Infinity of a Different Size

[ChristineM]

What do you mean by 'imaginary infinities'?

Even in the definition of integration, there is an infinite limit used. And integration is necessary for any physics that uses calculus.

A book on solid state physics will typically talk about an infinite lattice of atoms, for example. This simplifies the calculations and gives accurate results, It is certainly NOT a sign that something is wrong.

In cosmology, it is typical to have space infinite in extent, especially for flat or negatively curved space.

In studies of wave propagation, it is typical to consider the far-field, which is done by taking infinite limits. Again, this is not a sign that something has gone wrong.

Once again, I would bet that almost every advanced physics book uses some sort of infinite limit in its calculations.

Imaginary: in our imagination.

I believe referal to infinate is simply a cop put for very many/ very big/ very small. Im talking really here.

And infinite limits seem to me to be an oxymoron.

 

[Polymath257]

Imaginary: in our imagination.

I believe referal to infinate is simply a cop put for very many/ very big/ very small. Im talking really here.

And infinite limits seem to me to be an oxymoron.

Limits as in calculus.

I would also point out that you are using a very outdated definition of 'infinity' as 'not limited'. That was based on some serious misunderstandings of the possibilities and isn't used by mathematicians any longer.

To describe a square wave in terms of 'pure' tone requires an infinite sum. This is common and standard and not a sign of anything wrong.

Most boundary value problems require some sort of infinite expansion. You will see it in any book on electromagnetism.

 

[ChristineM]

Does the number 2? No. But they are very useful for modeling what we see and predicting new things about the universe around us.

2
There ya go, yes it exists.

I am not saying infinity is not a useful tool, i am saying i doubt it can exist in reality. Though the word is used to denote very big and very small. These quantities can (in reality) be added to or subtracted from.

 

[Debater Slayer]

Imaginary: in our imagination.

I believe referal to infinate is simply a cop put for very many/ very big/ very small. Im talking really here.

And infinite limits seem to me to be an oxymoron.

There are infinities that converge into finite quantities. Convergent series can be summed to infinity and produce a finite value because the convergence "eventually" reaches a negligible amount.

I talked about this in more detail here:

Stumbling Intuition #3: Summing to Infinity

Infinity is an indispensably useful concept in mathematics, engineering, and science. It has practical applications. Much like complex/imaginary numbers, it's a wildly counterintuitive concept that still has immense practical utility. Imaginary numbers are central to the development of electrical and electronic systems as well as the analysis thereof, for example.

 

[ChristineM]

Limits as in calculus.

I would also point out that you are using a very outdated definition of 'infinity' as 'not limited'. That was based on some serious misunderstandings of the possibilities and isn't used by mathematicians any longer.

To describe a square wave in terms of 'pure' tone requires an infinite sum. This is common and standard and not a sign of anything wrong.

Most boundary value problems require some sort of infinite expansion. You will see it in any book on electromagnetism.

And there is my problem i am using a definition of infinite that is "limitless or endless in space, extent, or size; impossible to measure or calculate." From the current OED.

 

[Polymath257]

And there is my problem i am using a definition of infinite that is "limitless or endless in space, extent, or size; impossible to measure or calculate." From the current OED.

While this is the relevant article for mathematics:

Infinite set - Wikipedia

Another for cardinalities in general:

Cardinal number - Wikipedia

This also includes some of the rules for calculating with them.

And an article for the alephs:

Aleph number - Wikipedia

The problem is that the OED definition hasn't really caught up to what mathematics has achieved. I have an old thread on this. I'll see if I can find it.

 

[Polymath257]

2
There ya go, yes it exists.

I am not saying infinity is not a useful tool, i am saying i doubt it can exist in reality. Though the word is used to denote very big and very small. These quantities can (in reality) be added to or subtracted from.

The number 2 is just as abstract as aleph_0. It is a useful abstraction for many things, just like infinite limits.

And yes, infinite sets can be added to and subtracted from. Depending on the size of the set added or subtracted, you can get interesting results. The infinite cardinals can certainly be calculated with and are a staple of modern mathematics, like the number 2.

Here's a question: does the number pi exist? or is it simply a figment of our imagination? How about the cube root of 3?

 

[Revoltingest]

While this is the relevant article for mathematics:

Infinite set - Wikipedia

Another for cardinalities in general:

Cardinal number - Wikipedia

This also includes some of the rules for calculating with them.

And an article for the alephs:

Aleph number - Wikipedia

The problem is that the OED definition hasn't really caught up to what mathematics has achieved. I have an old thread on this. I'll see if I can find it.

Also, dictionaries usually don't address complexities
& technical jargon. The basic definition works...but
it just doesn't explore all the ramifications.

 

[Debater Slayer]

The number 2 is just as abstract as aleph_0. It is a useful abstraction for many things, just like infinite limits.

And yes, infinite sets can be added to and subtracted from. Depending on the size of the set added or subtracted, you can get interesting results. The infinite cardinals can certainly be calculated with and are a staple of modern mathematics, like the number 2.

Here's a question: does the number pi exist? or is it simply a figment of our imagination? How about the cube root of 3?

I have used this example before when talking about the relationship between math and reality: math is to physical phenomena what sheet music is to musical instruments.

 

[Polymath257]

Also, dictionaries usually don't address complexities
& technical jargon. The basic definition works...but
it just doesn't explore all the ramifications.

It has some fundamental issues. For example, look at a half line starting at one point and going off from there in some direction. is it limited by the point and thereby not infinite, or is it unlimited in one direction and hence infinite.

If you look at a quarter plane, is it finite because it is bounded by two lines or infinite because it goes on forever in some directions?

Or take the real numbers between 0 and 1. Is it finite because it is bounded by 0 and 1 or is it infinite because no finite collection of individual points will exhaust all the numbers?

Does the fact that infinite cardinalities calculated with mean that they are not actually infinite?

So the usual definition is rather incoherent when looked at closer.

It gets worse when you get into the old notions of potential and actual infinities. Those have mostly been discarded in today's mathematics (although some remnants remain).

 

[ChristineM]

The number 2 is just as abstract as aleph_0. It is a useful abstraction for many things, just like infinite limits.

And yes, infinite sets can be added to and subtracted from. Depending on the size of the set added or subtracted, you can get interesting results. The infinite cardinals can certainly be calculated with and are a staple of modern mathematics, like the number 2.

Here's a question: does the number pi exist? or is it simply a figment of our imagination? How about the cube root of 3?

I can have 2 of something really, i cannot have infinite quantities of the same thing.

Yes pi exists, but back to my original statement, can you imagine it? I think you can imagine some of it, but not all of it.

 

[Revoltingest]

It has some fundamental issues. For example, look at a half line starting at one point and going off from there in some direction. is it limited by the point and thereby not infinite, or is it unlimited in one direction and hence infinite.

Applying the dictionary definition, a line segment
has an infinite number of dimensionless points.
It's about what one focuses upon.

If you look at a quarter plane, is it finite because it is bounded by two lines or infinite because it goes on forever in some directions?

Aspects of it fit the definition.

Or take the real numbers between 0 and 1. Is it finite because it is bounded by 0 and 1 or is it infinite because no finite collection of individual points will exhaust all the numbers?

Does the fact that infinite cardinalities calculated with mean that they are not actually infinite?

Some aspects aren't amenable to calculation.

So the usual definition is rather incoherent when looked at closer.

It gets worse when you get into the old notions of potential and actual infinities. Those have mostly been discarded in today's mathematics (although some remnants remain).

The definition is good enuf for non-technical
usage, yet still comport (but not be useful)
for technical usage.

 

[Polymath257]

I have used this example before when talking about the relationship between math and reality: math is to physical phenomena what sheet music is to musical instruments.

There is also the issue that the 'laws of physics' are abstractions and primarily in our imaginations.

 

[ChristineM]

The definition is good enuf for non-technical
usage,

A can go with that.

 

[Polymath257]

I can have 2 of something really, i cannot have infinite quantities of the same thing.

Yes pi exists, but back to my original statement, can you imagine it? I think you can imagine some of it, but not all of it.

What do you mean all of it? It is a number between 3 and 4. A nice, finite number with an infinite decimal expansion. I can imagine it as easily as i can imagine 3/7 and perhaps better. it is the length around a circle of radius 1. Seems straightforward to me.

Given that no two things in reality are identical, can you really have two of something? or do you have multiple ones of things? Are you sure 2 isn't just in your imagination?

And are you sure there are not infinitely many galaxies? I don't know of any cosmologist that would commit one way or the other to that one.

 

[Valjean]

I guess that depends somewhat on what you mean by the word 'comprehend'. I understand the properties; I can predict new characteristics; I teach about it on a regular basis; I can distinguish different sizes of infinity from others; etc.

What else is required for comprehension?

Picturing in the mind?
I can picture a cube. I can't picture a tesseract.

 

[Polymath257]

Picturing in the mind?
I can picture a cube. I can't picture a tesseract.

Well, I don't get distinct mental pictures at all. I get sort of brief flashes. And I can get a flash for a tesseract as readily as I can for a cube. It takes some practice and some drawing, but it is possible.

Look up aphantasia.

 

[ChristineM]

What do you mean all of it? It is a number between 3 and 4. A nice, finite number with an infinite decimal expansion. I can imagine it as easily as i can imagine 3/7 and perhaps better. it is the length around a circle of radius 1. Seems straightforward to me.

Given that no two things in reality are identical, can you really have two of something? or do you have multiple ones of things? Are you sure 2 isn't just in your imagination?

And are you sure there are not infinitely many galaxies? I don't know of any cosmologist that would commit one way or the other to that one.

All of it, every digit that is the number pi. Every single one of them to infinity

Of course its in my imagination, just like infinity

I don't know how many galaxies there are, thought the universe is thought to be finite, so is an infinite number of galaxies allowable in a finite volume?

 

[ChristineM]

@Polymath257

Is it Possible to Imagine Infinity in Our Minds?

I think it explains my view far better than i can explain it myself...

For many of us, it’s easy to understand the concept of infinity, but we can’t comprehend how “big” or “never-ending” it is, because our perception of time always has a beginning and an end — minutes, days, years, lifespans.​

 

[Valjean]

Sometimes it's hard to grok complex geometrical concepts: