The concept of infinity

[taeto]

I am sorry, but I feel that I really need this explained.

Prompted by a very nice account that I read on math stack exchange from a parent who tried to answer questions from a 5-year old child about "infinity". The reactions turned into suggestions about whole numbers and "numbers that are larger than all numbers that you can imagine", and similarly. Numbers?

I looked up the wikipedia page on "infinity". Similar story. The first sentences establish that "infinity" is a philosophical concept concerning something without any bound. Then it is said that "modern mathematics uses the general concept of infinity..."

I may be a terrible example of a parent to a 5-year old. But I do know that mathematics in no way uses the philosophical concept of "infinity" (or indeed any other purely philosophical concepts).

The infinity symbol \(\infty\) is used in mathematics to abbreviate expressions involving limits, typically in summations and integrals. There is no "concept" involved, since you could freely choose to replace the symbol by the more tedious original limit expressions without any change of the meaning. The symbol is also used in situations, in topology, elliptic curves, and real and complex analysis, etc., when it is practical or necessary to add an additional element to an existing structure to obtain a larger structure with desired properties. The "extra" element is traditionally named using the symbol \(\infty\). There is no concept whatsoever involved in this choice, since you could use any other symbol desired. The statement "mathematics uses a concept of infinity" seems a basic misunderstanding, from my traditional mathematical point of view. 

Maybe someone can point to a usage of "infinity" in mathematics, which is actually related to the philosophical concept and not just as a choice of a symbolism?

 

 

[Strange]

31 minutes ago, taeto said:

But I do know that mathematics in no way uses the philosophical concept of "infinity" (or indeed any other purely philosophical concepts).

It could be argued that numbers (and everything else in mathematics) are purely philosophical concepts. 

[studiot]

54 minutes ago, taeto said:

the philosophical concept of "infinity"

Can you say what you mean by "the philosophical concept of infinity" ?

[QuantumT]

Since this is posted in the philosophical section, I though I might add my philosophical POV to it.

To me infinity is the void where the big bang happened. The void beyond the boundaries of our universe.
I imagine it as just one endless nothing. And our universe is just one tiny exception to that.

[Strange]

6 minutes ago, QuantumT said:

Since this is posted in the philosophical section, I though I might add my philosophical POV to it.

To me infinity is the void where the big bang happened. The void beyond the boundaries of our universe.
I imagine it as just one endless nothing. And our universe is just one tiny exception to that.

That is not philosophy. And it is pretty much nonsense as cosmology (there is no void or boundary). It is also completely off topic.

[QuantumT]

13 minutes ago, Strange said:

That is not philosophy. And it is pretty much nonsense as cosmology (there is no void or boundary). It is also completely off topic.

Sorry! (Tough crowd...)

[taeto]

1 hour ago, Strange said:

It could be argued that numbers (and everything else in mathematics) are purely philosophical concepts. 

Very well then. Everything in mathematics is philosophical. 

But the converse does not seem to apply without restriction. Is everything philosophical also a mathematical concept? 

1 hour ago, studiot said:

Can you say what you mean by "the philosophical concept of infinity" ?

What I have is from the Wikipedia page:  "Infinity is a concept describing something without any bound, or something larger than any natural number."

Since this is not a mathematical explanation, I take it to be an explanation of the philosophical concept. The sentence following it states that "Philosophers have speculated about the nature of the infinite..." 

Maybe "Infinity is a concept describing something without any bound, or something larger than any natural number" is not even a philosophically valid statement, in which case I admit to be barking up the wrong tree. I simply am not sufficiently familiar with philosophy to tell. Although then the sentence about "Philosophers have speculated..." would seem largely irrelevant if that is the case.

[studiot]

53 minutes ago, taeto said:

Very well then. Everything in mathematics is philosophical. 

But the converse does not seem to apply without restriction. Is everything philosophical also a mathematical concept? 

What I have is from the Wikipedia page:  "Infinity is a concept describing something without any bound, or something larger than any natural number."

Since this is not a mathematical explanation, I take it to be an explanation of the philosophical concept. The sentence following it states that "Philosophers have speculated about the nature of the infinite..." 

Maybe "Infinity is a concept describing something without any bound, or something larger than any natural number" is not even a philosophically valid statement, in which case I admit to be barking up the wrong tree. I simply am not sufficiently familiar with philosophy to tell. Although then the sentence about "Philosophers have speculated..." would seem largely irrelevant if that is the case.

It is difficult to know what  reply as I am surprised you don't already know much of this stuff.

Anyway here goes

Aristotle was the main philosopher in question. They did not have a well developed theory of limits in those days, but they introduced the notions of

Real infinity and potential infinity.

Real infinity is what would happen if you actually went on for ever with some process eg adding 1.

But they were frightened of the implications of this so they said that in the real universe there are no real infinities only potential infinities, since nothing goes on for ever.

So to them the potential infinity is what you would get if you went on adding 1 for ever.
But it did not exist in the real universe for them.
You could only tend towards it.

 

Now this is not a limit because it is divergent.

Limits are convergent.

But limits were only beginning to be introduced in the 17th century.

The infinity occurs in the indexing variable in a limit "for I = 1 to infinity"

When Cantor wanted to study infinities he did not use the term, he introduced the term transfinite instead.

 

How are we doing, am I telling you stuff you already know?

[taeto]

10 hours ago, studiot said:

How are we doing, am I telling you stuff you already know?

Doing fine, thanks a lot!

It is not quite that I do not know stuff. Maybe I know quite a lot. But I have trouble to look up the right pieces of stuff and put them together into some appropriate picture which it seems everyone else can see.

In the explaining infinity to a 5 year old situation, the focus turned to explaining how for every number you can add one to it and get a larger number. This is covered by integer arithmetic. There is no object of integer arithmetic that is larger than any integer or has no bound; nothing can be infinite. The "set of all natural numbers" belongs to set theory, not to arithmetic. 

You will have to add a concept of repeating the addition, and make a mental picture of "repeating forever". Is it correct to say that this already presupposes the concept of time being able to extend infinitely into the future? In which case the "infinity" stuff is something that you have to add extra into the mathematics and which is not already present from the outset?

The "potential infinity" construction is well-liked by cranks especially. You save the day by adding "potential" in front of the despised and/or scary "infinity". How does this actually work semantically: is it a similar kind of construction like when you add "real" in front of "number", to get "real number", which is an object altogether not the same as a "number" (by default a natural number) that is "real"? If so, then introducing "potential infinity" in order to explain "infinity" is a red herring, just like you do not explain natural numbers by making reference to real numbers. And if the construction is to make a new concept "potential infinity" starting from the concept "infinity", thus meaning "the kind of infinity that is potential", then it also does not help to explain how to come up with "infinity" in the first place. 

Sorry for rambling. Maybe at least I get my confusion across.

[studiot]

49 minutes ago, taeto said:

Sorry for rambling. Maybe at least I get my confusion across.

Not a problem, infinity is a big subject.

 

I would suggest that convergent infinity getting closer and closer to some finite object (it doesn't have to be a number, in fact I wouldn't recommend it at that age) is too subtle for a 5 year old.

But the idea of going on and on (for ever), or divergent infinity is much easier to grasp.

How about asking about divergent infinity?

How many steps to grandma's or to the corner shop or whatever?

Then ask how many to the next village / town?

Then ask how many to Copenhagen etc

Then how many to the Moon.

I think there are infinity/eternity  stories in Viking mythology (about Loki's punishment) and Hans Anderson that you might like to invoke (simplified) as well.

 

Physicists use infinity when discussing lens theory in real world optics.

Have you seen this?

Edited February 18, 2019 by studiot

[Sensei]

13 minutes ago, studiot said:

I would suggest that convergent infinity getting closer and closer to some finite object (it doesn't have to be a number, in fact I wouldn't recommend it at that age) is too subtle for a 5 year old. 

...or make Möbius strip from piece of paper, and ask child where it has the beginning and the end..

https://en.wikipedia.org/wiki/Möbius_strip

 

[taeto]

6 minutes ago, studiot said:

Not a problem, infinity is a big subject.

And yet I seem blind to it .

8 minutes ago, studiot said:

But the idea of going on and on (for ever), or divergent infinity is much easier to grasp.

Unless this idea presupposes that "infinity" is already available as the amount of time, or the number of steps,  that something can take. 

10 minutes ago, studiot said:

Physicists use infinity when discussing lens theory in real world optics.

Have you seen this?

I have not. But maybe physicists have their own idea of the concept, separate from philosophers? Is it to "use infinity" to integrate products involving wave functions over \(\mathbb{R}^3\) to calculate the expected value of an observable in QM? If so, then I have seen that a lot.

 Maybe I should explain why "something larger than any natural number" is a meaningless explanation in mathematics. I assume that natural numbers are ordered in their natural sequence. Also I assume that \(\{1\}\) is not a natural number (as e.g. in Von Neumann's classical construction of \(\mathbb{N}\) ). Then let \(\infty = \{1\}\), and extend the natural ordering \(<\) of \(\mathbb{N}\) to \(\mathbb{N}\cup \{\infty\}\) by letting \(n < \infty\) for every natural number \(n.\) Then it is clear that \(\infty = \{1\}\) satisfies the requirement for being infinity. You can replace \(\{1\}\) by anything that is not itself a natural number. 

[studiot]

I am confused.

Are we preparing ideas to put to a 5 year old or helping a 35 year old with some maths?

[taeto]

A 5 year old who sincerely asks for explanation of the meaning of infinite things in mathematics should preferably be given at least some bits of actual information. It seems that the arithmetic is basic and a fine place to start. 

I worry about how the philosophical stuff sounds circular and worthless. Some people have to look up things on wikipedia or the like, especially if they are not familiar with the basics, and even so, if they look for some further idea to spice up an explanation. But materials should at least be currently relevant, not only having had possible value in ancient times. 

If the 5 year old asks about the source of combustion, I would feel it is misguided to explain the phlogiston theory as my answer.

[studiot]

So you want to teach a bit of basic maths.

My posts were directed towards this in all sincerity.
That was why is suggested introducing infinity through 'eternity'

Also you could introduce infinity through the counting route one, two, many, (so many it doesn't matter how many = infinity)
This is the basis of the optics I was talking about. Infinity is so far away that rays are parallel and you are considering what happens when you put [math]1/\infty [/math] into the lens formula (which I assume you know).
This happens a lot in Science and Engineering for instance the idea of infinite dilution in Chemistry.

 

By 5 years they should be doing some counting and you could introduce the idea of putting in corrrespondence.
In truth they play (and love) games based on this from about 3.

If you can get over the idea of one-to-one correspondence and remainders that would be a real bonus in basics.

[Sirjon]

Infinity, in layman's term is philosophically mean... toward perfection....

A perfect telescope - that can see the object in an infinite distance

A perfect computer - that can process infinite information per given lapse of time

A perfect chip (IC) - that can house infinite numbers of electronic components in a very tiny chip

A perfect clock - that can do infinite cycles to speed up processing (in a computer)

No offense... I hope people here will not ask me what is your proof or evidence, I just use my common sense and this is simply my humble way of 'expressing my thought' and share it, with your permission, if I may.

[DrP]

2 hours ago, Sirjon said:

Infinity, in layman's term is philosophically mean... toward perfection....

Not to any definition I've seen of it.

2 hours ago, Sirjon said:

No offense... I hope people here will not ask me what is your proof or evidence, I just use my common sense and this is simply my humble way of 'expressing my thought' and share it, with your permission, if I may.

No offence taken - it is just wrong.

2 hours ago, Sirjon said:

A perfect telescope - that can see the object in an infinite distance

A perfect computer - that can process infinite information per given lapse of time

A perfect chip (IC) - that can house infinite numbers of electronic components in a very tiny chip

A perfect clock - that can do infinite cycles to speed up processing (in a computer)

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?

[wtf]

On 2/18/2019 at 5:27 AM, taeto said:

A 5 year old who sincerely asks for explanation of the meaning of infinite things in mathematics should preferably be given at least some bits of actual information. It seems that the arithmetic is basic and a fine place to start. 

I worry about how the philosophical stuff sounds circular and worthless. Some people have to look up things on wikipedia or the like, especially if they are not familiar with the basics, and even so, if they look for some further idea to spice up an explanation. But materials should at least be currently relevant, not only having had possible value in ancient times. 

If the 5 year old asks about the source of combustion, I would feel it is misguided to explain the phlogiston theory as my answer.

taeto, Have you heard about Cantor's theory of transfinite numbers? Mathematical infinity is basic to modern math. 

As @studiot asked, it's not clear if you're asking about mathematical infinity, or how to explain philosophical infinity to a five year old. If the former I can be of service; if the latter, probably not. 

[Eise]

12 hours ago, wtf said:

how to explain philosophical infinity to a five year old

What about (some variation of) the Hilbert hotel?

[Carrock]

14 hours ago, wtf said:

...how to explain philosophical mathematical infinity to a five year old....

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

 

 

[wtf]

12 hours ago, Eise said:

What about (some variation of) the Hilbert hotel?

For a five year old? Maybe, if you want to confuse them. Most adults have trouble with that fable. How could there be such a hotel? If all the rooms are full, how could everyone move to a new room? If there are infinitely many guests, where do new guests come from? What about the propagation speed of signaling? How could you notify the people in the faraway rooms to move? Etc. etc. etc. I've seen all of these issues causing confusion in discussions of Hilbert's hotel.

Hilbert, by the way, only mentioned the story once in his life, in a lecture given to the general public. He never wrote about it or mentioned it  again. It was only resurrected by George Gamow in a popularized math book. And Gamow, let it be noted, was a physicist and not a mathematician.

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. What makes the bowling ball push down on the sheet? Meta-gravity? No, the answer is that the rubber sheet and bowling ball picture is just a story; a visualization for amateurs that is not actually a physics demonstration. It's not meant to be thought about too deeply. Likewise Hilbert's hotel, which generally causes more confusion than enlightenment. 

Not to mention that sophists like William Lane Craig have used Hilbert's hotel to make theological points to credulous and naive audiences.

And for a five year old? What are they supposed to learn from such a distorted and confusing story? You really think they're going to understand that the story is not about hotels and guests, but is rather an illustration of the fact that an infinite set can be in bijection with a proper subset of itself? I think not.

Well now you've heard my rant about Hilbert's hotel, and thanks for asking! 

 

Edited March 21, 2019 by wtf

[beecee]

3 hours ago, wtf said:

 It's like the rubber sheet and bowling ball visualization of relativistic gravity. What makes the bowling ball push down on the sheet? Meta-gravity? No, the answer is that the rubber sheet and bowling ball picture is just a story; a visualization for amateurs that is not actually a physics demonstration. It's not meant to be thought about too deeply.

What makes the bowling ball push down on the rubber sheet? I could also ask what makes Earth push down on spacetime. It's not gravity...gravity is the effect we feel from when mass affects the geometry of flat spacetime. It's an analogy, rather then just a story to show us in 2D what happens to spacetime in 3D or 4D including time. But yes, analogies in near all cases are limited in what they illustrate. I like the John Wheeler description..."Space tells matter how to move; Matter tells space how to curve"

But yes, infinity is a concept that is hard to get one's head around, despite a couple of knowledgable members here once trying to explain it to me. The same for defining nothing.  It actually seems like I have been around for infinity, and when I finally kick the bucket, I'll be dead for infinity!

[wtf]

55 minutes ago, beecee said:

What makes the bowling ball push down on the rubber sheet? I could also ask what makes Earth push down on spacetime.

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.

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.

 

55 minutes ago, beecee said:

But yes, infinity is a concept that is hard to get one's head around

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. 

https://en.wikipedia.org/wiki/Ultrafinitism

 

Edited March 21, 2019 by wtf

[beecee]

13 minutes ago, wtf said:

Did you utterly miss the point of my post? Or did I utterly miss the point of yours? One of those must be true in order for the quoted sentence to have been typed.

Just clarifying the usefulness of analogies made in your rant.

[wtf]

15 minutes ago, beecee said:

Just clarifying the usefulness of analogies made in your rant.

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.

ps -- It's CERTAINLY not gravity! After all a bowling ball does not distort a steel plate, and the gravitational attraction between the ball and the plate is STRONGER than between the ball and the sheet because the steel plate is more massive.

No, the reason a bowling ball distorts a rubber sheet is because of gravity AND the molecular structure of rubber, whatever makes it stretchy. That's MUCH more complicated than gravity! 

You may have seen that Feynmann video on Youtube where he talks about why his chair holds him up, and in the end it really comes down to the weird forces acting inside the atom. 

 

 

Edited March 22, 2019 by wtf