The concept of infinity

[beecee]

37 minutes ago, wtf said:

I hope you understood my point, which was: The bowling ball and rubber sheet story is not physics, just as Hilbert's hotel is not mathematic, and sometimes popularized metaphors can confuse as well as enlighten.

That's what I said [But yes, analogies in near all cases are limited in what they illustrate.] as opposed to your total criticism of analogies. 

Quote

Your post was interesting about gravity. I'm aware your bolded words pass for conventional wisdom among the physicists. but "A causes B and B causes A" does not strike me as much of an explanation of anything.

That's about all we know with regards to why warped/curved/twisted spacetime, exhibits gravity. This may help.....https://www.youtube.com/watch?v=MO0r930Sn_8

 

Quote

On the contrary. Can you search your mind and your soul, and tell me whether you have an intuitive idea of the endless sequence of counting numbers 0, 1, 2, 3, 4, ...? If yes, you have comprehended infinity. It's more sensible to imagine the sequence going on forever; than to imagine it suddenly stopping at some point. Infinity is an idea that's baked in to our brain. Agreed that the formal study of mathematical or physical or philosophical infinity is complicated. But the intuition is obvious even to children. You can count forever. And even if the physical world couldn't allow it; you can still count forever in principle. And I would contend that this is obvious to virtually everyone who thinks about it, even for a moment, unless one is a committed ultrafinitist. 

I strongly disagree. Your counting "analogy" has a flaw rather then a limitation....You first need to begin counting. The application of infinity is difficult to comprehend. Sure a child may blithely accept the finality of a fairy tale and Prince Charming and his bride living forever and ever amen. Acceptance is not understanding. I accept according to the evidence and validation of GR that mass/energy curves spacetime...I don't know why though. I accept that the universe/space/time as we know them, arose from nothing, as the only true scientific answer. It was nothing [defined as the quantum foam] that has existed [if we can use that word] for eternity. That of course needs "nothing" to be defined as quantum foam. But it certainly has some basis in scientific thinking and application. https://www.astrosociety.org/publication/a-universe-from-nothing/

19 minutes ago, wtf said:

Useful, sure. But potentially misleading. Why do bowling balls distort a rubber sheet? It's a pretty good question actually. One not addressed by the analogy.

Agreed...In the bowling ball/Rubber sheet analogy it certainly is gravity and elasticity as you say, but he point of the analogy was to show how mass distorts spacetime, which we don't really know why...The raisin loaf is another more illustrative analogy. When giving an analogy to facilitate understanding, then we also make sure that it is understood how such analogies are limited to various extents. As a born and bred lay person, that gained my interest in science in later life, analogies served a purpose. The same goes for pop science docos. Limited, yes, sometimes not exactly correct, yes, but if the individual watching them, has any real interest, he will like I did, take it further and check it out in greater depth. 

Edited March 22, 2019 by beecee

[wtf]

12 minutes ago, beecee said:

I strongly disagree. 

Do you claim to have no intuition of the endlessness of the counting numbers?

Sticking to infinity for the moment since I'm not qualified to talk about the physics of gravity. But note that even though I have little knowledge of the physics of gravity; I have a strong intuition and experience of gravity from birth and probably from before that, since a fetus probably knows up from down. Just as one has an intuition of infinity or even an experience of it from contemplating the counting numbers; even without having technical knowledge of the theory.

Edited March 22, 2019 by wtf

[beecee]

Just now, wtf said:

Do you claim to have no intuition of the endlessness of the counting numbers?

I claim like most all people, not to fully understand the application of infinity, no beginning, no end. Perhaps I'm not as bright as you. 

[Carrock]

18 minutes ago, beecee said:

The application of infinity is difficult to comprehend.

Can you tell me which part of "infinity for five year olds," showing that there's twice as many integers as there are integers, or as many even numbers as integers or whatever, is beyond your comprehension?

Or, as you likely meant, beyond the comprehension of a five year old?

Edited March 22, 2019 by Carrock
x-posted with beecee's post

[wtf]

1 hour ago, beecee said:

I claim like most all people, not to fully understand the application of infinity, no beginning, no end. Perhaps I'm not as bright as you. 

I can only say that I myself have a clear and strong intuition of infinity. Whether I acquired that at an early age, or only feel that way because I had it beaten into me in math class, I honestly can't say. 

 

1 hour ago, Carrock said:

Can you tell me which part of "infinity for five year olds," showing that there's twice as many integers as there are integers, or as many even numbers as integers or whatever, is beyond your comprehension?

 

As Wittgenstein might have said: Whereof one cannot speak without snark; one must thereof put a sock in it. I shall heed his advice.

Uh ... which is it? Are there twice as many or the same as? Perhaps it's beyond YOUR comprehension. Damn, the snark leaked out anyway. Only happened because I read your post a second time and realized you yourself are fuzzy on the example.

1 hour ago, Carrock said:

Or, as you likely meant, beyond the comprehension of a five year old?

The story of Hilbert's hotel is beyond the comprehension of most adults. I don't know the five year old referenced by the OP.

Edited March 22, 2019 by wtf

[Eise]

11 hours ago, wtf said:

For a five year old? Maybe, if you want to confuse them.

Well, isn't it confusing? You say it already: most adults get confused when discussing infinity, even if it is in one of the simplest examples, mathematically, with natural numbers. 

11 hours ago, wtf said:

Hilbert's hotel is a fable, a story for the tourists. It's not a mathematical argument. It's like the rubber sheet and bowling ball visualization of relativistic gravity.

I don't agree. The rubber sheet and bowling ball are real things that can be shown, but are a bad illustration of relativistic gravity. Infinity in natural numbers is mathematically real, but the Hilbert Hotel isn't. However, I still think it is a nice illustration to show that our conception of 'infinity' as 'a great number' is wrong. And there is a direct relationship with the examples of the Hilbert Hotel (the hotel is full, but you still can add (1) one guest, (2) an infinite number of guests, etc) and the mathematical operation (1 + infintiy = infinity, infinity + infinity = infinity, etc). You do not need a mathematical argument: you only need a 1 to 1 relationship between the mathematical operations and the example. (In this of course the rubber sheet analogy miserably fails: it already presupposes gravitation).

OTOH, I was never in the situation to explain infinity to somebody. So I have no empirical and didactic experience with it. 

7 hours ago, beecee said:

The same for defining nothing.

Infinity and nothing have nothing in common. 

(If you understand that sentence, you have understood 'nothing'. But can still be confused about 'infinity'... )

7 hours ago, wtf said:

sometimes popularized metaphors can confuse as well as enlighten

In what way exactly then is the Hilbert Hotel confusing? That is, more confusing than the abstract concept of 'infinity'?

7 hours ago, wtf said:

On the contrary.

Then it should be easy to explain to a 5 year old... So taeto's problem isn't a problem at all?

 

Edited March 22, 2019 by Eise

[wtf]

54 minutes ago, Eise said:

In what way exactly then is the Hilbert Hotel confusing? That is, more confusing than the abstract concept of 'infinity'?

I stated the points of confusion in my earlier post. I have nothing to add. And needless to say, my feelings about the subject won't alter the popularity of Hilbert's hotel.

[Sirjon]

On 3/21/2019 at 12:56 AM, DrP said:

Not to any definition I've seen of it.

No offence taken - it is just wrong.

Where did you get these definitions from? Did you make this up? I see where you are coming from when you say 'a perfect chip houses infinite number of component...'  but a could say that a perfect chip doesn't NEED an infinite amount.   Where did you get the definition of this perfect chip from?  Made up?

Haven't you not observing it?

One example is the circumference of a circle. We can 'perfectly' draw a circle by knowing the radius and yet, we can only get the estimate value of circumference because the phi is a non-terminating, non-ending series of numbers toward infinity. So, by way of saying infinity, we can do it vice versa, a perfect circle but undetermined exact value of the circumference or imperfect 'thing' toward perfection thru perceiving infinite elements to make it perfect. 

Well if you want to say, "I jut made  it up", YES it is, through analyzing how I conceive and observe things?  Now, let me return back to you, regarding your own 'opinion' , based on the what you learned, Do you think, HOW MANY, EXACTLY, OR ROUGHLY,  THE NUMBERS OF  ELECTRONIC COMPONENTS do we need to create a perfect computer? 

[DrP]

1 minute ago, Sirjon said:

Do you think, HOW MANY, EXACTLY, OR ROUGHLY,  THE NUMBERS OF  ELECTRONIC COMPONENTS do we need to create a perfect computer?

It would depend on how you choose to define a 'perfect' computer - I do not know but it has nothing to do with infinity unless you constrict it with the made up definition of a perfect computer.

Your association of infinity with 'perfection' is made up woo I think.

 

 

[Sirjon]

1 minute ago, DrP said:

It would depend on how you choose to define a 'perfect' computer - I do not know but it has nothing to do with infinity unless you constrict it with the made up definition of a perfect computer.

Your association of infinity with 'perfection' is made up woo I think.

 

 

Perfect. Now you made a good example of the word 'infinity'... 

Well if you read it back my original statement, ... I just SHARE my thought' regarding the concept of Infinity as something not so scientific but more on its philosophical aspect. Butif you insist that I need to support my 'opinion or idea, then I no longer know what is the dividing line between Science and Philosophy . 

To add to this:

A 'perfect computer[' will able to give the exact value of phi!

[Carrock]

15 hours ago, wtf said:

16 hours ago, Carrock said:

Can you [beecee] tell me which part of "infinity for five year olds," showing that there's twice as many integers as there are integers, or as many even numbers as integers or whatever, is beyond your comprehension?

Or, as you likely meant, beyond the comprehension of a five year old?

As Wittgenstein might have said: Whereof one cannot speak without snark; one must thereof put a sock in it. I shall heed his advice.

Uh ... which is it? Are there twice as many or the same as? Perhaps it's beyond YOUR comprehension. Damn, the snark leaked out anyway. Only happened because I read your post a second time and realized you yourself are fuzzy on the example.

 

Did you read this earlier post, referenced in the post you quoted?[pedantic snark] Nothing in the rules against using onsite references.[/pedantic snark]

On 3/21/2019 at 8:57 AM, Carrock said:

I'll bite... assuming she's learned her two times tables and how to divide/multiply by two..

Ask her what is the 2^nd even number.

Then 4^th, 5^th etc and ask her if she sees a pattern....

If she does, ask what is the 43^rd even number, reassuring her that doing it the easy way without counting each number isn't cheating, but higher maths. Do a few more calculations until she's comfortable with the idea.

Then ask her for the biggest 'real' whole number she can think of. And of course the [biggest 'real' whole number she can think of]^th even number.

Shouldn't take her very long to realise* that the set of positive integers can be placed in one to one correspondence with the set of positive even integers.

And so on....

 

*informally

Anything fuzzy here?

The later post was inspired by a third party story of [appeal to authority] R. Feynman [/appeal to authority] showing a child 'there are more numbers than there are numbers.'

I think children are as good as adults at learning something new, especially when, as I was careful to exemplify, they already have all the basic maths they need to understand a new concept. It takes a certain perverse sort of ignorance/laziness to teach children infinity is 'difficult,' compared to e.g. division.

[beecee]

13 hours ago, Eise said:

Infinity and nothing have nothing in common. 

Other then perplexing humanity since humanity evolved sufficiently to ask those questions. "Nothing" as I have mentioned previously does need redefining.

On the question in hand, I asked myself what is the opposite of infinity. The following would align closest to my views.....

https://www.scienceabc.com/eyeopeners/what-is-the-opposite-of-infinity.html

No, the answer isn’t zero. Infinity is the largest number there is, so the opposite of infinity would be the smallest number there is. Zero would mean nothing, so what we’re looking for is a number just greater than zero. However, as we’ll find out, determining this number isn’t as simple as pointing to the number 1.

Infinities are weird

Infinity has baffled humanity since antiquity. One must realize that infinity is not a concrete number, but rather an idea; it exists only in abstraction. Infinity cannot be a concrete number, say, x, because we can, by the logic of addition, add 1 to x and create a new infinity. We can then add another 1 to create a larger infinity. We can, in fact, add infinity to infinity to create perhaps the infinity of all infinities, but then we can add to this infinity another 1 and…you know the drill.

The microscopic realm isn’t any different. The opposite of infinity is called infinitesimal, and its nature is equally bizarre. Unlike whole numbers, real numbers aren’t rigid. Their splintered nature allows us to find and create infinite numbers between any two numbers. A number can be combined as many times as it can be divided. There could be a hundred numbers between 0 and 1, from 0.01-0.99, or even millions, one just has to add zeroes after the decimal point — divide it increasingly to create new numbers. So, while 0.00000000000000001 seems infinitesimal, one can just divide it by 10 to create a new infinitesimal — 0.000000000000000001.

So, infinitesimal, like infinity, exists only in abstraction, yet its uncertain nature is very disconcerting not just for mathematicians, but also physicists.

 

more at link....

 

 

[studiot]

16 hours ago, Eise said:

Infinity and nothing have nothing in common.

Have you thought about L'Hopital's rule?

[math]\frac{\infty }{\infty }and\frac{0}{0}[/math]

 

[wtf]

18 hours ago, Eise said:

Well, isn't it confusing? You say it already: most adults get confused when discussing infinity, even if it is in one of the simplest examples, mathematically, with natural numbers. 

I don't agree. The rubber sheet and bowling ball are real things that can be shown, but are a bad illustration of relativistic gravity. Infinity in natural numbers is mathematically real, but the Hilbert Hotel isn't. However, I still think it is a nice illustration to show that our conception of 'infinity' as 'a great number' is wrong. And there is a direct relationship with the examples of the Hilbert Hotel (the hotel is full, but you still can add (1) one guest, (2) an infinite number of guests, etc) and the mathematical operation (1 + infintiy = infinity, infinity + infinity = infinity, etc). You do not need a mathematical argument: you only need a 1 to 1 relationship between the mathematical operations and the example. (In this of course the rubber sheet analogy miserably fails: it already presupposes gravitation).

OTOH, I was never in the situation to explain infinity to somebody. So I have no empirical and didactic experience with it. 

Infinity and nothing have nothing in common. 

(If you understand that sentence, you have understood 'nothing'. But can still be confused about 'infinity'... )

In what way exactly then is the Hilbert Hotel confusing? That is, more confusing than the abstract concept of 'infinity'?

Then it should be easy to explain to a 5 year old... So taeto's problem isn't a problem at all?

 

You made some good points, let me take another run at a reply. 

 

> Well, isn't it confusing? You say it already: most adults get confused when discussing infinity, even if it is in one of the simplest examples, mathematically, with natural numbers. 

This was in reference to my claim that Hilbert's hotel would confuse a five year old. I guess I don't agree that we should tell a story that confuses the issue. But I don't know anything about five year olds. So that part of the conversation I should try to avoid.

 

> I don't agree. The rubber sheet and bowling ball are real things that can be shown, but are a bad illustration of relativistic gravity.

Yay!! I'm glad you agree with me. At least about the bowling ball. How many times have we seen the picture of the bowling ball and rubber sheet. It doesn't stand up to scrutiny though.

Now I do take your point that the Hilbert hotel story is at least mathematically accurate. I just object to the extraneous details of a hotel with guests in it, because I have seen many people get confused about those physical elements.

> Infinity in natural numbers is mathematically real, but the Hilbert Hotel isn't. However, I still think it is a nice illustration to show that our conception of 'infinity' as 'a great number' is wrong. And there is a direct relationship with the examples of the Hilbert Hotel (the hotel is full, but you still can add (1) one guest, (2) an infinite number of guests, etc) and the mathematical operation (1 + infintiy = infinity, infinity + infinity = infinity, etc). You do not need a mathematical argument: you only need a 1 to 1 relationship between the mathematical operations and the example. (In this of course the rubber sheet analogy miserably fails: it already presupposes gravitation).

Well here I really disagree with you. Let me first admit that almost everyone agrees with you and not with me. Hilbert's hotel is very popular. 

My problem with it (for adults) is that it brings up all kinds of irrelevant objections like, If the hotel is full how can everyone move to a new room? Do they all move at once or what? What happens to the last guy? Oh there is no last guy? Why not? And if there are infinitely many guests where do you get new guests? I've personally seen these types of questions completely derail discussions of Hilbert's hotel.

I much prefer a simple mathematical example, bijecting the natural numbers to the even natural numbers. The evens are, on the one hand, a proper subset; but on the other hand, clearly in one-to-one correspondence with the whole set of naturals. From which we conclude that, unlike a finite set, an infinite set can be in bijection with one of its proper subsets, Which we then turn around and make that the very definition of an infinite set. A set is infinite if it happens that it can be placed in bijection with one of its proper subsets.

I would MUCH rather have to explain and defend and explain again that paragraph, which is actual mathematics; than to have to parry questions about hotels and guests and how many maids does it take to clean all the rooms. (Only one. She cleans the first room in half a minute, the second room in 1/4 minute, and so forth. She cleans the whole hotel in exactly one minute). 

I personally don't like clouding up the explanation of what is an infinite set, with confusion about how such a hotel could exist and whether the guests all move and once or one after another, and where do they move to. Those are good questions, but they are completely off in a wrong direction if you're trying to explain mathematical infinity.

That's my opinion on the matter. But truly, if anyone's bothered by it, please be relieved to know that everyone agrees with you and not me. The story is very popular. 

>  In what way exactly then is the Hilbert Hotel confusing?

Everything implied by "hotel." It's a building, it has rooms, there are people in it. Leads to many irrelevant issues.

> That is, more confusing than the abstract concept of 'infinity'?

I think simply noting that the even numbers are in in bijection with the natural numbers is a far simpler observation to make.

> Then it should be easy to explain to a 5 year old...  

I honestly I don't believe that. But as I'm on record saying that the Hilbert hotel story shouldn't even be told to adults; I feel even more strongly that it's not suitable for children.

Just show people the sequence of natural numbers and show how you can biject the evens to the whole set. THEN when they understand that, you can tell them the story about the hotel. If you do it the other way around, confusion inevitably results. 

Edited March 23, 2019 by wtf

[studiot]

10 hours ago, wtf said:

That's my opinion on the matter. But truly, if anyone's bothered by it, please be relieved to know that everyone agrees with you and not me.

 

I happen to agree with you, Hilbert's Hotel is far to subtle for the mathematical knowledge and maturity of the average 5 year old.

However I also think that (I certainly find this) we are constantly overestimating the capability of students, because we misremember just what we knew at their age.
There were two recent threads where 16year olds were asking for essay help with mathematical subjects. I certainly offered too much depth.
 

So I think this is too much depth for a 5 year old.

10 hours ago, wtf said:

I much prefer a simple mathematical example, bijecting the natural numbers to the even natural numbers. The evens are, on the one hand, a proper subset; but on the other hand, clearly in one-to-one correspondence with the whole set of naturals. From which we conclude that, unlike a finite set, an infinite set can be in bijection with one of its proper subsets, Which we then turn around and make that the very definition of an infinite set. A set is infinite if it happens that it can be placed in bijection with one of its proper subsets.

I would MUCH rather have to explain and defend and explain again that paragraph, which is actual mathematics; than to have to parry questions about hotels and guests and how many maids does it take to clean all the rooms. (Only one. She cleans the first room in half a minute, the second room in 1/4 minute, and so forth. She cleans the whole hotel in exactly one minute). 

Although I agree it is mathematically far superior.

There is also a philosophical point about Hilbert.
No definition of 'full' is offered in relation to the word infinity and it all hinges on that missing definition.

Incidentally I have now received an interlibrary copy of Bell and I have mixed reactions.
I find I a pretty up to date compact grand survey and it would be a useful compact source book for facts.
But therin also lies its main weakness. It is disappointing in that it mostly stops tantalisingly short of detail in depth.
 

 

[Eise]

On 3/22/2019 at 11:50 PM, studiot said:

Have you thought about L'Hopital's rule?

\frac{\infty }{\infty }and\frac{0}{0}

Is the number 0 'nothing'?

For the rest, it was mainly a joke of course. But with one important 'morale', beecee:

The word 'nothing' has a very clear meaning. You could understand any sentence with 'nothing' in it. The confusion comes when it is substantivied as something absolute.

However, nothing in its normal use points to that it is some metaphysical category. (1)

Just look carefully if I would change the sentence just a little bit:

'Nothing' in its normal use points to that it is some metaphysical category. (2)

(1) is a plain sentence, but the 'its' refers the 'nothing' 2 sentences before. In sentence (2) 'its' refers to the 'Nothing' in the same sentence, and therefore it is metaphysical BS. 

Edited March 24, 2019 by Eise

[studiot]

6 hours ago, Eise said:

Is the number 0 'nothing'?

Smart move mister. +1

I did assume nothing was meant to mean mathematical zero, though I do know the difference.

[ecstaticdancer]

Infinity is not real in the sense that a brick is real. It is a mathematical extrapolation of the natural numbers. This should be explained to any 5 year old.

Surprised to not notice any reference to the Planck length, if the infinitely small is not in our Universe then why should the infinitely large?

[zapatos]

23 minutes ago, ecstaticdancer said:

if the infinitely small is not in our Universe then why should the infinitely large?

Is that some fundamental law of the universe? That if one extreme is not possible, then neither is the other? 

[Strange]

45 minutes ago, ecstaticdancer said:

Infinity is not real in the sense that a brick is real.

Five is not real in the sense that a brick is real. (In other words, I'm not sure that is a very useful statement.)

On the other hand, infinity is just as real (whatever that means) as five.

46 minutes ago, ecstaticdancer said:

It is a mathematical extrapolation of the natural numbers.

There are a few problems with this simple statement.

It could be taken to imply that infinity is one of the natural numbers (the last one?). It is more accurate to say that it is the cardinality ("size") of the set of natural numbers.

And there is more than one infinity; there is another (larger) infinity that is the cardinality of the reals.

49 minutes ago, ecstaticdancer said:

Surprised to not notice any reference to the Planck length, if the infinitely small is not in our Universe then why should the infinitely large?

Again, there seem to be multiple problems with this.

The Planck length is not "infinitely small" (several theories in physics require entities smaller than the Planck length).

Also, as zapatos notes, this appears to be a non sequitur. There may be a minimum size for things in the universe (or not) but that has not connection to whether the universe is infinite or not.

[ecstaticdancer]

2 hours ago, zapatos said:

Is that some fundamental law of the universe? That if one extreme is not possible, then neither is the other

Not a law of the Universe, but has a symmetry. The natural numbers are reflected into the interval between O and 1 by taking the reciprocal. 1 stays the same and 0 becomes the reciprocal of infinity, aleph 0 if I remember Cantor correctly.

[zapatos]

4 minutes ago, ecstaticdancer said:

Not a law of the Universe, but has a symmetry.

My face has symmetry but that does not imply anything about the size of the universe.

[ecstaticdancer]

1 hour ago, Strange said:

The Planck length is not "infinitely small" (several theories in physics require entities smaller than the Planck length)

Exactly, the Planck length is about 1.6x10^-35m, a long way from 0.

I am not aware of the theories you mention, names? references? This is the reason I'm here...

[Strange]

21 minutes ago, ecstaticdancer said:

I am not aware of the theories you mention, names? references? This is the reason I'm here...

Pretty much all theories of quantum gravity: string theory, causal dynamical triangulation, loop quantum gravity, etc.

[ecstaticdancer]

3 hours ago, Strange said:

Pretty much all theories of quantum gravity: string theory, causal dynamical triangulation, loop quantum gravity, etc.

Thanks for that, I thought I might have missed something, but no. All those theories are as speculative as mine!

At least mine is falsifiable... just need someone to do the experiments.