality is non-local, although this is not what we perceive. Instantaneous actions between entangled particles can (probably) occur over boundless distances. Something links these particles. Would it be totally illogical to suggest that they are linked by their wave nature, and that this wave nature is their normal and permanent state? If this were the case, it might be only our restricted 4D view of reality that obliges us to interpret their state as particulate under certain conditions. If this were the case, it would follow that their mass must be "supplied" by observation.


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Originally Posted By: Bill S.
Lack of response to the last post suggests that fellow posters might consider this thread as deceased, perhaps with some justification.

I guess other would-be posters are less motivated than yourself. My (perhaps wrong) impression is that you'd prefer to regard infinity and eternity as real. Regarding science: if, in discovering how the universe works, it turns out that some things are theoretically infinite then so be it. My own intuition says that infinity and eternity are real, but that's not science - neither is it anti-science.


"Time is what prevents everything from happening at once" - John Wheeler
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It's not so much a matter of preferring to regard infinity (& eternity) as being real. It's just that if it is not real, there must have been a time when there was nothing, in which case, there should surely still be nothing now; but manifestly there is something.


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Originally Posted By: Bill S.
It's not so much a matter of preferring to regard infinity (& eternity) as being real. It's just that if it is not real, there must have been a time when there was nothing, in which case, there should surely still be nothing now; but manifestly there is something.


Agreed. This is Occam’s Razor at it’s very best. This is why I am firmly encamped in the reality of infinity. Theories that imply spontaneous existence are not palatable to me.

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Good to see you again KG, and not just because you agree. smile
It seems that it is not too difficult to get people to agree that there can never have been a time when there was absolutely nothing; the problem appears to come if you try to look at what the something that has always existed might be.


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"the problem appears to come if you try to look at what the something that has always existed might be."

It is MOTS (more of the same). smile


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That's one of the problems with infinity; it's all MOTS! As child, that was something I found slightly worrying about the idea of eternal life.

That should bring Rev into this thread. laugh


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