# Logical argument using infinity

Discussion in 'General Philosophy' started by arfa brane, Mar 19, 2019.

1. ### Neddy BateValued Senior Member

Messages:
2,333
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.

3. ### NotEinsteinValued Senior Member

Messages:
1,986
Ah, OK, I understand. Technically, that's a shell, not a sphere; a sphere is solid.

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

Write4U likes this.

5. ### Write4UValued Senior Member

Messages:
18,607
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?......

Last edited: Mar 28, 2019

7. ### NotEinsteinValued Senior Member

Messages:
1,986
I don't know; I haven't engaged in that particular line of reasoning.

8. ### Write4UValued Senior Member

Messages:
18,607
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?

9. ### arfa branecall me arfValued Senior Member

Messages:
7,660

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.

10. ### Write4UValued Senior Member

Messages:
18,607
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

11. ### SpeakpigeonValued Senior Member

Messages:
1,123
Nop...
Curiously, there don't seem to be any specific word for a spherical solid.
EB

12. ### NotEinsteinValued Senior Member

Messages:
1,986
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!

13. ### SpeakpigeonValued Senior Member

Messages:
1,123
Tautological.
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 likes this.
14. ### Write4UValued Senior Member

Messages:
18,607
I think you kinda missed the point, but yes, that too.......

Last edited: Mar 29, 2019
15. ### arfa branecall me arfValued Senior Member

Messages:
7,660
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.

16. ### Write4UValued Senior Member

Messages:
18,607
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....

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

Messages:
37,110

18. ### SpeakpigeonValued Senior Member

Messages:
1,123
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...
So, which one?
And the word "solid" doesn't help:
So, I guess mathematicians can live with a degree of ambiguity...
EB

19. ### SpeakpigeonValued Senior Member

Messages:
1,123
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.
Should be compulsory reading for all would-be philosophers of the infinite.
EB

Write4U likes this.
20. ### James RJust this guy, you know?Staff Member

Messages:
37,110
https://en.wikipedia.org/wiki/Ball_(mathematics)

21. ### BaldeeeValued Senior Member

Messages:
2,121
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.

22. ### SpeakpigeonValued Senior Member

Messages:
1,123
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

23. ### SpeakpigeonValued Senior Member

Messages:
1,123
Write4U likes this.