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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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Bill S. Offline OP
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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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