Logical argument using infinity

[Neddy Bate]

How does anything not infinite approach infinity, though...

The phrase "approach infinity" is not a very good one, and unfortunately I tend to use it because it was taught to me when I learned about mathematical limits. A better phrase would probably be "increase without bound".

For every circle of finite radius that you can imagine, there is another larger circle of finite radius that you can imagine, (simply add one to the radius, for example). Let your larger circle become the first circle, and then repeat this process over and over again.

The large circle never reaches infinite radius, and so in that sense, the phrase "approach infinity" can seem misleading if it suggests eventually reaching infinity. That's why the phrase "increase without bound" is probably a better one.

 

[NotEinstein]

No, I just visualize a sphere as being an infinite collection of equal circumferences in all 3 dimensions, equi-distant from the center, completely enclosing the volume of the sphere?

Ah, OK, I understand. Technically, that's a shell, not a sphere; a sphere is solid.

Any single measurement of a circumference of a sphere yields a circle, no?

I wouldn't word it that way myself, but yes, that is correct.

 

[Write4U]

Not being able to find a physical case of an infinite circle doesn't mean one can't be constructed in abstract, so no this part of your reasoning doesn't hold up.

According to the argument put forth it is logically and mathematically impossible to construct an circle from infinitely long straight lines.

I submit it is impossible to make an arc from a 1 D straight line. It never closes and immediately fails the definition of being a circle.

I know you can make an abstract parabola from a bunch of straight lines.....

Where does the reasoning fail? There can never be any proof, can there?......

 

[NotEinstein]

According to the argument put forth it is logically impossible to construct an infinite circle from straight lines.
Where does the reasoning fail?

I don't know; I haven't engaged in that particular line of reasoning.

 

[Write4U]

Would it be incorrect to say that "infinity" is by definition formless. I cannot visualize any object (concrete or abstract) which would retain it's shape when extended into infinity.

And IMO, that would be logical as infinity is not a true number (value) that can be assigned to any shape.

Anyone?

 

[arfa brane]

These two images are a conceptual representation of the infinite plane as a light blue disc, with a "boundary at infinity" as a dashed line.

But the dashed line can't be a circle with a finite radius, although it obviously goes around the infinite plane (an infinity paradox?).
The right image is supposed to indicate what happens to a circle with a point on the infinite line.

 

[Write4U]

These two images are a conceptual representation of the infinite plane as a light blue disc, with a "boundary at infinity" as a dashed line.

But the dashed line can't be a circle with a finite radius, although it obviously goes around the infinite plane (an infinity paradox?).
The right image is supposed to indicate what happens to a circle with a point on the infinite line.

But you aretrying to represent an image which is not presentable.
Is that an image of the shape of infinity? Seems pretty bounded to me.

Can we draw an infinite triangle? A square? A circle?

IMO, Infinity is formless. There are no arcs, no angles, no planes, no spaces. There is only infinity. It has no defined property other than that it never ends. The Hilbert Hotel has nothing to do with hostelry, it has to do with the concept of never running out of rooms even while the hotel is fully occupied.

I believe the only symbolic representation of infinity can be found in the Mandelbrot Fractal;

Mandelbrot set images may be created by sampling the complex numbers and testing, for each sample point {\displaystyle c}

, whether the sequence {\displaystyle f_{c}(0),f_{c}(f_{c}(0)),\dotsc }

goes to infinity (in practice -- whether it leaves some predetermined bounded neighborhood of 0 after a predetermined number of iterations).
https://en.wikipedia.org/wiki/Mandelbrot_set

 

[Speakpigeon]

Technically, that's a shell, not a sphere; a sphere is solid.

Nop...

Sphere
Mathematics A three-dimensional surface, all points of which are equidistant from a fixed point.

Curiously, there don't seem to be any specific word for a spherical solid.
EB

 

[NotEinstein]

Nop...

Curiously, there don't seem to be any specific word for a spherical solid.

Huh, interesting. Seems like I picked up the physicist jargon version of the word, instead of the official one. I'll have to un-learn that one. Thanks for the correction!

 

[Speakpigeon]

Any single measurement of a circumference of a sphere yields a circle, no?

Tautological.

Circumference
1. The boundary line of a circle.

A circumference is a circle.
I guess what you may mean is that any planar section of a sphere is a circle.
Although, it can be reduced to a point.
But a point is also a circle unto itself. No kidding.
EB

 

[Write4U]

Tautological.

Circumference, noun

1. the enclosing boundary of a curved geometric figure, especially a circle.
   synonyms: perimeter, border, boundary;
   • the distance around something.

A circumference is a circle.
I guess what you may mean is that any planar section of a sphere is a circle.
Although, it can be reduced to a point.
But a point is also a circle unto itself. No kidding.
EB

I think you kinda missed the point, but yes, that too.......

 

[arfa brane]

. . . an image which is not presentable.
Is that an image of the shape of infinity? Seems pretty bounded to me.

Well exactly. How can you represent an infinite plane as a disc? The dashed line is straight--the line at infinity--but "goes around" the blue disc. Obviously it's more of a diagram or a kind of topological representation.

On the left the red circle is localised, somewhere in the infinite plane (forget about where it is in the image), because of its finite radius. Consider pairs of lines tangent to this circle intersecting somewhere else, either in the plane or at the boundary.

 

[Write4U]

How can you represent an infinite plane as a disc

That was the question. Can infinity be represented at all?

As soon as you try to draw a representation, you create a finite form which can never be representative of something infinite.

I am not trying to be argumentative, but just voicing my inability to visualize a representation of infinity.

Even the symbol we use for infinity is not infinite except for the endless repetitive travel along a single line, which according to the proposed result of infinitely reduction in arcs, should consist of straight lines.

It just doesn't make sense. Which leads me to suspect, infinity is altogether formless and anything with a form is by definition not infinite. I suspect I'm not alone....

The Infinite,
Working with the infinite is tricky business. Zeno’s paradoxes first alerted philosophers to this in 450 B.C.E. when he argued that a fast runner such as Achilles has an infinite number of places to reach during the pursuit of a slower runner. Since then, there has been a struggle to understand how to use the notion of infinity in a coherent manner. This article concerns the significant and controversial role that the concepts of infinity and the infinite play in the disciplines of philosophy, physical science, and mathematics.

https://www.iep.utm.edu/infinite/

 

[James R]

Curiously, there don't seem to be any specific word for a spherical solid.

How about "ball"?

 

[Speakpigeon]

How about "ball"?

A ball can have an empty interior.
Still, I guess you could say that a ball is a spherical body with possibly some empty region inside.
But I was thinking of mathematical terms.
It's not even clear what a sphere is for mathematicians...

Sphere
(Mathematics) A perfect round solid in which all points on its surface are equidistant from a fixed point called the center.

Sphere
Mathematics A three-dimensional surface, all points of which are equidistant from a fixed point.

So, which one?
And the word "solid" doesn't help:

Solid
b. Mathematics Of or relating to three-dimensional geometric figures or bodies.

So, I guess mathematicians can live with a degree of ambiguity...
EB

 

[Speakpigeon]

That was the question. Can infinity be represented at all? As soon as you try to draw a representation, you create a finite form which can never be representative of something infinite. I am not trying to be argumentative, but just voicing my inability to visualize a representation of infinity. Even the symbol we use for infinity is not infinite except for the endless repetitive travel along a single line, which according to the proposed result of infinitely reduction in arcs, should consist of straight lines. It just doesn't make sense. Which leads me to suspect, infinity is altogether formless and anything with a form is by definition not infinite. I suspect I'm not alone....

An infinite is an unbounded quantity. It seems clear that no one can represent infinity in some analogous or homologous way, for example with a little drawing of it, or in imagination. But we can conceive of it, something we do just by talking of unbounded quantities.
I agree that attempting to represent infinity with a little drawing is risky but I provided an example of a representation of a circle with an infinite radius earlier in this thread. It's not entirely analogous but I think it does the job.

https://www.iep.utm.edu/infinite/

Should be compulsory reading for all would-be philosophers of the infinite.
EB

 

[James R]

A ball can have an empty interior.
Still, I guess you could say that a ball is a spherical body with possibly some empty region inside.
But I was thinking of mathematical terms.
It's not even clear what a sphere is for mathematicians...

https://en.wikipedia.org/wiki/Ball_(mathematics)

 

[Baldeee]

A ball can have an empty interior.
Still, I guess you could say that a ball is a spherical body with possibly some empty region inside.
But I was thinking of mathematical terms.

The term "ball" is a mathematical term, though - for the space bounded by the sphere (where "sphere" is reference to the surface).
But you're correct that there appears to be no specific single word for a solid ball.

Edit: Ah, JamesR beat me to it.

 

[Speakpigeon]

These two images are a conceptual representation of the infinite plane as a light blue disc, with a "boundary at infinity" as a dashed line.
But the dashed line can't be a circle with a finite radius, although it obviously goes around the infinite plane (an infinity paradox?).
The right image is supposed to indicate what happens to a circle with a point on the infinite line.

I would draw a parallel between a circle with an infinite radius and a point at infinity or indeed an infinite number.
If we think of the Integers, all integers are finite numbers by definition in the sense that for each integer there is a greater integer. So, there is no infinite integer. I think the same reasoning applies to circles in the plane. For any point on the plane, there is a point which is further away from origin. So, there is no point in the plane at infinity. And, similarly, there is no circle at infinity. This seems to come as a direct consequence of the definition of infinity as not bounded.
However, an interval of Reals, say between 0 and 1, is bounded and yet we normally conceive of it as having an infinity of points. So, here we have an example of a bounded infinity. Paradoxical, sure, but not contradictory: it is logically possible.
Crucially, and unlike the situation with the Integers, there is, in our conception of the Reals, an actual infinity Real numbers between 0 and 1. The analogy with the question of the circle at infinity is then that we can conceive of an actual circle with an infinite radius, just not one in the plane.
EB

 

[Speakpigeon]

https://en.wikipedia.org/wiki/Ball_(mathematics)

Balls!
You're right!
Thanks.
EB