Is infinity an illusion?

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Offline Bill S

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Is infinity an illusion?
« on: 23/08/2013 19:18:28 »
The Infinity Illusion                 New Scientist 17.08.2013.


Over 40 years ago I had a long discussion with a maths teacher about infinity.  It culminated in his conceding that the series of whole numbers, although apparently unbounded in both directions, was not an example of true infinity.  Now, in 2013, more than 40 years on, could the scientific/mathematical community be moving in that direction?  Imagine my elation when I saw, in the New Scientist an article suggesting that some physicists were trying to remove infinity from scientific – and even mathematical – calculations. 

Max Tegmark, it seems, regards infinity as “…the ultimate untested assumption.”  He says: “All of our problems with inflation and the measure problem come immediately from our assumption of the infinite.”

This must raise the question: How can we test for infinity?  Surely such a test would require an infinite amount of information. 

Later in the article we are told that “The largest number of informational bits a universe of our size can hold is 10^122.”   

Nobel laureate, David Wineland, says: “Certainly we need nothing like that number of bits to record the outcome of experiments.”  Is he saying that experimental science has no need for infinity?  The article points out that “…even the best device will not measure with infinite accuracy….”.

Raphael Bousso says:  “I don’t think anyone likes infinity.  It’s not the outcome of any experiment.”

Mathematician Norman Wildberger says of “potential” infinity:  “This type of infinity allows us to add 1 to any number without fear of hitting the end of the number line, but is never actually reached itself.  That is a long way from accepting “actual” infinity – one that has already been reached and conveniently packaged as a mathematical entity we can manipulate in equations.”

I doubt that Wildberger is, here, criticising Cantor who appears to have achieved precisely that “packaging” in set theory.  However, the fact remains that Cantor’s infinities are “mathematical” infinities, and run into problems with “absolute” infinity”. 

“For the past decade he [Wildberger] has been working on a new, infinity-free of trigonometry and Euclidian geometry.”  He is working to counter the fact that angles are related, via circles, to pi, with its endless digits following the decimal point.  I lack the maths to evaluate Wildberger’s work, but Doron Zeilberger say of it: “Everything is made completely rational.  It’s a beautiful approach.” 

If Wildberger’s approach is revolutionary, surely, Zeilberger’s must be calculated to set the cat among the pigeons. 

Zeilberger, it seems, wants, not only to get rid of infinity; he “wants to dispose of potential infinity as well.” Zeilberger believes there is a largest number.  “Start at 1 and just keep on counting and eventually you will hit a number you cannot exceed – a kind of speed of light for mathematics.”  Zeilberger’s answer to the question as to what happens if you add 1 to this number is that it acts like a computer which has a maximum number it can process.  If you add 1 to it, it either gives you an “error” message, or resets the number to zero.  Apparently, Zeilberger favours the latter option.
 
My own feeling about this is that it draws too heavily on the analogy between a computer and the Universe.  Who, or what, would generate the “error” message, or reset the number?  Zeilberger’s assertion that this largest number is “…so big you could never reach it” seems a bit of a cop out.  Consider what he has actually said: “…eventually you will hit a number you cannot exceed” and “…you could never reach it”.

Tegmark points out that “the calculations and simulations that physicists use to check a theory against the hard facts of the world can all be done on a finite computer. “That” he says “already shows that we don’t need the infinite for anything we’re doing.”  He continues: “There’s absolutely no evidence whatsoever that nature is doing it any differently, that nature needs to process an infinite amount of information.”

Physicist Seth Lloyd points out that “We have no evidence that the universe behaves as though it were a classical computer, and plenty of evidence that it behaves like a quantum computer.”  This reintroduces infinity, because: “If you really wanted to specify the full state of one qubit, it would require an infinite amount of information.” 

Obviously, we do not have an infinite amount of anything at our disposal, so the problem seems academic.

Set theorist, Hugh Woodin suggests separating the two issues of physical and mathematical infinities.  He says: “It may well be that physics is completely finite, but in that case, our conception of set theory represents the discovery of a truth that is somehow far beyond the physical universe.”

Obviously there is more in this article than I have mentioned here, but hopefully others will read it and raise different points.  I have added a few of my own thoughts above, but I would be delighted to have some other people’s input before risking swamping the subject with my own crackpottery.

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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #1 on: 24/08/2013 00:00:21 »
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“For the past decade he [Wildberger] has been working on a new, infinity-free of trigonometry and Euclidian geometry.”  He is working to counter the fact that angles are related, via circles, to pi, with its endless digits following the decimal point.  I lack the maths to evaluate Wildberger’s work, but Doron Zeilberger say of it: “Everything is made completely rational.  It’s a beautiful approach.” 

The "endless digits" business is irrelevant to infinity. Pi is not infinite: it is the quotient of two finite quantities and therefore finite. It just happens that pi is irrational, that is it can't be expressed as the quotient of integers. No big deal: neither can e or log2, both of which are finite, irrational, wholly defined, and extremely useful.

There's no doubt that zero is important and useful. But a lot of useful physics is done in inverse space, so why can't we have a symbol for 1/0? 
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #2 on: 24/08/2013 14:31:13 »
Quote from: alancalverd
The "endless digits" business is irrelevant to infinity.

Would you also apply this to the “endless digits” of number lines?  I look for clarification here because it is so rare to find someone who doesn’t insist that these are examples of infinite series.


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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #3 on: 25/08/2013 01:42:06 »
There's a distinction between the trivial fact that you cannot express pi, e or log2 with a finite number of digits, and the significant fact that all three quantities are finite.

In geometry and electrical engineering we often quantify rotation in multiples or fractions of pi, and e turns up pretty regularly, which helps to make physics rational since exp(ix) = cos x + i sin x , so  e^(pi i) = -1. and. e^(2 pi i) = e^0 = 1 - all very useful identities. But log2 remains stubbornly irrational. 

Not too sure what is meant by number lines. As far as I can gather, these are arbitrary samples of the rational set, used for elementary teaching. The rational set is by definition a countably infinite set of countably infinite sets, which gives it several interesting properties, if you are interested in that sort of thing. In practice it means that (a) the longer you are prepared to wait for a jpeg image to upload and download, the closer it can resemble the object, and (b) you can image as many objects as you like without repeating yourself.
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #4 on: 25/08/2013 20:58:23 »
I'm not sure if my lack of maths is an advantage or a drawback when thinking about infinity.  In most scientific discussion it is a serious disadvantage, but perhaps infinity is sufficiently divorced from science to make a difference. 

For example: Woodin is a mathematician and he obviously sees infinity as part of mathematics, whereas he is quite prepared to remove infinity from physics.  My own feeling is that mathematics in general, and set theory in particular, may cope very well with “infinities”, but they are ultimately finite.  Physical reality, on the other hand, must have its basis in infinity, or there would be no physical reality.

When Woodin says “It may well be that physics is completely finite”.  I would agree, as long as he is talking about our finitely bounded understanding of physics.

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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #5 on: 26/08/2013 00:08:14 »
Consider an everyday concept in classical electrostatics. Potential is the work done to add unit charge: one volt = 1 joule per coulomb.   

The simplest case is to visualise a point charge - say an electron, at a position r = 0. The electric field of an isolated point charge in space extends in all directions with a local strength of 1/r^2 (by Cavendish's experiment). This function is nonzero for any finite r. Now we measure the potential at that point by bringing up another test charge (another electron) and measuring the force required to do so as it approaches the target. The integral of (force x distance) from r to 0 is the work done. Clearly the integral depends on the initial value of r, but there can only be one value for the potential at r = 0, so the definition of potential must be expanded to "work done to bring an additional unit charge from infinity"   

Thus in at least one small area of physics even if the universe is finite, our understanding of it uses the simple infinity defined as the value of r at which 1/r = 0.   
« Last Edit: 26/08/2013 00:17:27 by alancalverd »
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #6 on: 26/08/2013 02:28:49 »
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the definition of potential must be expanded to "work done to bring an additional unit charge from infinity".

The bit where I am lost is the idea of bringing a unit charge from infinity.  At least I can pretend to understand the rest, but bringing some from infinity.......??   

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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #7 on: 26/08/2013 08:03:23 »
Go back to the preliminary definition: work done to add unit charge. Now clearly I have to do more work to move a charge one meter in a radial field than to move it one centimeter, but due to the 1/r2 relationship the difference gets less as the distances increase. So the reference origin is where 1/r = 0.

This distinguishes between theoretical physics (the definition of potential) and experimental physics (the measurement of potential). Noting that it would take a very long time to move a charge from infinity, we cunning engineering types measure potential difference with reference to something handy like a spike in the surface of the earth or the negative terminal of the battery.   
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #8 on: 26/08/2013 15:34:03 »
Let’s see if I have this right.

1/r is the reciprocal of r.

If 1/r = 0,    r = infinity.

In order to work with this rather inconvenient infinity, those who work with practical reality perform a sort of “renormalisation” and leave infinity to the theorists.


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Offline JP

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Re: Is infinity an illusion?
« Reply #9 on: 26/08/2013 17:22:37 »
I'm not sure if my lack of maths is an advantage or a drawback when thinking about infinity.  In most scientific discussion it is a serious disadvantage, but perhaps infinity is sufficiently divorced from science to make a difference. 

For example: Woodin is a mathematician and he obviously sees infinity as part of mathematics, whereas he is quite prepared to remove infinity from physics.  My own feeling is that mathematics in general, and set theory in particular, may cope very well with “infinities”, but they are ultimately finite.  Physical reality, on the other hand, must have its basis in infinity, or there would be no physical reality.

When Woodin says “It may well be that physics is completely finite”.  I would agree, as long as he is talking about our finitely bounded understanding of physics.


I think its fair to say that in mathematics, infinity is fine and well.  Mathematicians can talk easily about infinite sets because those sets are ideas, not physical objects.  Infinite series consist of infinite numbers of elements but can sum to finite values.


In physics, one has to be a bit more careful.  Infinity most often means "really large."  It's generally easier to do calculus and other calculations with infinity than inserting some arbitray "really large" number and it introduces negligible error.


In some cases such as the size of the universe, some theorists take it literally.  This is where I hedge my bets since you'd have to come up with a testable hypothesis to distinguish between an infinite universe or not.  As far as I know the question isn't settled.  Clearly no experiment has directly measured an infinite number of things.


This is why it's so important to be precise: infinity definitely exists in mathematics, is used but doesn't claim to be a real object in much of classical physics, and may or may not exist in the universe at all in cosmology or quantum mechanics.

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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #10 on: 26/08/2013 20:48:51 »
Let’s see if I have this right.

1/r is the reciprocal of r.

If 1/r = 0,    r = infinity.

In order to work with this rather inconvenient infinity, those who work with practical reality perform a sort of “renormalisation” and leave infinity to the theorists.



Not at all. We make r as large as we need to for the practical purpose. This sort of thing turns up in many aspects of practical science (remember "infiinte dilution" in physical chemistry?) , and our brother engineers are perfectly happy making optical systems focussed at infinity, for instance. 

I have no idea what you mean by "renormalisation" - would appreciate an explanation.
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #11 on: 26/08/2013 21:52:26 »
As I understand it, renormalisation (with a z if you are on the West side of the Atlantic) is a technique for getting rid of unwanted infinities in QM calculations. 

If I remember rightly infinity on a camera’s distance scale originally represented anything greater than one hundred times the focal length of the lens, and may still be used in that sense on occasions.  Obviously this is only loosely connected to anything that might be considered genuinely infinite, and probably is not what the NS article suggests may be an illusion.

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #12 on: 26/08/2013 21:57:45 »
Quote from: JP
Infinite series consist of infinite numbers of elements but can sum to finite values.

So mathematical infinities exist.  I would not question that, but, as Barrow points out, mathematical and physical existence are not necessarily the same thing. 

“Gradually mathematicians lighted upon a new concept of existence.  Mathematical ‘existence’ meant only logical self-consistency and this neither required nor needed physical existence to complete it.  If a mathematician could write down a set of non-contradictory axioms and rules for deducing true statements from them, then those statements would be said to ‘exist’.”  John Barrow


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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #13 on: 28/08/2013 09:24:04 »
If I remember rightly infinity on a camera’s distance scale originally represented anything greater than one hundred times the focal length of the lens, and may still be used in that sense on occasions.  Obviously this is only loosely connected to anything that might be considered genuinely infinite, and probably is not what the NS article suggests may be an illusion.

100f makes sense for a terrestrial camera, where your horizon is a few miles and your resolution is limited by camera shake, but not for an astronomical telescope. Fortunately, classical geometry comes to our aid: "rays from infinity are parallel" so we can build and test a scope on the ground before putting it into orbit. Except of course for Hubble MkI!   
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Offline JP

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Re: Is infinity an illusion?
« Reply #14 on: 28/08/2013 13:37:48 »
If I remember rightly infinity on a camera’s distance scale originally represented anything greater than one hundred times the focal length of the lens, and may still be used in that sense on occasions.  Obviously this is only loosely connected to anything that might be considered genuinely infinite, and probably is not what the NS article suggests may be an illusion.

100f makes sense for a terrestrial camera, where your horizon is a few miles and your resolution is limited by camera shake, but not for an astronomical telescope. Fortunately, classical geometry comes to our aid: "rays from infinity are parallel" so we can build and test a scope on the ground before putting it into orbit. Except of course for Hubble MkI!   

Yep.  And this is the type of infinity that many physicists (and probably all engineers) use.  It's a nice shorthand for "very big."  It would be a waste of time to figure out what "100 f" or another suitably large number was for every case.  So long as the error introduced between using that number and infinity for distance is significantly smaller than the tolerances in the design, it can be done without too much of a problem.

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #15 on: 28/08/2013 17:54:18 »
Max Tegmark says:

"When quantum mechanics was discovered,we realised that classical mechanics was just an approximation. I think another revolution is going to take place, and we'll see that continuous quantum mechanics is itself just an approximation to some deeper theory, which is totally finite."

If a physicist at MIT is seriously making a statement like this, should it not stir some scientific thoughts on a science discussion forum?

Of course, it takes more than one physicist's early suspicions to cause a revolution, but David Hilbert's assertion: "No one shall dispel us from the paradise Cantor has created" could begin to look a little insecure.




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Offline JP

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Re: Is infinity an illusion?
« Reply #16 on: 28/08/2013 18:25:46 »
I agree with most of what you say, Bill, but as a professional physicist I like to be much pickier and more precise about using the term "infinity."    :p  As I pointed out above, infinity certainly exists in the world of thought of mathematical theory.  Infinite sets and infinite series are ideas in mathematics and can be manipulated.  Infinity is used in physics as a stand-in for "very large," and it comes into play with "very small" (infinitesimally small) and we can use the mathematical tools for handling infinity in these cases to produce useful results, often with much less work than would be involved if we tried to plug in large or small numbers.

This discussion seems to be entirely about whether there can physically exist situations consisting of an infinite number of things, an infinitely-sized object (including the entire universe) or an infinitelimally small object.  To me, the most testable one of these ideas is the infinitesimally small--if space is continuous, then no matter what tiny distance you pick, I can divide it in half.  This means there is no smallest size and infinitesimal distances are real things.  If, on the other hand, there is some limit to how far you can divide up a distance before you get into discrete units of space, then infinitesimal distances are just an approximation.  They work well because these distances are so small compared to what we're looking at that we don't notice the error introduced by this approximation.

I think this is what Max Tegmark is getting at.  The problem with an in-depth discussion of this is that there isn't a lot of scientific evidence one way or the other since our measurements aren't precise enough.  If there is discretization, the obvious place to look is near the Planck length, since that's where we expect quantum effects on space-time to become important.  I don't see a reason to discount this idea, and I'm actually fairly partial to it.  At the same time, there's also no reason to jump into the no-infinity boat until there's evidence to refute it.  Right now, infinity serves us well whether or not we think it really describes a physical object.
« Last Edit: 28/08/2013 18:34:19 by JP »

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #17 on: 31/08/2013 17:31:02 »
I have no quarrel with infinite sets/series as mathematical tools.  As far as the extremely small is concerned, I am fine with “infinitesimally small”.  I wish scientists would use it instead of “infinitely small”, which, in my opinion, is tantamount to saying it cannot be further divided, even in principle.  If space is continuous, this must be the same as saying it is nonexistent.

Unfortunately, there seems to be no term, equivalent to “infinitesimal”, to cover things that are “sort of” infinitely large.     

I would certainly not “jump into the no-infinity boat”.  Without infinity, what would I argue about?  :)

Seriously, though, I see no realistic way round the idea that something must be eternal/infinite, otherwise we would not be here.  This has caused me to do a lot of thinking about infinity, which might well be considered as being philosophy rather than science.  I would not argue with that, except to say that if it is something so fundamental to our existence, then perhaps it has as much right to a place in scientific thought as does the underlying “reality” of QM.


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Offline JP

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Re: Is infinity an illusion?
« Reply #18 on: 02/09/2013 16:44:36 »
I have no quarrel with infinite sets/series as mathematical tools.  As far as the extremely small is concerned, I am fine with “infinitesimally small”.  I wish scientists would use it instead of “infinitely small”, which, in my opinion, is tantamount to saying it cannot be further divided, even in principle.  If space is continuous, this must be the same as saying it is nonexistent.
Most scientists who are being precise would use "infinitesimally small" instead of "infinitely small."  I don't think "infinitely small" means much in a technical sense, while infinitesimal has a precise meaning.  I don't read too many pop-sci books, but I wouldn't be surprised if they abuse terminology by saying "infinitely small." 

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Unfortunately, there seems to be no term, equivalent to “infinitesimal”, to cover things that are “sort of” infinitely large.     
That's because "sort of" infinitely large isn't precise.  Do you mean "big enough that we can treat it as infinity in our equations"?  When we simply call it infinity and acknowledge that we mean "big enough that we can treat it as infinity."  That might not be the best way to express things for a layperson, but its efficient and universally understood by physicists.

In pop-sci books, the standard should probably be to write all this out before introducing the word "infinity" so readers understand how it's being used (an approximation or an absolute?"

Quote
I would certainly not “jump into the no-infinity boat”.  Without infinity, what would I argue about?  :)

Seriously, though, I see no realistic way round the idea that something must be eternal/infinite, otherwise we would not be here.  This has caused me to do a lot of thinking about infinity, which might well be considered as being philosophy rather than science.  I would not argue with that, except to say that if it is something so fundamental to our existence, then perhaps it has as much right to a place in scientific thought as does the underlying “reality” of QM.

Well, the issue is that QM provides testable predictions.  Testing whether "infinity" is real or not is probably not feesable in the near future.  That doesn't make it unscientific to discuss it, but it puts it on less of a hard-science ground than QM and limits what scientific details we can actually add to the discussion. 

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Offline yor_on

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Re: Is infinity an illusion?
« Reply #19 on: 02/09/2013 19:11:46 »
Infinity is not a definition making sense. everything having a stop make sense, even when we don't know where that stop exist. But infinity make sense, considering our universe. It's definitely not defined, neither from relativity, nor from QM.
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #20 on: 04/09/2013 21:09:38 »
Quote from: JP
  I don't read too many pop-sci books, but I wouldn't be surprised if they abuse terminology by saying "infinitely small."

Hunting through pop-sci books for examples would be very time consuming and of little ultimate value, but Paul Davies, John Gribbin , Peter Cattermole and Stuart Clark come readily to mind.

Quote
Do you mean "big enough that we can treat it as infinity in our equations"?  When we simply call it infinity and acknowledge that we mean "big enough that we can treat it as infinity."

That’s fair enough, and I how I now accept it when I read “infinite/infinity” in the context of science or maths.  I wish the authors of pop-sci books would make this clear, as there must be a risk that their readers will conclude that they are talking about some sort of “absolute” infinity, when they are talking about an approximation.  It is easy for “hitch-hikers”, like myself, to believe we have something sussed, just because an expert has said it.  As a child, someone said to me: “Question everything, not just because you think the person saying it is wrong, but because you may have misunderstood it.  Never believe you understand something until you know you understand it”.  I still try to maintain that attitude, but sometimes it annoys the hell out of people!

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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #21 on: 04/09/2013 21:27:59 »
Yep.  And this is the type of infinity that many physicists (and probably all engineers) use.  It's a nice shorthand for "very big." 

No. Infinity isn't shorthand for"very big"  but for "bigger than anything you can define".

Which makes an interesting point. I mentioned parallel rays of light as having their common source at infinity. So where is the common source for converging rays? 
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #22 on: 05/09/2013 03:34:14 »
Quote from: alancalverd
So where is the common source for converging rays? 

Do converging rays have a common source?

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Offline JP

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Re: Is infinity an illusion?
« Reply #23 on: 05/09/2013 04:17:26 »
Yep.  And this is the type of infinity that many physicists (and probably all engineers) use.  It's a nice shorthand for "very big." 

No. Infinity isn't shorthand for"very big"  but for "bigger than anything you can define".

Which makes an interesting point. I mentioned parallel rays of light as having their common source at infinity. So where is the common source for converging rays? 

If you read what I wrote carefully, you'll notice that I said a certain use of infinity is shorthand for very big, and it indeed it.  The parallel ray assumption is used all the time in practice and no one literally assumes that a planet or star is really infinitely far from the telescope.  It's even used commonly in designing consumer cameras for use in photography on earth where distances are in tens or hundreds of meters.  All that matters is that distances are significantly larger than the camera aperture so that the actual angle of rays is negligibly different from parallel--at least compared to other sources of error in the design.  I can guarantee this is true because I am a trained optical physicist who does this type of calculation on a daily basis. 

All applied physics involves approximations, and infinity used as a stand-in for "very big" pops up ubiquitously in most branches of physics (except for the most fundamental questions) and engineering.  It is regularly used anywhere where the error introduced by letting "very large" be infinity is small compared to other sources of error. 

I brought this up to point out to Bill that infinity is a useful concept, even if we don't take it literally.  It comes with a host of mathematical tools that simplify things greatly.  Try computing the electric field emitted by an oscillating dipole on earth by rigorously accounting for all matter on earth, all planets, moons, the sun, all stars in our galaxy, all other galaxies, etc. and compare this to just assuming the source radiates away to infinity to see the use of this.  Unless there are other sources/sinks nearby this dipole, the two calculations will give nearly identical results, and the one using infinities will actually be do-able.  :)

I wholeheartedly agree that the question of whether physical infinities exist is separate and interesting, but infinity is a real concept in mathematics and is extremely useful in practice, regardless of whether it is physically realizable or not.

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Offline JP

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Re: Is infinity an illusion?
« Reply #24 on: 05/09/2013 04:22:40 »
Quote from: JP
  I don't read too many pop-sci books, but I wouldn't be surprised if they abuse terminology by saying "infinitely small."

Hunting through pop-sci books for examples would be very time consuming and of little ultimate value, but Paul Davies, John Gribbin , Peter Cattermole and Stuart Clark come readily to mind.

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Do you mean "big enough that we can treat it as infinity in our equations"?  When we simply call it infinity and acknowledge that we mean "big enough that we can treat it as infinity."

That’s fair enough, and I how I now accept it when I read “infinite/infinity” in the context of science or maths.  I wish the authors of pop-sci books would make this clear, as there must be a risk that their readers will conclude that they are talking about some sort of “absolute” infinity, when they are talking about an approximation.  It is easy for “hitch-hikers”, like myself, to believe we have something sussed, just because an expert has said it.  As a child, someone said to me: “Question everything, not just because you think the person saying it is wrong, but because you may have misunderstood it.  Never believe you understand something until you know you understand it”.  I still try to maintain that attitude, but sometimes it annoys the hell out of people!


I agree with you that pop-sci should be as rigorous as possible to as not to confuse people.  To be honest, I don't read much of it since I'm so busy keeping up on technical literature in my field that my pleasure reading tends to be as non-sciencey as possible (fantasy or history).  I did read a book recently, "The Disappearing Spoon" that was quite good.  But even though I enjoyed it, and most of it involved chemistry that was outside my field, I noticed one rather glaring error in its physics.  It compared the idea of an electron orbiting an atom without spiraling into the nucleus as being as shocking as if a radio worked indefinitely without needing its battery replaced.  To a physicist the former is possible, if implausible, since it conserves energy.  The latter is impossible since sound waves leaving the radio take away energy, so that a radio that works forever without replacing the batteries generates infinite power--it's basically a perpetual motion machine.

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #25 on: 05/09/2013 22:13:10 »
Quote from: JP
....the idea of an electron orbiting an atom without spiraling into the nucleus....

For a long time that gave me problems; then I, sort of, got my head round the idea that it was the quantization of energy that prevented spiraling.  However, there was still the problem that if the electron was orbiting, it must be accelerating, so where was the energy coming from?  I suppose it's blatant thread drift to talk about it here, but am I right in thinking that we should no longer think of the electron as orbiting, but rather as being in any, or all, of an "infinite" (See! I just brought it back on topic) range of positions at the same time?


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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #26 on: 05/09/2013 23:02:12 »
Yes.
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #27 on: 06/09/2013 14:08:17 »
Great, but does that mean that the electron/wave is stationary, or is that too intuitive for thinking about QM?

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #28 on: 07/09/2013 21:02:25 »
Quote from: JP
In some cases such as the size of the universe, some theorists take it literally.  This is where I hedge my bets since you'd have to come up with a testable hypothesis to distinguish between an infinite universe or not.  As far as I know the question isn't settled. 

I run into a problem here.  If the Universe started at the BB, how could it be infinite?

I have been assured that it could have been infinite from the start.  However, I have yet to find an explanation as to how something infinitesimally small could be infinite.

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #29 on: 07/09/2013 21:23:32 »
While looking at the question of infinity I came across the Hartle-Hawking State.

http://en.wikipedia.org/wiki/Hartle%E2%80%93Hawking_state 

A few questions spring to mind.

Planck epoch = 0 to 10^-43 s.  This is a period of time; therefore time must have started at t=0.  Therefore time exists during the Planck epoch.  Do I have the right impression?

They suggest that “….if we could travel backward in time toward the beginning g of the universe, we would note that quite near what might have otherwise been the beginning, time gives way to space such that at first there is only space and no time.”

Does this statement imply that we would see that the condition of “space and no time” exists after t=0, or that we would be able to see beyond the BB?

 “….that at first there is only space and no time”.

Is this before or after the BB?

If there was a time when there was space but no time, how could this condition change?  Surely time is needed in order to allow change.  If there were a condition in which there were space and no time, there would be only space now, unless some outside influence “created” time.

They propose that “the universe is infinitely finite”!   I’m told I am the one with odd ideas about infinity. 

Is there a difference between “infinitely finite” and “unbounded”?

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Offline Pmb

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Re: Is infinity an illusion?
« Reply #30 on: 08/09/2013 05:37:35 »
Quote from: Bill S
Over 40 years ago I had a long discussion with a maths teacher about infinity.  It culminated in his conceding that the series of whole numbers, although apparently unbounded in both directions, was not an example of true infinity.
Do you still believe this? If so, why? It certainly isn't true. Infinity quite litteraly means without bound. In this case for the sequence (not series) of whole numbers, by which I assume that you're referring to a sequence of numbers which increases without bound, then it's an infinite sequence. I don't understand how anyone could conclude otherwise. Please explain.

Quote from: Bill S
Imagine my elation when I saw, in the New Scientist an article suggesting that some physicists were trying to remove infinity from scientific – and even mathematical – calculations. 
Please post the reference to this article. I'd like to read it. I assume you've read it. Would that be a correct assumption?

Quote from: Bill S
This must raise the question: How can we test for infinity?  Surely such a test would require an infinite amount of information. 
That depends highly on what it is that we're trying to measure as being infinite. Can you give an illustrative example?

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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #31 on: 08/09/2013 10:04:27 »
Great, but does that mean that the electron/wave is stationary, or is that too intuitive for thinking about QM?

The key phrase in your question was "think of". By no means the same as "is".

The wave function eloquently and adequately describes the behaviour of electrons in and around atoms, but an isolated electron  in a vacuum behaves delightfully like a charged particle. What it is, is an electron. 

As for measuring infinities, by definition, you can't.
« Last Edit: 08/09/2013 10:08:21 by alancalverd »
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Offline Bill S

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Re: Is infinity an illusion?
« Reply #32 on: 08/09/2013 18:09:18 »
Quote from: Pmb
Do you still believe this? If so, why? It certainly isn't true. Infinity quite litteraly means without bound. In this case for the sequence (not series) of whole numbers, by which I assume that you're referring to a sequence of numbers which increases without bound, then it's an infinite sequence. I don't understand how anyone could conclude otherwise. Please explain.

I feel sure you did not intend starting an etymological diversion, but if you go back to the Latin root, infinite means without end.   A sequence, such as the whole numbers, has no end, so if it is something that is real, in itself, then it is infinite, within the strict literal meaning of the word, but in both archaic and modern usage the word infinite has acquired wider significance. 

However, the sequence of numbers is something which has no existence other than as a tool for counting objects.  3 x nothing and 300 x nothing are the same thing.  3 and 300 have significance only in respect of the things they enumerate.  3 apples exist, 300 apples exist; an infinite number may exist, but has no real meaning; an infinite number of apples cannot exist in any context that we could recognise as real.

An infinite series (sequence) exists only in the (presumably finite) mind of the mathematician as a useful concept.

I am not questioning the use or value of infinities in mathematics, nor the value, in calculations, of treating infinity as though it were a number.  I simply call attention to the fact that infinity is not a number, and that outside mathematical usage, the concept of infinity is of singular importance in thinking about the origin of the Universe.



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Offline Bill S

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Re: Is infinity an illusion?
« Reply #33 on: 08/09/2013 18:43:10 »
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Please post the reference to this article. I'd like to read it. I assume you've read it. Would that be a correct assumption?

 The article appeared in the New Scientist 17.08.2013.  As far as I am aware, the full article is available on line only to subscribers.  As a non-subscriber, I have access only to a hard copy.  I have not offered to scan and post this as I'm sure it would infringe copyright. 

Your assumption is correct, I have read the article; several times.  Your local library might well have back numbers of NS.

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #34 on: 08/09/2013 20:40:45 »
Quote from: Pmb
That depends highly on what it is that we're trying to measure as being infinite. Can you give an illustrative example?

Surely, whatever we are trying to measure will be either finite, in which case we will, in principle, be able to measure it, or it will be infinite, in which case we will be unable to measure it with any finite measuring tool. 

“Illustrative example”?  Suppose my back garden (yard)  is 20x15m and I want to turf it.  I can measure it, and I will know how much turf to order, and how much this will cost.

Now suppose I decide my garden has infinite length and width.  With only finite measuring instruments, how can I measure it?  How can I know how much turf to order?

You might argue that I need an infinite amount of turf.  True as that might be, mathematically; what does it mean in the real world?

If I go to a supplier and ask for an infinite amount of turf, I suspect I would exit his office very quickly.

However, if he is anxious for business and thinks “I might do well out of this nutter”; how does he work out how much to charge me?  Łinfinite, perhaps?  What chance does he have of getting that?  In fact, all this example illustrates is that discussing the concept of an infinite amount leads to talking rubbish.


I hope this will prompt you to provide an example of how this “depends highly on what it is that we're trying to measure as being infinite”.  I'm doing my best to see the other side of this, but it's hard going.  :)




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Offline Bill S

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Re: Is infinity an illusion?
« Reply #35 on: 08/09/2013 20:53:13 »
Quote from: alancalverd
The key phrase in your question was "think of". By no means the same as "is".

Although I agree with your statement, I have to say you have lost me.  I think I may have expressed my meaning badly.  I’ll try again.

If we regard the electron as being in any, or all, of an "infinite" range of positions at the same time, do we regard the electron as being in motion, or stationary?  I then wondered if that question, in itself, was too intuitive to have a place in our thinking about QM.


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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #36 on: 08/09/2013 21:24:13 »
Disregarding "regard" for the time being, the wave function does not go to zero anywhere, so there is an infinitesimal probability that whatever it describes could turn up anywhere at all. So in principle, however large you make your search radius, it could be outside.

Turfing the lawn is an interesting question. If you rotate a hyperbola around the y axis you will generate a solid with a finite volume but infinite area, so a finite amount of water will flood a hyperbolic lawn, but you can never buy enough turf to cover it.   
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Offline dlorde

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Re: Is infinity an illusion?
« Reply #37 on: 08/09/2013 23:08:30 »
Quote from: Bill S
Imagine my elation when I saw, in the New Scientist an article suggesting that some physicists were trying to remove infinity from scientific – and even mathematical – calculations. 
Please post the reference to this article. I'd like to read it...
This is as close as you'll get without a sub.: Infinity's end: Time to ditch the never-ending story?
It talks about the Tegmark's dissatisfaction with the 'measure problem' (that an infinite multiverse predicts everything), and confuses the size of the observable universe with the size of the universe, concluding that 'there's no room for infinity'. It mentions the problem of being unable to measure to infinite precision/accuracy, and of Wildberger's "rational geometry" that, replaces angles with a "spread" defined as a rational output extracted from mathematical vectors representing two lines in space. It mentions Zeilberger idea that the number line is a loop (i.e. that there is a biggest number that 'overflows' when you add 1!). All in all, it's a bit of a hashed together article about opinions on the fringes, although, to be fair, it does give some space to those who don't have a problem with infinity, or think the problem lies elsewhere.

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Offline dlorde

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Re: Is infinity an illusion?
« Reply #38 on: 08/09/2013 23:21:53 »
If we regard the electron as being in any, or all, of an "infinite" range of positions at the same time, do we regard the electron as being in motion, or stationary? 
As I understand it, the wave function doesn't tell you that the electron is everywhere at the same time, but just tells you the probability of detecting it at any given point. What the electron is doing in between measurements is anyone's guess. As the wave function evolves, so the probability of detecting the electron at any given point may change. I expect you may infer, from successive measurements, that an electron has moved, although presumably that involves certain assumptions... 

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Offline dlorde

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Re: Is infinity an illusion?
« Reply #39 on: 08/09/2013 23:23:05 »
Turfing the lawn is an interesting question. If you rotate a hyperbola around the y axis you will generate a solid with a finite volume but infinite area, so a finite amount of water will flood a hyperbolic lawn, but you can never buy enough turf to cover it.   
You could paint it green instead, by flooding it with green paint!  ;)

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Offline JP

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Re: Is infinity an illusion?
« Reply #40 on: 08/09/2013 23:29:34 »
Disregarding "regard" for the time being, the wave function does not go to zero anywhere, so there is an infinitesimal probability that whatever it describes could turn up anywhere at all. So in principle, however large you make your search radius, it could be outside.

Just to nitpick--there can be zeros in the wave function, but they are essentially nodes of a standing wave (like the points in a standing wave on a string that don't move).  You're absolutely right that in principle, the wave function does extend out to infinity, though after a point there would be such a small chance of finding the electron that you'll never see it there in practice.

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #41 on: 09/09/2013 18:16:42 »
Quote from: alancalverd
If you rotate a hyperbola around the y axis you will generate a solid with a finite volume but infinite area

I could do with an explanation, here, please.  ???

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Offline dlorde

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Re: Is infinity an illusion?
« Reply #42 on: 09/09/2013 21:27:46 »
Quote from: alancalverd
If you rotate a hyperbola around the y axis you will generate a solid with a finite volume but infinite area

I could do with an explanation, here, please.  ???
Torricelli's Trumpet (Gabriel's Horn).

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Offline alancalverd

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Re: Is infinity an illusion?
« Reply #43 on: 09/09/2013 23:50:38 »
Apologies! Of course, you have to rotate it around an asymptote, not the y axis. Pretty neat trick, though, ain't it?
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Re: Is infinity an illusion?
« Reply #44 on: 10/09/2013 13:30:56 »
Quote from: JP
In some cases such as the size of the universe, some theorists take it literally.  This is where I hedge my bets since you'd have to come up with a testable hypothesis to distinguish between an infinite universe or not.  As far as I know the question isn't settled. 

I run into a problem here.  If the Universe started at the BB, how could it be infinite?

I have been assured that it could have been infinite from the start.  However, I have yet to find an explanation as to how something infinitesimally small could be infinite.
Bill, what are you saying was infinitesimally small?  the universe or the observable universe?


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Offline Bill S

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Re: Is infinity an illusion?
« Reply #45 on: 10/09/2013 15:28:53 »
Torricelli’s Trumpet is another example of a mathematical infinity, with which I have no argument.  I have a little trouble understanding how the volume could be finite; unless there is a mathematical “finiteness” similar to a mathematical “infinity”.  I’m very willing to accept that that may just be my lack of understanding, though.

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #46 on: 10/09/2013 15:37:53 »
Quote from: lean bean
Bill, what are you saying was infinitesimally small?  the universe or the observable universe?

We may be slipping into a terminological pitfall, here.  What do cosmologists believe began its existence at the BB; the universe or the observable universe?

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lean bean

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Re: Is infinity an illusion?
« Reply #47 on: 10/09/2013 18:42:31 »

We may be slipping into a terminological pitfall, here.  What do cosmologists believe began its existence at the BB; the universe or the observable universe?

What began its existence at the BB was a determinable space and time universe, before that equations breakdown. I think It's the density of matter/energy that brings about a breakdown of equations (singularity) not whether the universe is finite or infinite in size.
I know you like the following link...  http://www.astro.ucla.edu/~wright/infpoint.html
:)
You may remember my reply to you here..  Reply #24.
http://www.thenakedscientists.com/forum/index.php?topic=48385.0





« Last Edit: 10/09/2013 18:50:10 by lean bean »

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Offline JP

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Re: Is infinity an illusion?
« Reply #48 on: 10/09/2013 21:13:35 »
This comes back to the original point of the post.  We got to the idea of "big bang" because we can only measure the universe where we have measuring devices (for cosmology, the relevant devices are mostly particle colliders and telescopes).  This limits the data we can collect.  From this data, we can figure out physical laws governing the universe.  By extrapolating back in time, we can try to figure out what the universe was like in the past.  This assumes, of course, that laws don't change in time and we're also limited to theories that match our current evidence.  For conditions far outside of what we can observe, we can't be sure our models are correct.

What happens when we extrapolate back far enough is that the models start predicting ever increasing density--a density that tends to infinity at some point in the past.  Most scientists take Bill's point of view here and believe that this infinity is an indication of a flaw in the model, not something that physically existed. 


« Last Edit: 10/09/2013 23:50:09 by JP »

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Offline Bill S

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Re: Is infinity an illusion?
« Reply #49 on: 10/09/2013 22:26:41 »
Quote from: lean bean
You may remember my reply to you here..  Reply #24.

Glad you mentioned that.  I'd lost track of that thread.  Going back over it may save some repetition.  :)